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

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/2359706 A Browsable Format for Proof Presentation Article · February 1970 Source: CiteSeer CITATIONS READS 11 43 3 authors , including: Jim Grundy Apple Inc. 43 PUBLICATIONS 454 CITATIONS SEE PROFILE All content following this page was uploaded by Jim Grundy on 29 November 2014. The user has requested enhancement of the downloaded file.

A Bro wsable F ormat for Pro of Presen tation Jim Grundy T urku Cen tre for Computer Science TUCS T ec hnical Rep ort No �� June ���� ISBN ������������� ISSN ���������

Abstract The pap er describ es a format for presen ting pro ofs called structur e d c alcula� tional pr o of � The format resem bles calculational pro of� a st yle of reasoning p opular among computer scien tists� but extended with structuring facilities� A protot yp e to ol has b een dev elop ed whic h allo ws readers to in teractiv ely bro wse pro ofs presen ted in this format via the w orld wide w eb� The abilit y to bro wse a pro of increases its readabilit y � and hence its v alue as a pro of� Computers ha v e b een used for some time to b oth construct and c hec k math� ematical pro ofs� but using them to enhance the readabilit y of pro ofs is a relativ ely no v el application� This pap er w as originally presen ted at the symp osium on L o gic� Mathematics and the Computer The reference is as follo ws� Jim Grundy � A bro wsable format for pro of presen tation� In Christo�er Gefw ert� P ekk a Orp onen and Jouk o Sepp� anen �editors�� L o gic� Mathematics and the Computer � F oundations� History� Philosophy and Applic ations � v olume �� of the Finnish Arti�cial In telligence So ciet y Symp osium Series� Helsinki� ��� June ����� pages �������� Keyw ords� calculational pro of� pro of bro wsing� pro of presen tation TUCS Researc h Group Programming Metho dology Researc h Group

� In tro duction It is not usually p ossible� nor indeed desirable� to presen t a pro of in complete detail� P artly this is due to the limited n um b er of pages a v ailable to an y author publishing in a journal or conference pro ceedings� More signi�can tly � ho w ev er� it is due to the fact that to presen t an y non trivial argumen t in complete detail�righ t do wn to the lev el of the basic axioms and inference rules of the logic�w ould render it unreadable� A t what lev el of detail should a pro of b e presen ted� If an author presen ts a pro of in to o �ne a detail it will b e di�cult to read� and hence uncon vincing� Including to o little detail will also mak e a pro of uncon vincing� Judging the righ t lev el of detail for a pro of is one of the things that distinguishes a go o d author� Go o d authors kno w their audience�what they �nd ob vious� and what they will �nd in teresting�and presen t their pro ofs accordingly � Ho w ev er� since not all readers are the same� ev en the b est author cannot presen t a pro of in a w a y that is optimal for its whole audience� P erhaps the solution is to tak e the problem out of the hands of authors� and to let readers decide for themselv es ho w m uc h detail they need to see in pro of� Of course� this is not p ossible with pro ofs presen ted on pap er� but the increasing p opularit y of electronic publishing ma y so on mak e suc h considerations less imp ortan t� This pap er describ es a structured format for presen ting pro ofs whic h� with the aid of computer supp ort� allo ws pro ofs to b e view ed at v arious lev els of detail� � A structured pro of format This section describ es a simple structured format for presen ting pro ofs� The basic ideas are due to Ralph Bac k and Joakim v on W righ t� and a more complete description is b eing prepared b y the three of us �BGvW���� The structured nature of the format allo ws pro ofs presen ted in it to b e bro wsed at v arying degrees of detail� By w a y of in tro duction� consider the pro of b elo w� The problem� to pro v e that � � is ev en� w as part of the ���� Finnish high�sc ho ol general math� k k ematics matriculation exam� � � k � � k even � f Extract the common factor� g � � k � � k � ��� even � f Distribute o v er � � g even � k � even � k � �� even � f Since even � i � �� is � � even i �� g �

� � � � even � k k even � f The de�nition of square� g � � � even � k � k �� k even � f Distribute o v er � � g even � � � even � � k k k even even f � Disjunction is idemp oten t� g � � � even � k k even f � La w of the excluded middle� g � The format this pro of is presen ted in is kno wn as c alculational pr o of �GS�� �� and is p opular among computer scien tists� It is also quite similar to the w a y in whic h a high�sc ho ol mathematics studen t migh t presen t it� except p erhaps for the descriptiv e commen ts� The p opularit y of this pro of format can b e explained b y its readabilit y and uniformit y � One dra wbac k of calculational pro of is that there is no w a y to decom� p ose a large pro of in to smaller ones� Large argumen ts are usually presen ted semi�formally as a collection of calculational pro ofs stitc hed together with informal text� Natural deduction �Gen�� �� another common pro of metho d� is particularly go o d at decomp osing large pro ofs in to smaller ones� Ho w� ev er� natural deduction pro ofs are seldom as easy to read as calculational ones� Can w e in v en t a pro of format that com bines the structuring facilities of natural deduction with the readabilit y of calculational pro of � The pro of just presen ted is not large enough to require structuring� nev� ertheless� it can b e used to demonstrate the concept� Steps ��� of the pro of transform the same sub expression� the righ t disjunct� It can b e adv an ta� geous to think ab out these steps as a single separate subpro of whic h trans� forms � k � � �� in to � � even �� Using the structured calculational pro of k even format of Bac k� Grundy and v on W righ t �BGvW���� w e can re�express the pro of to mak e the separate subpro of explicit� � even � k � � k � f Extract the common factor� g � even � k � � k � ��� � f Distribute o v er � � g even � k � even � k � �� even � f Simplify the righ t disjunct� g � � even � k � �� � f Since � i � �� is � � even i �� g even � � � even k � � f The de�nition of square� g �