A Browsable Format for Proof Presentation Article February 1970 - - PDF document

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A Browsable Format for Proof Presentation

Article · February 1970

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SLIDE 2 A Bro wsable F

• rmat

for Pro

• f

Presen tation Jim Grundy T urku Cen tre for Computer Science TUCS T ec hnical Rep

• rt

No

• June
• ISBN
• ISSN

SLIDE 3 Abstract The pap er describ es a format for presen ting pro

• fs

called structur e d c alcula tional pr

• f

The format resem bles calculational pro

• f

a st yle

• f

reasoning p

• pular

among computer scien tists but extended with structuring facilities A protot yp e to

• l

has b een dev elop ed whic h allo ws readers to in teractiv ely bro wse pro

• fs

presen ted in this format via the w

• rld

wide w eb The abilit y to bro wse a pro

• f

increases its readabilit y

• and

hence its v alue as a pro

• f

Computers ha v e b een used for some time to b

• th

construct and c hec k math ematical pro

• fs

but using them to enhance the readabilit y

• f

pro

• fs

is a relativ ely no v el application This pap er w as

• riginally

presen ted at the symp

• sium
• n

L

• gic

Mathematics and the Computer The reference is as follo ws Jim Grundy

• A

bro wsable format for pro

• f

presen tation In Christoer Gefw ert P ekk a Orp

• nen

and Jouk

• Sepp

anen editors L

• gic

Mathematics and the Computer

• F
• undations

History Philosophy and Applic ations v

• lume
• f

the Finnish Articial In telligence So ciet y Symp

• sium

Series Helsinki

• June
• pages
• Keyw
• rds

calculational pro

• f

pro

• f

bro wsing pro

• f

presen tation TUCS Researc h Group Programming Metho dology Researc h Group

SLIDE 4

• In

tro duction It is not usually p

• ssible

nor indeed desirable to presen t a pro

• f

in complete detail P artly this is due to the limited n um b er

• f

pages a v ailable to an y author publishing in a journal

• r

conference pro ceedings More signican tly

• ho

w ev er it is due to the fact that to presen t an y non trivial argumen t in complete detailrigh t do wn to the lev el

• f

the basic axioms and inference rules

• f

the logicw

• uld

render it unreadable A t what lev el

• f

detail should a pro

• f

b e presen ted If an author presen ts a pro

• f

in to

• ne

a detail it will b e dicult to read and hence uncon vincing Including to

• little

detail will also mak e a pro

• f

uncon vincing Judging the righ t lev el

• f

detail for a pro

• f

is

• ne
• f

the things that distinguishes a go

• d

author Go

• d

authors kno w their audiencewhat they nd

• b

vious and what they will nd in terestingand presen t their pro

• fs

accordingly

• Ho

w ev er since not all readers are the same ev en the b est author cannot presen t a pro

• f

in a w a y that is

• ptimal

for its whole audience P erhaps the solution is to tak e the problem

• ut
• f

the hands

• f

authors and to let readers decide for themselv es ho w m uc h detail they need to see in pro

• f

Of course this is not p

• ssible

with pro

• fs

presen ted

• n

pap er but the increasing p

• pularit

y

• f

electronic publishing ma y so

• n

mak e suc h considerations less imp

• rtan

t This pap er describ es a structured format for presen ting pro

• fs

whic h with the aid

• f

computer supp

• rt

allo ws pro

• fs

to b e view ed at v arious lev els

• f

detail

• A

structured pro

• f

format This section describ es a simple structured format for presen ting pro

• fs

The basic ideas are due to Ralph Bac k and Joakim v

• n

W righ t and a more complete description is b eing prepared b y the three

• f

us BGvW The structured nature

• f

the format allo ws pro

• fs

presen ted in it to b e bro wsed at v arying degrees

• f

detail By w a y

• f

in tro duction consider the pro

• f

b elo w The problem to pro v e that k

• k

is ev en w as part

• f

the

• Finnish

highsc ho

• l

general math ematics matriculation exam even k

• k
• f

Extract the common factorg even k

• k
• f

Distribute even

• v

er

• g

even k

• evenk
• f

Since eveni

• is

even ig

SLIDE 5 even k

• even

k

• f

The denition

• f

squareg even k

• evenk

k

• f

Distribute even

• v

er

• g

even k

• even

k

• even

k

• f

Disjunction is idemp

• ten

tg even k

• even

k

• f

La w

• f

the excluded middleg

• The

format this pro

• f

is presen ted in is kno wn as c alculational pr

• f

GS

• and

is p

• pular

among computer scien tists It is also quite similar to the w a y in whic h a highsc ho

• l

mathematics studen t migh t presen t it except p erhaps for the descriptiv e commen ts The p

• pularit

y

• f

this pro

• f

format can b e explained b y its readabilit y and uniformit y

• One

dra wbac k

• f

calculational pro

• f

is that there is no w a y to decom p

• se

a large pro

• f

in to smaller

• nes

Large argumen ts are usually presen ted semiformally as a collection

• f

calculational pro

• fs

stitc hed together with informal text Natural deduction Gen

• another

common pro

• f

metho d is particularly go

• d

at decomp

• sing

large pro

• fs

in to smaller

• nes

Ho w ev er natural deduction pro

• fs

are seldom as easy to read as calculational

• nes

Can w e in v en t a pro

• f

format that com bines the structuring facilities

• f

natural deduction with the readabilit y

• f

calculational pro

• f
• The

pro

• f

just presen ted is not large enough to require structuring nev ertheless it can b e used to demonstrate the concept Steps

• f

the pro

• f

transform the same sub expression the righ t disjunct It can b e adv an ta geous to think ab

• ut

these steps as a single separate subpro

• f

whic h trans forms even k

• in

to even k

• Using

the structured calculational pro

• f

format

• f

Bac k Grundy and v

• n

W righ t BGvW w e can reexpress the pro

• f

to mak e the separate subpro

• f

explicit evenk

• k
• f

Extract the common factorg evenk k

• f

Distribute even

• v

er

• g

even k

• evenk
• f

Simplify the righ t disjunctg

• evenk
• f

Since even i

• is

even ig even k

• f

The denition

• f

squareg

SLIDE 6 even k k

• f

Distribute even

• v

er

• g

even k

• even

k

• f

Disjunction is idemp

• ten

tg even k

• even

k

• even

k

• f

The la w

• f

the excluded middleg

• The

basic structuring step

• f

the format is to iden tify a subterm

• f

the larger term to b e transformed and separate

• its

transformation in to a nested subpro

• f

The example has just

• ne

nested subpro

• f

but in general there ma y b e pro

• fs

within pro

• fs

and so

• n

When w e remo v e a term from its con text w e lo

• se

information F

• r

example the term even x is not as meaningful in isolation as it is in a con text lik e x

• even

x T

• coun

ter this dra wbac k w e extend the format b y allo wing a subpro

• f

to b egin with a list

• f

assumptions enclosed in angled brac k ets that follo w from its con text These assumptions and those

• f

an y enclosing pro

• fs

ma y b e used throughout the subpro

• f

F

• r

example using the format w e can simplify x

• even

x as follo ws x

• even

x

• f

Simplify the consequen tg

• hx
• i

even x

• f

F rom the assumptiong even

• f

Prop ert y

• f

eveng

• x
• f

Prop ert y

• f

g

• Quan

tication places restrictions

• n

the inheritance

• f

assumptions but w e need not discuss that here

• Soundness

and completeness The pro

• fs

presen ted in this pap er are necessarily informal but the struc tured calculational pro

• f

format can also b e used to presen t completely for mal pro

• fs

It should b e noted that for suc h pro

• fs

the format is just that

SLIDE 7 a format not a new metho d

• f

pro

• f

There is a straigh tforw ard trans lation b et w een the structured calculational format and natural deduction The existence

• f

the translation describ ed elsewhere GruT A

• establishes

the soundness

• f

structured calculational pro

• f

It has also b een sho wn GruT A

• b

y induction

• v

er the structure

• f

natural deduction pro

• fs

that ev ery result whic h can b e established using natural deduction can also b e established b y structured calculation This amoun ts to a completeness result for those logics where natural deduction is complete and a kind

• f

relativ e completeness for

• ther

logics

• Pro
• f

bro wsing The structure

• f

the prop

• sed

pro

• f

format admits the p

• ssibilit

y

• f

pro

• f

bro wsing to increase readabilit y

• A

large pro

• f

con taining man y subpro

• fs

can app ear daun ting to a reader Rather than presen ting the whole pro

• f

at

• nce

it can b e initially presen ted with all its subpro

• fs

hidden If the reader is in terested in a subpro

• f

they can select the commen t that describ es it The rst la y er

• f

that subpro

• f

will then b e rev ealed In this w a y the reader can see not

• nly

the individual steps

• f

a pro

• f

but also the structure

• f

the pro

• f

as a whole F urthermore the reader need

• nly

rev eal as m uc h

• f

the pro

• f

as is necessary for them to b e con vinced

• f

its v alidit y

• The

pro

• f

that k

• k

is ev en from section

• is

not large enough to require bro wsing nev ertheless w e can use it to illustrate the concept Figure

• presen

ts t w

• views
• f

this pro

• f

A pro

• f

bro wser w

• uld

initially presen t the view

• n

the left If the reader w ere in terested in seeing more details they could select the underlined commen t The bro wser w

• uld

then rev eal the hidden subpro

• f

th us sho wing the reader the righ thand view By selecting the dot marking the b eginning

• f

the subpro

• f

the reader could cause the bro wser to collapse its presen tation and return to the view

• n

the left A protot yp e pro

• f

bro wser has b een implemen ted to illustrate these ideas The input to the protot yp e is a single do cumen t whic h ma y con tain sev eral pro

• fs

with accompan ying text and graphics The

• utput

is a bro wsable do cumen t whic h ma y b e view ed

• n

the w

• rld

w eb w eb b y an

• rdinary

w eb bro wser By follo wing the links in the w eb do cumen t the reader is able to bro wse the structure

• f

the pro

• fs

it con tains In gure

• w

e see the expanded pro

• f

as displa y ed b y the Netscap e Na vigator w eb bro wser

• This

article presen ts structured calculational pro

• f

using an alternativ e notation called window infer enc e whic h is b etter suited for in teractiv e use with a mec hanised reasoning to

• l
• The

markup language used to describ e do cumen ts

• n

the w

• rld

wide w eb do es not

SLIDE 8 even k

• k
• f

Extract the common factorg even k

• k
• f

Distribute even

• v

er

• g

even k

• even

k

• f

Simplify the righ t disjunct g even k

• even

k

• f

The la w

• f

the excluded middleg

• even

k

• k
• f

Extract the common factorg even k

• k
• f

Distribute even

• v

er

• g

even k

• even

k

• f

Simplify the righ t disjunctg

• even

k

• f

Since even i

• is

even ig even k

• f

The denition

• f

squareg even k

• k
• f

Distribute even

• v

er

• g

even k

• even

k

• f

Disjunction is idemp

• ten

tg even k

• even

k

• even

k

• f

The la w

• f

the excluded middleg

• Figure
• Tw
• Views
• f

the Example Pro

• f
• Historical

p ersp ectiv e Computers ha v e b een used for some time to b

• th

disco v er and c hec k mathe matical pro

• fs

T

• da

y

• automated

reasoning is a vibran t eld in its

• wn

righ t The application

• f

computers to the presen tation and distribution

• f

pro

• fs

has ho w ev er b een largely limited to adv ances in computer t yp esetting Y et in some w a ys the presen tation and distribution

• f

a pro

• f

can b e almost as imp

• rtan

t as its existence In

• rder

for a result to b e generally accepted it m ust not

• nly

b e pro v ed but that pro

• f

m ust b e eectiv ely comm unicated to

• thers

There are t w

• dimensions

to the notion

• f

pro

• f

On

• ne

hand a pro

• f

is itself a mathematical

• b

ject a tree structure with the result at its ro

• t

inference rules as its branc hes and axioms at its lea v es It is to this dimension

• f

pro

• f

that computer supp

• rt

has traditionally b een applied Ho w ev er a pro

• f

is also a so cial pro cess that exists in the in teraction whic h tak es place b et w een the author and reader via the prin ted page If the authors argumen t do es not con vince the reader then in

• ne

resp ect it is not a pro

• f

at least curren tly supp

• rt

standard mathematical notation but future v ersions are exp ected to do so

SLIDE 9 Figure

• Bro

wsing a Pro

• f

not for that reader Increasing the readabilit y

• f

a pro

• f

then in some w a ys mak es it more

• f

a pro

• f

The tec hnique

• f

in teractiv ely bro wsing pro

• fs

describ ed here is a p

• ten

tial application

• f

computer supp

• rt

to this

• ther

dimension

• f

pro

• f

The great ma jorit y

• f

pro

• fs

presen ted to da y are still published

• n

pap er F

• r

suc h pro

• fs

the structured calculational format describ ed here ma y still b e

• f

some help with presen tation but in teractiv e bro wsing is clearly imp

• s

sible Ho w ev er the recen t adv en t

• f

electronic journals ma y help to c hange that The Journal

• f

Universal Computer Scienc e MS

• for

example al ready publishes pap ers as electronic do cumen ts that ma y b e view ed

• n

the w

• rld

wide w eb Journals lik e this could b e used to publish bro wsable pro

• fs

using the system describ ed in section

SLIDE 10

• Conclusions

and related w

• rk

The pap er has adv

• cated

writing pro

• fs

in a structured calculational format together with in teractiv e bro wsing as a w a y to increase their readabilit y

• The

example pro

• f

con tained in the pap er w as b y necessit y

• to
• small

to really b enet from bro wsing A t an y rate the pro

• f

w as not bro wsable since it w as presen ted

• n

pap er Ho w ev er the utilit y and readabilit y

• f

the prop

• sed

format and bro wsing tec hnique has b een demonstrated b y using it to solv e a signican t and un biased set

• f

problems The

• Finnish

high sc ho

• l

general mathematics matriculation exam Bro wsable v ersions

• f

the solutions to these problems are a v ailable

• n

the w eb for in terested readers to assess

• The

most notable sources

• f

inspiration for the pro

• f

format describ ed in the pap er are the calculational pro

• f

format p

• pular

among computer scien tists GS the windo w inference st yle

• f

reasoning prop

• sed

b y Robinson and Staples RS and later generalised b y Grundy GruT A and natural deduction Gen Another system

• f

pro

• f

presen tation with similar aims is the structured pro

• f

notation

• f

Lamp

• rt

Lam

• Lamp
• rt

also discusses the idea

• f

pro

• f

bro wsing although without as concrete a prop

• sal

as pre sen ted here References BGvW Ralph Bac k Jim Grundy

• and

Joakim v

• n

W righ t The pre sen tation and dissemination

• f

pro

• f

Unpublished Man uscript

• Ab
• Ak

ademi Univ ersit y

• Departmen

t

• f

Computer Science Lem mink aisenk atu A

• T

urku Finland Ma y

• Gen

Gerhard Gen tzen Un tersuc h ungen

• ub

er das logisc he Sc hliessen In v estigations in to logical deduction Mathematische Zeitschrift

• T

ranslated in Szab

• Sza

pp

• GruT

A Jim Grundy

• T

ransformational hierarc hical reasoning The Com puter Journal to app ear GS Da vid Gries and F red B Sc hneider A L

• gic

al Appr

• ach

to Dis cr ete Math c hapters

• pages
• T

exts and Monographs in Computer Science Springer V erlag New Y

• rk
• The

solutions are a v ailable at httpwwwab

• jgrundyscho
• lmathhtml
• Y
• u

will need a bro wser that supp

• rts

tables and can con trol their alignmen t

SLIDE 11 Lam Leslie Lamp

• rt

Ho w to write a pro

• f

The A meric an Mathemat ic al Monthly

• AugustSeptem

b er

• MS

Mermann Maurer and Klaus Sc hmaranz JUCS

• The

next generation

• f

electronic journal publishing Journal

• f

Universal Computer Scienc e

• No

v em b er

• RS

P eter J Robinson and John Staples F

• rmalizing

a hierarc hical structure

• f

practical mathematical reasoning Journal

• f

L

• gic

and Computation

• F

ebruary

• Sza

M E Szab

• editor

The Col le cte d Pap ers

• f

Gerhar d Gentzen Studies in Logic and the F

• undations
• f

Mathematics North Hol land Amsterdam

SLIDE 12

SLIDE 13 T urku Cen tre for Computer Science Lemmink

• aisenk

atu

• FIN

T urku Finland httpwwwtucsab

• Univ

ersit y

• f

T urku

• Departmen

t

• f

Mathematical Sciences

• Ab
• Ak

ademi Univ ersit y

• Departmen

t

• f

Computer Science

• Institute

for Adv anced Managemen t Systems Researc h T urku Sc ho

• l
• f

Economics and Business Administration

• Institute
• f

Information Systems Science

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