The Higher-Order Prover Leo-III Alexander Steen 1 , 2 , jww. C. Benzmüller and M. Wisniewski 1 Freie Universität Berlin Matryoshka Workshop 2018, Amsterdam 1This author has been supported by the DFG under grant BE 2501/11-1 (Leo-III). 2This author has been supported by the Volkswagenstiftung (project CRAP).
T alk outline 1. Higher-Order Logic (HOL) 2. The Leo-III Prover 3. Automation of Non-Classical Logics 4. Summary 5. Live Demo (optional) , The Higher-Order Prover Leo-III, Matryoshka 2018 2
Higher Order Logic (HOL) Based on Church’s ”Simple type theory” (typed λ -calculus) [Church,1940] More specifically: Extentional T ype Theory (ExTT) [Henkin, JSL, 1950] Syntax ◮ Simple types T generated by base types and → ◮ T ypically, base types are o and i ◮ T erms defined by ( τ, ν ∈ T ) s, t ::= c τ ∈ Σ | X τ ∈ V ◮ Primitive logical connectives ( τ ∈ T ) ¬ o → o , ∨ o → o → o , Π τ ( τ → o ) → o , = τ � � ⊆ Σ τ → τ → o , The Higher-Order Prover Leo-III, Matryoshka 2018 3
Higher Order Logic (HOL) Based on Church’s ”Simple type theory” (typed λ -calculus) [Church,1940] More specifically: Extentional T ype Theory (ExTT) [Henkin, JSL, 1950] Syntax ◮ Simple types T generated by base types and → ◮ T ypically, base types are o and i ◮ T erms defined by ( τ, ν ∈ T ) s, t ::= c τ ∈ Σ | X τ ∈ V ◮ Primitive logical connectives ( τ ∈ T ) ¬ o → o , ∨ o → o → o , Π τ ( τ → o ) → o , = τ � � ⊆ Σ τ → τ → o , The Higher-Order Prover Leo-III, Matryoshka 2018 3
Higher Order Logic (HOL) Based on Church’s ”Simple type theory” (typed λ -calculus) [Church,1940] More specifically: Extentional T ype Theory (ExTT) [Henkin, JSL, 1950] Syntax ◮ Simple types T generated by base types and → ◮ T ypically, base types are o and i T ype of truth-values ◮ T erms defined by ( τ, ν ∈ T ) s, t ::= c τ ∈ Σ | X τ ∈ V ◮ Primitive logical connectives ( τ ∈ T ) ¬ o → o , ∨ o → o → o , Π τ ( τ → o ) → o , = τ � � ⊆ Σ τ → τ → o , The Higher-Order Prover Leo-III, Matryoshka 2018 3
Higher Order Logic (HOL) Based on Church’s ”Simple type theory” (typed λ -calculus) [Church,1940] More specifically: Extentional T ype Theory (ExTT) [Henkin, JSL, 1950] Syntax ◮ Simple types T generated by base types and → ◮ T ypically, base types are o and i T ype of individuals ◮ T erms defined by ( τ, ν ∈ T ) s, t ::= c τ ∈ Σ | X τ ∈ V ◮ Primitive logical connectives ( τ ∈ T ) ¬ o → o , ∨ o → o → o , Π τ ( τ → o ) → o , = τ � � ⊆ Σ τ → τ → o , The Higher-Order Prover Leo-III, Matryoshka 2018 3
Higher Order Logic (HOL) Based on Church’s ”Simple type theory” (typed λ -calculus) [Church,1940] More specifically: Extentional T ype Theory (ExTT) [Henkin, JSL, 1950] Syntax ◮ Simple types T generated by base types and → ◮ T ypically, base types are o and i ◮ T erms defined by ( τ, ν ∈ T ) s, t ::= c τ ∈ Σ | X τ ∈ V ◮ Primitive logical connectives ( τ ∈ T ) ¬ o → o , ∨ o → o → o , Π τ ( τ → o ) → o , = τ � � ⊆ Σ τ → τ → o , The Higher-Order Prover Leo-III, Matryoshka 2018 3
Higher Order Logic (HOL) Based on Church’s ”Simple type theory” (typed λ -calculus) [Church,1940] More specifically: Extentional T ype Theory (ExTT) [Henkin, JSL, 1950] Syntax ◮ Simple types T generated by base types and → ◮ T ypically, base types are o and i ◮ T erms defined by ( τ, ν ∈ T ) s, t ::= c τ ∈ Σ | X τ ∈ V | ( λX τ . s ν ) τ → ν | ( s τ → ν t τ ) ν ◮ Primitive logical connectives ( τ ∈ T ) ¬ o → o , ∨ o → o → o , Π τ ( τ → o ) → o , = τ � � ⊆ Σ τ → τ → o , The Higher-Order Prover Leo-III, Matryoshka 2018 3
Higher Order Logic (HOL) Based on Church’s ”Simple type theory” (typed λ -calculus) [Church,1940] More specifically: Extentional T ype Theory (ExTT) [Henkin, JSL, 1950] Syntax ◮ Simple types T generated by base types and → ◮ T ypically, base types are o and i ◮ T erms defined by ( τ, ν ∈ T ) s, t ::= c τ ∈ Σ | X τ ∈ V | ( λX τ . s ν ) τ → ν | ( s τ → ν t τ ) ν ◮ Primitive logical connectives ( τ ∈ T ) ¬ o → o , ∨ o → o → o , Π τ ( τ → o ) → o , = τ � � ⊆ Σ τ → τ → o , The Higher-Order Prover Leo-III, Matryoshka 2018 3
Higher Order Logic (HOL), cont. Semantics ◮ Leo-III automates HOL with Henkin semantics ◮ Some valid axioms (axiom schemes): ◮ Boolean Extensionality := ∀ P o . ∀ Q o . ( P ⇔ Q ) ⇒ P = o Q EXT o ◮ Functional Extensionality := ∀ F ντ . ∀ G ντ . ( ∀ X τ . F X = ν G X ) ⇒ F = ντ G EXT ντ ◮ T ype-restricted comprehension COM τ,ν := ∀ G ν . ∃ F ντ n . ∀ X n . F X n = G ν ◮ Further semantics exist: ◮ Without Extensionality � Elementary T ype Theory [Andrews, 1974] ◮ Intermediate systems [Benzmüller et al.,2004] ◮ Andrews’ v -complexes [Andrews, 1971] ◮ Intensional models [Muskens, 2007] , The Higher-Order Prover Leo-III, Matryoshka 2018 4
Higher Order Logic (HOL), cont. Semantics ◮ Leo-III automates HOL with Henkin semantics ◮ Some valid axioms (axiom schemes): ◮ Boolean Extensionality := ∀ P o . ∀ Q o . ( P ⇔ Q ) ⇒ P = o Q EXT o ◮ Functional Extensionality := ∀ F ντ . ∀ G ντ . ( ∀ X τ . F X = ν G X ) ⇒ F = ντ G EXT ντ ◮ T ype-restricted comprehension COM τ,ν := ∀ G ν . ∃ F ντ n . ∀ X n . F X n = G ν ◮ Further semantics exist: ◮ Without Extensionality � Elementary T ype Theory [Andrews, 1974] ◮ Intermediate systems [Benzmüller et al.,2004] ◮ Andrews’ v -complexes [Andrews, 1971] ◮ Intensional models [Muskens, 2007] , The Higher-Order Prover Leo-III, Matryoshka 2018 4
Higher Order Logic (HOL), cont. Semantics ◮ Leo-III automates HOL with Henkin semantics ◮ Some valid axioms (axiom schemes): ◮ Boolean Extensionality := ∀ P o . ∀ Q o . ( P ⇔ Q ) ⇒ P = o Q EXT o ◮ Functional Extensionality := ∀ F ντ . ∀ G ντ . ( ∀ X τ . F X = ν G X ) ⇒ F = ντ G EXT ντ ◮ T ype-restricted comprehension COM τ,ν := ∀ G ν . ∃ F ντ n . ∀ X n . F X n = G ν ◮ Further semantics exist: ◮ Without Extensionality � Elementary T ype Theory [Andrews, 1974] ◮ Intermediate systems [Benzmüller et al.,2004] ◮ Andrews’ v -complexes [Andrews, 1971] ◮ Intensional models [Muskens, 2007] , The Higher-Order Prover Leo-III, Matryoshka 2018 4
Higher Order Logic (HOL), cont. Semantics ◮ Leo-III automates HOL with Henkin semantics ◮ Some valid axioms (axiom schemes): ◮ Boolean Extensionality := ∀ P o . ∀ Q o . ( P ⇔ Q ) ⇒ P = o Q EXT o ◮ Functional Extensionality := ∀ F ντ . ∀ G ντ . ( ∀ X τ . F X = ν G X ) ⇒ F = ντ G EXT ντ ◮ T ype-restricted comprehension COM τ,ν := ∀ G ν . ∃ F ντ n . ∀ X n . F X n = G ν ◮ Further semantics exist: ◮ Without Extensionality � Elementary T ype Theory [Andrews, 1974] ◮ Intermediate systems [Benzmüller et al.,2004] ◮ Andrews’ v -complexes [Andrews, 1971] ◮ Intensional models [Muskens, 2007] , The Higher-Order Prover Leo-III, Matryoshka 2018 4
Higher Order Logic (HOL), cont. Semantics ◮ Leo-III automates HOL with Henkin semantics ◮ Some valid axioms (axiom schemes): ◮ Boolean Extensionality := ∀ P o . ∀ Q o . ( P ⇔ Q ) ⇒ P = o Q EXT o ◮ Functional Extensionality := ∀ F ντ . ∀ G ντ . ( ∀ X τ . F X = ν G X ) ⇒ F = ντ G EXT ντ ◮ T ype-restricted comprehension COM τ,ν := ∀ G ν . ∃ F ντ n . ∀ X n . F X n = G ν ◮ Further semantics exist: ◮ Without Extensionality � Elementary T ype Theory [Andrews, 1974] ◮ Intermediate systems [Benzmüller et al.,2004] ◮ Andrews’ v -complexes [Andrews, 1971] ◮ Intensional models [Muskens, 2007] , The Higher-Order Prover Leo-III, Matryoshka 2018 4
T alk outline 1. Higher-Order Logic (HOL) 2. The Leo-III Prover 3. Automation of Non-Classical Logics 4. Summary 5. Live Demo (optional) , The Higher-Order Prover Leo-III, Matryoshka 2018 5
Evolution of the Leo Provers LEO-I (1997–2006 at Saarbrücken/Birmingham) [ Benzmüller et al.,CADE,1998 ] ◮ Extensional higher-order RUE-resolution approach ◮ Pioneered higher-order—first-order cooperation (E prover) ◮ Hard-wired to the Ω MEGA proof assistant LEO-II (2006-2012 at Cambridge/Berlin) [ Benzmüller et al.,JAR,2015 ] ◮ Extensional higher-order RUE-resolution approach ◮ Primitive equality, first steps towards polymorphism and choice/description, ◮ Fostered & paralleled the development of TPTP THF (EU FP7 project) ◮ First CASC winner in THF category in 2010 , The Higher-Order Prover Leo-III, Matryoshka 2018 6
Evolution of the Leo Provers LEO-I (1997–2006 at Saarbrücken/Birmingham) [ Benzmüller et al.,CADE,1998 ] ◮ Extensional higher-order RUE-resolution approach ◮ Pioneered higher-order—first-order cooperation (E prover) ◮ Hard-wired to the Ω MEGA proof assistant LEO-II (2006-2012 at Cambridge/Berlin) [ Benzmüller et al.,JAR,2015 ] ◮ Extensional higher-order RUE-resolution approach ◮ Primitive equality, first steps towards polymorphism and choice/description, ◮ Fostered & paralleled the development of TPTP THF (EU FP7 project) ◮ First CASC winner in THF category in 2010 , The Higher-Order Prover Leo-III, Matryoshka 2018 6
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