Agent-based Proof Search with Indexed Formulas

Size: px

Start display at page:

Download "Agent-based Proof Search with Indexed Formulas"

Adela Reynolds
5 years ago
Views:

1 5 I7. - $ '! " gent-based Proof earch with Indexed ormulas Malte Hübner, erge utexier, and Christoph Benzmüller with thans to Voler orge aarland University, aarbrücen, Germany Calculemus 2002, Marseille, July 3rd 34 V2I 6 /R01 8 U#, I R()I*+ % V&

2 M } ƒ U; < >? Ž GRHIVJIK L IO P RQ D C I his is wor in progress 9dapting the suggestion mechanism proof search in COR :- to support based on a (dynamic) rewriting approach (developed by erge utexier) 9COR: new basic inference system for :MG that is hat is the tal about?

3 M } ƒ U; < >? or Ž GRHIVJIK L IO P RQ D C I ystem recently applied to interactive proof of no constraints on the order of quantifier elimination treats (propositional) logical aspects implicit; e.g, unwrapping of hypothesis calculi Our motivation: better suited for practical reasoning than conventional trong support by simplification; e.g., simplifies to Overall goal: rewrite problem formalization to et of rewrite rules dynamically created for each focus Rewriting and focus placement as basic constructs COR: new basis for MG

4 M } ƒ U; < >? GRHIVJIK L IO P RQ D C I Proof of is different! ž xample: ž How does it wor? Ž

5 M IO } ƒ < >? Ž GRHIVJIK L U; D C PMalte Hübner, erge utexier, and Christoph Benzmüller I

6 M } ƒ U; < >? ª ³ ª ««Ž GRHIVJIK L IO P RQ D C I implification: ote: Identical proof for. simplifies to ª ± ² ª ocus on C «ª Internal representation: Indexed formula tree How does it wor?

7 M } ƒ U; < >? GRHIVJIK L IO P RQ D C I º-related subformulas belong to the same context r rewriting ž µ ¹ ž µ Indexed formula tree Ž

8 M } ƒ U; < >? À Á Â Á Ž GRHIVJIK L IO P RQ D C I more information is available in each proof state big steps rather than reasoning about logical details dvantages for proof search: Bactracing ocus choice and rule selection are interdependent Rules of the form»x¼ ½ ¾ Ã are generated dynamically Challenge: choice of focus earch pace: focus) ew system well suited for interaction with humans (but tae care with Properties of COR

9 M } ƒ U; < >? Ž GRHIVJIK L IO P RQ D C I dapt and employ sugggestion mechanism Ä- filters and heuristically orders the instances of applicable rules computes instances of additional tactics and methods computes instances of applicable rewrite rules maes suggestions on how to place focus o support interactive and automated proof search we want a mechanism that: Proof earch

10 M } ƒ U; < >? Æ Ë Çin partial proof; Ç Â Ã GRHIVJIK L IO P RQ D C I Heuristically order applicable rules Certain patterns of instantiations are considered to be applicable Communication (about dependencies between parameters) via blacboards Distributed search for instantiations of distributed computation of instantiations for Â Ë Å Â ÅÆ È É Ê ÌË Ë Å Â ÅÆ Key idea: - Ž

11 M } ƒ < >? GRHIVJIK L IO P U; D C I RQ Ž

12 M } ƒ U; < >? Ž GRHIVJIK L IO P RQ D C I Proposal is to split wor - one agent that employs O-unification - one or several woring with HO-unification Model one generic rewrite agent that handles all rules sequentially: + no dynamic creation of agents - no distribution of complex instantiation criteria (HO-unification) - dynamic creation of agents for each rule + distribution of search for instances Model all (dynamic) rewrite rules as agents in Ä-: How to search for applicable instances in formula tree? eeded: suggestion of applicable rewrite rule instances for current focus Rewriting with dynamically generated rewrite rules instead of D pproach for Rewrite Rules

13 M } ƒ U; < >? Ç Ç Â Ž GRHIVJIK L IO P RQ D C I Ç À ÅÍ ÅÏ Å Ã ransform each applicable instantiation pattern in rewrite rule: If is the set of parameters for which instantiations were found Å Â ÅÎ gents have to search for instantiations in indexed formula tree Use Ä- as it is pplicable partial argument instantiations have to be cast into rewrite rules for application in the system È É Ê Ë Æ Ë Å Â ÅÍ ÅÎ ÅÏ ÅÐ Å Ã tatic tactics/methods still specified and modeled as before: pproach for actics/methods

14 M } ƒ U; < >? ½ Ø Ø Ý GRHIVJIK L IO P RQ D C I»Ú Ý À Ý Ø Ý À Ú Ý ocusing on Ý should yield the following rewrite rules: ÇÑ Ø Ý Ó ÕÔ Ö¾ Å ÂßÑ Ø Ú ÅÍ Ñ à xample: ÙØ Ú Û Ü Þ Ø Ý ÇÑ Ó ÕÔ Ö¾ Å ÂÒÑ ÅÍ Ñ pproach for actics/methods Ž

15 M } ƒ U; < >? Ž GRHIVJIK L IO P RQ D C I user model needed for optimal support? Problem: focus placement can easily become to complicate for humans Criteria of human mathematicians Criteria from theoretical investigations for heuristics/strategies uggests to consider certain foci and excludes other foci from the search ocus gent: pproach for ocus Placement

16 M } ƒ U; < >? Ž GRHIVJIK L IO P RQ D C I mixed-initiative proof search uture wor: utomation of proof search based on Ä- and Possible evaluation: Get a novice user (e.g. maths student) to replay his blacboard proof of a mathematical problem such as irrationnality of in the system as adequately and as comfortably as possible Current focus of wor: Interactive theorem proving Difficulty: interplay between focus placement, dynamic generated rewrite rules, tactics/methods, and bactracing daptation of Ä- to support interactive proof search in COR inding up

Framework for functional tree simulation applied to 'golden delicious' apple trees

Purdue University Purdue e-pubs Open Access Theses Theses and Dissertations Spring 2015 Framework for functional tree simulation applied to 'golden delicious' apple trees Marek Fiser Purdue University