smttotptp a converter for theorem proving formats
play

SMTtoTPTP - A Converter for Theorem Proving Formats Peter - PowerPoint PPT Presentation

Sep 12, 2022 •11 likes •391 views

SMTtoTPTP - A Converter for Theorem Proving Formats Peter Baumgartner Introduction TPTP (Thousands of Problems for Theorem Proving) Languages : clause logic, [typed]FOL[+arithmetics], HOL Problem library : > 20k problems Infrastructure :

  1. SMTtoTPTP - A Converter for Theorem Proving Formats Peter Baumgartner

  2. Introduction TPTP (Thousands of Problems for Theorem Proving) Languages : clause logic, [typed]FOL[+arithmetics], HOL Problem library : > 20k problems Infrastructure : utilities, solutions to problems SMT-LIB Language : sorted FOL + background theories (e.g., arithmetics, arrays) Problem library : > 100k problems Infrastructure : utilities SMTtoTPTP Translation SMT-LIB problems ⇒ TPTP problems Who bene fi ts? (Remark: “sort” = “type” in this talk) 2

  3. Who Bene fi ts? Maintainers of TPTP problem collections SMTtoTPTP makes it easy to add existing SMT-LIB benchmarks to TPTP Developers of TPTP theorem provers SMTtoTPTP provides a front-end for problems written in SMT-LIB Users of SMT solvers SMTtoTPTP provides the link to (also) use TPTP theorem provers Rest of this talk Example SMT-LIB ⇒ TPTP transformation SMTtoTPTP algorithm 3

  4. SMT-LIB Scripts (set-logic UFLIA) (declare-sort Color 0) (declare-fun red () Color) (declare-sort Pair 2) (define-sort Int-Pair (S) (Pair Int S)) (declare-fun get-int ((Int-Pair Color)) Int) (declare-fun int-color-pair (Int Color) (Pair Int Color)) (assert (forall ((i Int) (c Color)) (= (get-int (int-color-pair i c)) i))) (check-sat) 4

  5. SMT-LIB Scripts Uninterpreted function symbols + LIA (set-logic UFLIA) (declare-sort Color 0) (declare-fun red () Color) (declare-sort Pair 2) (define-sort Int-Pair (S) (Pair Int S)) (declare-fun get-int ((Int-Pair Color)) Int) (declare-fun int-color-pair (Int Color) (Pair Int Color)) (assert (forall ((i Int) (c Color)) (= (get-int (int-color-pair i c)) i))) (check-sat) 4

  6. SMT-LIB Scripts Uninterpreted function symbols + LIA (set-logic UFLIA) 0-ary sort Color (declare-sort Color 0) (declare-fun red () Color) (declare-sort Pair 2) (define-sort Int-Pair (S) (Pair Int S)) (declare-fun get-int ((Int-Pair Color)) Int) (declare-fun int-color-pair (Int Color) (Pair Int Color)) (assert (forall ((i Int) (c Color)) (= (get-int (int-color-pair i c)) i))) (check-sat) 4

  7. SMT-LIB Scripts Uninterpreted function symbols + LIA (set-logic UFLIA) 0-ary sort Color (declare-sort Color 0) Color -constant red (declare-fun red () Color) (declare-sort Pair 2) (define-sort Int-Pair (S) (Pair Int S)) (declare-fun get-int ((Int-Pair Color)) Int) (declare-fun int-color-pair (Int Color) (Pair Int Color)) (assert (forall ((i Int) (c Color)) (= (get-int (int-color-pair i c)) i))) (check-sat) 4

  8. SMT-LIB Scripts Uninterpreted function symbols + LIA (set-logic UFLIA) 0-ary sort Color (declare-sort Color 0) Color -constant red (declare-fun red () Color) 2-ary sort Pair (declare-sort Pair 2) (define-sort Int-Pair (S) (Pair Int S)) (declare-fun get-int ((Int-Pair Color)) Int) (declare-fun int-color-pair (Int Color) (Pair Int Color)) (assert (forall ((i Int) (c Color)) (= (get-int (int-color-pair i c)) i))) (check-sat) 4

  9. SMT-LIB Scripts Uninterpreted function symbols + LIA (set-logic UFLIA) 0-ary sort Color (declare-sort Color 0) Color -constant red (declare-fun red () Color) 2-ary sort Pair (declare-sort Pair 2) Macro Sort ↦ Sort (define-sort Int-Pair (S) (Pair Int S)) (declare-fun get-int ((Int-Pair Color)) Int) (declare-fun int-color-pair (Int Color) (Pair Int Color)) (assert (forall ((i Int) (c Color)) (= (get-int (int-color-pair i c)) i))) (check-sat) 4

  10. SMT-LIB Scripts Uninterpreted function symbols + LIA (set-logic UFLIA) 0-ary sort Color (declare-sort Color 0) Color -constant red (declare-fun red () Color) 2-ary sort Pair (declare-sort Pair 2) Macro Sort ↦ Sort (define-sort Int-Pair (S) (Pair Int S)) (declare-fun get-int ((Int-Pair Color)) get-int : Int) ( Pair Int Color ) ↦ Int (declare-fun int-color-pair (Int Color) (Pair Int Color)) (assert (forall ((i Int) (c Color)) (= (get-int (int-color-pair i c)) i))) (check-sat) 4

  11. SMT-LIB Scripts Uninterpreted function symbols + LIA (set-logic UFLIA) 0-ary sort Color (declare-sort Color 0) Color -constant red (declare-fun red () Color) 2-ary sort Pair (declare-sort Pair 2) Macro Sort ↦ Sort (define-sort Int-Pair (S) (Pair Int S)) (declare-fun get-int ((Int-Pair Color)) get-int : Int) ( Pair Int Color ) ↦ Int (declare-fun int-color-pair (Int Color) (Pair Int Color)) (Well-sorted) input formula (assert (forall ((i Int) (c Color)) (= (get-int (int-color-pair i c)) i))) (check-sat) 4

  12. SMT-LIB Scripts Uninterpreted function symbols + LIA (set-logic UFLIA) 0-ary sort Color (declare-sort Color 0) Color -constant red (declare-fun red () Color) 2-ary sort Pair (declare-sort Pair 2) Macro Sort ↦ Sort (define-sort Int-Pair (S) (Pair Int S)) (declare-fun get-int ((Int-Pair Color)) get-int : Int) ( Pair Int Color ) ↦ Int (declare-fun int-color-pair (Int Color) (Pair Int Color)) (Well-sorted) input formula (assert (forall ((i Int) (c Color)) (= (get-int (int-color-pair i c)) i))) (check-sat) 4 Translation into TPTP? Compatibility with TPTP format?

  13. SMT-LIB ⇒ TPTP: (In)Compatibilities ( ✓ = compatible ✗ = incompatible) Sorts ✓ SMT-LIB arithmetic sorts ≈ TPTP arithmetic sorts ✗ SMT-LIB: n-ary user sorts ≠ TPTP: 0-ary user sorts Overloaded operators ✓ SMT-LIB equality = TPTP equality = : S × S ↦ Bool for any sort S ✓ SMT-LIB arithmetic operators ≈ TPTP arithmetic operators ✗ SMT-LIB overloaded array operators (prede fi ned) (declare-sort Array 2) select : ( Array S T) × S ↦ T for any sorts S and T store : ( Array S T) × S × T ↦ ( Array S T) ⇒ It is the types that require the most attention in the transformation 5

  14. Example SMT-LIB ⇒ TPTP (set-logic UFLIA) tff('Color', type, 'Color': $tType). (declare-sort Color 0) (declare-fun red () Color) tff('Pair', type, (declare-sort Pair 2) 'Pair[Int,Color]': $tType). (define-sort Int-Pair (S) (Pair Int S)) tff(get_int, type, get_int: 'Pair[Int,Color]' > $int). (declare-fun get-int ((Int-Pair Color)) Int) tff(int_color_pair, type, int_color_pair: (declare-fun int-color-pair ($int * 'Color') > 'Pair[Int,Color]'). (Int Color) (Pair Int Color)) (assert tff(formula, axiom, (forall ((i Int) (c Color)) ( ! [I:$int, C:'Color'] : (= (get-int (get_int(int_color_pair(I, C)) (int-color-pair i c)) = I))). i))) (check-sat) 6

  15. Example SMT-LIB ⇒ TPTP Color ↝ ‘Color’ (set-logic UFLIA) tff('Color', type, 'Color': $tType). (declare-sort Color 0) (declare-fun red () Color) tff('Pair', type, (declare-sort Pair 2) 'Pair[Int,Color]': $tType). (define-sort Int-Pair (S) (Pair Int S)) tff(get_int, type, get_int: 'Pair[Int,Color]' > $int). (declare-fun get-int ((Int-Pair Color)) Int) tff(int_color_pair, type, int_color_pair: (declare-fun int-color-pair ($int * 'Color') > 'Pair[Int,Color]'). (Int Color) (Pair Int Color)) (assert tff(formula, axiom, (forall ((i Int) (c Color)) ( ! [I:$int, C:'Color'] : (= (get-int (get_int(int_color_pair(I, C)) (int-color-pair i c)) = I))). i))) (check-sat) 6

  16. Example SMT-LIB ⇒ TPTP Color ↝ ‘Color’ (set-logic UFLIA) Constant red : unused hence forget tff('Color', type, 'Color': $tType). (declare-sort Color 0) (declare-fun red () Color) tff('Pair', type, (declare-sort Pair 2) 'Pair[Int,Color]': $tType). (define-sort Int-Pair (S) (Pair Int S)) tff(get_int, type, get_int: 'Pair[Int,Color]' > $int). (declare-fun get-int ((Int-Pair Color)) Int) tff(int_color_pair, type, int_color_pair: (declare-fun int-color-pair ($int * 'Color') > 'Pair[Int,Color]'). (Int Color) (Pair Int Color)) (assert tff(formula, axiom, (forall ((i Int) (c Color)) ( ! [I:$int, C:'Color'] : (= (get-int (get_int(int_color_pair(I, C)) (int-color-pair i c)) = I))). i))) (check-sat) 6

  17. Example SMT-LIB ⇒ TPTP Color ↝ ‘Color’ (set-logic UFLIA) Constant red : unused hence forget tff('Color', type, 'Color': $tType). (declare-sort Color 0) Instance (Pair Int Color) ↝ ‘Pair[Int,Color]’ (declare-fun red () Color) tff('Pair', type, (declare-sort Pair 2) 'Pair[Int,Color]': $tType). (define-sort Int-Pair (S) (Pair Int S)) tff(get_int, type, get_int: 'Pair[Int,Color]' > $int). (declare-fun get-int ((Int-Pair Color)) Int) tff(int_color_pair, type, int_color_pair: (declare-fun int-color-pair ($int * 'Color') > 'Pair[Int,Color]'). (Int Color) (Pair Int Color)) (assert tff(formula, axiom, (forall ((i Int) (c Color)) ( ! [I:$int, C:'Color'] : (= (get-int (get_int(int_color_pair(I, C)) (int-color-pair i c)) = I))). i))) (check-sat) 6

  18. Example SMT-LIB ⇒ TPTP Color ↝ ‘Color’ (set-logic UFLIA) Constant red : unused hence forget tff('Color', type, 'Color': $tType). (declare-sort Color 0) Instance (Pair Int Color) ↝ ‘Pair[Int,Color]’ (declare-fun red () Color) tff('Pair', type, (declare-sort Pair 2) 'Pair[Int,Color]': $tType). (define-sort Int-Pair (S) (Pair Int S)) tff(get_int, type, get_int: 'Pair[Int,Color]' > $int). (declare-fun get-int ((Int-Pair Color)) Int) tff(int_color_pair, type, int_color_pair: (declare-fun int-color-pair ($int * 'Color') > 'Pair[Int,Color]'). (Int Color) (Pair Int Color)) (assert tff(formula, axiom, (forall ((i Int) (c Color)) ( ! [I:$int, C:'Color'] : (= (get-int (get_int(int_color_pair(I, C)) (int-color-pair i c)) = I))). i))) (check-sat) 6

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

On Theorem Proving for Program Checking Historical perspective and recent developments Maria

On Theorem Proving for Program Checking Historical perspective and recent developments Maria

Outline Introduction Where is theorem proving in program checking Inside theorem proving Big and little engines together: a new theorem proving style Decision procedures with speculative inferences Current and future challenges On Theorem

944 views • 69 slides
Visual theorem proving with the Incredible Proof Machine The idea Theorem Proving without

Visual theorem proving with the Incredible Proof Machine The idea Theorem Proving without

Interactive Theorem Proving, Nancy August 22, 2016 Joachim Breitner Karlsruhe Institute of Technology University of Pennsylvania, Philadelphia Visual theorem proving with the Incredible Proof Machine The idea Theorem Proving without Syntax

579 views • 43 slides
Automated Theorem Proving 1/4: Introduction and Propositional Theorem Proving A.L. Lamprecht

Automated Theorem Proving 1/4: Introduction and Propositional Theorem Proving A.L. Lamprecht

Automated Theorem Proving 1/4: Introduction and Propositional Theorem Proving A.L. Lamprecht Course Program Semantics and Verfication 2020, Utrecht University September 21, 2020 Lecture Notes Automated Reasoning by Gerard A.W. Vreeswijk.

609 views • 40 slides
Automated Theorem Proving 2/4: First-Order Theorem Proving A.L. Lamprecht Course Program

Automated Theorem Proving 2/4: First-Order Theorem Proving A.L. Lamprecht Course Program

Automated Theorem Proving 2/4: First-Order Theorem Proving A.L. Lamprecht Course Program Semantics and Verfication 2020, Utrecht University September 23, 2020 Lecture Notes Automated Reasoning by Gerard A.W. Vreeswijk. Available for

1.03k views • 54 slides
Theorem-Proving Environments Nathan Ng CSC2547: Learning to Search Theorem Proving What is a

Theorem-Proving Environments Nathan Ng CSC2547: Learning to Search Theorem Proving What is a

Theorem-Proving Environments Nathan Ng CSC2547: Learning to Search Theorem Proving What is a theorem? Statement proven based on basis of previously established statements Premise: If I attend UofT, I am a student Premise: I attend

375 views • 25 slides
Artificial Intelligence in Theorem Proving Cezary Kaliszyk VTSA Overview Last Lecture theorem

Artificial Intelligence in Theorem Proving Cezary Kaliszyk VTSA Overview Last Lecture theorem

Artificial Intelligence in Theorem Proving Cezary Kaliszyk VTSA Overview Last Lecture theorem proving problems premise selection deep learning for theorem proving state estimation Today automated reasoning learning in classical ATPs

1.18k views • 97 slides
Health Warning Principles, Techniques, Applications Theorem Proving is addictive Gerwin Klein

Health Warning Principles, Techniques, Applications Theorem Proving is addictive Gerwin Klein

W HAT YOU WILL LEARN how to use a theorem prover background, how it works how to prove and specify NICTA Advanced Course Slide 1 Slide 3 Theorem Proving Health Warning Principles, Techniques, Applications Theorem Proving is

262 views • 9 slides
Artificial Intelligence in Theorem Proving Cezary Kaliszyk VTSA 2019 Computer Theorem Proving

Artificial Intelligence in Theorem Proving Cezary Kaliszyk VTSA 2019 Computer Theorem Proving

Artificial Intelligence in Theorem Proving Cezary Kaliszyk VTSA 2019 Computer Theorem Proving Computer used to automate reasoning in a logic Traditionally part of artificial intelligence (not machine learning) Field of research since the

1.25k views • 101 slides
Formal Verification Methods 4: Theorem Proving John Harrison Intel Corporation Need for

Formal Verification Methods 4: Theorem Proving John Harrison Intel Corporation Need for

Formal Verification Methods 4: Theorem Proving Formal Verification Methods 4: Theorem Proving John Harrison Intel Corporation Need for general theorem proving Set theory vs. higher-order logic Herbrand-based approaches

475 views • 18 slides
Learning theorem proving through self-play Stanisaw Purga Overview AlphaZero Proving

Learning theorem proving through self-play Stanisaw Purga Overview AlphaZero Proving

Learning theorem proving through self-play Stanisaw Purga Overview AlphaZero Proving game adjusting MCTS for proving game some results 2019-10 1 Neural black box game state S move policy expected outcome R n v

455 views • 45 slides
Saturation-based Theorem Proving and ML Course Machine Learning and Reasoning 2020 MLR 2020 1 1

Saturation-based Theorem Proving and ML Course Machine Learning and Reasoning 2020 MLR 2020 1 1

First-order Logic and Theorem Proving Saturation-based Proving Further Tuning and the Role of Strategies Summary Saturation-based Theorem Proving and ML Course Machine Learning and Reasoning 2020 MLR 2020 1 1 Czech Technical Univeristy in

913 views • 64 slides
Mechanical Theorem Proving in Tarskis Geometry. Julien Narboux under the supervision of Hugo

Mechanical Theorem Proving in Tarskis Geometry. Julien Narboux under the supervision of Hugo

Mechanical Theorem Proving in Tarskis Geometry. Julien Narboux under the supervision of Hugo Herbelin LIX, INRIA Futurs, Ecole Polytechnique 31/08/2006, Pontevedra, Spain Outline 1 Interactive proof / Automated theorem proving 2

1.1k views • 73 slides
Proving In fi nite Satis fi ability Peter Baumgartner Joshua Bax Goal Theorem Proving in

Proving In fi nite Satis fi ability Peter Baumgartner Joshua Bax Goal Theorem Proving in

Proving In fi nite Satis fi ability Peter Baumgartner Joshua Bax Goal Theorem Proving in Hierarchic Combinations of Speci fi cations Dis Background theory linear integer arithmetic Data structures ? Conjecture list axioms/arrays

443 views • 15 slides
CO3 a COnverter for proving COnfluence of COnditional term rewriting systems Ver. 1.1 Naoki

CO3 a COnverter for proving COnfluence of COnditional term rewriting systems Ver. 1.1 Naoki

CO3 a COnverter for proving COnfluence of COnditional term rewriting systems Ver. 1.1 Naoki Nishida Makishi Yanagisawa Karl Gmeiner Nagoya University UAS Technikum Wien CoCo 2014 Vienna, July 13, 2014 Target and Main

230 views • 5 slides
Theorem Proving and the Real Numbers Applications and Challenges Lawrence C Paulson 1.

Theorem Proving and the Real Numbers Applications and Challenges Lawrence C Paulson 1.

Theorem Proving and the Real Numbers Applications and Challenges Lawrence C Paulson 1. Interactive Theorem Proving A partial, biased history A UTOMATH L. S. van Benthem Jutting, Checking Landaus Grundlagen in the A UTOMATH

920 views • 37 slides
Planning and Theorem Proving Slides by Svetlana Lazebnik, 9/2016 with modifications by Mark

Planning and Theorem Proving Slides by Svetlana Lazebnik, 9/2016 with modifications by Mark

Planning and Theorem Proving Slides by Svetlana Lazebnik, 9/2016 with modifications by Mark Hasegawa-Johnson, 2/2020 CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=18656684 Planning and Theorem Proving Examples Automatic

666 views • 40 slides
Propositional Logic: Completeness of Formal Deduction Alice Gao Lecture 10 CS 245 Logic and

Propositional Logic: Completeness of Formal Deduction Alice Gao Lecture 10 CS 245 Logic and

Propositional Logic: Completeness of Formal Deduction Alice Gao Lecture 10 CS 245 Logic and Computation Fall 2019 1 / 37 Learning Goals By the end of this lecture, you should be able to their defjnitions. theorem. CS 245 Logic and

600 views • 37 slides
Normal Forms for Boolean Expressions A NORMAL FORM defines a class expressions s.t. a. Satisfy

Normal Forms for Boolean Expressions A NORMAL FORM defines a class expressions s.t. a. Satisfy

Normal Forms for Boolean Expressions A NORMAL FORM defines a class expressions s.t. a. Satisfy certain structural properties b. Are usually universal: able to express every boolean function 1. Disjunctive Normal Form (DNF) - Sum Of Products of

1.01k views • 68 slides
Review. Ford-Fulkerson algorithm for max-flow: repeatedly augment flow along paths in the

Review. Ford-Fulkerson algorithm for max-flow: repeatedly augment flow along paths in the

Review. Ford-Fulkerson algorithm for max-flow: repeatedly augment flow along paths in the residual graph If capacities are integers, F-F finds an integer valued flow Running time: O((m+n)OPT), where OPT is the value of the max flow

836 views • 30 slides
Special topics on theorem proving and static analysis: Introduction Gang Tan CSE 597 Fall 2016

Special topics on theorem proving and static analysis: Introduction Gang Tan CSE 597 Fall 2016

Special topics on theorem proving and static analysis: Introduction Gang Tan CSE 597 Fall 2016 Penn State University 1 Critical Software Systems Software is ubiquitous E-commerce E-voting Airplane control software Fly

237 views • 23 slides
CPSC 121: Models of Computation should be able to: Identify the form of statement the strategy

CPSC 121: Models of Computation should be able to: Identify the form of statement the strategy

Pre-Class Learning Goals By the start of class, for each proof strategy below, you CPSC 121: Models of Computation should be able to: Identify the form of statement the strategy can prove. Sketch the structure of a proof that uses the

311 views • 15 slides
CS70: Lecture 2. Outline. Quick Background and Notation. Direct Proof. Theorem: For any a , b , c

CS70: Lecture 2. Outline. Quick Background and Notation. Direct Proof. Theorem: For any a , b , c

CS70: Lecture 2. Outline. Quick Background and Notation. Direct Proof. Theorem: For any a , b , c Z , if a | b and a | c then a | ( b c ) . Integers closed under addition. Proof: Assume a | b and a | c Today: Proofs!!! b = aq and c = aq

162 views • 4 slides
Defining the Ethereum Virtual Machine for Interactive Theorem Provers Yoichi Hirai Ethereum

Defining the Ethereum Virtual Machine for Interactive Theorem Provers Yoichi Hirai Ethereum

Overview Some Technicality Own Evaluation Summary Defining the Ethereum Virtual Machine for Interactive Theorem Provers Yoichi Hirai Ethereum Foundation Workshop on Trusted Smart Contracts Malta, Apr. 7, 2017 1/32 Yoichi Hirai Defining

538 views • 32 slides
4.5: Proving the Correctness of Grammars In this section, we consider techniques for proving the

4.5: Proving the Correctness of Grammars In this section, we consider techniques for proving the

4.5: Proving the Correctness of Grammars In this section, we consider techniques for proving the correctness of grammars, i.e., for proving that grammars generate the languages we want them to. 1 / 21 Definition of Suppose G is a grammar and

324 views • 21 slides

More recommend