NonT erminating Pro cesses in the Situation Calculus Giusepp - - PDF document

SLIDE 1 NonT erminating Pro cesses in the Situation Calculus Giusepp e De Giacomo Dipartimen to di Informatica e Sistemistica Univ ersit a di Roma La Sapienza degiacomodisuniromait Eugenia T erno vsk aia Departmen t

• f

Computer Science Univ ersit y

• f

T

• ron

to eugeniacstorontoedu Ra y Reiter Departmen t

• f

Computer Science Univ ersit y

• f

T

• ron

to reitercstorontoedu

• In

tro duction By their v ery design man y rob

• t

con trol programs are nonterminating T

• giv

e a simple example

• ne

w e shall use in this pap er

• an
• ce

coeedeliv ery rob

• t

migh t b e implemen ted as an innite lo

• p

in whic h the rob

• t

resp

• nds

to exogenous requests for coee that are main tained

• n

a queue Since a future coee request is alw a ys p

• ssible

the program nev er terminates As is the case for more con v en tional programs w e w an t some reliabilit y assurances for rob

• t

con trollers This pap er describ es the approac h b eing tak en b y

• ur

Cognitiv e Rob

• tics

Group to expressing and pro ving prop erties

• f

nonterminating programs expressed in GOLOG a high lev el logic program ming language for mo deling and implemen ting dynamical systems The kinds

• f

prop erties w e ha v e in mind are traditional in computer science liv eness fairness etc W e dier from the classical approac hes LS

• Cou
• MP
• for

reasons dictated b y the follo wing c haracteristics

• f

GOLOG

• T
• write

a GOLOG program the programmer rst axiomatizes the primitiv e actions

• f

the appli cation domain using rst

• rder

logic These actions ma y also include exogenous ev en ts

• Next

she describ es in GOLOG the complex b eha viors her rob

• t

is to exhibit in this domain This GOLOG program is in terpreted b y means

• f

a form ula this time in second

• rder

logic

• Finally
• a

suitable theorempro v er executes the program Because these features are all represen ted in classical second

• rder

logic it is natural to express and pro v e prop erties

• f

GOLOG programs including nonterminating

• nes

in the v ery same logic This approac h to program pro

• fs

has the adv an tage

• f

logical uniformit y and the a v ailabilit y

• f

classical pro

• f

theory

• It

also pro vides a v ery ric h language with whic h to express program prop erties as w e shall see in this pap er Moreo v er it pro vides for pro

• fs
• f

programs with incomplete initial state the normal situation in rob

• tics

where the agen t do es not ha v e complete information ab

• ut

the w

• rld

it inhabits Finally

• this

approac h gracefully accommo dates exogenous ev en t

• ccurrences

and pro

• fs
• f

program prop erties in their presence

• F
• rmal

Preliminaries

• The

Situation Calculus The situation calculus is a second

• rder

language sp ecically designed for represen ting dynamically c hang ing w

• rlds

All c hanges to the w

• rld

are the result

• f

named actions A p

• ssible

w

• rld

history

• whic

h is simply a sequence

• f

actions is represen ted b y a rst

• rder

term called a situation The constan t S

• is

used to denote the initial situation namely the empt y history

• There

is a distinguished binary function sym b

• l

do do s denotes the successor situation to s resulting from p erforming the action

• Actions

ma y b e parameterized F

• r

example putx y

• migh

t stand for the action

• f

putting

• b

ject x

• n
• b

ject y

• in

whic h case doputA B

• s

denotes that situation resulting from placing A

• n

B when the history is s Notice that in the situation calculus actions are denoted b y rst

• rder

terms and situations w

• rld

histories are also rst

• rder

terms F

• r

example doputdow nA dow al k L dopick upA S

SLIDE 2 is a situation denoting the w

• rld

history consisting

• f

the sequence

• f

actions pic kupA w alkL put do wnA Notice that the sequence

• f

actions in a history

• in

the

• rder

in whic h they

• ccur

is

• btained

from a situation term b y reading

• the

actions from righ t to left The situation calculus has a distin guished predicate sym b

• l

Poss the in tended meaning

• f

P

• ssa

s is that it is p

• ssible

to p erform the action a in situation s Relations functions whose truth v alues function v alues v ary from situation to situation are called r elational functional uents

• They

are denoted b y predicate function sym b

• ls

taking a situation term as their last argumen t F

• r

example hasC

• f

f ee p s is a relational uen t whose in tended meaning is that p erson p has coee in situation s r

• botLocations

is a functional uen t denoting the rob

• ts

lo cation in situation s When formalizing an application domain

• ne

m ust sp ecify certain axioms

• A

ction pr e c

• ndition

axioms

• ne

for eac h primitiv e action These c haracterize the relation P

• ss

and giv e the preconditions for the p erformance

• f

an action in a situation In a rob

• t

coee deliv ery setting suc h an axiom migh t b e P

• ssg

iv eC

• f

f ee per son s

• hol

ding C

• f

f ee s

• r
• botLocations
• f

f ice per son This sa ys that the preconditions for the rob

• t

to giv e coee to p erson p are that the rob

• t

is carrying coee and the rob

• ts

lo cation is ps

• ce
• Suc

c essor state axioms

• ne

for eac h uen t These capture the causal la ws

• f

the domain together with a solution to the frame problem Rei F

• r
• ur

coee deliv ery rob

• t

the follo wing is an example P

• ssa

s

• hol

ding C

• f

f ee doa s

• a
• pick

upC

• f

f ee

• hol

ding C

• f

f ee s

• per

sona

• g

iv eC

• f

f ee per son In

• ther

w

• rds

pro vided the action a is p

• ssible

the rob

• t

will b e holding a cup

• f

coee after action a is p erformed i a is the action

• f

the rob

• t

pic king up the coee

• r

the rob

• t

is already holding coee and a is not the action

• f

the rob

• t

giving that coee to someone

• Unique

names axioms for the primitiv e actions stating that dieren t names for actions denote dieren t actions

• Axioms

describing the initial situation

• what

is true initially

• b

efore an y actions ha v e

• ccurred

This is an y nite set

• f

sen tences whic h men tion no situation term

• r
• nly

the situation term S

• Examples
• f

axioms for the initial situation for

• ur

coee deliv ery example are phasC

• f

f ee p S

• r
• botLocationS
• C

M

• These

ha v e the in tended reading that initially

• no
• ne

has coee and the rob

• t

is lo cated at the coee mac hine C M

• See

LRL

• for

a full description

• GOLOG

GOLOG LRL

• is

a situation calculusbased logic programming language that allo ws for dening complex actions using a rep ertoire

• f

user sp ecied primitiv e actions GOLOG pro vides the usual kinds

• f

imp erativ e programming language con trol structures as w ell as v arious forms

• f

nondeterminism Briey

• GO

LO G programs are formed b y using the follo wing constructs

• Primitive

actions a Do action a in the curren t situation Actually a is a pseudoaction

• btained

from an action b y suppressing the situation argumen t in eac h functional uen t The function as that giv en a pseudoaction a and a situation s returns the

• riginal

action see LRL

• T

est actions

• T

est the truth v alue

• f

expression

• in

the curren t situation As for primitiv e actions

• is

a pseudoform ula

• btained

from a situation calculus form ula b y suppressing all situation argumen ts The function s that giv en a pseudoform ula

• and

a situation s returns the

• riginal

form ula

• Se

quenc e

• Execute

program

• follo

w ed b y program

SLIDE 3

• Nondeterministic

action choic e

• j
• Execute
• r
• Nondeterministic

choic e

• f

ar guments

• z
• Nondeterministically

pic k a v alue for z

• and

for that v alue

• f

z

• execute

program

• Nondeterministic

r ep etition

• Execute
• a

nondeterministic n um b er

• f

times

• While

lo

• ps

while

• do
• endWhile
• whic

h is expressed as

• Conditionals

if

• then
• else
• whic

h is expressed as

• j
• Pr
• c

e dur es including recursion pro c P r

• cN

ame

• v
• P

r

• cN

ame endPro c

• Single

step seman tics for GOLOG In LRL

• GOLOG

programs are in terpreted b y means

• f

a sp ecial relation D

• s

s

• that

giv en a generally nondeterministic program

• and

a situation s returns a p

• ssible

situation s

• resulting

b y executing

• starting

from s Actually in LRL

• the

relation D

• is

not denoted b y a predicate but instead it is dened implicitly b y using macr

• s

exp ansion rules suc h as D

• s

s

• def
• s
• D
• s

s

• D
• s
• s
• D
• j
• s

s

• def
• D
• s

s

• D
• s

s

• D
• s

s

• def
• P
• P

s s

• where
• stands

for the conjunction

• f

s sP s s s s

• s
• P

s s

• D
• s
• s
• P

s s

• ne

for eac h construct in the language By using suc h macro expansions rules the relation D

• s

s

• for

the particular program

• is

dened b y a generally second

• rder

form ula

• s

s

• not

men tioning

• at

all This is v ery con v enien t since it completely a v

• ids

the in tro duction

• f

programs in to the language they are used

• nly

during the macro expansion pro cess to get the form ulas

• s

s

• corresp
• nding

to D

• s

s

• Observ

e ho w ev er that in this w a y programs cannot b e quan tied

• v

er b ecause they are not terms

• f

the language

• f

the situation calculus The kind

• f

seman tics D

• asso

ciates to programs whic h is based

• n

the complete ev aluation

• f

the program is sometimes called evaluation semantics Hen

• Suc

h a seman tics is not w ell suited to in terpret nonterminating programs lik e innite lo

• ps

since for suc h programs the ev aluation can nev er b e completed and a nal situation can nev er b e reac hed F

• r

nonterminating programs

• ne

needs to rely

• n

a seman tics that allo ws for in terpreting se gments

• f

pr

• gr

am exe cutions So w e adopt a kind

• f

seman tics called c

• mputational

semantics Hen whic h is based

• n

single steps

• f

computation

• r

tr ansitions

• A

step here is either a primitiv e

• r

a test action W e b egin b y in tro ducing t w

• sp

ecial relations Final and T r ans

• Final
• is

in tended to sa y that program

• is

in a nal state ie it ma y legally terminate in the curren t situation T r ans

• s
• s
• is

in tended to sa y that program

• in

situation s ma y legally execute

• ne

step ending in situation s

• with

program

• remaining

T

• follo

w this approac h it is necessary to quan tify

• v

er programs and so unlik e in LRL

• w

e need to enco de GO LO G programs as rstorder terms including in tro ducing constan ts denoting v ariables and so

• n

This is lab

• rious

but quite straigh tforw ard Lei

• W

e

• mit

all suc h details here and simply use programs within form ulas as if they w ere already rstorder terms Final and T r ans are denoted b y predicates dened inductiv ely

• n

the structure

• f

the rst argumen t It is con v enien t to include a sp ecial empt y program

• denoting

that nothing

• f

the program remains to b e p erformed

• Both

t yp es

• f

seman tics b elong to the family

• f

structural

• p

erational seman tics in tro duced in Plo

• W

e assume that the predicates in tro duced in this section including Final and T r ans

• cannot
• ccur

in tests hence disallo wing selfreference

SLIDE 4 The denition

• f

Final is as follo ws

• Final
• F
• F
• where
• stands

for the conjunction

• f

the univ ersal closure

• f

the follo wing clauses F

• F
• F
• F
• F
• F
• F
• j
• F
• F
• z
• F
• F
• P

r

• cN

ame

• F

P r

• cN

ame

• x
• Observ

e that b eing nal is a syn tactic prop ert y

• f

programs programs

• f

a certain form are considered to b e in a nal state Moreo v er b eing nal do es not dep end

• n

the

• b

jects the program deals with indeed Final

• z
• and

Final P r

• cN

ame

• x
• dep

end

• nly
• n
• and
• P

r

• cN

ame and not

• n

the particular v alues

• f

z and

• x

resp ectiv ely

• Observ

e that from the ab

• v

e denition w e get that primitiv e and test actions are nev er nal for all actions a Final a

• F

alse and for all tests

• Final
• F

alse

• The

denition

• f

T r ans is as follo ws

• s
• s
• T

r ans

• s
• s
• T
• T
• s
• s
• where
• stands

for the conjunction

• f

the univ ersal closure

• f

the follo wing clauses Poss as s

• T

a s

• doas

s s

• T
• s
• s

T

• s
• s
• T
• s
• s
• Final
• T
• s
• s
• T
• s
• s
• T
• s
• s
• T
• j
• s
• s
• T
• s
• s
• T
• j
• s
• s
• y

T

• z

y

• s
• s
• T
• z
• x

s

• s
• T
• s
• s
• T
• s
• s
• T
• P

r

• cN

ame

• v
• x
• s
• s
• T

P r

• cN

ame

• x

s

• s
• The

clauses dening T r ans c haracterize when a c

• ngur

ation

• s

can ev

• lv

e in a single step to a conguration

• s
• In

tuitiv ely they can b e read as follo ws

• a

s ev

• lv

es to

• dos

s pro vided as is p

• ssible

in s Observ e that after ha ving p erformed a nothing remains to b e p erformed

• s

ev

• lv

es to

• s

pro vided that s holds Otherwise it cannot pro ceed Observ e that in an y case the situation remains unc hanged

• s

can ev

• lv

e to

• s
• pro

vided that

• s

can ev

• lv

e to

• s
• Moreo

v er it can ev

• lv

e to

• s
• pro

vided that

• is

nal and

• s

can ev

• lv

e to

• s
• j
• s

can ev

• lv

e to

• s
• pro

vided that either

• s
• r
• s

can do so

• z
• s

can ev

• lv

e to

• s
• pro

vided that there exists a y suc h that

• z

y

• s

can ev

• lv

e to

• s
• z

is b

• und

b y

• in
• z
• and

is t ypically free in

• s

can ev

• lv

e to

• s
• pro

vided that

• s

can ev

• lv

e to

• s
• Observ

e that

• s

can also not ev

• lv

e at all since

• is

nal

• P

r

• cN

ame

• x
• s

can ev

• lv

e to

• s
• pro

vided that the b

• dy
• P

r

• cN

ame

• f

the pro cedure P r

• cN

ame with the actual parameters

• x

substituted for the formal parameters

• v
• can

do so The p

• ssible

congurations that can b e reac hed b y a program

• starting

in a situation s are those

• btained

b y rep eatly follo wing the transition relation denoted b y T r ans starting from

• s

ie those in

• Here
• z

y is the usual notion

• f

substitution in whic h the nondeterministic c hoice

• p

erator

• is

treated lik e a quan tier

SLIDE 5 the reexiv e transitiv e closure

• f

the transition relation Suc h a relation is denoted b y the reexiv e transitiv e closure

• f

T r ans T r ans

• dened

as

• s
• s
• T

r ans

• s
• s
• U
• U
• s
• s
• where
• stands

for the conjunction

• f

the univ ersal closure

• f

the follo wing clauses U

• s
• s

U

• s
• s
• T

r ans

• s
• s
• U
• s
• s
• Using

T r ans

• and

Final w e ma y denote the relation D

• as

follo ws D

• s

s

• def
• T

r ans

• s
• s
• Final
• In
• ther

w

• rds

D

• s

s

• holds

if it is p

• ssible

to rep eatedly singlestep the program

• btaining

a program

• and

a situation s

• suc

h that

• can

legally terminate in s

• Note

that this form ulation

• f

D

• is

equiv alen t to the

• ne

in LRL

• cf

Hen

• Exogenous

actions Exogenous action are primitiv e actions that are not under the con trol

• f

the program They are executed b y

• ther

agen ts in an async hronous w a y wrt the program T r ans can b e easily mo died to tak e in to accoun t exogenous actions as w ell It suce to add to the ab

• v

e denition a clause ha ving as a rst appro ximation the form Exoexo

• Poss

exo s

• T
• s
• doexo
• s

whic h sa ys that an y conguration

• s

can ev

• lv

e due to the

• ccurrence
• f

an exogenous action exo to

• doexo

s where the situation has c hanged but the program hasnt The ab

• v

e clause enables the

• ccurrence
• f

an exogenous action exo ev ery time the action preconditions for exo and hence Possexo

• s

are true Ho w ev er it is

• f

in terest to restrict further the actual

• ccurrence
• f

exo along a sequence

• f

transitions establishing some sort

• f

dynamics for exogenous actions Suc h a dynamics has a role similar to that

• f

programs for normal primitiv e actions although t ypically it is not strict enough to extract a program that implemen ts it Rather the dynamics

• f

exogenous actions has to b e sp ecied b y means

• f

suitable axioms A p

• ssible

w a y to follo w suc h a strategy is to in tro duce a sp ecial uen t D y naP

• ssexo

s and mo dify T r ans b y in tro ducing the follo wing renemen t

• f

the ab

• v

e clause Exoexo

• Poss

exo

• s
• D

y naP

• ss

exo

• s
• T
• s
• doexo
• s

Then

• ne

uses sp ecial axioms expressing the dynamics

• f

exogenous actions b y sp ecifying in whic h situa tions s along a sequence

• f

transitions D y naP

• ss

exo

• s

holds Suc h axioms ma y express sophisticated temp

• ral dynamic

la ws and t ypically they are going to b e second

• rder

Observ e that exo can actually

• ccur
• nly

if b

• th

P

• ssexo
• s

and D y naP

• ss

exo s hold in s

• Logical

represen tation

• f

inductiv e denitions and xp

• in

ts The relations T r ans and Final are dened inductiv ely

• Inductive

denitions Acz

• Mos
• are

broadly used in mathematical logic for dening sets F

• r

the past sev eral y ears they b ecame p

• pular

in computer science CC

• A

ruleb ase d inductive denition is a set R

• f

rules

• f

the form P c

• where

P is the set

• f

premises and c is the conclusion together with a closure condition a set Z is Rclosed if eac h rule in R whose premises are in Z also has its conclusion in Z

• A

set H

• inductively

dene d by R is giv en b y H

• T

fZ j Z is Rclosedg

• r

b y H

• S

fZ j Z is Rclosed g The former is called a p

• sitive

inductive denition

• f

H

• the

latter is called a ne gative inductive

• r

c

• inductive

denition

• f

H

• Let

U b e a set An

• p

er ator induc e d by an inductive denition is a total mapping !

• P
• w

U

• P
• w

U

• suc

h that !Z

• fc
• U

j P

• Z
• P

c

• Rg

That is ! is a mapping taking sets to sets Inductiv e denitions are strongly related to xp

• int

pr

• p

erties ie prop erties dened as solutions

• f

recursiv e equations Sp ecically

• p
• sitiv

e inductiv e denitions are related to least xp

• in

ts ie minimal

SLIDE 6 solution

• f

the recursiv e equations whereas negativ e inductiv e denitions are related to greatest xp

• in

ts ie maximal solutions

• f

the recursiv e equations Dynamic prop erties are t ypically xp

• in

t prop erties expressed as the least

• r

greatest solutions

• f

certain recursiv e logical equations eg see Sti Ev ery prop ert y denable as an extreme xp

• in

t m ust ha v e b y denition

• its
• wn

construction principle a recursiv e equation a xp

• in

t

• f

whic h is

• ur

prop ert y

• an

appropriate induction

• r

coinduction principle to guaran tee the minimalit y

• r

maximalit y

• f

the solution

• f

the recursiv e equation

• Construction

principle T

• dene

a set Z

• here

denoted b y a predicate Z

• x
• w

e need to sa y what its elemen ts are The c

• nstruction

principle tells us ho w to

• btain

these elemen ts recursiv ely

• xZ
• x
• Z
• x
• In

this case

• is

called a c

• nstructor

for Z

• An

y solution

• f

this recursiv e equation is called a xp

• int
• f

the

• p

erator

• The

KnasterT arski Theorem Kna

• T

ar

• guaran

tees that if the

• p

erator

• is

monotone the equation

• has

b

• th

a least and a greatest solution A sucien t condition for monotonicit y is that all

• ccurrence
• f

Z

• ccur

within a ev en n um b er

• f

negations

• This

condition is alw a ys satised in this pap er

• Induction

principle Least xp

• in

ts T

• guaran

tee that Z is the smallest solution w e apply the induction principle

• P
• xf
• y

P

• y
• P
• y
• Z
• x
• P
• x

g

• ie

whatev er solution P

• f

the recursiv e sp ecication w e tak e Z is included in it A set Z satisfying construction principle

• and

induction principle

• is

denoted b y

• P
• y

P

• y
• x
• and

it is called a le ast xp

• int
• f

an

• p

erator P

• y
• Note

that in

• P
• y

P

• y
• x
• the

predicate v ariable P and the individual v ariables

• y

are considered b

• unded

b y

• while

the individual v ariables

• x

are free Another view

• f
• P
• y

P

• y
• x
• is

that

• P
• y

P

• y
• is

the name

• f

a dened predicate and

• x

are its argumen ts W e can rewrite the induction principle

• in

the follo wing w a y

• xfZ
• x
• P
• y

P

• y
• P
• y
• P
• xg
• Notice

that implication in the

• pp
• site

direction follo ws from the construction principle

• W

e

• btain
• xf

P

• y

P

• y
• x
• P
• y

P

• y
• P
• y
• P
• x

g

• The

last sen tence is

• ften

considered as a formal denition

• f

a least xp

• in

t Observ e that it has exactly the form w e ha v e used to dene T r ans and Final as w ell as D

• s

s

• in

LRL

• Coinduction

principle Greatest xp

• in

ts T

• guaran

tee that Z is the biggest solution

• f
• w

e apply the c

• induction

principle P

• xf
• y

P

• y
• P
• y
• P
• x
• Z
• x

g

• ie

whatev er solution P

• f

the recursiv e sp ecication w e tak e Z includes it W e can rewrite the coinduction principle

• in

the follo wing w a y

• x

fP

• y

P

• y
• P
• y
• P
• x
• Z
• xg
• An

explicit expression for a greatest xp

• in

t can b e

• btained

in a similar w a y as w as done for a least xp

• in

t

• xf

P

• y

P

• y
• x
• P
• y

P

• y
• P
• y
• P
• xg
• The

last sen tence can b e tak en as a denition

• f

a greatest xp

• in

t

• In

terpreting

• as

an abbreviation for

• The

idea

• f

dening a least xp

• in

t using t w

• principles

construction and induction is from Heh

SLIDE 7

• Examples
• f

expressible dynamic prop erties With T r ans and Final in place a wide v ariet y

• f

dynamic prop erties can b e expressed b y relying

• n

second

• rder

form ulae expressing least and greatest xp

• in

t prop erties In particular prop erties expressible b y logics

• f

programs suc h as dynamic logics KT

• m

ucalculus P ar

• Sti

and temp

• ral

logics Eme

• can

b e rephrased in

• ur

setting Let us presen t some examples

• The

form ula Q

• s
• def
• P
• s
• s
• s
• T

r ans

• s
• s
• P
• s
• s
• where
• s
• are

individual v ariables denes a predicate Q

• s
• that

denotes the smallest set

• f

congurations C

• suc

h that a conguration

• s

b elongs to this set the predicate Q

• is

true

• n
• s

if and

• nly

if either

• is

true

• n
• s
• r

there exists a conguration

• s
• reac

hable in

• ne

step b y the relation T r ans whic h also b elongs to the set C

• In

this w a y the form ula expresses that from eac h conguration

• s
• n

whic h the sp ecied predicate is true there exists an execution path that ev en tually reac hes a conguration

• s
• n

whic h

• is

true As a sp ecial case b y taking

• s

def

• s
• Final
• ne

can express that there exists a terminating execution

• f

program

• starting

from situation s

• suc

h that

• is

true in the nal situation

• The

form ula Q

• s
• def
• P
• s

f

• s
• s
• T

r ans

• s
• s
• s
• T

r ans

• s
• s
• P
• s
• g
• s
• denes

a predicate Q

• s
• that

denotes the smallest set

• f

congurations C

• suc

h that the predicate is true

• n

conguration

• s

if and

• nly

if either

• is

true

• n
• s
• r

there exists a conguration

• s
• reac

hable in

• ne

step b y the relation T r ans

• and
• n

all suc h congurations the predicate is still true In this w a y the form ula expresses that from eac h conguration

• s
• n

whic h the sp ecied predicate is true all execution paths ev en tually reac h a conguration

• s
• n

whic h

• is

true

• The

form ula Q

• s
• def
• P
• s
• s
• s
• T

r ans

• s
• s
• P
• s
• s
• denes

a predicate Q

• s
• that

denotes the greatest set

• f

congurations C

• suc

h that the predicate is true

• n

conguration

• s

if and

• nly

if b

• th
• is

true

• n
• s

and the predicate is still true

• n

at least

• ne

conguration

• s
• reac

hable in

• ne

step b y the relation T r ans In this w a y the form ula expresses that from eac h conguration

• s
• n

whic h the sp ecied predicate is true there exists a nonterminating execution path along whic h

• is

alw a ys true As a sp ecial case b y

• s

def

• T

rue

• ne

can express that there exists a nonterminating execution path

• The

form ula Q

• s
• def
• P
• s
• s
• s
• T

r ans

• s
• s
• P
• s
• s
• denes

a predicate that denotes the greatest set

• f

congurations C

• suc

h that the predicate is true

• n

conguration

• s

if and

• nly

if b

• th
• is

true

• n
• s

and the predicate is still true

• n

eac h conguration

• s
• reac

hable in

• ne

step b y the relation T r ans In this w a y the form ula expresses that from eac h conguration

• s
• n

whic h the sp ecied predicate is true along all execution paths

• is

alw a ys true As a sp ecial case b y

• s

def

• Final
• s
• T

r ans

• s
• s
• ne

can express that all execution paths are nonterminating and no nal state is ev er reac hed

SLIDE 8

• Example

A Coee Deliv ery Rob

• t

Here w e describ e a rob

• t

whose task is to deliv er coee in an

• ce

en vironmen t The rob

• t

can carry just

• ne

cup

• f

coee at a time and there is a cen tral coee mac hine from whic h it gets the coee The rob

• t

receiv es asynchr

• nous

requests for coee from emplo y ees These requests are put in a queue The rob

• t

con tin uously tak es the rst request from the queue and serv es coee to the sp ecied p erson The use

• f

the queue guaran tees that all requests will in fact b e serv ed implemen ting a fair serving p

• licy
• Represen

tation

• f

the queue As usual to dene an abstract data t yp e w e need to sp ecify the domain

• f

its values and its functions and pr e dic ates The domain

• f

v alues for queues is constructed inductiv ely from the constan t nil and the functor cons

• as

follo ws

• q

I sQueue q

• Q
• Qq
• where
• stands

for the conjunction

• f

Qnil

• f
• r

Qr

• Qconsf
• r
• The

functions and predicates for queues are the usual f ir st deq ueue

• enq

ueue

• and

isE mpty

• They

are dened in

• ur

setting as follo ws f

• r

f ir stconsf

• r
• f

unspecified for nil

• f
• r

deq ueue consf

• r
• r

unspecified for nil

• penq

ueue nil

• p
• consp

nil

• p

f

• r

enq ueue consf

• r
• p
• cons

f

• enq

ueue r

• p

q isE mpty q

• q
• nil

T

• these

w e add the function l eng th

• that

returns the length

• f

the queue and the predicate isF ul l since w e are going to need queues

• f

a b

• unded

length l eng th nil

• f
• r

l eng th consf

• r
• "

l eng th r

• q

isF ul lq

• l

eng thq

• W

e enforce unique name assumption for terms built from nil and cons

• but
• b

viously not for those built with the functions deq ueue

• enq

ueue

• and

l eng th

• F
• rmalization
• f

the Example Primitiv e Actions

• r

eq uestC

• f

f ee per son A request for coee is receiv ed from the emplo y ee per son This action is an exo genous

• ne

ie an action not under the con trol

• f

the rob

• t

pExor eq uestC

• f

f ee p holds

• sel

ectR eq uest per son The rst request in the queue is selected and the emplo y ee per son that made that request will b e serv ed

• pick

upC

• f

f ee

• The

rob

• t

pic ks up a cup

• f

coee from the coee mac hine

• g

iv eC

• f

f ee per son The rob

• t

giv es a cup

• f

coee to per son

• star

tGo l

• c
• l
• c
• The

rob

• t

starts to go from lo cation l

• c
• to

l

• c
• endGo

l

• c
• l
• c
• The

rob

• t

ends its pro cess

• f

going from lo cation l

• c
• to

l

• c
• Equiv

alen tly

• q
• I

sQueue q

• Qq

q

• nil
• f
• r

q

• cons

f

• r
• Qr

q

SLIDE 9 Fluen ts

• q

ueue s A functional uen t denoting the queue

• f

requests in situation s

• r
• botLocation

s A functional uen t denoting the rob

• ts

lo cation in situation s

• hasC
• f

f ee per son s per son has coee in s

• g
• ing

l

• c
• l
• c
• s

In situation s the rob

• t

is going from l

• c
• to

l

• c
• hol

ding C

• f

f ee s In situation s the rob

• t

is holding a cup

• f

coee Situation Indep enden t Predicates and F unctions

• f

f ice per son Denotes the

• ce
• f

per son

• C

M

• Constan

t denoting coee mac hines lo cation

• S

ue M ar y

• B

il l

• J
• e

Constan ts denoting p eople Primitiv e Action Preconditions P

• ssr

eq uestC

• f

f ee p s

• isF

ul lq ueue s P

• sssel

ectR eq uest p s

• isE

mpty q ueue s

• p
• f

ir stq ueue s P

• sspick

upC

• f

f ee

• s
• hol

ding C

• f

f ee s

• r
• botLocations
• C

M P

• ssg

iv eC

• f

f ee per son s

• hol

ding C

• f

f ee s

• r
• botLocations
• f

f ice per son P

• ssstar

tGo l

• c
• l
• c
• s
• l
• l
• g
• ing

l

• l
• s
• l
• c
• l
• c
• r
• botLocations
• l
• c
• P
• ssendGol
• c
• l
• c
• s
• g
• ing

l

• c
• l
• c
• s

Successor State Axioms P

• ssa

s

• q

ueue doa s

• q
• pa
• r

eq uestC

• f

f ee p

• q
• enq

ueue q ueue s p

• pa
• sel

ectR eq uest p

• q
• deq

ueue q ueue s p

• pa
• r

eq uestC

• f

f ee p

• a
• sel

ectR eq uest p

• q
• q

ueue s P

• ssa

s

• hasC
• f

f eeper son doa s

• a
• g

iv eC

• f

f ee per son

• hasC
• f

f ee per son s P

• ssa

s

• r
• botLocation

doa s

• l
• c
• l
• c
• a
• endGol
• c
• l
• c
• r
• botLocations
• l
• c
• l
• c
• l
• c
• a
• endGol
• c
• l
• c
• P
• ssa

s

• g
• ing

l

• l
• doa

s

• a
• star

tGo l

• l
• g
• ing

l

• l
• s
• a
• endGol
• l
• P
• ssa

s

• hol

ding C

• f

f ee doa s

• a
• pick

upC

• f

f ee

• hol

ding C

• f

f ee s

• per

sona

• g

iv eC

• f

f ee per son Additional Axioms

• sI

sQueue q ueue s the values

• f

q ueue

• are

queues Unique names axioms stating that the follo wing terms together with those formed from nil and cons

• see

ab

• v

e are pairwise unequal S ue M ar y

• B

il l

• J
• e

C M

• f

f ice S ue

• f

f ice M ar y

• f

f ice B il l

• f

f ice J

• e
• The

rst axiom is not strictly necessary

• w

e add it for sak e

• f

clarit y

SLIDE 10 Initial Situation r

• botLocationS
• C

M

• hol

ding C

• f

f ee S

• l
• l
• g
• ing

l

• l
• S
• phasC
• f

f ee p S

• q

ueue S

• nil

Rob

• ts

GOLOG Program The rob

• t

execute the program D el iv er C

• f

f ee dened as follo ws note the suppressed situation argumen t in primitiv e and test actions pro c D el iv er C

• f

f ee while T rue do if isE mpty q ueue

• then
• psel

ectR eq uest p S er v eC

• f

f ee p else T rue skip endWhile endPro c pro c S er v eC

• f

f ee p Goto C M

• pick

upC

• f

f ee

• Gotoof

f ice p g iv eC

• f

f ee p endPro c pro c Gotol

• c

star tGo r

• botLocation

l

• c

endGor

• botLocation
• l
• c

endPro c Dynamics

• f

Exogenous Actions Along all p

• ssible

ev

• lutions
• f

an y program

• starting

from S

• in

to an y conguration in a nite n um b er

• f

transitions a situation s is reac hed where someb

• dy

ma y request coee D y naP

• ss

holds pro vided that it is p

• ssible

to request coee ie that also P

• ss

holds

• sT

r ans

• S
• s
• E

xoLaw s

• s

E xoLaw s

• s
• def
• E
• s

fpD y naP

• ssr

eq uestC

• f

f ee p s

• s
• T

r ans

• s
• s
• E
• s
• g
• s
• Reasoning

Next w e sho w some dynamic prop erties

• f

the

• v

erall system the program plus the exogenous actions First it is easy to see from its structure that the program D el iv er C

• f

f ee will nev er reac h a nal conguration

• sT

r ans

• D

el iv er C

• f

f ee

• S
• s
• Final
• A

more complex prop ert y that is p

• ssible

to sho w is the follo wing ev ery request for coee so

• ner
• r

later will b e serv ed F

• rmally
• the

fairness prop ert y F air D el iv er C

• f

f ee

• S
• holds

where F air

• s
• def
• p
• sT

r ans

• s
• dor

eq uestC

• f

f ee p s

• E

v entual l y S er v edp

• dor

eq uestC

• f

f ee p s and E v entual l y S er v edp

• s
• def
• P
• s

fs

• s
• dosel

ectR eq uest p s

• s
• T

r ans

• s
• s
• s
• T

r ans

• s
• s
• P
• s
• g
• s
• It

is also p

• ssible

to sho w that there exists an innite execution path where no coee is ev er serv ed P

• ssibl

y Al w ay sI dl eD el iv er C

• f

f ee

• S
• where

P

• ssibl

y Al w ay sI dl e

• s
• def
• A

s fp s

• s
• dosel

ectR eq uest p s

• s
• T

r ans

• s
• s
• A
• s
• g
• s
• Ho

w ev er b y the fairness prop ert y ab

• v

e this means that no requests for coee w ere made along that execution path

SLIDE 11

• Conclusion

and further w

• rk

In this pap er w e ha v e giv en an accoun t

• f

nonterminating programs in the Situation Calculus The framew

• rk
• btained

is quite p

• w

erful It allo ws the sp ecication

• f

the dynamic system b y mo deling

• ne

agen t with a program and external ev en ts b y suitable dynamic la ws extensions to m ultiple agen ts are also p

• ssible

see DGLL for hin ts Observ e that although related this framew

• rk

is more general than that t ypically considered in program v erication where exogenous actions that are sp ecied b y dynamic la ws axioms are not allo w ed There are man y directions for further researc h Among these w e men tion the dev elopmen t

• f

systematic tec hniques for v erication suc h as suitable induction principles References Acz P

• Aczel

An in tro duction to inductiv e denitions In J Barwise editor Handb

• k
• f

Mathematic al L

• gic

pages

• Elsevier
• CC

P

• Cousot

and R Cousot Inductiv e denitions seman tics and abstract in terpretation In Confer enc e R e c

• r

d

• f

the

• th

A CM SIGA CTSIGMODSIGAR T Symp

• sium
• n

Principles

• f

Pr

• gr

amming L anguages pages

• New

Y

• rk

USA

• A

CM Press Cou P

• Cousot

Metho ds and logics for pro ving programs In J v an Leeu w en editor Handb

• k
• f

The

• r

etic al Computer Scienc e pages

• DGLL

G De Giacomo H J Lev esque and Y Lesp

• erance

Reasoning ab

• ut

concurren t executions prioritized in terrupts and exogenous actions in the situation calculus Submitted

• Eme

E A Emerson Automated temp

• ral

reasoning ab

• ut

reactiv e systems In L

• gics

for Concur r ency Structur e versus A utomata n um b er

• in

Lecture Notes in Computer Science pages

• SpringerV

erlag

• Heh

ECR Hehner A pr actic al the

• ry
• f

pr

• gr

amming SpringerV erlag

• Hen

M Hennessy

• The

Semantics

• f

Pr

• gr

amming L anguages John Wiley

• Sons
• Kna

B Knaster Un th

• eor
• eme

sur les fonctions densem bles A nn So c Polon Math

• KT

D Kozen and J Tiuryn Logics

• f

programs In J v an Leeu w en editor Handb

• k
• f

The

• r

etic al Computer Scienc e pages

• Lei

D Leiv an t Higher

• rder

logic In Handb

• k
• f

L

• gic

in A rticial Intel ligenc e and L

• gic

Pr

• gr

amming v

• lume
• pages
• Clarendon

Press

• LRL
• HJ

Lev esque R Reiter Y Lesp

• erance

F Lin and R Sc herl GOLOG a logic programming language for dynamic domains J

• f

L

• gic

Pr

• gr

amming

• T
• app

ear LS J Lo ec kx and K Sieb er F

• undation
• f

Pr

• gr

am V eric ation T eubnerWiley

• New

Y

• rk
• Mos

YN Mosc ho v akis Elementary Induction

• n

A bstr act Structur es Amsterdam North Holland

• MP

Z Manna and A Pn ueli The T emp

• r

al L

• gic
• f

R e active and Concurr ent Systems V

• l
• Springer

V erlag

• P

ar D P ark Fixp

• in

t induction and pro

• fs
• f

program prop erties In Machine Intel ligenc e v

• lume
• pages
• Edin

burgh Univ ersit y Press

• Plo

G Plotkin A structural approac h to

• p

erational seman tics T ec hnical Rep

• rt

D AIMIFN Com puter Science Dept Aarh us Univ Denmark

• Rei

R Reiter The frame problem in the situation calculus a simple solution sometimes and a complete ness result for goal regression In Vladimir Lifsc hitz editor A rticial Intel ligenc e and Mathematic al The

• ry
• f

Computation Pap ers in Honor

• f

John McCarthy pages

• Academic

Press San Diego CA

• Sti

C Stirling Mo dal and temp

• ral

logics for pro cesses In L

• gics

for Concurr ency Structur e versus A utomata n um b er

• in

Lecture Notes in Computer Science pages

• SpringerV

erlag

• T

ar B T arski A latticetheoretical xp

• in

t theorem and its applications Pacic J Math