Agent-Based HOL Reasoning 1
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1 Agent-Based HOL Reasoning 1 Alexander Steen Max Wisniewski Christoph Benzmüller Freie Universität Berlin 5th International Congress on Mathematical Software (ICMS 2016) 1 This work has been supported by the DFG under grant BE 2501/11-1 (Leo-III).
2 Talk outline 1. Automated Theorem Proving 2. Higher-Order Logic 3. Agents 4. Leo-III Prover, Agent-Based HOL Reasoning, ICMS
3 Automated Theorem Proving (ATP) ATP as computer-assisted reasoning No human interaction needed Autonomous proof search Output of result/proof Translate problem to formal representation... underlying theory as axioms... claim/goal as conjecture System result: Theorem, counter satisfiable,... and a proof? Further approaches: Interactive Theorem Proving, Semi-Interactive... Agent-Based HOL Reasoning, ICMS
4 Automated Theorem Proving (ATP) ATP as computer-assisted reasoning No human interaction needed Autonomous proof search Output of result/proof Translate problem to formal representation... underlying theory as axioms... claim/goal as conjecture System result: Theorem, counter satisfiable,... and a proof? Further approaches: Interactive Theorem Proving, Semi-Interactive... Agent-Based HOL Reasoning, ICMS
5 Automated Theorem Proving (ATP) ATP as computer-assisted reasoning No human interaction needed Autonomous proof search Output of result/proof Translate problem to formal representation... underlying theory as axioms... claim/goal as conjecture System result: Theorem, counter satisfiable,... and a proof? Further approaches: Interactive Theorem Proving, Semi-Interactive... Agent-Based HOL Reasoning, ICMS ,
6 Recent successes in Automated Deduction Proof of Four color theorem [AppelHaken, 1976] [Gonthier, 2005] Proof of Kepler s conjecture (Flyspeck project) [Hales et al., 2014] Formal inspection (verification) of Gödel s Ontological Argument and various versions of it [BenzmüllerWoltzenlogel Paleo, IJCAI,2016] Agent-Based HOL Reasoning, ICMS
7 Formalization: Hands on More concrete application example Cantor s surjective theorem: There exists no surjective function f from a set to its power set What is an appropriate formalization? Can it be proved automatically? Agent-Based HOL Reasoning, ICMS
8 Formalization: Hands on More concrete application example Cantor s surjective theorem: There exists no surjective function f from a set to its power set What is an appropriate formalization? Can it be proved automatically? Agent-Based HOL Reasoning, ICMS
9 Higher Order Logic (HOL) Due to Alonzo Church s Simple type theory (a typed λ-calculus) [Church, J.Symb.L., 1940] Simple types T generated by base types and Typically, base types are o and i Agent-Based HOL Reasoning, ICMS
10 Higher Order Logic (HOL) Due to Alonzo Church s Simple type theory (a typed λ-calculus) [Church, J.Symb.L., 1940] Simple types T generated by base types and Typically, base types are o and i Type of truth-values Agent-Based HOL Reasoning, ICMS
11 Higher Order Logic (HOL) Due to Alonzo Church s Simple type theory (a typed λ-calculus) [Church, J.Symb.L., 1940] Simple types T generated by base types and Typically, base types are o and i Type of individuals Agent-Based HOL Reasoning, ICMS
12 Higher Order Logic (HOL) Due to Alonzo Church s Simple type theory (a typed λ-calculus) [Church, J.Symb.L., 1940] Simple types T generated by base types and Typically, base types are o and i Terms defined by (α T) s, t ::=p α X α Formulae of HOL are those terms with type o Semantics well-understood (not mentioned here) Agent-Based HOL Reasoning, ICMS
13 Higher Order Logic (HOL) Due to Alonzo Church s Simple type theory (a typed λ-calculus) [Church, J.Symb.L., 1940] Simple types T generated by base types and Typically, base types are o and i Terms defined by (α T) s, t ::=p α X α (λx α.s β ) α β (s α β t α ) β Formulae of HOL are those terms with type o Semantics well-understood (not mentioned here) Agent-Based HOL Reasoning, ICMS
14 Higher Order Logic (HOL) Due to Alonzo Church s Simple type theory (a typed λ-calculus) [Church, J.Symb.L., 1940] Simple types T generated by base types and Typically, base types are o and i Terms defined by (α T) s, t ::=p α X α (λx α.s β ) α β (s α β t α ) β ( o o s o ) o (s o o o o t o ) o ( α X α.s o ) o Formulae of HOL are those terms with type o Semantics well-understood (not mentioned here) Agent-Based HOL Reasoning, ICMS
15 Higher Order Logic (HOL) Due to Alonzo Church s Simple type theory (a typed λ-calculus) [Church, J.Symb.L., 1940] Simple types T generated by base types and Typically, base types are o and i Terms defined by (α T) s, t ::=p α X α (λx α.s β ) α β (s α β t α ) β ( o o s o ) o (s o o o o t o ) o Π α (α o) o (λx α. s o ) α o Formulae of HOL are those terms with type o Semantics well-understood (not mentioned here) Agent-Based HOL Reasoning, ICMS
16 Higher Order Logic (HOL) Due to Alonzo Church s Simple type theory (a typed λ-calculus) [Church, J.Symb.L., 1940] Simple types T generated by base types and Typically, base types are o and i Terms defined by (α T) s, t ::=p α X α (λx α.s β ) α β (s α β t α ) β ( o o s o ) o (s o o o o t o ) o ( α X α.s o ) o Formulae of HOL are those terms with type o Semantics well-understood (not mentioned here) Agent-Based HOL Reasoning, ICMS
17 Why HOL? Why HOL? (Pragmatically) expressive language Concise expressions Higher-order quantification Anonymous functions ( built-in comprehension) Allows easy representation of common notions (e.g. induction principle) Allows encoding of various non-classical logics (e.g. modal logics, free logic,...) Previous example There exists no surjective function from a set to its power set Agent-Based HOL Reasoning, ICMS
18 Why HOL? Why HOL? (Pragmatically) expressive language Concise expressions Higher-order quantification Anonymous functions ( built-in comprehension) Allows easy representation of common notions (e.g. induction principle) Allows encoding of various non-classical logics (e.g. modal logics, free logic,...) Previous example There exists no surjective function from a set to its power set Agent-Based HOL Reasoning, ICMS
19 Why HOL? Why HOL? (Pragmatically) expressive language Concise expressions Higher-order quantification Anonymous functions ( built-in comprehension) Allows easy representation of common notions (e.g. induction principle) Allows encoding of various non-classical logics (e.g. modal logics, free logic,...) Previous example There exists no surjective function from a set to its power set Agent-Based HOL Reasoning, ICMS
20 Why HOL? Why HOL? (Pragmatically) expressive language Concise expressions Higher-order quantification Anonymous functions ( built-in comprehension) Allows easy representation of common notions (e.g. induction principle) Allows encoding of various non-classical logics (e.g. modal logics, free logic,...) Previous example There exists no surjective function from a set to its power set F ι (ι o). Y ι o. X ι. F X = Y Agent-Based HOL Reasoning, ICMS
21 Why HOL? Why HOL? (Pragmatically) expressive language Concise expressions Higher-order quantification Anonymous functions ( built-in comprehension) Allows easy representation of common notions (e.g. induction principle) Allows encoding of various non-classical logics (e.g. modal logics, free logic,...) Previous example There exists no surjective function from a set to its power set F ι (ι o). Y ι o. X ι. F X = Y Agent-Based HOL Reasoning, ICMS
22 Why HOL? Why HOL? (Pragmatically) expressive language Concise expressions Higher-order quantification Anonymous functions ( built-in comprehension) Allows easy representation of common notions (e.g. induction principle) Allows encoding of various non-classical logics (e.g. modal logics, free logic,...) Previous example There exists no surjective function from a set to its power set F ι (ι o). Y ι o. X ι. F X = Y Agent-Based HOL Reasoning, ICMS
23 Call the Agents Reasoning in HOL is involved Complicated routines (some of those undecidable) Different (orthogonal) reasoning techniques Tableaux methods (Satallax) Resolution methods (LEO-II) First-order encoding methods (tptp_isabelle) Paramodulation methods (?) Not (yet) clear, if there s a superior approach Agent-Based HOL Reasoning, ICMS
24 Call the Agents Reasoning in HOL is involved Complicated routines (some of those undecidable) Different (orthogonal) reasoning techniques Tableaux methods (Satallax) Resolution methods (LEO-II) First-order encoding methods (tptp_isabelle) Paramodulation methods (?) Not (yet) clear, if there s a superior approach Agent-Based HOL Reasoning, ICMS
25 Call the Agents Reasoning in HOL is involved Complicated routines (some of those undecidable) Different (orthogonal) reasoning techniques Tableaux methods (Satallax) Resolution methods (LEO-II) First-order encoding methods (tptp_isabelle) Paramodulation methods (?) Not (yet) clear, if there s a superior approach Why not combine them all? Have the best of all worlds! Motivation of Leo-III Employ specialists as independent agents Let them find a proof cooperatively Independence allows parallel execution. Agent-Based HOL Reasoning, ICMS
26 Architecture of Leo-III Shared blackboard architecture Independent agents collaboratively acting on it Agents may be internal or external Coordination of agents by combinatorical auction game scheduling Agent-Based HOL Reasoning, ICMS
27 Current state of Leo-III Built around a paramodulation-based HOL calculus... currently as sequential loop At any point the loop may request assistance from external agents Internal agents Normalization variants Relevance filtering (pre-processing) Sequential loops Goal state checker External agents Leo-II Satallax exemplary every TPTP-compatible prover can be used Agent-Based HOL Reasoning, ICMS
28 Current state of Leo-III Built around a paramodulation-based HOL calculus... currently as sequential loop At any point the loop may request assistance from external agents Internal agents Normalization variants Relevance filtering (pre-processing) Sequential loops Goal state checker External agents Leo-II Satallax exemplary every TPTP-compatible prover can be used Agent-Based HOL Reasoning, ICMS
29 Current state of Leo-III Built around a paramodulation-based HOL calculus... currently as sequential loop At any point the loop may request assistance from external agents Internal agents Normalization variants Relevance filtering (pre-processing) Sequential loops Goal state checker External agents Leo-II Satallax exemplary every TPTP-compatible prover can be used Agent-Based HOL Reasoning, ICMS
30 First assessment First assessment is positive... Competed in this year s CADE ATP System Competition (CASC) Experimental paramodulation calculus seems feasible Cooperation with external provers beneficial Agent-Based HOL Reasoning, ICMS
31 First assessment First assessment is positive... Competed in this year s CADE ATP System Competition (CASC) Experimental paramodulation calculus seems feasible Cooperation with external provers beneficial... but Poor parameter settings No first-order cooperation yet A lot of things to do until competitive Agent-Based HOL Reasoning, ICMS
32 Live Demo Demo Agent-Based HOL Reasoning, ICMS
33 Modifications Sequential loop agents only for the current version Split internal saturation into more fine-grained agent tasks Use machine learning techniques for coordination of agents, search space traversal Include more specialist systems/external agents: First-order provers SAT/SMT solvers for internal reasoning assistance Consistency checkers (e.g. nitpick) Unification specialists Agent-Based HOL Reasoning, ICMS
34 Modifications Sequential loop agents only for the current version Split internal saturation into more fine-grained agent tasks Use machine learning techniques for coordination of agents, search space traversal Include more specialist systems/external agents: First-order provers SAT/SMT solvers for internal reasoning assistance Consistency checkers (e.g. nitpick) Unification specialists Agent-Based HOL Reasoning, ICMS
35 Perspective Leo-III s underlying datastructure/architecture framework as a stand-alone package For the implementation of HO reasoners on top of it Easily extensible by agent-based design Already available! On the long run: Reasoning-as-a-service for domain specific tools Many non-classical logics through internal embedding into HOL Machine learning components could also be re-used by applications Natural language processing Algebra systems,... Agent-Based HOL Reasoning, ICMS
36 Perspective Leo-III s underlying datastructure/architecture framework as a stand-alone package For the implementation of HO reasoners on top of it Easily extensible by agent-based design Already available! On the long run: Reasoning-as-a-service for domain specific tools Many non-classical logics through internal embedding into HOL Machine learning components could also be re-used by applications Natural language processing Algebra systems,... Agent-Based HOL Reasoning, ICMS
37 Conclusion and Further work Conclusion Presented Leo-III as an agent-based blackboard proving system Based on (polymorphic) higher-order logic Flexible (ad-hoc) inclusion of specialist systems LeoPARD accessible for own developments Further work Split internal saturation into more fine-grained agent tasks Do experiments to find good parameters Lots of technical improvements Implementation of/connection to more external specialists Better proof output Agent-Based HOL Reasoning, ICMS
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