Utilizing Church s Type Theory as a Universal Logic 1

Transcription

1 Utilizing Church s Type Theory as a Universal Logic 1 Christoph Benzmüller Freie Universität Berlin Presentation at TU Wien, October 31, This work has been funded by the DFG under grants BE 2501/6-1, BE 2501/8-1 and BE 2501/9-1 C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

2 My sincere apologies... for not visiting earlier! C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

3 Presentation Overview Core questions of my current research: 1 Classical Higher-order Logic (HOL) as Universal Logic? 2 HOL Provers & Model Finders as Generic Reasoning Tools? 3 Integration of Specialist Reasoners (if available)? C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

4 Presentation Overview Core questions of my current research: 1 Classical Higher-order Logic (HOL) as Universal Logic? 2 HOL Provers & Model Finders as Generic Reasoning Tools? 3 Integration of Specialist Reasoners (if available)? Talk Outline: Classical Higher-order Logic HOL & HOL-ATP Examples of Natural Fragments of HOL: Quantified Multimodal Logics (QMLs) Quantified Conditional Logics (QCLs) if time permits Reasoning about Logics (and their Combinations) Evaluation of HOL-ATPs for reasoning within QMLs Short Demonstration Conclusion C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

5 My Motivation Automated Reasoning within and about Expressive Ontologies? Expressive Ontologies: SUMO (Adam Pease) or Cyc (Doug Lenat) They have often been advertised as first-order ontologies, but they are not! They contain higher-order constructs They contain modal operators holdsduring knows believes... Limited automation with traditional FOL-ATPs Hypothesis: We can do better with HOL-ATPs [BenzmüllerPease, J. Web Semantics, 2012] C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

6 HOL & HOL-ATP (Classical Higher-order Logic/Church s Type Theory) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

7 What is HOL? (Church s Type Theory, Alonzo Church, 1940) Expressivity FOL HOL Example Quantification over - Individuals X p(f(x)) - Functions - F p(f(a)) - Predicates/Sets/Rels - P P(f(a)) Unnamed - Functions - (λx X) - Predicates/Sets/Rels - (λx X a) Statements about - Functions - continuous(λx X) - Predicates/Sets/Rels - reflexive(= ) Powerful abbreviations - reflexive =λr λx R(X, X) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

8 What is HOL? (Church s Type Theory, Alonzo Church, 1940) Expressivity FOL HOL Example Quantification over - Individuals X ι p ι o (f ι ι (X ι )) - Functions - F ι ι p ι o (F ι o (a ι )) - Predicates/Sets/Rels - P ι o P ι o (f ι ι (a ι )) Unnamed - Functions - (λx ι X ι ) - Predicates/Sets/Rels - (λx ι ι X ι ι ι ι p a) ι ) Statements about - Functions - continuous (ι ι) o (λx ι X ι ) - Predicates/Sets/Rels - reflexive (ι ι o) o (= ι ι o ) Powerful abbreviations - reflexive (ι ι o) o =λr (ι ι o) λx ι R Simple Types: Prevent Paradoxes and Inconsistencies C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

9 What is HOL? (Alonzo Church, 1940) Simple Types α ::= ι o α 1 α 2 C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

10 What is HOL? (Alonzo Church, 1940) Simple Types α ::= ι o α 1 α 2 Individuals Booleans (True and False) Functions C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

11 What is HOL? (Alonzo Church, 1940) Simple Types α ::= ι µ o α 1 α 2 Possible worlds Individuals Booleans (True and False) Functions C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

12 What is HOL? (Alonzo Church, 1940) Simple Types α ::= ι µ o α 1 α 2 HOL Syntax s, t ::= c α X α Constant Symbols Variable Symbols (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 ( X α t o ) o C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

13 What is HOL? (Alonzo Church, 1940) Simple Types α ::= ι µ o α 1 α 2 HOL Syntax s, t ::= c α X α Constant Symbols Variable Symbols Abstraction Application (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 ( X α t o ) o C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

14 What is HOL? (Alonzo Church, 1940) Simple Types α ::= ι µ o α 1 α 2 HOL Syntax s, t ::= c α X α Constant Symbols Variable Symbols Abstraction Application Logical Connectives (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 ( X α t o ) o C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

15 What is HOL? (Alonzo Church, 1940) Simple Types α ::= ι µ o α 1 α 2 HOL Syntax s, t ::= c 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 ( X α t o ) o }{{} (Π (α o) o (λx α t o)) o C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

16 What is HOL? (Alonzo Church, 1940) Simple Types α ::= ι µ o α 1 α 2 HOL Syntax s, t ::= c 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 o) o (λx α t o )) o HOL is (meanwhile) well understood - Origin [Church, J.Symb.Log., 1940] - Henkin-Semantics [Henkin, J.Symb.Log., 1950] [Andrews, J.Symb.Log., 1971, 1972] - Extens./Intens. [BenzmüllerEtAl., J.Symb.Log., 2004] [Muskens, J.Symb.Log., 2007] HOL with Henkin-Semantics: semi-decidable & compact (like FOL) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

17 EU Project THFTPTP: An Infrastructure for HOL-ATP Results of the EU Project THFTPTP Collaboration with Geoff Sutcliffe, Chad Brown and others Results THF0 syntax for HOL Online access to provers Library with example problems (e.g. entire TPS library) and results Ontology and syntax for proof results International CASC competition for HOL-ATP Various tools Improved availability and robustness of HOL-ATPs: TPS, LEO-II, Isabelle, Satallax, Refute, Nitpick, agsyhol [SutcliffeBenzmüller, J. Formalized Reasoning, 2010] [BenzmüllerRabeSutcliffe, IJCAR, 2008] C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

18 CASC Competitions in THF (HOL) Category since C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

19 CASC Competitions in THF (HOL) Category since LEO-II 1.2 solved 56% more than previous winner C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

20 CASC Competitions in THF (HOL) Category since LEO-II 1.2 solved 56% more than previous winner Satallax 2.1 solved 21% more than previous winner C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

21 CASC Competitions in THF (HOL) Category since LEO-II 1.2 solved 56% more than previous winner 2012 Satallax 2.1 solved 21% more than previous winner Isabelle-HOT solved 35% more than previous winner C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

22 CASC Competitions in THF (HOL) Category since C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

23 CASC Competitions in THF (HOL) Category since LEO-II cooperates with FOL prover E C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

24 CASC Competitions in THF (HOL) Category since Satallax cooperates with SAT solver Minisat C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

25 CASC Competitions in THF (HOL) Category since Isabelle-HOT cooperates with various FOL provers (sledgehammer) and SMT solvers (smt) and even with LEO-II and Satallax C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

26 Natural Fragments of HOL: Quantified Multimodal Logics C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

27 Combining the Kripke View and the Tarski View on Logics C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

28 Combining the Kripke View and the Tarski View on Logics C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

29 Combining the Kripke View and the Tarski View on Logics C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

30 Combining the Kripke View and the Tarski View on Logics C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

31 Combining the Kripke View and the Tarski View on Logics C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

32 Combining the Kripke View and the Tarski View on Logics C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

33 Combining the Kripke View and the Tarski View on Logics C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

34 (Multi-) Modal Logics in HOL Syntax: s, t ::= P s s t r s... Syntax - formulas s Kripke Semantics - worlds w - accessibility relations r HOL explicit transformation First Order Logic e.g. work of Ohlbach Not Needed! C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

35 (Multi-) Modal Logics in HOL Syntax: s, t ::= P s s t r s... Syntax - formulas s Kripke Semantics - worlds w - accessibility relations r HOL terms s ι o terms w ι terms r ι ι o C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

36 (Multi-) Modal Logics in HOL Quantifiers Syntax: s, t ::= P s s t r s... Syntax - formulas s Kripke Semantics - worlds w - accessibility relations r HOL terms s ι o terms w ι terms r ι ι o Syntax of embedded logic as abbreviations of HOL-terms P = P ι o = λs ι o λw ι (S W) = λs ι o λt ι o λw ι (S W) (T W) = λr ι ι o λs ι o λw ι V ι (R W V) (S V) [BenzmüllerPaulson, Log.J.IGPL, 2010], [BenzmüllerPaulson, Logica Universalis, 2012] C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

37 (Multi-) Modal Logics in HOL Quantifiers Conditional Logics... Syntax: s, t ::= P s s t r s... Syntax - formulas s Kripke Semantics - worlds w - accessibility relations r HOL terms s ι o terms w ι terms r ι ι o Syntax of embedded logic as abbreviations of HOL-terms P = P ι o = λs ι o λw ι (S W) = λs ι o λt ι o λw ι (S W) (T W) = λr ι ι o λs ι o λw ι V ι (R W V) (S V) ( p ), µ = λq µ (ι o) λw ι X µ (Q X W) [BenzmüllerPaulson, Log.J.IGPL, 2010], [BenzmüllerPaulson, Logica Universalis, 2012] C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

38 (Multi-) Modal Logics in HOL Quantifiers Syntax: s, t ::= P s s t r s... Syntax - formulas s Kripke Semantics - worlds w - accessibility relations r HOL terms s ι o terms w ι terms r ι ι o Syntax of embedded logic as abbreviations of HOL-terms P = P ι o = λs ι o λw ι (S W) = λs ι o λt ι o λw ι (S W) (T W) = λr ι ι o λs ι o λw ι V ι (R W V) (S V) ( p ), µ = λq µ (ι o) λw ι X µ (Q X W) f = λs ι o λt ι o λw ι V ι (f W S V) (T V) [BenzmüllerGenovese, NCMPL, 2011], [BenzmüllerGabbayGenoveseRispoli, Logica Universalis, 2012] C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

39 Embedding Meta-Level Notions Validity valid = λϕ ι o W ι ϕ W Similar: Satisfiability, Countersatisfiability, Unsatisfiability C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

40 Embedding Meta-Level Notions Validity valid = λϕ ι o W ι ϕ W Similar: Satisfiability, Countersatisfiability, Unsatisfiability Soundness and Completeness Theorem = ϕ iff = HOL validϕ ι o Consequence: Automation for free in HOL-ATPs! C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

41 Can Peter retire happy? Chris thinks that Peter can retire happy, if he knows that HOL-ATP is fostered by someone knowledgechris ( knowledgepeter X.fosters(X, holatp) canretirehappy(peter)) Peter knows that Chris fosters HOL-ATP knowledgepeter fosters(chris, holatp) Peter knows that Chad fosters HOL-ATP knowledgepeter fosters(chad, holatp) Peter knows that other persons do foster HOL-ATP Chris thinks that Peter can retire happy knowledgechris canretirehappy(peter) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

42 Can Peter retire happy? TPS says yes! Chris thinks that Peter can retire happy, if he knows that HOL-ATP is fostered by someone knowledgechris ( knowledgepeter X.fosters(X, holatp) canretirehappy(peter)) Peter knows that Chris fosters HOL-ATP knowledgepeter fosters(chris, holatp) Peter knows that Chad fosters HOL-ATP knowledgepeter fosters(chad, holatp) Peter knows that other persons do foster HOL-ATP Chris thinks that Peter can retire happy knowledgechris canretirehappy(peter) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

43 Automating Peter s Retirement Example Ax1 valid knowledgechris ( knowledgepeter X.fosters(X, holatp) canretirehappy(peter)) Ax2 Ax3 further axioms valid knowledgepeter fosters(chris, holatp) valid knowledgepeter fosters(chad, holatp) knowledgechris, knowledgepeter are S4 operators Conj valid knowledgechris canretirehappy(peter) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

44 Automating Peter s Retirement Example Ax1 valid knowledgechris ( knowledgepeter X.fosters(X, holatp) canretirehappy(peter)) Ax2 Ax3 further axioms valid knowledgepeter fosters(chris, holatp) valid knowledgepeter fosters(chad, holatp) knowledgechris, knowledgepeter are S4 operators Conj valid knowledgechris canretirehappy(peter) W ι knowledgechris canretirehappy(peter) W expanded abbreviation C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

45 Automating Peter s Retirement Example Ax1 valid knowledgechris ( knowledgepeter X.fosters(X, holatp) canretirehappy(peter)) Ax2 Ax3 further axioms valid knowledgepeter fosters(chris, holatp) valid knowledgepeter fosters(chad, holatp) knowledgechris, knowledgepeter are S4 operators Conj valid knowledgechris canretirehappy(peter) W ι knowledgechris canretirehappy(peter) W W ι V ι (knowledgechris W V) canretirehappy(peter) W expanded abbreviation C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

46 Automating Peter s Retirement Example Ax1 valid knowledgechris ( knowledgepeter X.fosters(X, holatp) canretirehappy(peter)) Ax2 Ax3 further axioms valid knowledgepeter fosters(chris, holatp) valid knowledgepeter fosters(chad, holatp) knowledgechris, knowledgepeter are S4 operators Conj valid knowledgechris canretirehappy(peter) W ι knowledgechris canretirehappy(peter) W W ι V ι (knowledgechris W V) canretirehappy(peter) W W ι V ι (knowledgechris W V) (canretirehappy peter W) expanded abbreviation C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

47 Translating Kripke style semantics to HOL Kripke style semantics M, w = P arbitrary M, w = s iff not M, w = s M, w = s t iff M, w = s or M, w = t Semantic embedding in HOL P = P ι o = λs ι o λw ι (S W) = λs ι o λt ι o λw ι (S W) (T W) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

48 Translating Kripke style semantics to HOL Kripke style semantics M, w = P arbitrary M, w = s iff not M, w = s M, w = s t iff M, w = s or M, w = t M, w = r s iff M, v = s for all v such that r(w, v) Semantic embedding in HOL P = P ι o = λs ι o λw ι (S W) = λs ι o λt ι o λw ι (S W) (T W) = λr ι ι o λs ι o λw ι V ι (R W V) (S V) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

49 Translating Kripke style semantics to HOL Kripke style semantics M, w = P arbitrary M, w = s iff not M, w = s M, w = s t iff M, w = s or M, w = t M, w = r s iff M, v = s for all v such that r(w, v) Semantic embedding in HOL P = P ι o = λs ι o λw ι (S W) = λs ι o λt ι o λw ι (S W) (T W) = λr ι ι o λs ι o λw ι V ι (R W V) (S V) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

50 Translating Kripke style semantics to HOL Kripke style semantics M, w = P arbitrary M, w = s iff not M, w = s M, w = s t iff M, w = s or M, w = t M, w = r s iff M, v = s for all v such that r(w, v) Semantic embedding in HOL P = P ι o = λs ι o λw ι (S W) = λs ι o λt ι o λw ι (S W) (T W) r = λs ι o λw ι V ι (r W V) (S V) To model r as T, S4 operator etc. add axioms like (reflexive r), etc. C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

51 Translating Kripke style semantics to HOL Kripke style semantics higher-order selection function! M, w = P arbitrary M, w = s iff not M, w = s M, w = s t iff M, w = s or M, w = t M, w = r s iff M, v = s for all v such that r(w, v) M, w = s f t iff M, v = t for all v f(w,[s]) with [s] = {u M, u = s} Semantic embedding in HOL P = P ι o = λs ι o λw ι (S W) = λs ι o λt ι o λw ι (S W) (T W) = λr ι ι o λs ι o λw ι V ι (R W V) (S V) f = λs ι o λt ι o λw ι V ι (f W S V) (T V) Add respective axioms for f C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

52 Quantified Modal Logics: Varying and Cumulative Domain Constant Domain Π = λq λw ι X µ (Q X W) Y s = ΠλY s [BenzmüllerPaulson, Logica Universalis, 2012] C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

53 Quantified Modal Logics: Varying and Cumulative Domain Constant Domain Varying and Cumulative Domain Π = λq λw ι X µ (Q X W) Y s = ΠλY s Π var = λq λw ι X µ (exinw X W) (Q X W) A: W ι X µ (exinw X W) B(c): W ι (exinw c W) B(f): W ι (exinw t 1 W)... (exinw t n W) (exinw(f t 1,... t n ) W) [BenzmüllerOttenRaths, ECAI 2012] C. Benzmüller [BenzmüllerPaulson, UtilizingLogica Church s Universalis, Type Theory 2012] as a Universal Logic TU Wien,

54 Quantified Modal Logics: Varying and Cumulative Domain Constant Domain Varying and Cumulative Domain Π = λq λw ι X µ (Q X W) Y s = ΠλY s Π var = λq λw ι X µ (exinw X W) (Q X W) A: W ι X µ (exinw X W) B(c): W ι (exinw c W) B(f): W ι (exinw t 1 W)... (exinw t n W) (exinw(f t 1,... t n ) W) C: X µ, V ι, W ι (exinw X V) (r V W) (exinw X W) [BenzmüllerOttenRaths, ECAI 2012] C. Benzmüller [BenzmüllerPaulson, UtilizingLogica Church s Universalis, Type Theory 2012] as a Universal Logic TU Wien,

55 Natural Fragments of HOL: Quantified Conditional Logics This work extends [BenzmüllerGenoveseGabbayRispoli, AMAI, 2012 (arxiv: v3)] [BenzmüllerPaulson, Logica Universalis, 2012 (arxiv: v1)] C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

56 Quantified Conditional Logics Motivation Theory for (Reasoning with) Counterfactual Conditionals If I had continued with competitive long-distance running in 1992, I would have won the Olympic Games in Problem: non-truth-functionality of counterfactual conditional statements Solution (Stalnaker and Thomason) selection function semantics (a possible world semantics, extension of modal logics) [Stalnaker68] If A then B }{{} (A B) is true in world w iff B is true for all v f(w, A) idea: f selects worlds that are very similar/close to the actual world w many closely related theories: [Lewis73, Pollock76, Chellas75] C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

57 Quantified Conditional Logic Syntax ϕ,ψ ::= P ϕ ϕ ψ ϕ ψ C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

58 Quantified Conditional Logic Syntax Propositional Variables (PV) Individual Variables (IV) Constants (Sym) ϕ,ψ ::= P ϕ ϕ ψ ϕ ψ P.ϕ X.ϕ k(x 1,...,X n ) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

59 Quantified Conditional Logic Syntax Propositional Variables (PV) Individual Variables (IV) Constants (Sym) ϕ,ψ ::= P ϕ ϕ ψ ϕ ψ P.ϕ X.ϕ k(x 1,...,X n ) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

60 Quantified Conditional Logic Syntax Propositional Variables (PV) Individual Variables (IV) Constants (Sym) ϕ,ψ ::= P ϕ ϕ ψ ϕ ψ P.ϕ X.ϕ k(x 1,...,X n ) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

61 Quantified Conditional Logic Syntax Propositional Variables (PV) Individual Variables (IV) Constants (Sym) ϕ,ψ ::= P ϕ ϕ ψ ϕ ψ P.ϕ X.ϕ k(x 1,...,X n ) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

62 Quantified Conditional Logic Syntax Propositional Variables (PV) Individual Variables (IV) Constants (Sym) ϕ,ψ ::= P ϕ ϕ ψ ϕ ψ P.ϕ X.ϕ k(x 1,...,X n ) Logical Connectives and Quantifiers (others may be defined as usual) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

63 Quantified Conditional Logic Syntax Propositional Variables (PV) Individual Variables (IV) Constants (Sym) ϕ,ψ ::= P ϕ ϕ ψ ϕ ψ P.ϕ X.ϕ k(x 1,...,X n ) Conditional (modal) operator C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

64 Quantified Conditional Logic Semantics ϕ,ψ ::= P ϕ ϕ ψ ϕ ψ P.ϕ X.ϕ k(x 1,...,X n ) Interpretation is a structure M = S, f, D, Q, I with S set of possible worlds f : S 2 S 2 S is the selection function D is a non-empty set of individuals (the first-order domain) Q is a non-empty collection of subsets of S (the propositional domain) I is a classical interpretation function where for each n-ary predicate symbol k, I(k, w) D n Variable Assignment g = g iv, g pv g iv : IV D maps individual variables to objects in D g pv : PV Q maps propositional variables to sets of worlds in Q C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

65 Quantified Conditional Logic Semantics ϕ,ψ ::= P ϕ ϕ ψ ϕ ψ P.ϕ X.ϕ k(x 1,...,X n ) Interpretation is a structure M = S, f, D, Q, I with S set of possible worlds f : S 2 S 2 S is the selection function D is a non-empty set of individuals (the first-order domain) Q is a non-empty collection of subsets of S (the propositional domain) I is a classical interpretation function where for each n-ary predicate symbol k, I(k, w) D n Variable Assignment g = g iv, g pv g iv : IV D maps individual variables to objects in D g pv : PV Q maps propositional variables to sets of worlds in Q C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

66 Quantified Conditional Logic Semantics Satisfiability M, g, s = ϕ defined as: M, g, s = P iff s g(p) M, g, s = k(x 1,...,X n ) iff s g(x 1 ),...,g(x n ) I(k, w) M, g, s = ϕ iff not M, g, s = ϕ M, g, s = ϕ ψ iff M, g, s = ϕ or M, g, s = ψ { [ϕ] }} { M, g, s = ϕ ψ iff M, g, v = ψ for all v f(s, {u M, g, u = ϕ}) M, g, s = X ϕ iff M,[d/X]g, s = ϕ for all d D M, g, s = P ϕ iff M,[p/P]g, s = ϕ for all p Q Validity M = ϕ iff for all worlds s and assignments g holds M, g, s = ϕ = ϕ iff ϕ is valid in every model M C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

67 Quantified Conditional Logic Semantics Satisfiability M, g, s = ϕ defined as: M, g, s = P iff s g(p) M, g, s = k(x 1,...,X n ) iff s g(x 1 ),...,g(x n ) I(k, w) M, g, s = ϕ iff not M, g, s = ϕ M, g, s = ϕ ψ iff M, g, s = ϕ or M, g, s = ψ { [ϕ] }} { M, g, s = ϕ ψ iff M, g, v = ψ for all v f(s, {u M, g, u = ϕ}) M, g, s = X ϕ iff M,[d/X]g, s = ϕ for all d D M, g, s = P ϕ iff M,[p/P]g, s = ϕ for all p Q Validity M = ϕ iff for all worlds s and assignments g holds M, g, s = ϕ = ϕ iff ϕ is valid in every model M C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

68 Quantified Conditional Logic Semantics Satisfiability M, g, s = ϕ defined as: M, g, s = P iff s g(p) M, g, s = k(x 1,...,X n ) iff s g(x 1 ),...,g(x n ) I(k, w) M, g, s = ϕ iff not M, g, s = ϕ M, g, s = ϕ ψ iff M, g, s = ϕ or M, g, s = ψ { [ϕ] }} { M, g, s = ϕ ψ iff M, g, v = ψ for all v f(s, {u M, g, u = ϕ}) M, g, s = X ϕ iff M,[d/X]g, s = ϕ for all d D M, g, s = P ϕ iff M,[p/P]g, s = ϕ for all p Q Validity M = ϕ iff for all worlds s and assignments g holds M, g, s = ϕ = ϕ iff ϕ is valid in every model M C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

69 Quantified Conditional Logic Semantics Satisfiability M, g, s = ϕ defined as: M, g, s = P iff s g(p) M, g, s = k(x 1,...,X n ) iff s g(x 1 ),...,g(x n ) I(k, w) M, g, s = ϕ iff not M, g, s = ϕ M, g, s = ϕ ψ iff M, g, s = ϕ or M, g, s = ψ { [ϕ] }} { M, g, s = ϕ ψ iff M, g, v = ψ for all v f(s, {u M, g, u = ϕ}) M, g, s = X ϕ iff M,[d/X]g, s = ϕ for all d D M, g, s = P ϕ iff M,[p/P]g, s = ϕ for all p Q Validity M = ϕ iff for all worlds s and assignments g holds M, g, s = ϕ = ϕ iff ϕ is valid in every model M C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

70 Quantified Conditional Logic Semantics Satisfiability M, g, s = ϕ defined as: M, g, s = P iff s g(p) M, g, s = k(x 1,...,X n ) iff s g(x 1 ),...,g(x n ) I(k, w) M, g, s = ϕ iff not M, g, s = ϕ M, g, s = ϕ ψ iff M, g, s = ϕ or M, g, s = ψ { [ϕ] }} { M, g, s = ϕ ψ iff M, g, v = ψ for all v f(s, {u M, g, u = ϕ}) M, g, s = X ϕ iff M,[d/X]g, s = ϕ for all d D M, g, s = P ϕ iff M,[p/P]g, s = ϕ for all p Q Validity M = ϕ iff for all worlds s and assignments g holds M, g, s = ϕ = ϕ iff ϕ is valid in every model M C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

71 Quantified Conditional Logic Normality Above semantics of enforces normality property: if ϕ and ϕ are equivalent, then they index the same formulas wrt. C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

72 Quantified Conditional Logic Normality Above semantics of enforces normality property: if ϕ and ϕ are equivalent, then they index the same formulas wrt. The axiomatic counterpart of the normality condition given by rule (RCEA) ϕ ϕ (RCEA) (ϕ ψ) (ϕ ψ) Above semantics forces also the following rules to hold: (ϕ 1... ϕ n) ψ (RCK) (ϕ 0 ϕ 1... ϕ 0 ϕ n) (ϕ 0 ψ) ϕ ϕ (RCEC) (ψ ϕ) (ψ ϕ ) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

73 Quantified Conditional Logic Normality Above semantics of enforces normality property: if ϕ and ϕ are equivalent, then they index the same formulas wrt. The axiomatic counterpart of the normality condition given by rule (RCEA) ϕ ϕ (RCEA) (ϕ ψ) (ϕ ψ) Above semantics forces also the following rules to hold: (ϕ 1... ϕ n) ψ (RCK) (ϕ 0 ϕ 1... ϕ 0 ϕ n) (ϕ 0 ψ) ϕ ϕ (RCEC) (ψ ϕ) (ψ ϕ ) Logic CK: minimal logic closed under rules RCEA, RCEC and RCK. In what follows only logic CK and its extensions are considered. C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

74 Quantified Conditional Logics as Fragments of HOL Kripke style semantics (higher-order) selection function! M, g, s = P iff s g(p) M, g, s = k(x 1,...,X n ) iff s g(x 1 ),...,g(x n ) I(k, w) M, g, s = ϕ iff not M, g, s = ϕ M, g, s = ϕ ψ iff M, g, s = ϕ or M, g, s = ψ { [ϕ] }} { M, g, s = ϕ ψ iff M, g, v = ψ for all v f(s, {u M, g, u = ϕ}) M, g, s = X ϕ iff M,[d/X]g, s = ϕ for all d D M, g, s = P ϕ iff M,[p/P]g, s = ϕ for all p Q Semantic embedding: ML HOL terms of type ι o P = λw ι (P ι o W) = P ι o k(x 1,...,X n ) = λw ι (k µ n (ι o) Xµ 1...Xµ) n W = λϕ ι o λw ι (ϕw) = λϕ ι o λψ ι o λw ι (ϕw) (ψ W) = λϕ ι o λψ ι o λw ι V ι (f W ϕv) (ψ V) µ (Π µ ) = λq µ (ι o) λw ι X µ (Q X W) p (Π p ) = λq (ι o) (ι o) λw ι P ι o (Q P W) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

75 Quantified Conditional Logics as Fragments of HOL Kripke style semantics (higher-order) selection function! M, g, s = P iff s g(p) M, g, s = k(x 1,...,X n ) iff s g(x 1 ),...,g(x n ) I(k, w) M, g, s = ϕ iff not M, g, s = ϕ M, g, s = ϕ ψ iff M, g, s = ϕ or M, g, s = ψ { [ϕ] }} { M, g, s = ϕ ψ iff M, g, v = ψ for all v f(s, {u M, g, u = ϕ}) M, g, s = X ϕ iff M,[d/X]g, s = ϕ for all d D M, g, s = P ϕ iff M,[p/P]g, s = ϕ for all p Q Semantic embedding: ML HOL terms of type ι o P = λw ι (P ι o W) = P ι o k(x 1,...,X n ) = λw ι (k µ n (ι o) Xµ 1...Xµ) n W = λϕ ι o λw ι (ϕw) = λϕ ι o λψ ι o λw ι (ϕw) (ψ W) = λϕ ι o λψ ι o λw ι V ι (f W ϕv) (ψ V) µ (Π µ ) = λq µ (ι o) λw ι X µ (Q X W) p (Π p ) = λq (ι o) (ι o) λw ι P ι o (Q P W) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

76 Quantified Conditional Logics as Fragments of HOL Kripke style semantics (higher-order) selection function! M, g, s = P iff s g(p) M, g, s = k(x 1,...,X n ) iff s g(x 1 ),...,g(x n ) I(k, w) M, g, s = ϕ iff not M, g, s = ϕ M, g, s = ϕ ψ iff M, g, s = ϕ or M, g, s = ψ { [ϕ] }} { M, g, s = ϕ ψ iff M, g, v = ψ for all v f(s, {u M, g, u = ϕ}) M, g, s = X ϕ iff M,[d/X]g, s = ϕ for all d D M, g, s = P ϕ iff M,[p/P]g, s = ϕ for all p Q Semantic embedding: ML HOL terms of type ι o P = λw ι (P ι o W) = P ι o k(x 1,...,X n ) = λw ι (k µ n (ι o) Xµ 1...Xµ) n W = λϕ ι o λw ι (ϕw) = λϕ ι o λψ ι o λw ι (ϕw) (ψ W) = λϕ ι o λψ ι o λw ι V ι (f W ϕv) (ψ V) µ (Π µ ) = λq µ (ι o) λw ι X µ (Q X W) p (Π p ) = λq (ι o) (ι o) λw ι P ι o (Q P W) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

77 Quantified Conditional Logics as Fragments of HOL Kripke style semantics (higher-order) selection function! M, g, s = P iff s g(p) M, g, s = k(x 1,...,X n ) iff s g(x 1 ),...,g(x n ) I(k, w) M, g, s = ϕ iff not M, g, s = ϕ M, g, s = ϕ ψ iff M, g, s = ϕ or M, g, s = ψ { [ϕ] }} { M, g, s = ϕ ψ iff M, g, v = ψ for all v f(s, {u M, g, u = ϕ}) M, g, s = X ϕ iff M,[d/X]g, s = ϕ for all d D M, g, s = P ϕ iff M,[p/P]g, s = ϕ for all p Q Semantic embedding: ML HOL terms of type ι o P = λw ι (P ι o W) = P ι o k(x 1,...,X n ) = λw ι (k µ n (ι o) Xµ 1...Xµ) n W = λϕ ι o λw ι (ϕw) = λϕ ι o λψ ι o λw ι (ϕw) (ψ W) = λϕ ι o λψ ι o λw ι V ι (f W ϕv) (ψ V) µ (Π µ ) = λq µ (ι o) λw ι X µ (Q X W) p (Π p ) = λq (ι o) (ι o) λw ι P ι o (Q P W) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

78 Quantified Conditional Logics as Fragments of HOL Kripke style semantics (higher-order) selection function! M, g, s = P iff s g(p) M, g, s = k(x 1,...,X n ) iff s g(x 1 ),...,g(x n ) I(k, w) M, g, s = ϕ iff not M, g, s = ϕ M, g, s = ϕ ψ iff M, g, s = ϕ or M, g, s = ψ { [ϕ] }} { M, g, s = ϕ ψ iff M, g, v = ψ for all v f(s, {u M, g, u = ϕ}) M, g, s = X ϕ iff M,[d/X]g, s = ϕ for all d D M, g, s = P ϕ iff M,[p/P]g, s = ϕ for all p Q Semantic embedding: ML HOL terms of type ι o P = λw ι (P ι o W) = P ι o k(x 1,...,X n ) = λw ι (k µ n (ι o) Xµ 1...Xµ) n W = λϕ ι o λw ι (ϕw) = λϕ ι o λψ ι o λw ι (ϕw) (ψ W) = λϕ ι o λψ ι o λw ι V ι (f W ϕv) (ψ V) µ (Π µ ) = λq µ (ι o) λw ι X µ (Q X W) p (Π p ) = λq (ι o) (ι o) λw ι P ι o (Q P W) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

79 Quantified Conditional Logics as Fragments of HOL Kripke style semantics (higher-order) selection function! M, g, s = P iff s g(p) M, g, s = k(x 1,...,X n ) iff s g(x 1 ),...,g(x n ) I(k, w) M, g, s = ϕ iff not M, g, s = ϕ M, g, s = ϕ ψ iff M, g, s = ϕ or M, g, s = ψ { [ϕ] }} { M, g, s = ϕ ψ iff M, g, v = ψ for all v f(s, {u M, g, u = ϕ}) M, g, s = X ϕ iff M,[d/X]g, s = ϕ for all d D M, g, s = P ϕ iff M,[p/P]g, s = ϕ for all p Q Semantic embedding: ML HOL terms of type ι o P = λw ι (P ι o W) = P ι o k(x 1,...,X n ) = λw ι (k µ n (ι o) Xµ 1...Xµ) n W = λϕ ι o λw ι (ϕw) = λϕ ι o λψ ι o λw ι (ϕw) (ψ W) = λϕ ι o λψ ι o λw ι V ι (f W ϕv) (ψ V) µ (Π µ ) = λq µ (ι o) λw ι X µ (Q X W) p (Π p ) = λq (ι o) (ι o) λw ι P ι o (Q P W) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

80 Quantified Conditional Logics as Fragments of HOL Kripke style semantics (higher-order) selection function! M, g, s = P iff s g(p) M, g, s = k(x 1,...,X n ) iff s g(x 1 ),...,g(x n ) I(k, w) M, g, s = ϕ iff not M, g, s = ϕ M, g, s = ϕ ψ iff M, g, s = ϕ or M, g, s = ψ { [ϕ] }} { M, g, s = ϕ ψ iff M, g, v = ψ for all v f(s, {u M, g, u = ϕ}) M, g, s = X ϕ iff M,[d/X]g, s = ϕ for all d D M, g, s = P ϕ iff M,[p/P]g, s = ϕ for all p Q Semantic embedding: ML HOL terms of type ι o P = λw ι (P ι o W) = P ι o k(x 1,...,X n ) = λw ι (k µ n (ι o) Xµ 1...Xµ) n W = λϕ ι o λw ι (ϕw) = λϕ ι o λψ ι o λw ι (ϕw) (ψ W) = λϕ ι o λψ ι o λw ι V ι (f W ϕv) (ψ V) µ (Π µ ) = λq µ (ι o) λw ι X µ (Q X W) p (Π p ) = λq (ι o) (ι o) λw ι P ι o (Q P W) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

81 Soundness and Completeness Validity defined as before valid = λϕ ι o W ι ϕ W Soundness and Completeness Theorem Proof Idea: = QCL ϕ iff = HOL validϕ ι o Explicate and analyze the relation between selection functions semantics and corresponding Henkin models; see paper for details. C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

82 Automation Quantified Conditional Logics in HOL ATPs Proof of the Barcan formula (confirms constant domain) ( X.ϕ ψ(x)) (ϕ X.ψ(X)) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

83 Automation Quantified Conditional Logics in HOL ATPs Proof of the Barcan formula (confirms constant domain) ( X.ϕ ψ(x)) (ϕ X.ψ(X)) valid( X.ϕ (ψ X)) (ϕ X.(ψ X)) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

84 Automation Quantified Conditional Logics in HOL ATPs Proof of the Barcan formula (confirms constant domain) ( X.ϕ ψ(x)) (ϕ X.ψ(X)) valid (Π µ λx ϕ (ψ X)) (ϕ Π µ λx (ψ X)) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

85 Automation Quantified Conditional Logics in HOL ATPs Proof of the Barcan formula (confirms constant domain) ( X.ϕ ψ(x)) (ϕ X.ψ(X)) W ι ( (Π µ λx ϕ (ψ X)) (ϕ Π µ λx (ψ X)))W C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

86 Automation Quantified Conditional Logics in HOL ATPs Proof of the Barcan formula (confirms constant domain) ( X.ϕ ψ(x)) (ϕ X.ψ(X)) W ι (λv ι (( (Π µ λx ϕ (ψ X)) V) ((ϕ Π µ λx (ψ X)) V))) W C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

87 Automation Quantified Conditional Logics in HOL ATPs Proof of the Barcan formula (confirms constant domain) ( X.ϕ ψ(x)) (ϕ X.ψ(X)) W ι ( (Π µ λx ϕ (ψ X)) W (ϕ Π µ λx (ψ X)) W) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

88 Automation Quantified Conditional Logics in HOL ATPs Proof of the Barcan formula (confirms constant domain) ( X.ϕ ψ(x)) (ϕ X.ψ(X))... by LEO-II or Satallax in 0.01 seconds C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

89 Automation Quantified Conditional Logics in HOL ATPs Proof of the Barcan formula (confirms constant domain) ( X.ϕ ψ(x)) (ϕ X.ψ(X))... by LEO-II or Satallax in 0.01 seconds Proof of the Converse Barcan formula (ϕ X.ψ(x)) ( X.ϕ ψ(x)) by LEO-II or Satallax in 0.01 seconds C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

90 Natural Fragments of HOL C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

91 Soundness and Completeness Results = ϕ iff = HOL validϕ ι o Propositional Multimodal Logics [BenzmüllerPaulson, Log.J.IGPL, 2010] Quantified Multimodal Logics [BenzmüllerPaulson, Logica Universalis, 2012] Propositional Conditional Logics [BenzmüllerEtAl., AMAI, 2012] Quantified Conditional Logics [BenzmüllerGenovese, NCMPL, 2011] Intuitionistic Logics: [BenzmüllerPaulson, Log.J.IGPL, 2010] Access Control Logics: [Benzmüller, IFIP SEC, 2009] Combinations of Logics: [Benzmüller, AMAI, 2011] C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

92 Why not throwing things together? Terms: m ::= c X (f m 1... m n ) Formulas: s, t ::= P (k m 1... m n ) s s t r s s f t X.s var X.s P.s Embedding in HOL: c = c µ X = X µ f = f µn µ P = P ι o k = k µn ι o r = k ι ι o (+axioms for r) f = f ι ι o (+axioms for f) = λs ι o λw ι (S W) = λs ι o λt ι o λw ι (S W) (T W) = λr ι ι o λs ι o λw ι V ι (R W V) (S V) = λf ι (ι o)ι o λs ι o λt ι o λw ι V ι (F W S V) (T V) Π = λq µ (ι o) λw ι X µ (Q X W) Π p = λq (ι o) (ι o) λw ι P ι o (Q P W) Π var/cumul = λq λw ι X µ (exinw X W) (Q X W)... further non-classical connectives, quantification over higher types, predicate abstraction, definite description... C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

93 Reasoning about Logics (and their Combinations) [Benzmüller, Festschrift Walther, 2010] [Benzmüller, AMAI, 2012] C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

94 Automating Meta-Properties of Logics in HOL Correspondences between axioms and semantic properties valid φ r φ r r φ (transitive r) Dependence/independence of axioms base modal logic K = axiom 4? Inclusion/non-inclusion relations between logics Is logic K45 (K+M+5) included in logic S4 (K+M+4)? (Relative) Consistency of logics and logic combinations Is logic S4 (K+M+4) consistent? Experiments: Modal Logics [Benzmüller, Festschrift Walther, 2010] Conditional Logics [Benzmüller, AMAI, 2012] C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

95 Automating Meta-Properties of Logics in HOL M S4 S5 = M5 MB5 M4B5 M45 M4B D4B D4B5 DB5 B = MB D D4 D5 D45 DB M K 4 5 B K4 K5 K45 KB5 K4B5 K4B K KB [Garson, Modal Logic, SEP 2009] C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

96 Automating Meta-Properties of Logics in HOL M.4 D S4.3.4 D D5.4.4 D S5 = M5 MB5 M4B5.2 M45.1 M4B.1.4 D4B.2 D4B5.1 DB5 B.6= MB fastest results by: DB.5 LEO-II Satallax TPS IsabelleP IsabelleN IsabelleM.1 K.1.1 K4.3 K K KB KB5.2 K4B5.1 K4B [Benzmüller, Festschrift Walther, 2010] C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

97 Automating Meta-Properties of Logics in HOL Semantic Conditions for Conditional Logic Axioms TPS ID Axiom A f A Condition f(w,[a]) [A] MP Axiom (A f B) (A B) Condition [A] f(w,[a]) CS Axiom (A B) (A f B) Condition w [A] f(w,[a]) {w} CEM Axiom (A f B) (A f B) Condition f(w,[a]) 1 AC Axiom (A f B) (A f C) (A C f B) Condition f(w,[a]) [B] f(w,[a B]) f(w,[a]) RT Axiom (A B f C) ((A f B) (A f C)) Condition f(w,[a]) [B] f(w,[a]) f(w,[a B]) CV Axiom (A f B) (A f C) (A C f B) Condition (f(w,[a]) [B] and f(w,[a]) [C] ) f(w,[a C]) [B] CA Axiom (A f B) (C f B) (A C f B) Condition f(w,[a B]) f(w,[a]) f(w,[b]) [BenzmüllerEtAl., AMAI, 2012] C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

98 Proofs and Countermodels at Meta-Level The correct interpretation of the proof task for MP is [ A, B.(A f B) (A B)] [ A, W.A (f W A)] versus (incorrect statement for MP) A, B.[((A f B) (A B)) W.A (f W A)] The former is provable. The latter is countersatisfiable; the countermodel reported by Nitpick is: choose D i = {i1}, A = {i1}, B = {i1}, W = i1, and { { f = i1 {i1} C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

99 Evaluation of HOL-ATPs for First-order Monomodal Logics [BenzmüllerOttenRaths, ECAI 2012] C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

100 QMLTP project: HOL-ATPs perform well for FML The QMLTP project: see Jens Otten and Thomas Raths, University of Potsdam infrastructure and benchmark library for testing and evaluating ATP systems for first-order modal logic collaborators: myself, Geoff Sutcliffe s TPTP project standardized extended TPTP syntax (called fml ) 600 problems in 11 problem domains 20 problems in first-order multimodal logic C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

101 See our paper at ECAI 2012: Theory & implementation of new provers for FML: embedding into higher-order logic (LEO-II & Satallax) a connection calculus based prover (MleanCoP) a sequent calculus based prover (MleanSeP) a tableau based prover (MleanTAP) an instantiation based prover (f2p-mspass) Moreover, we present a first comparative prover evaluation exploiting the new QMLTP library for FML C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

102 Experiment: 580 problems 5 logics 3 domain conditions 6 provers 600s tmo }{{} 8700 problems C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

103 Experiment: 580 problems 5 logics 3 domain conditions 6 provers 600s tmo Logic/ ATP system Domain f2p-mspass MleanSeP LEO-II Satallax MleanTAP MleanCoP v3.0 v1.2 v1.4.2 v2.2 v1.3 v1.2 K/varying K/cumul K/constant D/varying D/cumul D/constant T/varying T/cumul T/constant S4/varying S4/cumul S4/constant S5/varying S5/cumul S5/constant Strongest Prover! C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

104 Experiment: 580 problems 5 logics 3 domain conditions 6 provers 600s tmo Logic/ ATP system Domain f2p-mspass MleanSeP LEO-II Satallax MleanTAP MleanCoP v3.0 v1.2 v1.4.2 v2.2 v1.3 v1.2 K/varying K/cumul K/constant D/varying D/cumul D/constant T/varying T/cumul T/constant S4/varying S4/cumul S4/constant S5/varying S5/cumul S5/constant Second best provers; best coverage; strong recent improvement ( 25%) C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

105 Experiment: 580 problems 5 logics 3 domain conditions 6 provers 600s tmo Logic/ ATP system Domain f2p-mspass MleanSeP LEO-II Satallax MleanTAP MleanCoP v3.0 v1.2 v1.4.2 v2.2 v1.3 v1.2 K/varying K/cumul K/constant D/varying D/cumul D/constant T/varying T/cumul T/constant S4/varying S4/cumul S4/constant S5/varying S5/cumul S5/constant Results for 20 multimodal logic problems: LEO-II 15, Satallax 14 C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

106 Demo C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

107 Demo: HOL-ATPs as Universal easoners Analyzing example formula ( ( X.pX) Y. py qy) Z.qZ with HOL-ATPs: constant varying cumulative K CSA CSA??? D CSA CSA??? T THM THM THM S4 THM THM THM S5 THM THM THM CSA means Countersatisfiable, THM means Theorem C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

108 C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,

109 Conclusion Core Questions: 1 Classical Higher-order Logic (HOL) as Universal Logic? 2 HOL Provers & Model Finders as Generic Reasoning Tools? 3 Combinations with Specialist Reasoners (if available)? (1)&(2) are interesting and relevant: evidence given in talk!? (3) not further discussed: ongoing and future work Discussion Practical strength of approach? Collaboration with QMLTP project in Potsdam [BenzmüllerEtAl, ECAI, 2012] Scalability to large knowledge bases? How to deal with impredicativity? relevance filtering try to avoid C. Benzmüller Utilizing Church s Type Theory as a Universal Logic TU Wien,