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Four Institutions -- A Unified Presentation of Logical Systems for Specification

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SLIDE 2

Four Institutions

A Unified Presentation of Logical Systems for Specification

U. Wolter, R. Wess¨

aly, M. Klar, F. Cornelius

Bericht-Nr. 94-24

SLIDE 3 F

ur

Institutions { A Unied Presen tation

f

Logical Systems for Sp ecication 1 { Uw e W

lter,

Roland W ess aly , Marcus Klar, F elix Cornelius June 28, 1994 1 This w

rk

has b een partly supp

rted

b y the German Ministry

f

Researc h and T ec hnology (BMFT) as part

f

the pro ject \K

rSo

{ Korrekte Soft w are".

SLIDE 4 Abstract W e

er

a unied formal description

f

four imp

rtan

t logical systems used for sp ecication purp

ses

in computer science. F

r
ur

unied presen tation w e use the concept

f

institution de- v elop ed in [GB84 , GB92 ] whic h co v ers the follo wing comp

nen

ts

f

a logical system: signatures, sen tences, mo dels, and satisfaction

f

sen tences b y mo dels. W e presen t the institution

f

equational logic for total algebras (see [EM85 ]), the institution

f

algebraic systems with p

sitiv

e Horn form ulas (see [Mal73 ]), the institution

f

partial algebras with conditional existence equations (see [Rei87, W

l90,

Kn

u93]),

and the institution

f

! { con tin uous algebras (see [GTWW77 , W es94 ]). The pap er is the starting p

in

t for a series

f

forthcomimg pap ers where w e will in v estigate the compatibilit y and com bination

f

sp ecication tec hniques and languages

n

the basis

f

the underlying logical systems (see [WDC + 94 , W

l94]).

SLIDE 5 Con ten ts 1 In tro duction 3 2 Basic Notions 5 2.1 Set Theory : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.2 Category Theory : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.3 Institutions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 3 The Institution

f

Equational Logic 10 3.1 The Category S I GN

f

Signatures : : : : : : : : : : : : : : : : : : : : : : : : : : 10 3.2 The Mo del F unctor M

d

E Q : S I GN ! C AT

p

: : : : : : : : : : : : : : : : : : : 11 3.3 The Sen tence F unctor S en E Q : S I GN E Q ! S E T : : : : : : : : : : : : : : : : : : 13 3.4 The Satisfaction Condition : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 4 The Institution

f

Algebraic Systems 18 4.1 The Category S I GN AS

f

Signatures : : : : : : : : : : : : : : : : : : : : : : : : : 18 4.2 The F unctor M

d

AS : S I GN AS ! C AT

p

: : : : : : : : : : : : : : : : : : : : : : 19 4.2.1 The M

d

AS {image

f

signatures: the Categories

f

Algebraic Systems : : 19 4.2.2 The F

rgetful

F unctor M

d

AS () : : : : : : : : : : : : : : : : : : : : : : : 20 4.2.3 The Mo del F unctor M

d

AS : : : : : : : : : : : : : : : : : : : : : : : : : : 21 4.3 The Sen tence F unctor S en AS : S I GN AS ! S E T : : : : : : : : : : : : : : : : : : 21 4.3.1 F

rm

ulas for Algebraic Systems : : : : : : : : : : : : : : : : : : : : : : : : 21 4.3.2 T ranslation

f

F

rm

ulas

f

Algebraic Systems : : : : : : : : : : : : : : : : 23 4.3.3 The F unctor S en AS : S I GN AS ! S E T : : : : : : : : : : : : : : : : : : : 24 4.4 The Satisfaction Relation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 24 4.4.1 The Satisfaction Condition : : : : : : : : : : : : : : : : : : : : : : : : : : 25 5 The Institution

f

P artial Algebras 28 5.1 The Category S I GN P A : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 28 5.2 The Mo del F unctor M

d

P A : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 29 5.3 The Sen tence F unctor S en P A : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 32 5.4 The Satisfaction Condition in I P A : : : : : : : : : : : : : : : : : : : : : : : : : : 34 6 The Institution

f

Con tin uous Algebras 38 6.1 The Category S I GN : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 38 6.2 The Mo del F unctor M

d

C A : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 39 6.2.1 The Category M

d

C A ()

f

Con tin uous Algebras : : : : : : : : : : : : : : 39 6.2.2 The F

rgetful

F unctor M

d

C A () : : : : : : : : : : : : : : : : : : : : : : : 41 6.3 The Sen tence F unctor S en C A : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 42 6.3.1 The Set S en C A ( )

f

Conditional

Inequalities

: : : : : : : : : : : : : : 42 1

SLIDE 6 6.3.2 The T ranslation

f

Sen tences S en C A () : : : : : : : : : : : : : : : : : : : 45 6.4 The Satisfaction Condition : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 47 A P artial Ordered Sets 51 2

SLIDE 7 Chapter 1 In tro duction The presen t pap er has its ro

ts

in a n um b er

f

problems w e tac kled within the german pro ject \K

rSo
Korrekte

Soft w are". One

f

the main

b

jectiv es

f

this pro ject w as a framew

rk

and a metho dology for supp

rting

the dev elopmen t

f

correct soft w are b y means

f

the application

f

w ell-founded sp ecication tec hniques and languages. One

f
ur

ma jor concerns within K

rSo

w as the seman tical foundation

f

the general purp

se

sp ecication language Spectr um dened in [BF G + 93]. While w

rking

with Spectr um the follo wing situations sho w ed to b e t ypical for the practical use

f

sp ecication tec hniques and languages in soft w are dev elopmen t:

F
r

the sp ecication

f

a small lo cal part

f

a problem it is con v enien t to use a small and adequate sp ecication language with a restricted expressiv eness but with a w ell-understo

d

seman tics at hand

r

at least in mind. After solving the lo cal parts

f

the problem w e are forced to em b ed

r

to in tegrate, in a seman tically compatible w a y , all the dieren t lo cal solutions in to a general

ne

within a big sp ecication framew

rk,

e.g. Spectr um. Thereb y the simple seman tics

f

the sp ecications in the small languages shall b e transformed in to the naturally more complex seman tics in the big sp ecication framew

rk

(see [WDC + 94]).

W

e ha v e to sp ecify dieren t asp ects

f
ne

and the same problem in parallel , algebraic and

p

erational asp ects as w ell as asp ects

f

datao w, safet y and concurrency . F

r

eac h

f

these asp ects w e ha v e appropriate but quite dieren t sp ecication tec hniques whic h cannot

r

shall not b e agglomerated to

ne
v

ersized sp ecication language. In this situation w e ha v e to ensure that all the dieren t views to

ne

and the same problem are seman tically compatible

n

the

v

erlapping parts

f

the sp ecication languages in question.

W

e ha v e to implemen t sp ecications b y programs, i.e. w e ha v e to transform sp ecications in a sp ecication language and their mo del theoretic seman tics in to programs in a programming language and their

p

erational seman tics. Thereb y results concerning cor- rectness and comp

sitionalit

y should b e preserv ed b y this transformation. When lo

king

at these situations it b ecomes quite

b

vious that w e need a conceptual and theo- retical framew

rk

whic h enables us to describ e and to relate the logical systems underlying the dieren t sp ecication and programming languages. In the last decade the concept

f

institution, dev elop ed in [GB84 , GB92 ], has b een pro v en to b e appropriate for suc h a unied description

f

logical systems. But un til no w there are no comprehensiv e presen tations

f

the logical systems, w e had to deal with in the pro ject, using the concept

f

institution. Th us w e decided to pro duce the presen t pap er con taining explicit instan tiations

f

the concept

f

institution. 3

SLIDE 8 In the next step

f
ur
ngoing

w

rk

w e will in v estigate the relations b et w een the presen ted four logical systems. While ev aluating the examples w e will dev elop an appropriate theoreti- cal concept

f

maps b et w een institutions, to

.

In the literature, man y dieren t concepts are prop

sed,

e.g. b y Goguen and Burstall [GB84 , GB92 ], b y Meseguer [Mes89 ], b y Salibra and Scollo [SS92], b y Astesiano, Cerioli, and Meseguer [A C92 , Cer93 , CM93 ], and b y Kreo wski and Mossak

wski

[KM93 , Mos93 ]. But none

f

these concepts reects the complete transition from

ne

sp ecication language in to another

ne,

whic h is a t ypical situation in

ur

con text. The remaining part

f

the pap er is

rganized

as follo ws: Chapter 2 giv es the basic notions from set theory and category theory . Additionally the concept

f

institution is sk etc hed. Chapter 3 presen ts the w ell-kno wn institution

f

equational logic for total algebras (see [EM85 ]). This c hapter shall help the reader to b ecome familiar with the use

f

institutions for the presen tation

f

logical systems. Chapter 4 pro vides an exp

sition
f

the institution

f

algebraic systems in the sense

f

Malcev with p

sitiv

e Horn form ulas (see [Mal73 ]). In Chapter 5 w e dev elop the institution

f

partial algebras with conditional existence equations. Thereb y w e extend the framew

rks

in [Rei87], [W

l90],
r

[Kn

u93

] b y predicates. The crucial

bserv

ation is that predicates can b e considered as partial maps in to the empt y cartesian pro duct, i.e. in to a designated singleton set. In suc h a w a y predicates can b e naturally included in to the framew

rk
f

partial algebras. Moreo v er the institution

f

algebraic systems b ecomes a logical subsystem

f

suc h an extended institution

f

partial algebras. Finally , Chapter 6

ers

the institution

f

!

con

tin uous algebras. More results concerning this institution and the institutions

f
rdered

and pre-con tin uous algebras can b e found in [W es94 ] where w e ha v e in v estigated esp ecially the free constructions w.r.t. signature morphisms in these institutions. 4

SLIDE 9 Chapter 2 Basic Notions This c hapter pro vides the notions w e will frequen tly need in the subsequen t c hapters. First

f

all w e presume that basic set theory , lik e sets, mappings (functions), comp

sition
r

c haracteriza- tions

f

mappings (functions), lik e injectiv e, surjectiv e

r

bijectiv e are w ell{kno wn to the reader. With the notions `mapping' and `function' w e mean the same and use them in terc hangeable. No w w e in tro duce the necessities around S

indexed

sets, category theory and the concept

f

institution. The reader, who is familiar with these notions should skip this c hapter and use it

nly

when basic questions arise, but note that not ev ery question can b e answ ered b y basic notions. 2.1 Set Theory Let S b e an arbitrary set. An S

indexed

set A is a family A = (A(s) j s 2 S )

f

sets. An S

indexed

set A = (A(s) j s 2 S ) is a subset

f

an S

indexed

set B = (B (s) j s 2 S ), A

B

in sym b

ls,

if A(s)

B

(s) for all s 2 S . In consequence an S

indexed

mapping f : A ! B is a family f = (f (s) : A(s) ! B (s) j s 2 S )

f

mappings b et w een S

indexed

sets A and B . The comp

sition
f

S

indexed

mappings is dened comp

nen

t wise, i.e. for f : A ! B and g : B ! C w e get the S

indexed

mapping g

f

: A ! C , giv en b y g

f

= D ef ( g (s)

f

(s) : A(s) ! C (s) j s 2 S ). A binary S

indexed

relation R

A
B

b et w een t w

S
indexed

sets A and B is an S {indexed set R = (R(s) j s 2 S ) where R(s)

A(s)
B

(s) for ev ery s 2 S . By S

w

e denote the set

f

all nite w

rds
n

S , i.e. the set con taining the empt y w

rd
and

all sequences w = s 1 : : : s n with s i 2 S for 1

i
n.

F

r

an y S {indexed set A w e iden tify A() with a distinguished singleton set A() = D ef fg and w e set A(w)= D ef A(s 1 )

A(s

n ) for ev ery w = s 1 : : : s n 2 S

;

n

1

: No w an S

indexed

mapping f : A ! B can b e canonically extended for ev ery w 2 S

to

a mapping f (w) : A(w) ! B (w ) as follo ws f () ( )= D ef

2

B () ; i.e. f () : A() ! B () is the iden tical mapping

n

A() = B () = fg, and f (w)(a)= D ef (f (s 1 )(a 1 ); : : : ; f (s n ) (a n )) 2 B (w) for ev ery w = s 1 : : : s n 2 S

;

n

1

and ev ery a = (a 1 ; : : : a n ) 2 A(w). 5

SLIDE 10 2.2 Category Theory The title

f

this section promises p erhaps to m uc h. W e

nly

dene the t w

basic

concepts category and functor b et w een categories. F urthermore w e giv e some examples in

rder

to illustrate these abstract concepts a little and to pro vide necessary categories for the denitions in the next c hapters. The reader, who is in terested in a more detailed

r

motiv ated in tro duction to category theory , is referred to the standard literature. See for example MacLane, [Mac71 ], Barr, W ells, [BW90 ]

r

the more abstract b

k
f

Herrlic h and Strec k er, [HS73]. Our denitions are adopted from the article

f

P

ign
e

in the Handb

k
f

Logic in Computer Science, [P

i92

]. Denition 2.2.1 (Category) A category C consists

f
a

class

f
b

jects jC j ,

a

class

f

morphisms M

r

(C ),

a

source function sour ce : M

r

(C ) ! jC j,

a

target function tar g et : M

r

(C ) ! jC j,

a

function id : jC j ! M

r

(C ), assigning to eac h

b

ject C 2 jC j the iden tit y morphism id(C )

r

id C ,

a

partial function

:

M

r

(C )

M
r

(C ) ! M

r

(C ) , satisfying the follo wing prop erties:

for

ev ery morphisms f

g

2 M

r

(C ) w e ha v e f

g

dened ( ) tar g et(g ) = sour ce(f ) and then is sour ce(f

g

) = sour ce(g ) and tar g et(f

g

) = tar g et(f ).

sour

ce(id C ) = C = tar g et(id C ), for ev ery C 2 jC j ,

if

sour ce(f ) = A and tar g et(f ) = B then id B

f

= f and f

id

A = f ,

if

h

g

and g

f

are dened then h

(g
f

) = (h

g

)

f

. As usual w e write f : A ! B if f 2 M

r

(C ) with sour ce(f ) = A and tar g et(f ) = B . Denition 2.2.2 (F unctor) Let C 1 , C 2 b e categories. A functor F : C 1 ! C 2 is a pair (F O bj ; F M

r

)

f

mappings F M

r

: jC 1 j ! jC 2 j and F M

r

: M

r

(C 1 ) ! M

r

(C 2 ) satisfying

sour

ce(F M

r

(f )) = F O bj (sour ce(f )) for ev ery f 2 M

r

(C 1 ),

tar

g et(F M

r

(f )) = F O bj (tar g et(f )) for ev ery f 2 M

r

(C 1 )

F

M

r

(id C ) = id F O bj (C ) for ev ery C 2 jC j,

F

M

r

(g

f

) = F M

r

(g )

F

M

r

(f ) for ev ery f ; g 2 M

r

(C 1 ) if g

f

is dened. 6

SLIDE 11 Prop

sition

2.2.3 (Comp

sition
f

F unctors) L et C 1 , C 2 , C 3 b e c ate gories and let F : C 1 ! C 2 , G : C 2 ! C 3 b e functors. Then the c

mp
sition

G

F

: C 1 ! C 3 given by G

F

(C 1 ) = G(F (C 1 )) for every C 1 2 jC 1 j, G

F

(f 1 ) = G(F (f 1 )) for every f 1 2 M

r

(C 1 ), is again a functor. A t this p

in

t w e w an t to in tro duce the common con v en tion to drop all indices if the con text determines the missing indices. This is con v enien t esp ecially for the authors and w e hop e the readabilit y isn't decreased. F

r

instance, a functor F is a pair

f

mappings, where b

th

the

b

ject mapping and the morphism mapping, are referred b y F , instead

f

F O bj and F M

r

. Let's turn the atten tion to the promised examples, but men tion that w e consider essen tially the examples pro viding the basis for

ur

subsequen t w

rk.

T

see

the full p

w

er

f

category theory w e m ust refer the reader

nce

again to the literature. Example 2.2.4

The

category S E T is giv en b y jS E T j is the class

f

all sets, M

r

(S E T ) is the class

f

functions b et w een sets, id maps ev ery set to the iden tit y function, ; sour ce; tar g et are dened canonical.

The

p

w

erset functor P : S E T ! S E T is giv en b y P (M ) = fX j X

M

g for ev ery M 2 jS E T j, P (f )(X ) = f (X ) for ev ery f : M 1 ! M 2 , X 2 P (M 1 ).

F
r

ev ery category C there is the iden tit y functor id C : C ! C giv en b y id C (C ) = C for ev ery

b

ject C 2 jC j, id C (f ) = f for ev ery morphism f 2 M

r

(C ) .

The

category C AT is giv en b y jC AT j is the class

f

small categories, M

r

(C AT ) is the class

f

functors b et w een small categories, id maps ev ery category C to the iden tit y functor id C , ; sour ce; tar g et are dened canonical (see Prop

sition

2.2.3).

F
r

ev ery category C exists the

pp
site

(or dual) category C

p

giv en b y jC

p

j = jC j, M

r

(C

p

) = M

r

(C ), sour ce

p

(f ) = tar g et(f ) for ev ery f 2 M

r

(C

p

) tar g et

p

(f ) = sour ce(f ) for ev ery f 2 M

r

(C

p

), id

p

(C ) = id(C ) for ev ery C 2 jC j , f

p

g = D ef g

f

if g

f

dened in C . 2 2.3 Institutions The main purp

ses
f

this section are t w

fold.

W e w an t to motiv ate b

th

the necessit y

f

a formalization

f

logical systems and the adv an tages

f

c ho

sing

the notion `institution' as the appropriate

ne.

As men tioned in the in tro duction from the b eginning

f

the algebraic sp ecication approac h in the early sev en ties quite a lot

f

dev elop ers w

rk

ed in their

wn

directions, 7

SLIDE 12 formalized their

wn

kind

f

logic and their

wn

view to the relationship b et w een syn tax and seman tics in computer science. This div ersit y

f

eort lead to an amoun t

f

denitions, concepts and notational con v en tions that couldn't b e adv an tageous. F

r

instance, to describ e syn tax, in the literature exist among

thers

equational signatures, signatures with predicates

r

higher-

rder

signatures, all

f

them com bined with a sp ecial kind

f

logic to express prop erties, for example, equations, inequalities, Horn-clauses, rst-order

r

higher-order logic are used. As the coun terpart there are sev eral seman tical views. Mo dels could b e equipp ed with partial

r

total functions

r

higher-order functions as w ell as the domains

f

mo dels could b e function spaces, partial

rders
r
therwise

structured. This en umeration is certainly incomplete and rather co- inciden tal, to p

in

t

ut

that an abstract view to these concepts is desirable. Ho w ev er,

ne

need this amoun t

f

concepts b ecause

f

the div ersit y

f

applications and as men tioned, a general purp

se

language lik e Spectr um isn't adequate for smaller sp ecications. F rom this p

in

t

f

view Goguen and Burstall [GB84 , GB92] started to dev elop an abstract view at logical systems. They in tro duced the concept

f

institution. Institutions fo cus

n

the mo del theoretic asp ects

f

logical systems. Institutions consist

f

signatures and morphisms b et w een them; form ulas and the translations

f

form ulas along signature morphisms; mo dels and the restriction

f

mo dels in in v erse direction to signature morphisms; satisfaction

f

form ulas b y mo dels whic h has to b e compatible with translation and restriction. In the follo wing y ears sev eral concepts had b een prop

sed.

Beside the institution w e see essen- tially t w

ther

concepts. First

f

all there are sp ecication frames as dened b y Ehrig et. al. []. Sp ecication frames dier from institutions essen tially in the missing satisfaction condition and in gluing syn tax, i.e. the category

f

signatures and signature morphisms is replaced b y the category

f

sp ecications and sp ecication morphisms. The second new concept are `general logics' in tro duced b y Meseguer [Mes89 ] to co v er also pro

f

theoretic asp ects

f

logical systems. General logics enclose institutions, i.e. general logics are institutions pro vided with an en tailmen t system for sen tences. Wh y did w e prefer institutions? The rst reasons are

f

pragmatic nature. On

ne

hand seman tics

f

the sp ecication language Spectr um, as the

rigin
f
ur

tasks, is describ ed as an institution. On the

ther

hand w e pa y

ur

resp ect to Goguen and Burstall as w ell as to the wide acceptance. Moreo v er, in

ur
pinion

the satisfaction condition is an imp

rtan

t feature, b ecause it is reasonable to force syn tax and seman tics to satisfy certain compatibilit y conditions. Goguen and Burstall ha v e coined the slogan \T ruth is in v arian t under c hange

f

notation". ... F urther w e cut

ut

the en tailmen t system

f

Meseguer, since this could b e a ma jor fo cus

f

in terest in the future as w ell as the sp ecication frames

f

Ehrig. W e think that it remains a topic

f

researc h to clarify for what purp

ses

whic h notion is adequate. Our in v estigations will b e

ne

step in this direction. No w let us turn to the concrete denition

f

an institution. Denition 2.3.1 (Institution) An institution I consists

f
a

category S I GN , where the

b

jects are signatures and the morphisms are signature morphisms,

a

functor S en: S I GN ! S E T , assigning to ev ery signature

a

set S en()

f
sen

tences and to ev ery signature morphism

:
1

!

2

a translation S en() : S en( 1 )! S en( 2 )

f
1
sen

tences in to

2
sen

tences,

a

functor M

d

: S I GN ! C AT

p

, assigning to ev ery signature

a

category M

d(

)

f
mo

dels and

f
morphisms

and to ev ery signature morphism

:
1

!

2

a (forgetful) functor M

d()

: M

d(

2 )! M

d(

1 ), 8

SLIDE 13

a

satisfaction relation j =

jM
d(

)j

S

en() for ev ery

2

jj , suc h that the satisfaction condition A 2 j =

2

S en()(e 1 ) ( ) M

d()(A

2 ) j =

1

e 1 ; holds for ev ery signature morphism

:
1

!

2

, ev ery

2
mo

del A 2 and ev ery

1
sen

tence e 1 . 9

SLIDE 14 Chapter 3 The Institution

f

Equational Logic 3.1 The Category S I GN

f

Signatures In this subsection w e dene the category

f

signatures. Signature morphisms enable the translation

f

signatures. Denition 3.1.1 (Signature) A man y-sorted signature

=

(S ; O P ; dom ; co d) consists

f:
a

nite set S

f

sort sym b

ls,
a

nite set O P

f
p

eration sym b

ls,
dom

: O P ! S

,
co

d : O P ! S . The functions dom and co d are called domain and co domain function. Remark 3.1.2 1. It is

ften

not ne c essary to denote the functions dom and co d and in this c ase we write

=

(S ; O P ) inste ad

f
=

(S ; O P ; dom ; co d) assuming that the function dom and co d ar e implicitly given. 2. We write

p

: s 1 : : : s n ! s and

p

: w ! s if w = s 1 ; : : : ; s n = dom(op ) and s = co d (op ). Denition 3.1.3 (Signature Morphism) Giv en t w

signatures
1

= (S 1 ; O P 1 ; dom 1 ; co d 1 ) and

2

= (S 2 ; O P 2 ; dom 2 ; co d 2 ) . A signature morphism

:
1

!

2

is a pair ( S ,

O

P )

f

functions consisting

f:
S

: S 1 ! S 2 a map

f

sorts

O

P : O P 1 ! O P 2 a map

f
p

eration sym b

ls

suc h that for all

p

1 2 O P 1 : dom 2 ( O P (op 1 )) =

S

(dom 1 (op 1 )) and co d 2 ( O P (op 1 )) =

S

(co d 1 (op 1 )) Remark 3.1.4 1.

S

: S

1

! S

2

denotes the extension

f
S

: S 1 ! S 2

n

wor ds, i.e.:

S

( 1 ) =

2

10

SLIDE 15

S

(sw) =

S

(s) S (w ) 2. We also write (s) inste ad

f
S

(s), if s 2 S and (op) for

O

P (op ), if

p

2 O P . Denition and Prop

sition

3.1.5 (The Category S I GN ) The category S I GN has

signatures

as

b

jects,

signature

morphisms as morphisms.

The

iden tit y morphisms I d

:
!
are

pairs

f

iden tit y maps id

=

(id S ; id O P ) and

the

comp

sition
f

morphisms is the comp

sition
f

maps

n

the comp

nen

ts, i.e.

:
1

!

3

= D ef ( S

S

; O P

O

P ) for t w

signature

morphisms

:
1

!

2

and :

2

!

3

. Pro

f.

Because this denition based up

n

maps in the category S E T it is clear that S I GN forms a category . 2 3.2 The Mo del F unctor M

d

E Q : S I GN ! C AT

p

The F unctor M

d

E Q is the mo del functor in this institution. It in terprets eac h signature

b

y assigning it to the class

f

{algebras. F urther w e will see that eac h class

f

algebras w.r.t. a signature forms a category , so the co domain

f

M

d

E Q has to b e the quasi category C AT

p

. Denition 3.2.1 ({Algebra) Let

=

(S ; O P ) b e a signature. A {algebra A = (A(S ); A (O P ) ) is giv en b y

A(S

) = (A(s) j s 2 S ) is an S {indexed family

f

sets, called carriers.

A(O

P ) = (A(op ) j

p

2 O P ) is an O P {indexed family

f

total functions, with A(op) : A (w) ! A (s) for eac h

p

eration sym b

l
p

: w ! s 2 O P . Remark 3.2.2 1. The notation (A(S ); A (O P )) emphasizes that an algebr a is an interpr etation

f

the signatur e (S ; O P ) . 2. As r emarke d in Chapter 2 we write A (w) for the c artesian pr

duct

A(s 1 )

:

: :

A(s

n ) in c ase w = s 1 : : : s n . Denition 3.2.3 ({Homomorphism) Giv en

{algebras

A and B , a {homomorphism f : A ! B is an S {indexed family

f

maps f = (f (s) : A(s) ! B (s) j s 2 S ), suc h that f is compatible with the

p

erations, i.e. f (s)(A(op )(a 1 ; : : : ; a n )) = B (op)(f (s 1 )(a 1 ); : : : ; f (s n )(a n )) 11

SLIDE 16 for eac h n{ary

p

eration sym b

l
p

: s 1 : : : s n ! s 2 O P and all a i 2 A(s i ) , for

i
n.

This condition is also called homomorphism condition. Remark 3.2.4 As intr

duc

e d in Chapter 2 we denote f (w) : A (w) ! B (w) for f (w)(a)= D ef (f (s 1 )(a 1 ); : : : ; f (s n )(a n )) with a 2 A(w ) . Ther efor e the homomorphism c

ndition

c an b e shortly written: f (s)

A(op

) = B (op)

f

(w ) No w w e can dene the category

f

{algebras. Prop

sition

3.2.5 (The Category M

d

E Q () ) Given a signatur e . The c ate gory M

d

E Q () c

nsists
f
{algebr

as as

bje

cts,

{homomorphisms

as morphisms.

The

identity{homomorphism is the identity map

n

e ach c arrier and

the

c

mp
sition
f

two {homomorphisms is the c

mp
sition
f

maps, i.e. g

f

= (g (s)

f

(s) : A(s) ! C (s)js 2 S ) for f = (f (s) : A (s) ! B (s) j s 2 S ) and g = (g (s) : B (s) ! C (s) j s 2 S ) . Pro

f.

It is clear that the iden tit y homomorphism is really the iden tit y , b ecause the denition based up

n

maps. So it is

nly

left to pro v e that the comp

sition
f

t w

homomorphisms

is a homomorphism again. 2 Lemma 3.2.6 (Comp

sition
f

{Homomorphisms) Given two {homomorphisms f : A ! B and g : B ! C . The c

mp
sition

g

f

: A ! C is a {homomorphism. Pro

f.

W e ha v e to pro

f

that g

f

is compatible with the

p

erations. (g

f

)(s)

A(op

) = (g (s)

f

(s))

A(op)

= g (s)

B

(op)

f

(w ) = C (op )

g

(w )

f

(w ) = C (op )

(g
f

)(w ) for all s 2 S and

p

: w ! s 2 O P . 2 Denition 3.2.7 (F

rgetful

F unctor M

d

E Q ()) Giv en a signature morphism

:
1

!

2

, w e dene the forgetful functor M

d

E Q () : M

d

E Q ( 2 ) ! M

d

E Q ( 1 )

n
2

{mo dels and

2

{homomorphisms. 1. F

r

ev ery

2

{mo del A 2 w e get a

1

{mo del M

d

E Q () (A 2 ) where (M

d

E Q ()(A 2 ))(s 1 ) = D ef A 2 ( (s 1 )) for ev ery s 1 2 S 1 and (M

d

E Q ()(A 2 ))(op 1 ) = D ef A 2 ( (op 1 )) for ev ery

p

1 2 O P 1 2. F

r

ev ery

2

{homomorphism f 2 : A 2 ! B 2 w e

btain

a

1

{homomorphism M

d

E Q ()(f 2 ) : M

d

E Q () (A 2 ) ! M

d

E Q () (B 2 ) 12

SLIDE 17 dened b y (M

d

E Q ()(f 2 ))(s 1 ) = D ef f 2 ((s 1 )) for all s 1 2 S 1 . Pro

f.

F

r

ev ery signature morphism

:
1

!

2

the forgetful functor M

d

E Q () : M

d

E Q ( 2 ) ! M

d

E Q ( 1 ) dened ab

v

e consitutes a functor. 1. M

d

E Q () preserv es the iden tities: Let A 2 b e a

2

{Algebra with its iden tit y id A 2 : A 2 ! A 2 . F

r

all s 1 2 S 1 w e ha v e M

d

E Q () (id A 2 )(s 1 ) = id A 2 ((s 1 )) = id M

d

E Q ()(A 2 ) (s 1 ) 2. M

d

E Q () is compatible with comp

sition:

Giv en t w

2

{homomorphisms f 2 : A 2 ! B 2 and g 2 : B 2 ! C 2 w e ha v e for all s 1 2 S 1 (M

d

E Q ()(g 2

f

2 ))(s 1 ) = (g 2

f

2 )((s 1 )) = g 2 ( (s 1 ))

f

2 ((s 1 )) = (M

d

E Q ()(g 2 ))(s 1 )

(M
d

E Q ()(f 2 ))(s 1 ) 2 Denition and Prop

sition

3.2.8 (F unctor M

d

E Q ) W e can dene M

d

E Q as a functor M

d

E Q : S I GN ! C AT

p

:

F
r

all signatures

2

j S I GN j, M

d

E Q () is the category

f

all

{algebras

as dened in prop

sition

3.2.5.

F
r

all signature morphisms

:
1

!

2

2 M

r

(S I GN ), M

d

E Q ( ) : M

d

E Q ( 2 ) ! M

d

E Q ( 1 ) is the 'forgetful'{functor as dened ab

v

e. Pro

f.

W e ha v e to pro v e that the functor M

d

E Q preserv es the iden tit y morphisms and the comp

sition,

whic h follo ws immediately from denition 3.2.7. 2 3.3 The Sen tence F unctor S en E Q : S I GN E Q ! S E T In this subsection w e dene terms and sen tences w.r.t. a signature, where the sen tences are equations. The equations can b e translated along a signature morphism and so w e

btain

a functor. General Assumption 3.3.1 (Store

f

V ariable Sym b

ls)

A store

f

v ariable sym b

ls

is a coun table set

f

sym b

ls.

In the follo wing it will b e assumed, that the names

f

the v ariables are alw a ys distinguishable from all constan ts in an y

ther

sp ecication. This store is implicitly giv en for this institution and need normally no men tion. Denition 3.3.2 ({V ariable System) Let

=

(S ; O P ) b e a signature. A {v ariable system X for a store

f

v ariable sym b

ls
is

an S {sorted family

f

sets (X (s) j s 2 S ) with X (s)

for

all s 2 S and 13

SLIDE 18 s 1 6= s 2 = ) X (s 1 ) \ X (s 2 ) = ; for all s 1 ; s 2 2 S . Denition 3.3.3 (T erms

f

a Signature) Let

=

(S ; O P ) b e a signature and X a {v ariable system. T(; X ) = (T(; X )(s) j s 2 S ) is called family

f

{terms

v

er X and is dened for eac h s 2 S as the minimal set T(; X )(s) for whic h holds:

X

(s)

T(;

X )(s),

p

2 T( ; X )(s) for all

p

:

!

s 2 O P .

p(t

1 ; : : : ; t n ) 2 T(; X )(s) for ev ery

p

: s 1 : : : s n ! s 2 O P and all t i 2 T(; X )(s i ) . Denition 3.3.4 ({Equation) Let

=

(S ; O P ) b e a signature and X a {v ariable system for . A {equation is (X : l = r ) , where l, r 2 T(; X )(s) s 2 S . Eq(; X ) denotes the set

f

all {equations

v

er X and Eq( ) is the set

f

all equations for a giv en signature, i.e. Eq () = D ef S (Eq( ; X ) j X is a {v ariable system for

).

Signature morphisms pro vide the translation

f

signatures, and terms are build

v

er a signature so w e can dene the translation

f

terms. Denition 3.3.5 (T ranslation

f

T erms) Giv en t w

signatures
1

= (S 1 ; O P 1 ),

2

= (S 2 ; O P 2 ), a signature morphism

:
1

!

2

and X 1 a

1

{v ariable system. W e dene X 2 as a

2

{v ariable system b y X 2 (s 2 ) = D ef S (X 1 (s 1 ) j (s 1 ) = s 2 ) for all s 2 2 S 2 . No w the translation

:

T( 1 ; X 1 ) ! T( 2 ; X 2 )

f
1

{terms is giv en b y an S 1 {indexed family

f

maps (s 1 ) : T( 1 ; X 1 ) (s 1 ) ! T( 2 ; X 2 )((s 1 )) with

(s

1 )(x 1 ) = D ef x 1 for all x 1 2 X 1 (s 1 ),

(s

1 )(op 1 ) = D ef (op 1 ) for all

p

1 :

!

s 1 2 O P 1 ,

(s

1 )(op 1 (t)) = D ef (op 1 )( (w 1 )(t)) for all

p

1 : w 1 ! s 1 2 O P 1 and all t 2 T( 1 ; X 1 )(w 1 ) for all s 1 2 S 1 . Remark 3.3.6 1. Notic e that the union

f

variables is disjoint b e c ause

f

denition 3.3.2. If X 2 is dene d like ab

ve

then we may write (X 1 ) to ac c entuate that this variable system is induc e d by X 1 and . 2. If ther e is no ambiguity, we may write (t) for (s)(t) if t2 T( 1 ; X 1 ) (s). 14

SLIDE 19 Denition and Prop

sition

3.3.7 (The Sen tence F unctor S en E Q ) The functor S en E Q : S I GN E Q ! S E T is giv en b y: 1. On signatures: S en E Q ( ) = D ef Eq( ) 2. On signature morphisms: S en E Q ( :

1

!

2

) = S en E Q () : Eq( 1 ) ! Eq( 2 ) , where S en E Q ()(X : l 1 = r 1 ) = D ef ((X 1 ) : (l 1 ) = (r 1 )) Pro

f.

Again, w e ha v e to sho w that the iden tities and the comp

sition

are preserv ed. This follo ws immediately from denition. 2 The functor S en E Q : S I GN E Q ! S E T tak es eac h signature to the set

f

all equations

v

er this signature and eac h signature morphism to a morphism, whic h translates the equations along the signature morphism. 3.4 The Satisfaction Condition In this subsection w e w an t to pro v e that the syn tactical comp

nen

ts are compatible to the seman tics, b y sho wing the satisfaction condition. This is essen tial to yield an institution. Denition 3.4.1 (Assignmen t

f

V ariables) Giv en a {algebra A and X a {v ariable system. An assignmen t a : X ! A(S ) for X in A is an S {indexed family

f

maps a = (a(s) : X (s) ! A(s)js 2 S ) Denition 3.4.2 (Ev aluation

f

{T erms) F

r

a giv en {algebra A , T(; X ) and an assignmen t a : X ! A(S ) , the ev aluation

f

terms a : T(; X ) ! A(S ) is an S {indexed family

f

maps a = (a(s) : T(; X )(s) ! A (s)js 2 S ) dened for all s 2 S as follo ws:

a(s)(x

) = D ef a(s)(x) for all x 2 X (s),

a(s)(op

) = D ef A (op ) for all

p

:

!

s 2 O P ,

a(s)(op

(t)) = D ef A(op)(a (t)) for all

p

: w ! s 2 O P and all t 2 T(; X )(w ) . Denition 3.4.3 (Satisfaction Relation) F

r

ev ery signature

=

(S ; O P ) the satisfaction relation j =

jM
d

E Q ( )j

S

en E Q ( ), is dened for ev ery {algebra A and ev ery

{equation

(X : l = r ) b y Aj =

eq

( ) a(l) = a(r) for all assignmen ts a : X ! A(S ). Denition 3.4.4 (Corresp

nding

Assignmen ts) Giv en a signature morphism

:
1

!

2

, a

2

{mo del A 2 , a

1

{v ariable system X 1 and a

2

{v ariable system (X 1 ) as dened in denition 3.3.5. F

r

ev ery assignmen t a 1 : X 1 ! M

d

E Q ( )(A 2 )(S 1 ) there is an assignmen t a 2 : (X 1 ) ! A 2 (S 2 ), dened b y 15

SLIDE 20 a 2 (s 2 )(x ) = D ef a 1 (s 1 )(x ) for ev ery s 2 2 S 2 , where (s 1 ) = s 2 and x 2 X 1 (s 1 )

(X

1 )(s 2 ). In the same w a y for ev ery assignmen t a 2 : (X 1 ) ! A 2 (S 2 ) there is an assignmen t a 1 : X 1 ! M

d

E Q ()(A 2 )(S 1 ) giv en b y a 1 (s 1 )(x) = D ef a 2 ( (s 1 ))(x ) for ev ery s 1 2 S 1 , where x 2 X 1 (s 1 )

(X

1 )( (s 1 )). In this case w e call (a 1 , a 2 ) pair

f

corresp

ndig

assignmen ts. The situation ma y b e illustrated b y the follo wing diagram: 6 6 M

d

E Q ()(A 2 )(s 1 ) X 1 (s 1 ) A 2 ((s 1 )) =

(X

1 )((s 1 )) a 2 ((s 1 )) a 1 (s 1 ) Because

do

es not c hange the names

f

the v ariables and k eep them disjoin t and b ecause M

d

E Q () do es not mo dify the carriers, it is clear that a 2 maps essen tially as a 1 do es in resp ect to the

rigin
f

the v ariables. Ab

v

e w e ha v e seen that w e can form ulate corresp

nding

assignmen ts, thereb y w e can sho w that there is a similar relationship for the ev aluation

f

terms. Prop

sition

3.4.5 (Corresp

nding

Ev aluation) Given a signatur e morphism

:
1

!

2

, a

2

{mo del A 2 , and a

1

{variable system X 1 . F

r

every assignment a 1 : X 1 ! M

d

E Q () (A 2 )(S 1 ) we get the c

rr

esp

nding

assignment a 2 : (X 1 ) ! A 2 (S 2 ) and vic e versa (se e 3.4.4) with a 1 (s 1 )(t ) = a 2 ((s 1 ))( (s 1 )(t )) for al l s 1 2 S 1 , t 2 T( 1 ; X 1 )(s 1 ). Pro

f.

W e pro v e this b y structural induction

v

er the construction

f

the terms. F

r

all s 1 2 S 1 and all t 2 T( 1 ; X 1 )(s 1 ): 1. F

r

all v ariables t = x, with x 2 X 1 (s): a 1 (s 1 )(x ) = a 1 (s 1 )(x) (def.

f

a 1 ) = a 2 ( (s 1 ))(x ) (def.

f

a 2 ) = a 2 ( (s 1 ))(x ) (def.

f

a 2 ) = a 2 ( (s 1 ))( (s 1 )(x )) (def.

f

) 2. F

r

constan ts t =

p

1 , with

p

1 :

!

s 1 : a 1 (s 1 )(op 1 ) = M

d

E Q () (A 2 )(op 1 ) (def.

f

a 1 ) = A 2 ( (op 1 )) (def.

f

M

d

E Q ( )) = a 2 ((s 1 ))( (op 1 )) (def.

f

a 2 ) = a 2 ((s 1 ))( (s 1 )(op 1 )) (def.

f

) 16

SLIDE 21 3. F

r

comp

sed

terms t =

p

1 (r ), with

p

1 : w 1 ! s 1 and r 2 T( 1 ; X 1 )(w 1 ): a 1 (s 1 )(op 1 (r)) = M

d

E Q (A 2 ) (op 1 ) (a 1 (w)(r )) (def.

f

a 1 ) = M

d

E Q (A 2 ) (op 1 ) (a 2 ((w 1 ))( (w 1 )(r))) (induction) = A 2 ((op 1 )) (a 2 ( (w 1 ))( (w 1 )(r))) (def.

f

M

d

E Q () ) = a 2 ((s 1 )) ((op 1 )( (w 1 )(r))) (def.

f

a 2 ) = a 2 ((s 1 )) ((s 1 )(op 1 (r))) (def.

f

) 2 Theorem 3.4.6 (Satisfaction Condition) F

r

al l signatur e morpisms

:
1

!

2

, and al l

1

{e quation (X : l = r ) and every

2

{algebr a A 2 the fol lowing holds: A 2 j =

2

S en E Q ( )(eq ) ( ) M

d

E Q ( )(A 2 ) j =

1

eq Pro

f.

By prop

sition

3.4.5 w e kno w that the ev aluation

f

terms in b

th

algebras yield the same v alue: a 1 (s)(l) = a 2 ( (s))((s)(l )) resp ectiv ely a 1 (s )(r ) = a 2 ( (s))((s)(r)) where (a 1 , a 2 ) is a pair

f

corresp

nding

assignmen ts. The satisfaction condition follo ws immediately b y this and b y Denition 3.4.3. 2 Corollary 3.4.7 (Institution

f

Equational Logic) The institution I E Q

f

equational logic c

nsists
f
the

c ate gory S I GN (se e 3.1.5),

the

mo del functor M

d

E Q : S I GN E Q ! C AT

p

(se e 3.2.7),

the

sentenc e functor S en E Q : S I GN E Q ! S E T (se e 3.3.7),

the

family (j =

j

M

d

E Q () j S en E Q () j

2
n

)

f

satisfaction r elations (se e 3.4.3). Pro

f.

There is just to pro v e that the satisfaction condition holds. This is done in theorem 3.4.6. 2 17

SLIDE 22 Chapter 4 The Institution

f

Algebraic Systems 4.1 The Category S I GN AS

f

Signatures Denition 4.1.1 (Predicativ e Signature) A predicativ e signature

=

(S ; O P ; dom ; co d; P ; a rit y ) consists

f:
a

nite set

f

sort sym b

ls

S ,

a

nite set

f
p

eration sym b

ls

O P ,

a

domain function dom : O P ! S

,

where S

is

the set

f

nite w

rds
v

er S ,

a

co domain function co d : O P ! S ,

a

nite set

f

predicate sym b

ls

P and

an

arit y function a rit y : P ! S

,

suc h that O P \ P = ;. Remark 4.1.2 (Predicativ e Signature) 1. Whenever we talk ab

ut

elements

f

the fr e e monoid

ver

a set, e.g. S

,

we denote them by a single variable, e.g. w 2 S

inste

ad

f

denoting the elements

f

the wor d, sep ar ate d by dots: s 1 : : : s n 2 S

.

This abbr eviation also applies later to elements

f

the c artesian pr

duct.

2. In gener al we wil l write

p

: w ! s for an element

p

2 O P with dom (op ) = w and co d (op ) = s. Similarly we wil l write P : v whenever a rit y (P ) = v . 3. We wil l write

=

(S ; O P ; P ) for short, whenever this is appr

priate.

Denition 4.1.3 (Predicativ e Signature Morphism) Giv en t w

predicativ

e signatures:

i

= (S i ; O P i ; P i ), i = 1,2, a predicativ e signature morphism

:
1

!

2

is a triple ( S ,

O

P ,

P

)

f

functions

S

: S 1 ! S 2

O

P : O P 1 ! O P 2

P

: P 1 ! P 2 satisfying the follo wing compatibilit y conditions for eac h

p

: w ! s2 O P 1 ; P : v 2 P 1 ; s2 S ; w , v 2 S

1

: 18

SLIDE 23 (i) dom 2 ( O P (op )) =

S

(w ) (ii) co d 2 ( O P (op)) =

S

(s) (iii) a rit y 2 ( P (P )) =

S

(v ). Denition and F act 4.1.4 (The Category S I GN AS

f

Predicativ e Signatures) S I GN AS , consisting

f
predicativ

e signatures according to Denition 4.1.1 as

b

jects,

predicativ

e signature morphisms according to Denition 4.1.3 as morphisms forms a category , where

the

iden tities are triples

f

iden tities in the category S E T ,

the

comp

sition

is dened as the application

f

the comp

sition

in S E T to eac h comp

nen

t

f

the triples. 4.2 The F unctor M

d

AS : S I GN AS ! C AT

p

4.2.1 The M

d

AS {image

f

signatures: the Categories

f

Algebraic Systems Denition 4.2.1 (Algebraic System) Giv en a signature

=

(S ; O P ; P ) 2 j S I GN AS j , w e dene an algebraic

system

A = (A(S ) ; A(O P ); A (P )) as follo ws:

A(S

) = (A(s) j s 2 S ), an S

indexed

family

f

sets called carrier sets

f

eac h sort,

A(O

P ) = (A(op ) : A(w ) ! A(s) j

p

: w ! s 2 O P ), an O P

indexed

family

f

functions,

A(P

) = (A(P )

A(v

) j P : v 2 P ), a P

indexed

family

f

relations (subsets

f

the appropriate cartesian pro duct A(v )). Remark 4.2.2 (Algebraic System) Onc e again we fo cus

n

the use

f

variables for wor ds inste ad

f

listing anonymously their c

ntents:

A(v ) is an abbr eviation for the c artesian pr

duct

A(s 1 )

:

: :

A(s

n ) pr

vide

d that v = s 1 : : : s n (se e Chapter 2). Denition 4.2.3 (Homomorphism) Giv en a predicativ e signature

2

j S I GN AS j and t w

algebraic

{Systems A, B . An S

indexed

family

f

functions f = (f (s) : A(s) ! B (s) j s 2 S ) is called homomorphism f : A ! B , if for an y

p

: w ! s 2 O P , P : v 2 P the follo wing compatibilit y conditions hold: (i) f (s)

A

(op ) = B (op )

f

(w ) (ii) f (v )

A(P

)

B

(P )(

B

(v )) f (w ), f (v ) denote the usual extension

f

functions

n

sets to functions

n

the cartesian pro duct

f

sets, i.e. the parallel application. 19

SLIDE 24 Denition and Prop

sition

4.2.4 (The Category M

d

AS ()) Giv en a signature

2

j S I GN AS j w e dene the category

f

algebraic

systems

M

d

AS ( ) to comprise the class

f

all {Systems as

b

ject class and the homomorphisms according to Denition 4.2.3 to build the corresp

nding

morphism sets. Pro

f.

Firstly , the homomorphism b eing the iden tit y (in S E T )

n

an y carrier set is indeed an iden tit y; this is

b

vious. Secondly , the comp

sition
f

homomorphisms has to b e a homomorphism as w ell: Giv en {Systems A, B , C and t w

homomorphisms

f : A ! B , g : B ! C , w e dene g

f

: A ! C to b e the family

f

comp

sed

mappings in S E T . (i) (g

f

)(s)

A(op)

= g (s)

(f

(s)

A(op)

) Comp

nen

t wise comp

sition

= g (s)

(B

(op)

f

(w)) f is a homomorphism = C (op)

g

(w)

f

(w) g is a homomorphism = C (op)

(g
f

)(w) Comp

nen

t wise comp

sition

(ii) (g

f

)(v )

A(P

)

g

(v )

(f

(v )

A(P

)) g is a homomorphism

g

(v )

B

(P ) f is a homomorphism

C

(P ) g is a homomorphism for an y giv en

p

: w ! s 2 O P and P : v 2 P . 2 4.2.2 The F

rgetful

F unctor M

d

AS () Denition and Prop

sition

4.2.5 (The F

rgetful

F unctor M

d

AS () ) Giv en a predicativ e signature morphism

:
1

!

2

, w e dene the forgetful functor M

d

AS () : M

d

AS ( 2 ) ! M

d

AS ( 1 ) b et w een the mo del categories

f

algebraic systems

f

the corresp

nding

predicativ e signatures (in

pp
site

direction) as follo ws: 1.

n
b

jects (i.e. algebraic systems): Giv en an algebraic

2

{system A 2 , w e dene the algebraic

1
system

A 1 = def M

d

AS ()(A 2 ) to comprise the follo wing carrier sets,

p

erations and relations (predicates): (a) for ev ery sort s 1 2 S 1 : A 1 (s 1 ) = def A 2 ( S (s 1 )) (b) for ev ery

p

eration sym b

l
p

1 2 O P 1 : A 1 (op 1 ) = def A 2 ( O P (op 1 )) (c) for ev ery predicate sym b

l

P 1 2 P 1 : A 1 (P 1 ) = def A 2 ( P (P 1 )) 2.

n

morphisms (i.e. homomorphisms): Giv en a homomorphism f 2 : A 2 ! B 2 b et w een t w

algebraic
2
systems,

w e dene the

1
homomorphism

f 1 = def M

d

AS ( )(f 2 ): M

d

AS () (A 2 )!M

d

AS ( )(B 2 )

n

ev ery sort s 1 2 S 1 b y: f 1 (s 1 ) = def f 2 ( S (s 1 )) Pro

f.

1. W e ha v e to sho w that A 1 and f 1 are w ell-dened. This follo ws immediately from Denition 4.1.1, namely from the required compatibilit y

f
S

with

O

P and

P

, resp ectiv ely . 20

SLIDE 25 2. The rst functor prop ert y is: iden tities are preserv ed. That is in general: F (id A ) = id F (A) . Giv en an algebraic

2
system

A 2 and its corresp

nding

iden tit y homomorphism id A 2 : A 2 ! A 2 , w e conclude for ev ery sort s 1 2 S 1 : M

d

AS ( )(id A 2 )(s 1 ) = id A 2 ( S (s 1 )) (as dened ab

v

e) = id M

d

AS ()(A 2 ) (s 1 ) (denition ab

v

e, 1.(i)) 3. The second functor prop ert y is the comp

sition

compatibilit y: F (g

f

) = F (g )

F

(f ). Giv en t w

2
homomorphisms

f 2 : A 2 ! B 2 , g 2 : B 2 ! C 2 leading to the comp

sition

g 2

f

2 : A 2 !C 2 w e conclude for ev ery s 1 2 S 1 : M

d

AS () (g 2

f

2 )(s 1 ) = (g 2

f

2 )( S (s 1 )) (as dened ab

v

e) = g 2 ( S (s 1 ))

f

2 ( S (s 1 )) (Comp. in S E T ) = M

d

AS ()(g 2 )(s 1 )

M
d

AS ()(f 2 )(s 1 ) (as dened ab

v

e) 2 4.2.3 The Mo del F unctor M

d

AS Denition and F act 4.2.6 (The Mo del F unctor M

d

AS ) W e dene the mo del functor M

d

AS : S I GN AS ! C AT

p

as the functor mapping eac h

predicativ

e signature

to

M

d

AS (), the category

f

algebraic

systems

according to Denition 4.2.4.

predicativ

e signature morphism

:
1

!

2

to M

d

AS (), the forgetful functor according to Denition 4.2.5. 4.3 The Sen tence F unctor S en AS : S I GN AS ! S E T 4.3.1 F

rm

ulas for Algebraic Systems General Assumption 4.3.1 (V ariable Store, V ariable System) The store

f

v ariable sym b

ls
is

a coun table set whic h, for ev ery predicativ e signature , admits to c ho

se

a subset X (s)

for

ev ery s 2 S with the prop ert y that eac h v ariable sym b

l

can unam biguously b e assigned a concrete sort s 2 S . More formally this means, for an y t w

dieren

t sort sym b

ls

s 1 ,s 2 2 S : X (s 1 )\X (s 2 ) = ;, and, with X = def ( S X (s) j s 2 S ): X \(O P [ P ) = ;. W e call X = (X (s) j s 2 S ) a

v

ariable system. Denition 4.3.2 (T erms

f

a Signature) Giv en a predicativ e signature

=

(S ; O P ; P ) and a

v

ariable system X . W e call T(; X ) = (T(; X )(s) j s 2 S ) family

f

{terms

v

er X . It is dened for eac h s 2 S as the minimal set T(; X )(s) satisfying:

X

(s)

T(;

X ) (s) (every variable is a term)

F
r

eac h

p

:

!

s 2 O P w e ha v e

p

2 T(; X )(s) (every c

nstant

symb

l

is a term)

F
r

ev ery

p

: w ! s 2 O P and t2 T(; X )(w) w e ha v e

p(t)2

T(; X )(s) (c

mplex

terms) 21

SLIDE 26 Denition 4.3.3 (A tomic F

rm

ula) Giv en a predicativ e signature

and

a

v

ariable system X . W e dene t w

forms
f

atomic

form

ulas

v

er X : 1. W e assume the equation sym b

l

`=' to b e not included in P (otherwise w e had to rename the

ne

b y the

ther).

Let t 1 ,t 2 2 T(; X )(s), s 2 S , b e t w

terms
f

equal sort. Then t 1 =t 2 is an atomic

form

ula

v

er X called equation. 2. Giv en a predicate sym b

l

P : v 2 P and a (tuple-) term t 2 T( ; X )(v )

f

(tuple-) sort v . Then w e ha v e P (t) is an atomic

form

ula

v

er X . A tomic

form

ulas are called atoms for short. Remark 4.3.4 (A tomic F

rm

ula) T

indic

ate the variable system X the atomic formula is b ase d

n

we gener al ly write (X , at). This is imp

rtant

for the subse quent denitions that r ely

n

this

ne.

Ther e's is stil l

ne

subtle thing to mention: we use d a c

mma

in the ab

ve

notation to mer ely indic ate the variable system, the formula is based

n.

This do es not ne c essarily me an, that the formula is gener al ly quantife d

ver

al l variables to b e found in X . Whenever we want to expr ess that inste ad, we wil l use a c

lon
r

an explicit quantier! The arising pr

blem

is discusse d in R emark 4.3.6. Denition 4.3.5 (Univ ersal Horn F

rm

ula) Giv en a predicativ e signature , a

v

ariable system X and atomic

form

ulas

v

er X : at 1 ; : : : ; at n , at according to Denition 4.3.3. W e call f

=

(X : at 1 ; : : : ; at n = ) at ) univ ersal horn form ula

v

er

and

X . Remark 4.3.6 (Univ ersal Horn F

rm

ula)

This

is a p

int
f

p

ssible

extensions. Horn formulas ar e r ather r estrictive with r esp e ct to the expr essive p

wer

users want to have. We think

f

the fol lowing extension steps: Gentzen formulas (mor e than

ne

atom in the c

nclusion),

rst

r

der formulas including existential quantic ation.

The

ab

ve

denition r e quir es that any variable symb

l

use d in some at i , at is include d in X . The r everse, however, do es not hold. Particularly when tr ansforming formulas by me ans

f

lo gic al r e asoning, e.g. applic ation

f

a sp e cialization rule like a^ b! a, we

ften

c

me

to a state, wher e X c

ntains

`unuse d' variables. This implies an ee ct, which is gener al ly unintende d. In c ase this single variable b elongs to a sep ar ate sort, the r efer enc e d formula wil l b e satise d (se e Denition 4.4.5) in any algebr aic system, in which the c arrier set for this sort is empty. This le ads to sometimes r e quir eing al l c arrier sets to b e non empty.

Her

e again we put emphasis

n

the hand ling

f

variables (se e R emark 4.3.4 ab

ve).

The Denition

f

an \atomic formula" is not subsume d by the denition

f

a universal horn formula. If we want to expr ess an atomic formula as a valid formula in the se quel, we have to use it in the ab

ve

p attern, i.e. we have to r efer enc e to the variable system and we have to quantify gener al ly

ver

it (by syntactic al me ans

f

a c

lon!).

22

SLIDE 27 4.3.2 T ranslation

f

F

rm

ulas

f

Algebraic Systems Denition 4.3.7 (Induced V ariable System) Giv en t w

predicativ

e signatures

1

,

2

, a predicativ e signature morphism

:
1

!

2

and a

1
v

ariable system X 1 . W e call

(X

1 ) = ((X 1 )(s 2 ) j s 2 2 S 2 )

induced
2
v

ariable system with eac h comp

nen

t dened b y (X 1 )(s 2 ) = S fX 1 (s 1 ) j s 1 2 S 1 , (s 1 )=s 2 g Remark 4.3.8 (Induced V ariable System) If

S

is not surje ctive, the induc e d

2
variable

system (X 1 ) c

ntains

empty sets

f

variable symb

ls.

Denition 4.3.9 (T ranslation

f

T erms) Giv en t w

predicativ

e signatures

1

,

2

, a predicativ e signature morphism

:
1

!

2

and a

1
term

t 1 2 T( 1 ; X 1 )(s 1 ). W e dene the translation

f

t 1 w.r.t. , called (s 1 )(t 1 ) 2 T( 2 ;

(X

1 ))( S (s 1 )) recursiv ely b y (i) t 1 2 X 1 (s 1 )= ) (s 1 )(t 1 ) = t 1 (2 (X 1 )) (ii) t 1 :! s 1 2 O P 1 = ) (s 1 )(t 1 ) =

O

P (t 1 ) (iii) t 1

p

1 (t),

p

1 : w 1 ! s 1 2 O P 1 , t2 T( 1 ; X 1 )(w 1 ) = ) (s 1 )(op 1 (t)) =

O

P (op 1 )( (w 1 )(t )) Remark 4.3.10 (T ranslation

f

T erms) We c an extend the ab

ve

denition to an S 1

indexe

d map

p

erforming the tr anslation

f

terms w.r.t. :

=

((s 1 ) j s 1 2 S 1 ) (s 1 ) : T( 1 ; X 1 ) (s 1 )!T( 2 ;

(X

1 ))( (s 1 ) ) Denition 4.3.11 (T ranslation

f

A tomic F

rm

ulas) Giv en t w

predicativ

e signatures

1

,

2

, a predicativ e signature morphism

:
1

!

2

and an atomic

1
form

ula at 1 . W e dene the translation (at 1 )

f

at 1 w.r.t.

according

to the structure

f

Denition 4.3.3 as follo ws: (i) at 1 (t 1 = t 1 '), t 1 , t 1 ' 2 T( 1 ; X 1 )(s 1 ) = ) (at 1 ) = def ( (s 1 )(t 1 ) = (s 1 )(t 1 ')), (ii) at 1

P

1 (t v ), P 1 : v 1 2 P 1 , t v 2 T( 1 ; X 1 )(v 1 ) = ) (at 1 ) = def

P

(P 1 )( (v 1 )(t v )), using the translation

f

terms according to Denition 4.3.9 ab

v

e. Denition 4.3.12 (T ranslation

f

Univ ersal Horn F

rm

ulas) Giv en t w

predicativ

e signatures

1

,

2

, a predicativ e signature morphism

:
1

!

2

, a

1
v

ariable system X 1 and a univ ersal horn form ula f

1

= (X 1 : at 1 ; : : : ; at n = ) at )

v

er

1

and X 1 , w e dene the translation

f

f

1

w.r.t.

b

y: (f

1

) = ((X 1 ): (at 1 ); : : : ;

(at

n ) = ) (at)) using the Denitions 4.3.7 and 4.3.11. 23

SLIDE 28 4.3.3 The F unctor S en AS : S I GN AS ! S E T Denition and F act 4.3.13 (The F unctor S en AS : S I GN AS ! S E T ) W e dene the sen tence functor for algebraic systems: S en AS : S I GN AS ! S E T assigning to eac h predicativ e signature its set

f

admissible form ulas (sen tences), formally

n
b

jects and morphisms b y:

S

en AS () is the set

f

all univ ersal horn form ulas

v

er .

S

en AS () =

:

S en AS ( 1 ) ! S en AS ( 2 ) is the translation

f

univ ersal horn form ulas for an y giv en

:
1

!

2

.. 4.4 The Satisfaction Relation Denition 4.4.1 (Assignmen t

f

V ariables) Giv en a predicativ e signature 2 j S I GN AS j , an algebraic

system

A2 M

d

AS ( ) and a

v

ariable system X . An assignmen t

f

the v ariable sym b

ls

in X to v alues in A : a:X !A = (a(s) : X (s)!A(s) j s 2 S ) is a family

f

functions that assigns to eac h v ariable sym b

l

x

ne

v alue a(x) in the algebraic system. Denition 4.4.2 (Ev aluation

f

T erms) Giv en a predicativ e signature 2 j S I GN AS j , an algebraic

system

A2 M

d

AS ( ), a

v

ariable system X an assignmen t a:X !A and a

term

t2 T(; X )(s). The ev aluation

f

t under a: a:T( ; X ) ! A; a=(a(s) j s 2 S ) is dened recursiv ely according to Denition 4.3.2: (i) t2 X (s): a(s)(t ) = def a(s)(t) (ii) t:!s2 O P : a(s)(t ) = def A(t) (iii) t

p(t

w ) with

p

: w ! s2 O P , t w 2 T(; X )(w): a(s)(op (t w )) = def A(op)(a (w )(t w )) Remark 4.4.3 (Ev aluation

f

T erms)

We

explicitly assume t2 T(; X ) and the assignment a

p

er ating

n

X . The r e ason why we c an r estrict

urselves

that way is, that we

nly

admit `close d' formulas, i.e. close d under universal quantiing. Otherwise we had to intr

duc

e a notion

f

`solution ' and \solution sets", i.e. sets

f

assignments under which the formula b e c

mes

true.

Of

c

urse

t2 T(; X )

nly

me ans that X is the maximum set

f

symb

ls

that c an b e use d to build t. The pr

blems

with empty c arrier sets arise a se c

nd

time: If ther e ar e no assignments (they have to

p

er ate

n

the whole variable system!), then ther e ar e no evaluations, not even for c

nstant

symb

ls!

Denition 4.4.4 (Ev aluation

f

A toms) Giv en a predicativ e signature 2 j S I GN AS j , an algebraic

system

A2 M

d

AS ( ), a

v

ariable system X , an assignmen t a:X ! A and an atomic

form

ula at w.r.t

and

X . W e call at true under a, if: 24

SLIDE 29 (i) at(t 1 =t 2 ): a(s)(t 1 ) = a(s)(t 2 ), i.e. the ev aluation

f

the terms t 1 , t 2

f

sort s yields the same result in A, (ii) at

P

(t): a(v )(t)2 A(P ), i.e. the ev aluation

f

the term-tuple t

f

sort v in A yields an elemen t

f

the relation A (P ). Otherwise w e call at false under a. Denition 4.4.5 (Satisfaction Relation) Giv en a predicativ e signature 2 j S I GN AS j , an algebraic

system

A 2 M

d

AS (), a

v

ariable system X and a univ ersal horn form ula f

=

(X : at 1 ; : : : ; at n = ) at) w.r.t . A satises f

:

A j =

f
if

for every assignmen t a:X !A the follo wing holds: If all atoms at i , 1

i
n,

ev aluate to true under a, then at yields true under a, to

.

Remark 4.4.6 (Satisfaction Relation) A gain we mention the \empty c arrier sets"{pr

blem.

Whenever X c

ntains

variables to a sort s whose interpr etation A(s) is the empty set, the set

f

assignments is empty and, thus, the fol lowing formula satise d! 4.4.1 The Satisfaction Condition Denition 4.4.7 (Corresp

nding

Assignmen ts) Giv en a predicativ e signature morphism

:
1

!

2

, a

1
v

ariable system X 1 and a

2
algebraic

system A 2 . 1. Giv en an assignmen t a 2 : (X 1 )!A 2 , with (X 1 ) according to Denition 4.3.7, w e dene the corresp

nding

assignmen t a 1

a

2 : X 1 ! M

d

AS ()(A 2 ) dep ending

n

a 2 b y: a 1

a

2 (s 1 )(x 1 ) = def a 2 ( S (s 1 ))(x 1 ) for ev ery s 1 2 S 1 and x 1 2 X 1 (s 1 ). 2. Giv en an assignmen t a 1 :X 1 ! M

d

AS ( )(A 2 ) w e dene the corresp

nding

assignmen t a 2

a

1 :(X 1 )!A 2 dep ending

n

a 1 b y: a 2

a

1 (s 2 )(x 2 ) = def a 1 (s 1 )(x 2 ) for ev ery s 2 2 S 2 , s 1 2 S 1 , x 2 2 X 2 , pro vided that s 2 = S (s 1 ). Remark 4.4.8 (Corresp

nding

Assignmen ts) 1. a 1

a

2 is an S 1

sorte

d mapping. That is, we have to dene it for every s 1 2 S 1 . A nalo gously, a 2

a

1 has to b e dene d for every s 2 2 S 2 . 2. a 1

a

2 is wel l-dene d for every x 1 2 X 1 (s 1 ) . If

S

is not inje ctive, i.e. (s 1 ) = (s 1 ) = s 2 , the gener al assupmtion 4.3.1 makes sur e that (X 1 )( S (s 1 )) S (X 1 )( S (s 1 )) actual ly is a disjoint union. 3. a 2

a

1 is wel l-dene d for the fol lowing r e asons: again 4.3.1 assur es the unique existenc e

f

a sort, every variable b elongs to, even in the c ase

f

non-inje ctivity. Se c

nd

ly, in c ase

f

non surje ctivity we p

ssibly

do not nd a sort s 1 with

S

(s 1 ) = s 2 . This is no pr

blem

b e c ause in that c ase the variable set (X 1 )(s 2 ) is empty (se e Denition 4.3.7). That is, for those sorts, a 2

a

1 has to b e the empty mapping. 25

SLIDE 30 Summarizing w e conclude the existence

f

a corresp

nding

assignmen t to eac h

f

a 1 :X 1 ! M

d

AS ( )(A 2 ), a 2 :(X 1 )!A 2 , whic h is,

n

the lev el

f

the elements

f

X 1 , (X 1 ) and A 2 , M

d

AS ( )(A 2 ) essen tially the same. Ob viously , this corresp

ndence

is bijectiv e. No w, in Section 4.4 w e ha v e dened extensions

f

an assignmen t to terms, atoms (ev aluations) and form ulas. Giv en a pair

f

corresp

nding

assignmen ts (a 1 , a 2 ), w e can extend b

th
f

them yielding a notion

f

c

rr

esp

nding

evaluations (a 1 , a 2 ). Lemma 4.4.9 (Corresp

nding

Ev aluations) Given a pr e dic ative signatur e morphism

:
1

!

2

, a

1
variable

system X 1 , an algebr aic

2
system

A 2 2 j M

d

AS ( 2 ) j and a p air

f

c

rr

esp

nding

assignments (a 1 , a 2 ). They extend to a p air

f

corresp

nding

ev aluations (se e Denition 4.4.2): a 1 : T( 1 ; X 1 )!M

d

AS () (A 2 ) a 2 : T( 2 ;

(X

1 ))!A 2 They have for every s 1 2 S 1 , t 1 2 T( 1 ; X 1 )(s 1 ) the fol lowing pr

p

erty: a 1 (s 1 )(t 1 ) = a 2 ( S (s 1 ))( (s 1 )(t 1 )) Pro

f.

The lemma con tains a prop

sition
v

er terms t 1 2 T( 1 ; X 1 )(s 1 ) . That is, w e ha v e to pro v e it b y induction

v

er the structure

f

that term according to Denition 4.3.2: (i) t 1 2 X 1 : a 1 (s 1 )(t 1 ) = a 1 (s 1 )(t 1 ) 4:4:2, (i) = a 2 ( S (s 1 ))(t 1 ) 4:4:7, 1. = a 2 ( S (s 1 ))( (s 1 )(t 1 )) 4:3:9, (i) = a 2 ( S (s 1 ))( (s 1 )(t 1 )) 4:4:2, (i) (ii) t 1 :! s 1 2 O P 1 : a 1 (s 1 )(t 1 ) = M

d

AS ()(A 2 )(t 1 ) 4:4:2, (ii) = A 2 ( O P (t 1 )) 4:2:5, 1.(b) = A 2 ( (s 1 )(t 1 )) 4:3:9, (ii) = a 2 ( S (s 1 ))( (s 1 )(t 1 )) 4:1:3, (ii), and 4:4:2, (ii) (iii) t 1

p

1 (t w ) with

p

1 : w 1 ! s 1 2 O P 1 , t w 2 T( 1 ; X 1 )(w 1 ): a 1 (s 1 )(t 1 ) = a 1 (s 1 )(op 1 (t w )) t 1

p

1 (t w ) = M

d

AS ()(A 2 )(op 1 )(a 1 (w 1 )(t w )) 4:4:2, (iii) = M

d

AS ()(A 2 )(op 1 )(a 2 ( S (w 1 ))( (w 1 )(t w ))) b y induction h yp

thesis

= A 2 ( O P (op 1 ))(a 2 ( S (w 1 ))( (w 1 )(t w ))) 4:2:5, 1.(b) = a 2 ( S (s 1 ))( O P (op 1 )( (w 1 )(t w ))) 4:1:3, (ii), and 4:4:2, (iii) = a 2 ( S (s 1 ))( (s 1 )(op 1 (t w ))) 4:3:9, (iii) = a 2 ( S (s 1 ))( (s 1 )(t 1 )) t 1

p

1 (t w ) 2 Lemma 4.4.10 (A tomar Satisfaction Condition) Given a pr e dic ative signatur e morphism

:
1

!

2

, an algebr aic

2
system

A 2 , a

1
variable

system X 1 and an atomic

1
formula

at 1

ver

X 1 , the fol lowing two pr

p
sitions

ar e e quivalent: 1. at 1 is true under ev ery assignment a 1 :X 1 !M

d

AS (A 2 ) (i.e., M

d

AS (A 2 ) satises at 1 ). 26

SLIDE 31 2. (at 1 ) is true under ev ery assignment a 2 : (X 1 )!A 2 (i.e., A 2 satises (at 1 )). Pro

f.

The lemma con tains a prop

sition
v

er

1
atoms

at 1

v

er X 1 . That is, w e ha v e to pro v e it

v

er the structure

f

that atom according to Denition 4.3.3: (i) at 1

(t

1 = t 1 ') for giv en t 1 , t 1 ' 2 T( 1 ; X 1 )(s 1 ) : 1: ) 2: Giv en an assignmen t a 2 :(X 1 )!A 2 , w e ha v e to sho w that (at 1 ) is true under a 2 , that is: a 2 ( S (s 1 ))( (s 1 )(t 1 )) = a 2 ( S (s 1 ))( (s 1 )(t 1 ')). By Denition 4.4.7 and its follo wing remarks w e kno w that there is a unique assignmen t a 1 :X 1 ! M

d

AS (A 2 ) suc h that <a 1 , a 2 > are corresp

nding

assignmen ts. By (1.) w e kno w that at 1 is true under a 1 . This means: a 1 (s 1 )(t 1 ) = a 1 (s 1 )(t 1 ') By Lemma 4.4.9 w e conclude directly the prop

sition.

2: ) 1: This part

f

the pro

f

corresp

nds

exactly to the

ne

b efore. (ii) at 1

P

1 (t v ) for giv en P 1 : v 1 2 P 1 , t v 2 T( 1 ; X 1 ) (v 1 ): 1: ) 2: Giv en w.l.o.g. an assignmen t a 2 :(X 1 )! A 2 , w e ha v e to sho w that (at 1 ) is true under a 2 . Let a 1 :X 1 ! M

d

AS (A 2 ) denote the unique corresp

nding

assignmen t

f

a 2 . What w e ha v e to sho w, is: at 2 satises (P 1 (t v )) , at 2 satises

P

(P 1 )( (v 1 )(t v )) 4:3:11, (ii) , a 2 ( S (v 1 ))( (v 1 )(t v ))2 A 2 ( P (P 1 )) 4:4:4, (ii) , a 1 (s 1 )(t v )2 A 2 ( P (P 1 )) 4:4:9 , a 1 (s 1 )(t v )2 M

d

AS (A 2 )(P 1 ) 4:2:5, 1.(iii) , P 1 (t v ) is true under a 1 4:3:11 But this is what w e already kno w b y assumption! 2: ) 1: This part

f

the pro

f

corresp

nds

exactly to the

ne

b efore (P a y atten tion to the p

in

t wise equiv alences

f

the steps ab

v

e!). 2 Corollary 4.4.11 (The Satisfaction Condition) Given a pr e dic ative signatur e morphism

:
1

!

2

, an algebr aic

2
system

A 2 , a

1
variable

system X 1 and a universal horn formula f

1

= (X 1 : at 1 ; : : : ; at n = ) at ) , the fol lowing satisfaction condition holds: A 2 j =

2

(f

1

) ( ) M

d

AS () (A 2 ) j =

1

f

1

Pro

f.

The pro

f
f

this corollary is a direct consequence

f

the ab

v

e Lemma 4.4.10 and uses the same denitions and ideas. F

r

this reasons w e

mit

it here. 2 27

SLIDE 32 Chapter 5 The Institution

f

P artial Algebras 5.1 The Category S I GN P A Denition 5.1.1 (Signature) A signature

=

(S ; O P

;

dom ; co d) consists

f
a

nite set S

f

sort sym b

ls,
a

nite set O P

f
p

eration sym b

ls,
a

domain function dom : O P

!

S

,

and

a

co domain function co d : O P

!

S [ fg . Remark 5.1.2 1. Note, that we al low

to

b e a c

domain

for

p

er ation symb

ls

thus the c

nc

ept

f

signatur e use d in this chapter diers fr

m

the signatur es use d in the institutions I E Q and I C A . 2. If no c

nfusion

arises we wil l write

(S

; O P

)

inste ad

f

(S ; O P

;

dom ; co d ),

p

: s 1 : : : s n ! s and

p

: s 1 : : : s n !

r
p

: w ! v , if dom (op) = s 1 : : : s n = w and co d (op ) = s = v

r

co d (op ) =

=

v . 3. The

p

er ation symb

ls

c :

!

s ar e also c al le d constan t sym b

ls

and the

p

er ation symb

ls

p : w !

ar

e r eferr e d to as predicate sym b

ls.

Denition 5.1.3 (Signature Morphism) Let

1

= (S 1 ; O P

1

; dom 1 ; co d 1 ) and

2

= (S 2 ; O P

2

; dom 2 ; co d 2 ) b e signatures. A signature morphism

:
1

!

2

is a pair ( S ,

O

P )

f

functions

S

: S 1 ! S 2 and

O

P : O P

1

! O P

2

suc h that for ev ery

p

1 2 O P 1 : dom 2 ( O P (op 1 )) =

S

(dom 1 (op 1 )); co d 2 ( O P (op 1 )) =

S

(co d 1 (op 1 )); where

S

( 1 ) =

2

and

S

(w 1 ) =

S

(s 1 ) : : :

S

(s n ) for ev ery w 1 = s 1 : : : s n 2 S

1

. General Assumption 5.1.4 (Subscript) W e neglect subscripts resp ectiv ely sup erscripts if they are giv en b y the con text, i.e. w e write (s) instead

f
S

(s) and (op) instead

f
O

P (op), if p

ssible.

F urther the notation

:
1

!

2

implicitly supp

ses
i

= (S i ; O P

i

; dom i ; co d i ) for i = 1; 2. 28

SLIDE 33 Denition and Prop

sition

5.1.5 (Category S I GN P A ) The category S I GN P A

f

signatures has

signatures
=

(S ; O P

;

dom ; co d) as in denition 5.1.1 as

b

jects and

signature

morphisms

:
1

!

2

as in denition 5.1.3 as morphisms. The comp

sition
:
1

!

3
f

t w

signature

morphisms

:
1

!

2

and :

2

!

3

is dened comp

nen

t wise

=

D ef ( S

S

; O P

O

P ) th us for ev ery signature

the

iden tical signature morphism I d

:
!
is

giv en b y the pair I d

=

(id S ; id O P )

f

iden tical functions. Pro

f.

It can b e easily c hec k ed that

really

b ecomes a signature morphism and that I d

is

the iden tit y . 2 5.2 The Mo del F unctor M

d

P A Denition 5.2.1 (P artial

Algebra)

F

r

an y signature

=

(S ; O P

)

a partial algebra A = (A(S ); A (O P

))

is giv en b y

an

S

indexed

set A(S ) = (A(s) j s 2 S ), the carrier

f

A and b y

an

O P

indexed

family A(O P

)

= (A(op ) j

p

2 O P

)
f

partial

p

erations, where for ev ery

p

: w ! v 2 O P

the

partial

p

eration A(op) : A(w )

!

A(v ) is giv en b y the domain

f

denition domA(op)

A(w

) and b y a total function A(op ) : domA(op)

!

A(v ) : Remark 5.2.2 1. The c

nc

ept

f

p artial algebr as do es not for c e the c

nstants

to b e dene d in a p artial

algebr

a A. F

r

a c

nstant

symb

l

c :

!

s ther e exists a c

nstant

A(c)( ) 2 A(s)

nly

in c ase domA(c ) = A () = fg (cf. se ction 2.1). 2. F

r

a pr e dic ate symb

l

p : w !

2

O P

nly

the domain

f

denition domA(p ) is essential b e c ause A (p ) : domA(p )

!

A() is a c

nstant

function with A(p )(a) =

for

every a 2 domA(p) . 3. Note, that a p artial

p

er ation A(op) : A (w)

!

A(v ) c an b e se en a span A(w) i

domA(op

) A(op)

!

A(v )

f

total functions, wher e i : domA(op) , ! A (w) r epr esents the inclusion domA (op)

A(w).

A(op) is a total

p

eration if domA(op ) = A (w). Denition 5.2.3 (-Homomorphism) Let

=

(S ; O P

)

b e a signature and let A , B b e partial

algebras.

An S

indexed

family f = (f (s) : A(s) ! B (s) j s 2 S ) : A(S ) ! B (S )

f

total functions is a (w eak)

homomor-

phism, f : A ! B in sym b

ls,

i the follo wing t w

conditions

(D) and (H) are satised: (D) f (w)(domA (op))

domB

(op ) for ev ery

p

: w ! v 2 O P

.

(H) f (v )(A(op )(a)) = B (op )(f (w)(a)) for ev ery a 2 domA(op )

A(w)

,

p

: w ! v 2 O P

.

29

SLIDE 34 A

homomorphism

f is a full

homomorphism

i (D 1 ) f (w)(domA (op)) = f (w)(A(w ) ) \ B (op ) 1 (f (v )(A(v ))) for ev ery

p

: w ! v 2 O P

.

A

homomorphism

f is a closed

homomorphism

i (D 2 ) domA(op) = f (w ) 1 (domB(op) ) for ev ery

p

: w ! v 2 O P

.

Remark 5.2.4 1. If A(op) and B (op) ar e total, the c

nditions

(D), (D 1 ), and (D 2 ) ar e trivial ly satise d. 2. Note, that (D 1 ) implies (D), and that (H) and (D 2 ) to gether imply (D 1 ) thus every close d

homomorphism

is ful l and every ful l

homomorphism

satises c

ndition

(D). 3. F

r

pr e dic ates p : w !

2

O P

c
ndition

(D 1 ) b e c

mes

e quivalent to (D pr ed ) f (w)(domA(p )) = f (w ) (A(w ) ) \ domB(p) b e c ause B (p) 1 (f () (A() )) = B (p) 1 (B () ) = B (p) 1 (fg ) = domB (p) . F

r

arbitr ary

p

: w ! v 2 O P

(D

2 ) implies (D pr ed ) and (D pr ed ) to gether with (H) implies (D 1 ). 4. F

r

surjectiv e and full

homomorphisms

(D 1 ) is simplie d to the c

ndition

(D sur j ) f (w)(domA (op)) = B (op) 1 (f (v )(A (v ))) F

r

pr e dic ates p : w !

we

have f (w)(domA (p)) = domB (p ) in this c ase. In general full

homomorphims

are not closed under comp

sition.

But full and surjectiv e

homomorphims

are closed under comp

sition.

Prop

sition

5.2.5 (Comp

sition
f
Homomorphisms)

L et b e given two

homomorphisms

f : A ! B and g : B ! C . 1. The S

indexe

d identic al mapping id A = (id A(s) : A(s) ! A (s) j s 2 S ) : A(S ) ! A(S ) is a close d

homomorphism

id A : A ! A. 2. The c

mp
sition

g

f

: A(S ) ! C (S )

f

the underlying S

mappings

is also a

homo-

morphisms g

f

: A ! C . 3. g

f

is a ful l

homomorphisms

pr

vide

d that g is ful l and f is ful l and surje ctive. 4. g

f

is a close d

homomorphisms

pr

vide

d that g and f ar e close d. Denition and Prop

sition

5.2.6 (Category

f

P artial

Algebras)

F

r

ev ery signature

2

jS I GN P A j the category M

d

P A () has

partial
algebras

A as in Denition 5.2.1 as

b

jects and

homomorphisms

f : A ! B as in Denition 5.2.3 as morphisms. Pro

f.

According to Prop

sition

5.2.5 M

d

P A () really b ecomes a category . 2 No w w e consider the translation

f

categories

f

partial algebras w.r.t. signature morphisms. 30

SLIDE 35 Denition and Prop

sition

5.2.7 (F

rgetful

F unctor) F

r

ev ery signature morphism

:
1

!

2

w e can dene a forgetful functor M

d

P A () : M

d

P A ( 2 ) ! M

d

P A ( 1 ) , as follo ws:

n
b

jects: F

r

ev ery partial

2
algebra

A 2 w e ha v e a partial

1
algebra

M

d

P A () (A 2 ) giv en b y M

d

P A () (A 2 )(s 1 ) = D ef A 2 ((s 1 )) for ev ery s 1 2 S 1 , M

d

P A () (A 2 )(op 1 ) = D ef A 2 ((op 1 )) for ev ery

p

1 : w 1 ! v 1 2 O P

1

.

n

morphisms: F

r

ev ery

2
homomorphism

f 2 : A 2 ! B 2 w e ha v e a

1
homomorphism

M

d

P A ()(f 2 ): M

d

P A ()(A 2 ) ! M

d

P A ()(B 2 ) giv en b y M

d

P A () (f 2 )(s 1 ) = D ef f 2 ((s 1 )) for ev ery s 1 2 S 1 . Pro

f.

1. Iden tities are preserv ed: Let A 2 b e a

2
mo

del with iden tit y id A 2 : A 2 ! A 2 . Then w e ha v e for ev ery s 1 2 S 1 M

d

P A ()(id A 2 )(s 1 ) = id A 2 ((s 1 )) = id M

d

P A ()(A 2 ) (s 1 ). 2. Compatibilit y with comp

sition:

Let f 2 : A 2 ! B 2 and g 2 : B 2 ! C 2 b e t w

2
homomor-

phisms. Then w e ha v e for ev ery s 1 2 S 1 M

d

P A ()(g 2

f

2 )(s 1 ) = (g 2

f

2 )((s 1 )) = g 2 ((s 1 ))

f

2 ((s 1 )) = M

d

P A () (g 2 )(s 1 )

M
d

P A ()(f 2 )(s 1 ). 2 Denition and Prop

sition

5.2.8 (Mo del F unctor M

d

P A ) The pair (M

d

P A;O bj ; M

d

P A;M

r

)

f

mappings M

d

P A;O bj : jS I GN P A j ! jC AT

p

j and M

d

P A;M

r

: M

r

(S I GN P A ) ! M

r

(C AT

p

) giv en b y M

d

P A;O bj ( ) = D ef M

d

P A () for ev ery

2

jS I GN P A j , and M

d

P A;M

r

() = D ef M

d

P A () for ev ery

:
1

!

2

2 M

r

(S I GN P A ) denes the mo del functor M

d

P A : S I GN P A ! C AT

p

. Pro

f.

M

d

P A is w ell-dened according to 5.2.6 and 5.2.7. The functor prop erties M

d

P A (I d

)

= I d M

d

P A () for ev ery

2

jS I GN P A j and M

d

P A (

)

= M

d

P A ()

M
d

P A ( ) for all

:
1

!

2

, :

2

!

3

in S I GN P A are also ensured b y denition 5.2.6 and 5.2.7. 2 31

SLIDE 36 5.3 The Sen tence F unctor S en P A General Assumption 5.3.1 (Store

f

V ariable Sym b

ls)

The store

f

v ariable sym b

ls

is assumed to b e a coun table set

f

sym b

ls

dieren t from all

ther

sym b

ls

used in the institution I P A . Denition 5.3.2 (S

System
f

V ariables) F

r

a signature

=

(S ; O P

)

an S

system

X

f

v ariables is an S

indexed

set X = (X (s ) j s 2 S ) with X (s)

for

all s 2 S and s 1 6= s 2 = ) X (s 1 ) \ X (s 2 ) = ; for all s 1 ; s 2 2 S . Since w e allo w predicate sym b

ls

p : w !

w

e can not

nly

construct terms

f

regular sorts s 2 S but also terms

f

the `sort' . But

denotes

the empt y sequence in S

and

is not a sort sym b

l

itselfs. General Assumption 5.3.3 (Predicate Sym b

l)

The predicate sym b

l
is

assumed to b e a distinguished sym b

l

predened for the institution I P A . Giv en a set S

f

sort sym b

ls

w e denote b y S

=

S [ f g the extension

f

S b y

.

F urther the sort comp

nen

t : S 1 ! S 2

f

a signature morphism

:
1

!

2

will b e also considered as a mapping : S

1

! S

2

where ( ) =

.

F

r

a partial

algebra

A = (A(S ); A(O P

)

) w e set A( ) = D ef A() = fg (cf. section 2.1) and extend in suc h a w a y the S

indexed

set A(S ) = (A(s) j s 2 S ) canonically to an S

indexed

set A(S

)

= (A(v ) j v 2 S

).

Denition 5.3.4 (-T erm) F

r

a signature

=

(S ; O P

)

and an S

system

X

f

v ariables the S

indexed

set T(; X )

=

(T(; X )

(v

) j v 2 S

)
f

all

terms
v

er X is dened to b e the smallest S

indexed

set suc h that 1. X (s)

T(;

X )

(s)

for ev ery s 2 S , 2. hi 2 T( ; X )

(

), 3. c (t) 2 T(; X )

(s)

for ev ery c :

!

s 2 O P

,

s 2 S , and all t 2 T( ; X )

(

), 4. p(t 1 ; : : : ; t n ) 2 T(; X )

(

) for ev ery p : w !

2

O P

with

w = s 1 : : : s n , 1

n

and all t i 2 T(; X )

(s

i ) , 1

i
n.

5.

p(t

1 ; : : : ; t n ) 2 T(; X )

(s)

for ev ery

p

: w ! s 2 O P

with

w = s 1 : : : s n , 1

n

and all t i 2 T(; X )

(s

i ) , 1

i
n.

The terms in T(; X )

(

) are referred to as predicativ e

terms.

Remark 5.3.5 (Sp ecial T erms) 1. The

terms

p(t 1 ; : : : ; t n ) and

p(t

1 ; : : : ; t n ) in Denition 5.3.4 wil l b e also denote d by p(t) and

p(t)

r esp e ctively wher e t = (t 1 ; : : : ; t n ) 2 T(; X )

(w)

is the c

rr

esp

nding

n-tuple

f
terms.

32

SLIDE 37 2. A c c

r

ding to rule (2) and (3) ther e is a term c(hi ) 2 T(; X )

(s)

for every c

nstant

symb

l

c :

!

s . But additional ly we have for every pr e dic ative term p(t 1 ; : : : ; t n ) 2 T(; X )

()

a term c(p (t 1 ; : : : ; t n )) 2 T( ; X )

(s)

also with the symb

l

c

n

top. Semantic al ly this c an b e interpr ete d as a \r estricte d c

nstant",

i.e. a c

nstant
nly

dene d if p(t 1 ; : : : ; t n ) b e c

mes

true. 3. Contr ary to the institution I AS

f

A lgebr aic Systems pr e dic ative terms ar e al lowe d to b e- c

me

subterms

f
ther

terms. Denition 5.3.6 (Existence

Equation)

F

r

a signature

=

(S ; O P

)

an existence

equations

(X ; l e = r )

v

er an S

system

X

f

v ariables is giv en b y t w

terms

l ; r 2 T( ; X )

(v

), v 2 S

.

Note, that in l and r

nly
ccur

v ariables from X . But,

n

the

ther

side there can b e v ariables in X not

ccuring

in l and r . Denition 5.3.7 (Conditional Existence

Equation)

F

r

a signature

=

(S ; O P

)

and an S

system

X

f

v ariables a conditional existence

equation

cee

v

er X is cee = (X : l 1 e = r 1 ; : : : ; l n e = r n = ) l e = r ) , with

n

where (X ; l 1 e = r 1 ) ; : : : ; (X ; l n e = r n ) , and (X ; l e = r ) are existence

equations
v

er X . By Cee(; X ) w e denote the set

f

all conditional existence

equations
v

er X , and Cee( ) = D ef S (Cee( ; X ) j X

is

an S

system
f

v ariables). is the set

f

all conditional existence

equations.

W e dra w atten tion to the fact that an equation (X : l = r ) in the sense

f

the institution I E Q (cf. Denition 3.3.5) is not equiv alen t to an existence equation (X ; l e = r ). (X : l = r ) has to b e in terpreted as a conditional existence equation (X : ; = ) l e = r ) with an empt y premise. F

r

an algebra A the seman tics

f

(X ; l e = r ) will b e the set

f

solutions

f

(X ; l e = r ) in A. Con trary the seman tics

f

(X : l = r )

r

(X : ; = ) l e = r ) in A will b e a truth v alue. Denition 5.3.8 (T ranslation

f

T erms) Let b e giv en t w

signatures
1

= (S 1 ; O P

1

) ,

2

= (S 2 ; O P

2

) , a signature morphism

:
1

!

2

, and an S 1

system

X 1

f

v ariables. W e dene an S 2

system

X 2

f

v ariables as follo ws X 2 (s 2 ) = D ef S (X 1 (s 1 ) j s 1 2 S 1 ;

(s

1 ) = s 2 ) for all s 2 2 S 2 . F urther the translation

:

T( 1 ; X 1 )

!

T( 2 ; X 2 )

f
1
terms

is an S

1
indexed

family

f

maps (v 1 ) : T( 1 ; X 1 )

(v

1 ) ! T( 2 ; X 2 )

((v

1 )) , v 1 2 S

1

giv en b y 1. (s 1 )(x 1 ) = D ef x 1 for eac h x 1 2 X 1 (s 1 ), s 1 2 S 1 , 2. ( )(hi ) = D ef hi , 3. (s 1 )(c 1 (t)) = D ef (c 1 )( ( )(t )) for ev ery c 1 :

1

! s 1 2 O P

1

and all t 2 T( 1 ; X 1 )

(

), 4. ( )(p 1 (t)) = D ef (p 1 )( (w 1 )(t )) for ev ery p 1 : w 1 !

1

2 O P

1

with w 1 = s 1 : : : s n , 1

n

and all t 2 T( 1 ; X 1 )

(w

1 ). 33

SLIDE 38 5. (s 1 )(op 1 (t)) = D ef (op 1 )((w 1 )(t)) for ev ery

p

1 : w 1 ! s 1 2 O P

1

with w = s 1 : : : s n , 1

n

and all t 2 T( 1 ; X 1 )

(w

1 ). General Assumption 5.3.9 1. The comp

nen

ts

f

X 2 are pairwise disjoin t, th us X 2 really b ecomes an S 2

system
f

v ariables. X 2 will b e also denoted b y (X 1 ) to p

in

t

ut

that this system

f

v ariables is induced b y X 1 and

.

2. If there is no am biguit y , w e will write (t) instead

f

(w 1 )(t) if t 2 T( 1 ; X 1 )

(w

1 ). Denition and Prop

sition

5.3.10 (T ranslation

f

Sen tences) F

r

an y signature morphism

:
1

!

2

the translation

:

T( 1 ; X 1 )

!

T( 2 ; X 2 )

f

terms can b e extend to a translation

f

conditional existence equations

:

Cee ( 1 )

!

Cee ( 2 ) where for ev ery cee = (X 1 : l 1 e = r 1 ; : : : ; l n e = r n = ) l e = r ) 2 Cee ( 1 ) : (cee) = D ef ((X 1 ) : (l 1 ) e = (r 1 ); : : : ;

(l

n ) e =

(r

n ) = ) (l ) e = (r )) : Denition and Prop

sition

5.3.11 (Sen tence F unctor S en P A ) The sen tence functor S en P A : S I GN P A ! S E T is giv en b y: 1. S en P A () = D ef Cee( ) for ev ery signature

2

jS I GN P A j and 2. S en P A () = D ef

:

Cee ( 1 ) ! Cee( 2 ) for ev ery signature morphims

:
1

!

2

in S I GN P A . Pro

f.

W e ha v e to sho w that the iden tities and the comp

sition

are preserv ed b y S en P A . But, this follo ws immediately from the denitions and prop

sitions

ab

v

e. 2 The functor S en P A : S I GN P A ! S E T tak es eac h signature to the set

f

all conditional existence equations

v

er this signature and eac h signature morphism to a morphism, whic h translates the conditional existence equations along the signature morphism. 5.4 The Satisfaction Condition in I P A In this section w e consider the satisfaction

f

conditional existence equations in partial algebras. F urther w e sho w for signature morphisms the compatibilit y

f

satisfaction w.r.t. the translation

f

conditional existence equations in

ne

direction and the translation

f

partial algebras in the

ther

direction. Firstly w e dene the ev aluation

f

terms for an assignmen t

f

v ariables in a giv en partial algebra. Con trary to the institutions I E Q and I AS the ev aluation

f

terms in I P A b ecomes

b

viously partial, i.e. a term t will b e ev aluable for some assignmen ts and will b e not ev aluable for

ther

assignmen ts. Denition 5.4.1 (Assignmen t

f

V ariables) Let b e giv en a signature

=

(S ; O P

),

a partial

algebra

A, and an S

system

X

f

v ariables. An assignmen t a : X ! A (S ) for X in A is an S {indexed family

f

maps a = (a(s) : X (s) ! A(s) j s 2 S ): 34

SLIDE 39 Denition 5.4.2 (Ev aluable

{T

erms) Let b e giv en a signature

=

(S ; O P

),

a partial

algebra

A, and an S

system

X

f

v ariables. F

r

an assignmen t a : X ! A(S ) , the S

indexed

set T(a)

=

(T(a)

(v

) j v 2 S

)
T(;

X )

f

all a-ev aluable

terms

and the corresp

nding

partial ev aluation a: T(a)

!

A(S

)
f
terms,

i.e. the S

indexed

family

f

maps a = (a(v ) : T(a)

(v

) ! A(v ) j v 2 S

)

are inductiv ely dened as follo ws: 1. for ev ery s 2 S : (a) X (s)

T(a)
(s),

(b) a(s)(x ) = D ef a(s)(x) for eac h x 2 X (s) ; 2. (a) hi 2 T(a)

(

) , (b) a( )(hi ) = D ef

;

3. for ev ery c :

!

s 2 O P

,

s 2 S , and all t 2 T(; X )

(

): (a) c (t) 2 T(a)

(s)

i t 2 T(a)

(

) and domA(c ) = fg , (b) a(s)(c (t)) = D ef A(c )( ) if c(t) 2 T(a)

(s)

; 4. for ev ery p : w !

2

O P

with

w = s 1 : : : s n , 1

n

and all t i 2 T(; X )

(s

i ), 1

i
n

: (a) p(t 1 ; : : : ; t n ) 2 T(a)

(

) i t i 2 T(a)

(s

i ) for 1

i
n

and (a(s 1 )(t 1 ); : : : ; a(s n )(t n )) 2 domA(p ) , (b) a( )(p (t 1 ; : : : ; t n ) ) = D ef

if

p(t 1 ; : : : ; t n ) 2 T(a)

(

) ; 5. for ev ery

p

: w ! s 2 O P

with

w = s 1 : : : s n , 1

n

and all t i 2 T(; X )

(s

i ) , 1

i
n:

(a)

p(t

1 ; : : : ; t n ) 2 T(a)

(s)

i t i 2 T(a)

(s

i ) for 1

i
n

and (a(s 1 )(t 1 ); : : : ; a(s n )(t n )) 2 domA(op ) , (b) a(s)(op (t 1 ; : : : ; t n ) ) = D ef A(op )(a(s 1 )(t 1 ); : : : ; a(s n )(t n )) if

p(t

1 ; : : : ; t n ) 2 T(a)

(s

) . Note, that the single elemen t

in

A( ) = A () pla ys the r^

le
f

the truth v alue true. Denition 5.4.3 (Solution

f

Existence

Equation)

Let b e giv en a signature

=

(S ; O P

),

a partial

algebra

A, and an S

system

X

f

v ariables. An assignmen t a : X ! A (S ) for X in A is a solution

f

an existence

equation

(X ; l e = r ) with l ; r 2 T(; X )

(v

); v 2 S

in

A , A ; a j =

(X

; l e = r ) in sym b

ls,

i l and r are a-ev aluable to the same v alue, i.e. i l ; r 2 T(a)

(v

) and a(l ) = a(r ): 35

SLIDE 40 Denition 5.4.4 (Satisfaction Relation) F

r

ev ery signature

=

(S ; O P

)

the satisfaction relation j =

jM
d

P A ( )j

S

en P A ( ) is dened as follo ws: A conditional existence

equation

(X : l 1 e = r 1 ; : : : ; l n e = r n = ) l e = r ) 2 S en P A ( ) is satised in a partial

algebra

A 2 jM

d

P A ( )j , i.e. A j =

(X

: l 1 e = r 1 ; : : : ; l n e = r n = ) l e = r ) ; i for all assignmen ts a : X ! A(S ) : n ^ i=1 A ; a j =

(X

; l i e = r i ) = ) A ; a j =

(X

; l e = r ) : The compatibilit y

f

the satisfaction relations with signature morphisms is essen tially based

n

the translation

f

assignmen ts and

f

ev aluations along signature morphisms. Denition 5.4.5 (Corresp

nding

Assignmen ts) F

r

a signature morphism

:
1

!

2

, a partial

2
algebra

A 2 , and an S 1

system

X 1

f

v ariables let (X 1 ) b e the

2
system
f

v ariables according to 5.3.8 and 5.3.9. Then for ev ery assignmen t a 1 : X 1 ! M

d

P A () (A 2 )(S 1 ) there is an assignmen t a 2 : (X 1 ) ! A 2 (S 2 ) dened b y a 2 ((s 1 ))(x ) = D ef a 1 (s 1 )(x) for ev ery s 1 2 S 1 and all x 2 X 1 (s 1 )

(X

1 )((s 1 )). Con v ersely this equation denes for ev ery assignmen t a 2 : (X 1 ) ! A 2 (S 2 ) an assignmen t a 1 : X 1 ! M

d

P A () (A 2 )(S 1 ), i.e. a 1 (s 1 )(x ) = D ef a 2 ((s 1 ))(x ) for ev ery s 1 2 S 1 and all x 2 X 1 (s 1 )

(X

1 )((s 1 )). (a 1 ; a 2 ) is called a pair

f

corresp

nding

assignmen ts. Next w e can sho w that the translation

f

terms do es not c hange the results

f

ev aluations. Prop

sition

5.4.6 (Corresp

nding

P artial Ev aluations) L et b e given a signatur e morphism

:
1

!

2

, a p artial

2
algebr

a A 2 , an S 1

system

X 1

f

variables and a p air (a 1 ; a 2 )

f

c

rr

esp

nding

assignments. Then we

btain

due to Denition 5.4.2 a p air

f

corresp

nding

partial ev aluations a 1 : T(a 1 )

!

M

d

P A () (A 2 )(S

1

) and a 2 : T(a 2 )

!

A 2 (S

2

) such that for al l

1
terms

t 2 T( 1 ; X 1 )(s 1 ), s 1 2 S 1 : 1. t 2 T(a 1 )

(s

1 ) i (t) 2 T(a 2 )

((s

1 )), and 2. a 1 (s 1 )(t) = a 2 ((s 1 ))( (t )) if t 2 T(a 1 )

(s

1 ). Pro

f.

By structural induction according to the inductiv e denitions

f

the partial ev aluation

f

terms in 5.4.2 and

f

the translation

f

terms in 5.3.10. Compare the pro

fs
f

the analog results in the institutions I E Q , I AS , and I C A . 2 Theorem 5.4.7 (Satisfaction Condition) F

r

every signatur e morphism

:
1

!

2

, every p artial

2
algebr

a A 2 , and every c

nditional

existenc e

1
e

quation cee = (X 1 : l 1 e = r 1 ; : : : ; l n e = r n = ) l e = r ) the satisfaction c

ndition

A 2 j =

2

S en P A ()(cee ) ( ) M

d

P A () (A 2 ) j =

1

cee: holds (cf. Denition 2.3.1). 36

SLIDE 41 Pro

f.

Due to Denition 5.4.3 and Prop

sition

5.4.6 w e ha v e for eac h pair (a 1 ; a 2 )

f

corresp

nding

assignmen ts and eac h existence

1
equation

(X 1 ; l e = r ) A 2 ; a 2 j =

2

((X 1 );

(l

) e = (r ) ( ) M

d

P A () (A 2 ); a 1 j =

1

(X 1 ; l e = r ) th us the satisfaction condition follo ws immediately from Denition 5.4.4. 2 Denition and Prop

sition

5.4.8 (Institution I P A ) The follo wing four comp

nen

ts:

the

category S I GN P A due to Prop

sition

5.1.5,

the

mo del functor M

d

P A : S I GN P A ! C AT

p

due to Prop

sition

5.2.8,

the

sen tence functor S en P A : S I GN P A ! S E T due to Prop

sition

5.3.11, and

the

family (j =

j
2

jS I GN P A j)

f

satisfaction relations due to Denition 5.4.4 form the institution I P A

f

P artial Algebras. Pro

f.

The satisfaction condition holds due to Theorem 5.4.7. 2 37

SLIDE 42 Chapter 6 The Institution

f

Con tin uous Algebras 6.1 The Category S I GN Denition 6.1.1 (Signature) A signature

=

(S ; O P ; dom ; co d) consists

f
a

nite set

f

sort sym b

ls

S ,

a

nite set

f
p

eration sym b

ls

O P ,

a

domain function dom : O P ! S

,

and

a

co domain function co d : O P ! S . Remark 6.1.2 1. F

r

e ase

f

r e adability we intr

duc

e some abbr eviations. If no c

nfusion

arises

r

the missing information is cle ar by the c

ntext

we write

(S

; O P ) inste ad

f

(S ; O P ; dom ; co d),

p

: s 1 : : : s n ! s

r
p

: w ! s if dom (op) = s 1 : : : s n = w and co d(op) = s. 2. Pay attention to the functions dom and co d . We intr

duc

e d them to exclude

verlo

ading

f

the

p

er ations, although

verlo

ading is a desir able pr actic al fe atur e. Example 6.1.3 (Signature for Streams) A STREAM = f sort Stream, Elem &: Elem

Stream

! Stream Stream generated b y ?, & g 2 38

SLIDE 43 Denition 6.1.4 (Signature Morphism) Let

1

= (S 1 ; O P 1 ; dom 1 ; co d 1 ) and

2

= (S 2 ; O P 2 ; dom 2 ; co d 2 ) b e signatures. A signature morphism

:
1

!

2

is a pair ( S ,

O

P )

f

functions

S

: S 1 ! S 2 and

O

P : O P 1 ! O P 2 satisfying the follo wing compatibilit y conditions for ev ery

p

1 2 O P 1 : dom 2 ( O P (op 1 )) =

S

(dom 1 (op 1 )); co d 2 ( O P (op 1 )) =

S

(co d 1 (op 1 )); where

S

( 1 ) =

2

and

S

(w 1 ) =

S

(s 1 ) : : :

S

(s n ) for ev ery w 1 = s 1 : : : s n 2 S

1

. Remark 6.1.5 T

minimize

the numb er

f

unne c essary r ep etitions the r e ader should ke ep in mind that we ne gle ct subscripts r esp e ctively sup erscripts if they ar e given by the c

ntext.

In this p articular c ase we write (s) inste ad

f
S

(s) and (op) inste ad

f
O

P (op), if p

ssible.

F urther it should b e

bvious,

that writing

:
1

!

2

we assume that

i

= (S i ; O P i ; dom i ; co d i ) for i = 1,2 ar e implicitly given. A bbr eviations

f

this kind wil l b e made in the fol lowing without additional r emark. F act 6.1.6 (Category S I GN ) S I GN, consisting

f
signatures

as in denition 6.1.1 as

b

jects,

signature

morphisms as in denition 6.1.4 as morphisms forms a category , where

the

iden tities are pairs

f

iden tities,

the

comp

sition

is dened comp

nen

t wise. Pro

f.

F

llo

ws immediately from the prop erties in the category S E T and the compatibilit y

f

signature morphisms with the domain resp ectiv ely co domain functions. 2 6.2 The Mo del F unctor M

d

C A 6.2.1 The Category M

d

C A ()

f

Con tin uous Algebras Occasionally requiremen ts arise where the w ell-kno wn algebraic metho ds aren't sucen t. F

r

instance, dening algebraic seman tics

f

recursiv e program sc hemes

r

the description

f

innite

b

jects lik e streams. T

deal

prop erly with recursion, xp

in

t tec hniques are an usefull to

l,

esp ecially in functional programming. F

r

that purp

se,

the mo dels ha v e to b e equipp ed with complete partial

rders.

Ho w ev er, the algebraic approac h do es not imp

se

an y c hoice b et w een the v arious notions

f

completeness. Resp ecting the seman tic

f

Spectr um (see [GR92 ]) w e c hose !

complete

partial

rders

for the carriers, i.e. for ev ery coun table c hain exists a least upp er b

und,

and !

con

tin uous functions for the

p

erations, i.e. functions preserving least upp er b

unds.

Con tin uous mo dels are adv an tageous since all their nite elemen ts are denotable as usual and b eha viour

f

the innite elemen ts ma y b e inferred from the nite appro ximations b ecause the

p

erations are assumed to b e con tin uous. Denition 6.2.1 (!

Con

tin uous

mo

del) Let

=

(S ; O P ) b e a signature. An !

con

tin uous

mo

del A = (A(S ); A (O P ); v A ) is giv en b y 39

SLIDE 44

(A(S

), v A ) = ((A (s); v A(s) ) j s 2 S ) is an S

indexed

family

f

!

complete

partial

rders,
A(O

P ) = (A(op ) j

p

2 O P ) is an O P

indexed

family

f

!

con

tin uous functions with A(op) : A(s 1 )

:

: :

A(s

n ) ! A(s) if dom (op) = s 1 : : : s n and co d (op ) = s. Remark 6.2.2 1. F

r

the denition

f

!

c
mplete

p artial

r

ders (!

cp
)

and !

c
ntinuous

functions se e in the app endix denitions A:1:8 and A:1:14. However, it is worth noticing the existenc e

f

a le ast element ? A(s) for every sort s 2 S . 2. The domain

f

mo del functions is wel l-dene d, i.e. for every

p

: s 1 : : : s n ! s 2 O P is A(s 1 )

:

: :

A

(s n ) =: A(w ) an !

cp
with

the natur al

r

dering x v A(w ) y ( ) x i v A(s i ) y i for every i = 1; : : : ; n; wher e x = (x 1 ; : : : ; x n ), y = (y 1 ; : : : ; y n ). Se e also pr

p
sition

A:1:13 in the app endix. Denition 6.2.3 (!

Con

tin uous

morphism)

Let

=

(S ; O P ) b e a signature and A, B b e !

con

tin uous

mo

dels. An S

indexed

mapping f = (f (s) : A(s) ! B (s) j s 2 S ): A ! B is called !

con

tin uous

morphism,

if the follo wing three conditions hold:

f

(s) (A(op)(a 1 ; : : : ; a n ) ) = B (op )(f (s 1 ) (a 1 ), : : : , f (s n ) (a n )) for ev ery

p

: s 1 : : : s n ! s 2 O P and a i 2 A(s i ), i = 0; : : : ; n,

f

(s) is !

con

tin uous for ev ery s 2 S ,

f

(s) is strict for ev ery s 2 S , i.e. f (s)(? A(s) ) = ? B (s) . Remark 6.2.4 1. Me anwhile it should b e familiar to intr

duc

e the fol lowing abbr eviation: f (s) (A(op)(a)) = B (op )(f (w )(a)) for every

p

: w ! s, w = s 1 : : : s n , a = (a 1 ; : : : ; a n ) 2 A (w). 2. The strictness c

ndition

for !

c
ntinuous
morphisms

isn 't ne c essary at the moment, but r esp e cting pr

p

erties like initiality

f

the mo del

f

terms we chose morphisms as dene d. No w w e are in p

sition

to build the category M

d

C A ()

f

!

con

tin uous

mo

dels and !

con

tin uous

morphisms

for ev ery signature . Prop

sition

6.2.5 (The Category M

d

C A ( )) F

r

every signatur e

is

M

d

C A (), c

nsisting
f
!
c
ntinuous
mo

dels as

bje

cts,

!
c
ntinuous
morphisms

as morphisms, a c ate gory (the mo del c ate gory), wher e

the

identity is the identity

n

every c

mp
nent,
the

c

mp
sition

is dene d c

mp
nentwise.

Pro

f.

The prop

sition

follo ws immediately b y the prop erties in the category S E T and the fact that strict !

con

tin uous functions are closed under comp

sition.

See prop

sition

A.1.16 in the app endix. The asso ciativit y follo ws from the asso ciativit y

f

the comp

nen

ts. 2 40

SLIDE 45 6.2.2 The F

rgetful

F unctor M

d

C A () In this short subsection w e consider the translation

f

mo del categories with resp ect to signature morphisms to get in p

sition

for the denition

f

M

d

C A , the mo del functor. Denition 6.2.6 (The F

rgetful

F unctor M

d

C A () ) Let

:
1

!

2

b e a signature morphism. W e dene the forgetful functor, denoted b y M

d

C A (): M

d

C A ( 2 ) ! M

d

C A ( 1 ), as follo ws:

n
b

jects: F

r

ev ery

2
mo

del A 2 w e ha v e a

1
mo

del M

d

C A () (A 2 ) giv en b y M

d

C A ()(A 2 )(s 1 ) = A 2 ((s 1 )) for ev ery s 1 2 S 1 , M

d

C A ()(A 2 )(op 1 ) = A 2 ((op 1 )) for ev ery

p

1 : w 1 ! s 1 2 O P 1 , v M

d

C A ()(A 2 )(s 1 ) = v A 2 ((s 1 )) for ev ery s 1 2 S 1 .

n

morphisms: F

r

ev ery

2
morphism

f 2 : A 2 ! B 2 w e ha v e a

1
morphism

M

d

C A ()(f 2 ): M

d

C A () (A 2 ) ! M

d

C A () (B 2 ) giv en b y M

d

C A ()(f 2 )(s 1 ) = f 2 ((s 1 )) for ev ery s 1 2 S 1 . The denition

f

M

d

C A ()(A 2 ) is w ell-dened, since the !

completeness
f

the sorts is inher- ited as w ell as the con tin uit y

f

the

p

erations. Similarly w e conclude that the

1
morphism

M

d

C A () (f 2 ) in the ab

v

e denition is w ell-dened. Ho w ev er, the fact that M

d

C A () is indeed a functor is giv en in Prop

sition

6.2.7 F

r

every signatur e morphism

:
1

!

2

the for getful functor M

d

C A () : M

d

C A ( 2 ) ! M

d

C A ( 1 ) as dene d ab

ve

is a functor. Pro

f.

1. Iden tities are preserv ed: Let A 2 b e a

2
mo

del with iden tit y id A 2 : A 2 ! A 2 . Then w e ha v e for ev ery s 1 2 S 1 M

d

C A ()(id A 2 )(s 1 ) = id A 2 ((s 1 )) = id M

d

C A ()(A 2 ) (s 1 ). 2. Compatibilit y with comp

sition:

Let f 2 : A 2 ! B 2 and g 2 : B 2 ! C 2 b e t w

2
morphisms.

Then w e ha v e for ev ery s 1 2 S 1 M

d

C A () (g 2

f

2 )(s 1 ) = (g 2

f

2 )((s 1 )) = g 2 ((s 1 ))

f

2 ((s 1 )) = M

d

C A ()(g 2 )(s 1 )

M
d

C A ()(f 2 )(s 1 ). 2 Corollary 6.2.8 (The Mo del F unctor M

d

C A ) The denitions M

d

C A ( ) and M

d

C A () determine the mo del functor M

d

C A : S I GN ! C AT

p

. Pro

f.

Giv en b y the prop

sitions

6.1.6, 6.2.5 and 6.2.7. 2 41

SLIDE 46 6.3 The Sen tence F unctor S en C A In this section w e consider form ulas for ev ery signature and the translation

f

form ulas with resp ect to a giv en signature morphism, the necessary parts for the sen tence functor. 6.3.1 The Set S en C A ( )

f

Conditional

Inequalities

The con tin uous algebra approac h imp

ses

some requiremen ts

n

the set

f

form ulas. W e w an t to force the mo dels to satisfy additional prop erties. In the common approac h this is done b y equations

r

conditional equations, for instance. Ho w ev er, for

ur

purp

se

this isn't sucen t,

n

accoun t

f

the additional partial

rder.

By this means w e m ust b e able to describ e this partial

rder.

Ob viously w e need at least inequalities, describing whic h elemen ts are part

f

the relation. W e decided, ho w ev er, to go a step further. W e c hose the set

f

conditional inequalities as the appropriate set

f

form ulas for

ur

purp

ses.

Moreo v er,

n

accoun t

f

the an tisymmetrie

f

partial

rders

w e are able to express equalities as w ell as conditional equalities with conditional inequalities. The fundamen tal notions for ev ery kind

f

form ulas are v ariables and terms. General Assumption 6.3.1 The store

f

v ariable sym b

ls

is assumed to b e a coun table set

f

sym b

ls.

Remark 6.3.2 We intr

duc

e d the stor e

f

variable symb

ls

to get rid

f

any pr

blems

linke d with variables. Assuming an universe

f

variables, not r estricte d by the c

untability

as ab

ve,

set the

r

etic al pr

blems

arise, i.e. the wel l-known set versus class pr

blems.

Denition 6.3.3 (-V ariable-System) Let

b

e the store

f

v ariable sym b

ls.

A

v

ariable-system X for a giv en signature

=

(S ; O P ) is an S

indexed

family

f

sets (X (s) j s 2 S ) with X (s)

for

all s 2 S and s 1 6= s 2 = ) X (s 1 ) \ X (s 2 ) = ; for all s 1 ; s 2 2 S . Remark 6.3.4 Mostly we write just

variable-system

X and do not mention the stor e

f

variable names . One reason to deal with con tin uous algebras is the requiremen t

f

innite

b

jects, for example streams. In the traditional algebraic approac h

nly

nite terms app ear. F

r

con tin uous algebras, ho w ev er, it is not sucen t to deal with nite terms

nly

. There is

ne

simple argumen t. Assuming the requiremen t to build a mo del

f

terms it is necessary that the terms

f
ne

sort are extendable to an !

cp
,

dening an appropriate partial

rder.

The minimal condition for completeness is the least elemen t prop ert y , i.e. a complete partial

rder

is pro vided with a least elemen t, denoted b y ?. With the usual construction

f

terms there isn't an y term to pla y the part

f

the least elemen t. Hence w e ha v e to add a distinguishable term ? s for ev ery sort s 2 S , whic h b eha v es lik e a constan t sym b

l.

This means, terms are constructable up

n

? s as a sp ecial constan t. A t this p

in

t

ne

question arises naturally . When ? s b eha v es lik e a constan t, wh y isn't it a predened constan t sym b

l

for ev ery signature? The answ er is, b

th

w a ys are p

ssible.

So, wh y did w e c ho

se
ur

w a y

f

dropping ? s from the sky? The same question arises, talking ab

ut

relations lik e equalit y , inequalit y

ther

relations

r

constan ts with predened seman tic. Mostly these

b

jects are necessary for the construction

f

form ulas. F

rm

ulas are generated b y terms and 42

SLIDE 47 v arious relations, lik e for instance equations. These constructors

f

form ulas aren't included in the signatures, b ecause there is

nly
ne

w a y

f

in terpretation. Ev ery signature morphism m ust assign ? s to ? s , m ust assign .=. to .=. , etc. Equally the mo del functor is forced to assign ? s to the least elemen t

f

the corresp

nding

carrier in the mo del. F

r

this reasons w e decided to include ? s in the generation

f

terms. By this means ? s arises lik e a constan t sym b

l

in comp

sed

terms and the at

rder

isn't the least

rder,

as p erhaps exp ected, since the required monoton y

f

the

p

erations implies an induced

rder

generated b y ? v t. F

r

example w e ha v e ? s i v s i t i = )

p(t

1 ; : : : ; ? s i ; : : : ; t n ) v s

p(t

1 ; : : : ; t i ; : : : ; t n ) for ev ery

p

: s 1 : : : s n ! s 2 O P . By this means the nite terms fail to b e !

complete.

So, it is necessary to extend the nite terms in a minimal w a y with innite terms, i.e. w e add limits

f

!

c

hains. In the curren t pap er

nly

the institution

f

con tin uous algebras is to b e stated, ho w ev er, in a follo wing pap er will b e describ ed that the addition

f

innite terms is a more general construction. F

r

arbitrary partial

rders

it is p

ssible

to construct the !

completion.

Moreo v er, this will turn

ut

to b e a free construction. As men tioned w e ha v e to deal with innite terms. By that w e ha v e the problem to denotate them. W e c hose the w a y

f

describing terms as a subset (called !

con

tin uous

terms)
f

partial mappings b et w een I N ? + and O P [ X . F

r
ne-sorted

signatures this is due to [GTWW77 ]. F

r

man y-sorted signatures there are

nly

a few c hanges. Denition 6.3.5 (Cp

f

Sym b

ls)

Let

=

(S ; O P ) b e a signature and X a

v

ariable-system. Dene a at

rder
f

sym b

ls

Sym(s) := f? s ; x ;

p

j x 2 X (s ); co d(op) = sg for ev ery sort s 2 S , i.e. ? s v sy m;s x for all x 2 Sym(s) . Denition 6.3.6 (!

Con

tin uous

terms)

Let

=

(S ; O P ) b e a signature and X a

v

ariable-system. W e dene the S

indexed

family

f

!

con

tin uous

terms

T(; X ) 1 = (T( ; X ) 1 (s) j s 2 S ) for ev ery s 2 S as a subset

f

the partial mappings I N ? +

!

O P [ ? [ S X (I N ? + is an abbreviation for nite sequences

f

p

sitiv

e natural n um b ers and ? = f? s j s 2 S g. A partial mapping t: I N ? +

!

O P [ ? [ S X is an !

con

tin uous

term
f

sort s 2 S , i

t()

2 Sym(s),

v

i 2 def (t) = ) v 2 def (t), v 2 I N ? + , i 2 I N and t (v ) =

p

with dom (op) = s 1 : : : s n , i

n,
v

2 def (t), v 2 I N ? + and t(v ) =

p

with dom(op ) = s 1 : : : s n , i

n

= ) v i 2 def (t) and t(v i) 2 Sym(s i ). Remark 6.3.7 1. The denition

f

!

c
ntinuous
terms

is b ase d up

n

tr e es. Se e some articial examples in 6:3:8. The tr e e pr

p

erty is ensur e d in the se c

nd

c

ndition,

in p articular, the domain must b e pr ex-close d. The r

t

t() expr esses the sort

f

e ach term, while the thir d c

ndition

guar ante es that the images ar e c

mp

atible with sorts and

p

er ations. Pay attention to the domain c

nditions

expr esse d by the thir d c

ndition:

43

SLIDE 48 v i = 2 def (t) ( ) v = 2 def (t)

r

v 2 def (t), dom(t(v )) = s 1 : : : s n , i > n. 2. A n !

c
ntinuous
term

t is nite if jdef (t)j < 1. Ther efor e nite terms ar e dene d as usual, enriche d by a distinguishable symb

l

? s for every sort s 2 S . In the se quel we also use the usual denotation

f

nite terms if r e adability is impr

ve

d. 3. A n !

c
ntinuous
term

t is c al le d ground

term

if t 2 T(; ;) . Example 6.3.8 (!

Con

tin uous

terms)

Let

=

STREAM, stream = s, X (s) = fxg ? s & ? s x & & ? s x x . . . & & & x x x t t 1 t 2 t 3 The illustrated terms are dened b y t =

7!

? s t 1 =

7!

&; 11 7! ? s ; 12 7! x t 2 =

7!

&; 11 7! &; 12; 112 7! x; 111 7! ? s t 3 =

7!

&; 11; 111 7! &; 12; 112; 11 12 7! x ; 1111 7! ? s 2 Prop

sition

6.3.9 The !

c
ntinuous
terms

T(; X ) 1 (s) ar e !

c
mplete

for every s 2 S with r esp e ct to the fol lowing p artial

r

der v s : t 1 v s t 2 : ( ) def (t 1 )

def

(t 2 ); t 1 (v ) v sy m;s i t 2 (v ) for every v 2 def (t 1 ) with t 1 (v ) 2 Sym(s i ). Pro

f.

Eviden tly , v s denes a partial

rder.

No w, the pro

f

is divided in to three steps.

Denition
f

sup(G): Let G

T(;

X ) 1 (s) b e coun table directed. Dene sup(G) b y def (sup(G)) = [ t2G def (t) and sup (G )(v ) = max ft(v ) j t 2 G with v 2 def (t)g for ev ery v 2 def (sup(G )).

sup(G

) is w ell-dened: Since G is directed and v sy m;s is a at

rder

for ev ery s 2 S w e conclude for ev ery v 2 def (sup(G )): t 1 (v ) 6= ? s i ; t 2 (v ) 6= ? s i = ) t 1 (v ) = t 2 (v ), for all t 1 , t 2 2 G with v 2 def (t 1 ) \ def (t 2 ). Therefore is sup (G ) w ell-dened and sup (G ) 2 T(; X ) 1 (s).

sup(G

) is least upp er b

und:

By denition is def (t)

def

(sup(G)) and t (v ) v sy m;s i sup(G )(v ) for ev ery t 2 G and v 2 I N ? + with v 2 def (t) . By this means is sup(G) upp er b

und

for G. Moreo v er, sup (G ) is less than an y upp er b

und

b y construction. 44

SLIDE 49 2 The partial

rder
n

!

con

tin uous

terms

together with the canonical denition

f

the

p

erations

n

terms yield !

con

tin uous

p

erations and therefore the mo del

f

!

con

tin uous

terms.

Denition 6.3.10 (Conditional

Inequalit

y) Let

=

(S ; O P ) b e a signature and X a

v

ariable-sytem. A conditional

inequalit

y ciq

v

er X is ciq = (X : l 1

r

1 ; : : : ; l n

r

n = ) l

r

), where l i , r i 2 T(; X ) 1 (s i ), s i 2 S i , for i = 1; : : : ; n and l, r 2 T( ; X ) 1 (s), s 2 S . W e denote the set

f

conditional

inequalities
v

er X b y C I Q C A ( ; X ) and the set

f

all conditional

inequalities

b y C I Q C A ( ), where C I Q C A () = [ X

C

I Q C A ( ; X ) : Remark 6.3.11 Note, that

e

qualities (X : l = r ) and c

nditional
ine

qualities with empty pr emise (X : l

r

) , c al le d

ine

qualities, ar e c

ver

e d. 6.3.2 The T ranslation

f

Sen tences S en C A () Next w e dene for ev ery signature morphism a corresp

nding

translation

f

conditional

inequalities,

based up

n

the translation

f

!

con

tin uous

terms.

Denition 6.3.12 (Finite Appro ximation

f

!

Con

tin uous

terms)

Let t 2 T(; X ) 1 b e an !

con

tin uous

term.

W e dene the set

f

nite appro ximations

f

t, Fin(t ) , b y Fin(t) = fr v t j jdef (r)j < 1g. Lemma 6.3.13 Fin(t) is dir e cte d for every t 2 T(; X ) 1 (s). Pro

f.

Let t 1 , t 2 2 Fin(t). Dene r 2 T(; X ) 1 (s) b y def (r ) = def (t 1 ) [ def (t 2 ) and r (v ) = 8 > > > < > > > : max(t 1 (v ); t 2 (v )) ; v 2 def (t 1 ) \ def (t 2 ); t 1 (v ) ; v 2 def (t 1 ) n def (t 2 ); t 2 (v ) ; v 2 def (t 2 ) n def (t 1 ); undened ; else: Clearly r is w ell-dened and upp er b

und

for t 1 and t 2 . F urther r is nite and b y denition r v t, b ecause t 1 v t and t 2 v t . Hence is r 2 Fin(t ). 2 Lemma 6.3.14 F

r

every t 2 T(; X ) 1 (s) holds: t = sup (Fin(t) ). Pro

f.

Apparen tly holds sup(Fin(t)) v t, since t is an upp er b

und

for Fin(t). W e dene an !

c

hain (t n j n 2 I N)

f

nite cuts b y t n (v ) = 8 > > > < > > > : t(v ) ; jv j

n
1;

t(v ) ; jv j = n; t(v ) =

p

with

p

:

!

s; ? s i ; jv j = n; v = v i; t(v ) =

p

with

p

: s 1 : : : s n ! s; i

n;

undened ; else: 45

SLIDE 50 for ev ery n 2 I N and v 2 I N ? + . By denition w e ha v e t n v t n+1 and t n 2 Fin(t) for ev ery n 2 I N. Since T(; X ) 1 (s) is !

complete

exists sup (t n j n 2 I N). By denition is t = sup (t n j n 2 I N). Since (t n j n 2 I N)

Fin(t

) w e ha v e t = sup (t n j n 2 I N) v sup (Fin(t) ). Summarizing holds t = sup(Fin(t)). 2 Denition 6.3.15 (T erm T ranslation) Giv en a signature morphism

:
1

!

2

and a

1
v

ariable-system X 1

f

v ariables, there is an induced

2
v

ariable-system X 2

f

v ariables dened b y X 2 (s 2 ) = [ (s 1 )=s 2 X 1 (s 1 ) for ev ery s 2 2 S 2 : Hence follo ws the translation

f

!

con

tin uous

1
terms

as an S 1

indexed

map : T( 1 ; X 1 ) 1 ! T( 2 ; X 2 ) 1 ,

=

((s 1 ): T( 1 ; X 1 ) 1 (s 1 ) ! T( 2 ; X 2 ) 1 ((s 1 ))), dened for ev ery s 1 2 S 1 , t 2 T( 1 ; X 1 ) 1 (s 1 ) and v 2 I N ? + b y (s 1 )(t)(v ) = 8 > > > < > > > : ? (s 1 ) ; t(v ) = ? s 1 ; x ; t(v ) = x; x 2 X (s 1 ); (op 1 ) ; t(v ) =

p

1 ;

p

1 2 O P 1 ; undened ; else : Remark 6.3.16 1. Note, that by denition and X 1

T(

1 ; X 1 ) 1 it is p

ssible

to write X 2 = (X 1 ) F urther (X 1 ) satises the

2
variable-system

c

nditions,

i.e. variables

f

dier ent sorts ar e distinguishable. 2. The denition

f

!

c
ntinuous
1
terms

is wel l-dene d, i.e. the image (t) is an !

c
ntinuous

2

term,

sinc e by denition

f

signatur e morphisms the c

nditions
ne

and thr e e

f

the term denition ar e satise d and by denition

f

term tr anslation def (t) = def ( (t)) holds. Ther efor e the domain

f

(t) is pr ex-close d. F urther ar e images

f

innite terms again innite, as exp e cte d. The structur e is inherite d,

nly

the names change d. 3. F

r

nite terms the tr anslation

f

terms is exactly as usual. With the translation

f

terms w e are in p

sition

to dene the translation

f

form ulas, in particular

f

conditional

inequalities:

The last missing denition to determine the sen tence functor. Denition 6.3.17 (T ranslation

f

F

rm

ulas) Let

:
1

!

2

b e a signature morphism and ciq = (X 1 : l 1

r

1 ; : : : ; l n

r

n = ) l

r

) a conditional

1
inequalit

y . The translation with resp ect to

is

giv en b y (ciq ) = ((X 1 ) : (l 1 )

(r

1 ); : : : ;

(l

n )

(r

n ) = ) (l )

(r

)) : Note, that (ciq ) is clearly a

2
form

ula. Denition 6.3.18 (The Sen tence F unctor S en C A ) Dene the sen tence functor S en C A : S I GN ! S E T b y 46

SLIDE 51 S en C A () = C I Q C A ( ) for ev ery signature , S en C A () =

for

ev ery signature morphism long, where

is

dened as in denition 6.3.17. Prop

sition

6.3.19 S en C A : S I GN ! S E T is a functor. Pro

f.

F

llo

ws clearly b y denition

f
n

terms and form ulas. 2 6.4 The Satisfaction Condition In this section w e consider the satisfaction relation b et w een mo dels and conditional inequalities. F urther w e pro v e the compatibilit y

f

term translation and mo del translation resp ecting this relation. Firstly w e dene the ev aluation

f

terms in a giv en mo del based

n

an assignmen t to v ariables. This is done almost as usual. Only for innite terms there are some quite naturally extensions. Denition 6.4.1 (Assignmen t

f

V ariables) Giv en an !

con

tin uous

mo

del A and a

v

ariable-system X . An assignmen t a: X ! A(S ) for X in A(S ) is an S {indexed family

f

maps a = (a(s ) : X (s ) ! A(s) j s 2 S ). Denition 6.4.2 (Ev aluation

f

!

Con

tin uous

terms)

Giv en an !

con

tin uous

mo

del A, a

v

ariable-system X and an assignmen t a: X ! A(S ), w e dene the ev aluation

f

!

con

tin uous

terms

a: T( ; X ) 1 ! A (S ) as an S {indexed family

f

maps a = (a(s) : T(; X ) 1 (s) ! A(s) j s 2 S ) dened for ev ery s 2 S , t 2 T(; X ) 1 (s) as follo ws: 1. t nite: (with usual denotation

f

nite terms)

a(s)(?

s ) = ? A(s) ,

a(s)(x

) = a(s)(x) for ev ery x 2 X (s),

a(s)(op

) = A(op) for ev ery

p

:

!

s 2 O P ,

a(s)(op

(r)) = A(op)(a (w)(r )) for ev ery

p

: w ! s 2 O P and r 2 T(; X ) 1 (w ) with jr j < 1. 2. t innite: a(s)(t ) = sup a(s)(Fin(t) ), where a(s)(Fin(t)) = fa(s)(r) j r 2 Fin(t ) g: Prop

sition

6.4.3 The evaluation

f

!

c
ntinuous
terms

is wel l-dene d. Pro

f.

F

r

nite terms this is w ell-kno wn. F

r

innite terms t 2 T(; X ) 1 (s) w e ha v e to sho w that a(s)(Fin(t)) is coun table directed (see prop

sition

A.1.12). The coun tabilit y is clear. Eviden tly , a(s)(Fin(t)) is directed, if a(s) is monoton for nite terms, since Fin(t ) is directed b y lemma 6.3.13. Let t 1 v t 2 for t 1 , t 2 2 Fin(t ), i.e. t 1 (v ) v sy m;s i t 2 (v ) for ev ery v 2 def (t 1 ), where t 1 (v ) 2 f? s i ; x ;

p

j x 2 X (s i ); co d(op ) = s i g. Then w e ha v e t 1 (v ) = ? s i

r

t 1 (v ) = t 2 (v ) for ev ery v 2 def (t 1 ). By this means and a(s i )(? s i ) = ? A(s i ) follo ws the monoton y

f

a. 2 47

SLIDE 52 The ev aluation is also w ell-dened in the sense that it satises the !

con

tin uous

morphism

conditions. F urther the ev aluation will turn

ut

to b e the

nly

existing morphism with domain T(; ;) . So, T(; ;) is initial in M

d

C A () . Ho w ev er, this w asn't a pro

f,
nly

a notew

rth

y remark. Next w e consider the satisfaction relation b et w een !

con

tin uous !

con

tin uous

mo

dels and conditional

inequalities.

The denition is divided in to t w

steps.

The rst part constitutes this relation for a sub class

f

form ulas, for

inequalities,

i.e. conditional

inequalities

with empt y premise. By this means it is easy to dene the relation for conditional

inequalities.

Denition 6.4.4 (Satisfaction Relation) F

r

ev ery signature

=

(S ; O P ) there is a satisfaction relation j =

jM
d

C A ( )j

S

en C A ( ) based

n

v alidit y

f

assignmen ts for

inequalities

giv en b y A , a j =

(X

: l

r

) ( ) a(s)(l) v A(s) a(s)(r) for ev ery A 2 jM

d

C A ()j , for ev ery assignmen t a: X ! A(S ) and (X : l

r

) 2 S en C A ( ). Hence w e dene j =

for

conditional

inequalities

b y A j =

(X

: l 1

r

1 ; : : : ; l n

r

n = ) l

r

) ( ) V n i=1 A; a j =

(X

: l i

r

i ) = ) A ; a j =

(X

: l

r

); for ev ery assignmen t a : X ! A(S ) for ev ery ev ery conditional

inequalit

y (X : l 1

r

1 ; : : : ; l n

r

n = ) l

r

) 2 S en C A () and A 2 jM

d

C A () j. There remains to sho w the compatibilit y

f

satisfaction relation and translation

f

mo dels resp ectiv ely sen tences. As w e will see the main

bserv

ation is a bijectiv e dep endence b et w een the assignmen ts, resulting in a bijectiv e dep endence b et w een the ev aluations. Denition 6.4.5 (Corresp

nding

Assignmen ts) Giv en a signature morphism

:
1

!

2

, an !

con

tin uous

2
mo

del A 2 , a

1
v

ariable-system X 1 and the corresp

nding
2
v

ariable-system (X 1 ) as dened in denition 6.3.15. F

r

ev ery assignmen t a 1 : X 1 ! M

d

C A () (A 2 )(S 1 ) there is an assignmen t a 2 : (X 1 ) ! A 2 (S 2 ) dened b y a 2 (s 2 )(x ) = a 1 (s 1 )(x ) for ev ery s 2 2 S 2 , where s 1 2 S 1 with (s 1 ) = s 2 and x 2 X 1 (s 1 )

(X

1 )(s 2 ). Con v ersely for ev ery assignmen t a 2 : (X 1 ) ! A 2 (S 2 ) there is an assignmen t a 1 : X 1 ! M

d

C A () (A 2 )(S 1 ) dened b y a 1 (s 1 )(x) = a 2 ((s 1 ))(x ) for ev ery s 1 2 S 1 , where x 2 X 1 (s 1 )

(X

1 )((s 1 )). (a 1 ; a 2 ) is called pair

f

corresp

nding

assignmen ts. F

r

ev ery pair

f

corresp

nding

assignmen ts w e are able to pro v e that the ev aluations yield the same v alues, since the nite appro ximations

f

terms are preserv ed b y term translation resp ecting an arbitrary signature morphism, what is expressed in the follo wing lemma. Lemma 6.4.6 F

r

every signatur e morphism

:
1

!

2

and t 2 T( 1 ; X 1 ) 1 (s 1 ), s 1 2 S 1 , we have 48

SLIDE 53 (s 1 )(Fin(t)) = Fin((s 1 )(t )). Pro

f.

W e

mit

the simple pro

f.

2 Lemma 6.4.7 (Corresp

nding

Ev aluations) Given a signatur e morphism

:
1

!

2

, an !

c
ntinuous
2
mo

del A 2 , a

1
variable-system

X 1 and a p air

f

c

rr

esp

nding

assignments (a 1 ; a 2 ). F

r

every s 1 2 S 1 and t 2 T( 1 ; X 1 ) 1 (s 1 ) and every assignment a 1 : X 1 ! M

d

C A () (A 2 )(S 1 ) we have a 1 (s 1 )(t) = a 2 ((s 1 ))((s 1 )(t)): (a 1 ; a 2 ) is c al le d pair

f

corresp

nding

ev aluations. Pro

f.

1. Finite terms:

t

= ? s 1 : a 1 (s 1 )(? s 1 ) = ? M

d

C A ()(A 2 )(s 1 ) (def. a 1 ) = ? A 2 ((s 1 )) (def. M

d

C A () ) = a 2 ((s 1 ))(? (s 1 ) ) (def. a 2 ) = a 2 ((s 1 ))( (s 1 )(? s 1 ))) (def. )

t

= x, x 2 X (s 1 ): a 1 (s 1 )(x) = a 1 (s 1 )(x ) (def. a 1 ) = a 2 ((s 1 ))(x ) (def. a 2 ) = a 2 ((s 1 ))( (s 1 )(x)) (def.

and

a 2 )

t

=

p

1 ,

p

1 :

!

s 1 : a 1 (s 1 )(op 1 ) = M

d

C A ()(A 2 )(op 1 ) (def. a 1 ) = A 2 ((op 1 )) (def. M

d

C A ()) = a 2 ((s 1 ))( (s 1 )(op 1 )) (def.

and

a 2 )

t

=

p

1 (r ),

p

1 : w 1 ! s 1 , r 2 T( ; X ) 1 (w 1 ), jrj < 1: a 1 (s 1 )(op 1 (r)) = M

d

C A ()(A 2 )(op 1 )(a 1 (w 1 )(r)) (def. a 1 ) = M

d

C A ()(A 2 )(op 1 )(a 2 ((w 1 ))( (w 1 )(r))) (induction) = A 2 ((op 1 ))(a 2 ((w 1 ))( (w 1 )(r))) (def. M

d

C A ()) = a 2 ((s 1 ))((op 1 )((w 1 )(r))) (def. a 2 ) = a 2 ((s 1 ))( (s 1 )(op 1 (r))) (def. ) 2. Innite terms: a 1 (s 1 )(t) = sup a 1 (s 1 )(Fin(t)) (def. a 1 ) = sup a 2 ((s 1 ))( (s 1 )(Fin(t) )) (induction) = sup a 2 ((s 1 ))(Fin( (s 1 )(t)) ) (lemma 6.4.6) = a 2 ((s 1 ))( (s 1 )(t)) (def. a 2 ) 2 Theorem 6.4.8 (Satisfaction Condition) F

r

every signatur e morphism

:
1

!

2

, every !

c
ntinuous
2
mo

del A 2 , and every c

ndi-

tional

1
ine

quality ciq = (X 1 : l 1

r

1 ; : : : ; l n

r

n = ) l

r

) we have A 2 j =

2

S en C A ()(ciq) ( ) M

d

C A ()(A 2 ) j =

1

ciq : 49

SLIDE 54 Pro

f.

F

llo

ws directly b y the denition

f

the forgetful functor M

d

C A () , the denition

f

the translation

f

conditional

inequalities

S en C A () =

and

lemma 6.4.7. 2 Corollary 6.4.9 The c ate gories S I GN , S E T , C AT

p

with the functors M

d

C A (se e c

r
l

lary 6:2:8) and S en C A (se e pr

p
sition

6:3:19) form an institution in the sense

f

[GB92], c al le d the institution

f

con tin uous algebras with conditional inequalities. 50

SLIDE 55 App endix A P artial Ordered Sets Denition A.1.1 (P artial Order, T

tal

Order) A partial

rder

(p

set)

is a pair (M ; v M ), where M is a set and v M

M
M

is a reexiv e, (8 x 2 M : x v M x), an tisymmetric, (8 x,y 2 M : x v M y , y v M x ) y = x) and transitiv e, (8 x, y , z 2 M : x v M y , y v M z ) x v M z ). relation

n

M . A partial

rder

(M ; v M ) is a total

rder,

if for all x, y 2 M holds x v M y

r

y v M x. Denition A.1.2 (Flat Order) Let M b e an arbitrary set and ? M = 2 M . Dene canonical the at

rder
n

M [ f? M g b y x v y ( ) x = ? M

r

x = y , for all x, y 2 M [ f? M g . Denition A.1.3 (Cartesian Pro duct) Let (M i ; v M i ), i = 1, : : : ; n b e partial

rders.

The cartesian pro duct (M ; v M ) is dened b y

M

= M 1

:

: :

M

n ,

x

v M y ( ) x i v M i y i , for i = 1,: : : ; n , for ev ery x = (x 1 ; : : : ; x n ) , y = (y 1 ; : : : ; y n ) 2 M . It is easy to see that the cartesian pro duct

f

p

sets

yields a p

set.

Denition A.1.4 Let (M ; v M ) b e a partial

rder,

X

M

. W e dene that x 2 M is upp er b

und
f

X , if y v M x, for all y 2 X , lo w er b

und
f

X , if x v M y , for all y 2 X , least upp er b

und
f

X , if x is upp er b

und

and for all upp er b

unds

y 2 M hold: x v M y , greatest lo w er b

und
f

X , if x is upp er b

und

and for all upp er b

unds

y 2 M hold: y v M x, maxim um

f

X , if x is least upp er b

und
f

X and x 2 X , minim um

f

X , if x is greatest lo w er b

und
f

X and x 2 X . 51

SLIDE 56 W e denote ev ery least upp er b

und

for X b y sup M (X ), ev ery greatest lo w er b

und

for X b y inf M (X ), ev ery maxim um for X b y max M (X ) and ev ery minim um for X b y min M (X ). Lik e usual w e

mit

the indices if they are giv en b y the con text. F urther the set

f

all upp er b

unds
f

X is denoted b y X u and the set

f

all lo w er b

unds
f

X b y X l . Denition A.1.5 (W ell-Order) A relation v

n

a set M is a w ell-order if for ev ery non-empt y subset X

M

the minim um min (X ) exists. In particular ev ery w ell-order is a total

rder.

Denition A.1.6 (Subsets

f

a P artial Order) Let (M ; v M ) b e a partial

rder.

A subset C

M

is called

c

hain if C is a total

rder,
!
c

hain if C is a c hain suc h that for ev ery x 2 C

nly

for a nite n um b er

f

elemen ts y 2 C holds: y v M x.

directed

set if C 6= ; and for all x, y 2 C exist z 2 C with x v M z and y v M z . Remark A.1.7 1. Evidently every !

chain

is a chain and every chain is a dir e cte d set. 2. We c al l (M ; v M ) strict, if sup (;) = ? M exists. Denition A.1.8 (!

Cp
)

A partial

rder

(M ; v M ) is called !

cp
i

for ev ery !

c

hain C the least upp er b

und

sup (C ) exists. Remark A.1.9 1. Note, that for every !

cp
exists

the le ast element ? M , sinc e the empty set is an !

chain

(sup(;) = ? M ). 2. Every at

r

der is an !

cp
sinc

e chains c

ntain

at most two elements. Denition A.1.10 (Sub-!

Cp
)

Let (M ; v M ) b e an !

cp
and

X

M

. Then X is a sub-!

cp
f

M , if for ev ery !

c

hain C

X

the least upp er b

und

sup X (C ) exists suc h that sup X (C ) = sup M (C ). Remark A.1.11 Mention that ? X = ? M sinc e ; is an !

chain.

Prop

sition

A.1.12 (Characterisations

f

!

cp
's)

L et (M ; v M ) b e a p artial

r

der. Then the fol lowing c

nditions

ar e e quivalent: 1. F

r

every !

chain

in M exists the le ast upp er b

und.

2. F

r

every c

untable

chain in M exists the le ast upp er b

und.

3. F

r

every c

untable

dir e cte d subset

f

M exists the le ast upp er b

und.

52

SLIDE 57 Pro

f.

3) = ) 2): Eviden tly b y remark A.1.7. 2) = ) 1): Eviden tly b y remark A.1.7. 1) = ) 3): Let D

M

b e a coun table directed set and assume the existence

f

sup (C ) for ev ery !

c

hain C

M

. Cho

se

a w ell-order

n

D , i.e. D = fx n g n2 I N and x n+1 = min (G nfx ; : : : ; x n g). Denote an upp er b

und

(w.r.t. v M ) for nite X

M

b y u X . De- ne an !

c

hain C

M

and an !

c

hains G n , C n

M

inductiv ely b y

C

= G = fx g,

G

n+1 = G n [ fy n+1 ; u G n [fy n+1 g g, where y n+1 = min(G nG n ) w.r.t. ,

C

n+1 = C n [ u G n [fy n+1 g . By denition follo ws for ev ery n 2 I N

y

n

y

n+1 , y n 6= y n+1 ,

G

n

G

n+1 , G n nite directed w.r.t. v M ,

C

n

C

n+1 , C n nite c hain,

F
r

ev ery y 2 G exists a n 2 I N with y v M max C n . Therefore w e ha v e C = [ n2I N C n is an !

c

hain with sup C v M u for ev ery upp er b

und

u 2 G u , since C

G.

y v M sup C for ev ery y 2 G, since y v M max C n for appropriate n2 I N . Summarizing is sup C = sup G, i.e. for ev ery G

M

coun table directed exists the least upp er b

und.

2 Prop

sition

A.1.13 (Cartesian Pro duct

f

!

Cp
's)

The c artesian pr

duct

(M ; v M ) as dene d in denition A:1:3 is an !

cp
i

(M i ; v M i ) is an !

cp
for

every i = 1, : : : ; n: Pro

f.

The pro

f

is straigh t forw ard and therefore

mitted.

2 Denition A.1.14 (F unctions

n

Cp

's)

Let (M 1 ; v M 1 ) and (M 2 ; v M 2 ) b e partial

rders

and f : M 1 ! M 2 . Then f is monoton :( ) F

r

ev ery nite c hain C holds: sup f (C ) exists and f (sup(C )) = sup f (C ) f is !

con

tin uous :( ) (M 1 ; v M 1 ) and (M 2 ; v M 2 ) are !

cp
's

and for ev ery !

c

hain C holds: sup f (C ) exists and f (sup(C )) = sup f (C ) P a y atten tion to the denition

f

monoton functions. It is exactly as usual, i.e. f monoton :( ) x 1 v M 1 x 2 = ) f (x 1 ) v M 2 f (x 2 ) for all x 1 , x 2 2 M 1 , 53

SLIDE 58 b ecause fx 1 , x 2 g with x 1 v x 2 is a nite c hain. Remark A.1.15 Every !

c
ntinuous

function is monoton. Prop

sition

A.1.16 (Comp

sition)

L et (M 1 ; v M 1 ) and (M 2 ; v M 2 ) b e !

cp
's

and f : M 1 ! M 2 , g : M 2 ! M 3 b e !

c
ntinuous

functions. Then g f is !

c
ntinuous.

Pro

f.

Let C

M

1 b e an !

c

hain. Since f is monoton w e conclude that f (C ) is an !

c

hain. Then follo ws b y denition A.1.14 g (f (sup(C ))) = g (sup (f (C ))) (f !

con

tin uous) = sup (g (f (C ))) (g !

con

tin uous and f (C ) is c hain) 2 54

SLIDE 59 Biblio graph y [A C92] E. Astesiano and M. Cerioli. Relationships b et w een Logical Framew

rks.

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T ec hnisc he Univ ersit at M

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for Computing Scienc e. Series in Computer Science. Pren tice Hall In ternational, London, 1990. [Cer93] Maura Cerioli. R elationships b etwe en Lo gic al Formalisms. PhD thesis, Univ ersit a di Pisa{Geno v a{Udine, 1993. TD-4/93. [CKL93] F elix Cornelius, Marcus Klar, and Mic hael L

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KORSO: Ist{Analyse HDMS-A. T ec hnical Rep

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