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Csr2011 june18 15_45_avron

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  • 1. A Multiple-Conclusion Calculus for First-Order Gödel Logic Arnon Avron Ori Lahav Tel Aviv University CSR June 2011
  • 2. Gödel Logic In 1933, Gödel introduced a sequence {Gn } of n-valued matrices and used them to show some important properties of intuitionistic logic. In 1959, Dummett embedded all the Gn s in an infinite-valued matrix Gω . Gω is equivalent to G[0,1] , a natural matrix for the truth-values [0, 1]. The logic of G[0,1] is called Gödel logic. Gödel logic is perhaps the most important intermediate logic. Nowadays, Gödel logic is also recognized as one of the three most basic fuzzy logics.
  • 3. Many-Valued Semantics A structure M consists of: Non-empty domain D An interpretation I: I[c] ∈ D for every constant I[f ] ∈ D n → D for every n-ary function I[p] ∈ D n → [0, 1] for every n-ary predicate
  • 4. Many-Valued Semantics A structure M consists of: Non-empty domain D An interpretation I: I[c] ∈ D for every constant I[f ] ∈ D n → D for every n-ary function I[p] ∈ D n → [0, 1] for every n-ary predicate An M-evaluation is a function e: Assigning an element of D for every free variable Naturally extended to all terms (according to I)
  • 5. Many-Valued Semantics • M is defined as follows: e M p(t1 , . . . , tn ) e = I[p][e[t1 ], . . . , e[tn ]] M ⊥ e =0
  • 6. Many-Valued Semantics • M is defined as follows: e M p(t1 , . . . , tn ) e = I[p][e[t1 ], . . . , e[tn ]] M ⊥ e =0 M M M ψ1 ∨ ψ2 e = max{ ψ1 e , ψ2 e } ψ1 ∧ ψ2 M e = min{ M ψ1 e , M ψ2 e }
  • 7. Many-Valued Semantics • M is defined as follows: e M p(t1 , . . . , tn ) e = I[p][e[t1 ], . . . , e[tn ]] M ⊥ e =0 M M M ψ1 ∨ ψ2 e = max{ ψ1 e , ψ2 e } ψ1 ∧ ψ2 M e = min{ M ψ1 e , M ψ2 e } M 1 ψ1 M ≤ e ψ2 M e ψ1 ⊃ ψ2 e = M ψ2 e otherwise
  • 8. Many-Valued Semantics • M is defined as follows: e M p(t1 , . . . , tn ) e = I[p][e[t1 ], . . . , e[tn ]] M ⊥ e =0 M M M ψ1 ∨ ψ2 e = max{ ψ1 e , ψ2 e } ψ1 ∧ ψ2 M e = min{ M ψ1 e , M ψ2 e } M 1 ψ1 M ≤ e ψ2 M e ψ1 ⊃ ψ2 e = M ψ2 e otherwise M M ∀xψ e = inf{ ψ e[x:=d] | d ∈ D} M M ∃xψ e = sup{ ψ e[x:=d] | d ∈ D}
  • 9. Many-Valued Semantics • M is defined as follows: e M p(t1 , . . . , tn ) e = I[p][e[t1 ], . . . , e[tn ]] M ⊥ e =0 M M M ψ1 ∨ ψ2 e = max{ ψ1 e , ψ2 e } ψ1 ∧ ψ2 M e = min{ M ψ1 e , M ψ2 e } M 1 ψ1 M ≤ e ψ2 M e ψ1 ⊃ ψ2 e = M ψ2 e otherwise M M ∀xψ e = inf{ ψ e[x:=d] | d ∈ D} M M ∃xψ e = sup{ ψ e[x:=d] | d ∈ D} M is a model of a formula if ϕ if ϕ M = 1 for every e e
  • 10. Kripke-Style Semantics A frame is a tuple W = W , ≤, D, I, {Iw }w∈W where: W - nonempty set of worlds ≤ - linear order on W D - non-empty (constant) domain I - interpretation of constants and functions
  • 11. Kripke-Style Semantics A frame is a tuple W = W , ≤, D, I, {Iw }w∈W where: W - nonempty set of worlds ≤ - linear order on W D - non-empty (constant) domain I - interpretation of constants and functions Iw - is a predicate interpretation for every w ∈ W : Iw [p] ⊆ D n for every n-ary predicate p Persistence: Iu [p] ⊆ Iw [p] if u ≤ w
  • 12. Kripke-Style Semantics Satisfaction relation for a frame, a world, and an evaluation of the free variables: W, w, e p(t1 , . . . , tn ) iff e[t1 ], . . . , e[tn ] ∈ Iw [p] W, w, e ⊥
  • 13. Kripke-Style Semantics Satisfaction relation for a frame, a world, and an evaluation of the free variables: W, w, e p(t1 , . . . , tn ) iff e[t1 ], . . . , e[tn ] ∈ Iw [p] W, w, e ⊥ W, w, e ψ1 ∨ ψ2 iff W, w, e ψ1 or W, w, e ψ2 W, w, e ψ1 ∧ ψ2 iff W, w, e ψ1 and W, w, e ψ2
  • 14. Kripke-Style Semantics Satisfaction relation for a frame, a world, and an evaluation of the free variables: W, w, e p(t1 , . . . , tn ) iff e[t1 ], . . . , e[tn ] ∈ Iw [p] W, w, e ⊥ W, w, e ψ1 ∨ ψ2 iff W, w, e ψ1 or W, w, e ψ2 W, w, e ψ1 ∧ ψ2 iff W, w, e ψ1 and W, w, e ψ2 W, w, e ψ1 ⊃ ψ2 iff for every u ≥ w: W, u, e ψ1 or W, u, e ψ2
  • 15. Kripke-Style Semantics Satisfaction relation for a frame, a world, and an evaluation of the free variables: W, w, e p(t1 , . . . , tn ) iff e[t1 ], . . . , e[tn ] ∈ Iw [p] W, w, e ⊥ W, w, e ψ1 ∨ ψ2 iff W, w, e ψ1 or W, w, e ψ2 W, w, e ψ1 ∧ ψ2 iff W, w, e ψ1 and W, w, e ψ2 W, w, e ψ1 ⊃ ψ2 iff for every u ≥ w: W, u, e ψ1 or W, u, e ψ2 W, w, e ∀xψ iff W, w, e[x:=d] ψ for every d ∈ D W, w, e ∃xψ iff W, w, e[x:=d] ψ for some d ∈ D
  • 16. Kripke-Style Semantics Satisfaction relation for a frame, a world, and an evaluation of the free variables: W, w, e p(t1 , . . . , tn ) iff e[t1 ], . . . , e[tn ] ∈ Iw [p] W, w, e ⊥ W, w, e ψ1 ∨ ψ2 iff W, w, e ψ1 or W, w, e ψ2 W, w, e ψ1 ∧ ψ2 iff W, w, e ψ1 and W, w, e ψ2 W, w, e ψ1 ⊃ ψ2 iff for every u ≥ w: W, u, e ψ1 or W, u, e ψ2 W, w, e ∀xψ iff W, w, e[x:=d] ψ for every d ∈ D W, w, e ∃xψ iff W, w, e[x:=d] ψ for some d ∈ D W is a model of a formula if ϕ if W, w, e ϕ for every w and e
  • 17. Proof Theory Sonobe 1975 - first cut-free Gentzen-type sequent calculus Other calculi have been proposed later by Corsi, Avellone et al., Dyckhoff and others All of them use some ad-hoc rules of a nonstandard form A. 1991 - hypersequent calculus with standard rules: HG Extended to first-order Gödel logic by Baaz and Zach in 2000: HIF
  • 18. Hypersequents A sequent (Gentzen 1934) is an object of the form ϕ1 , . . . , ϕn ⇒ ψ1 , . . . , ψm Intuition: ϕ1 ∧ . . . ∧ ϕn ⊃ ψ1 ∨ . . . ∨ ψm Single-conclusion sequent: m ≤ 1
  • 19. Hypersequents A sequent (Gentzen 1934) is an object of the form ϕ1 , . . . , ϕn ⇒ ψ1 , . . . , ψm Intuition: ϕ1 ∧ . . . ∧ ϕn ⊃ ψ1 ∨ . . . ∨ ψm Single-conclusion sequent: m ≤ 1 A hypersequent is an object of the form s1 | . . . | sn where the si ’s are sequents Intuition: s1 ∨ . . . ∨ sn Single-conclusion hypersequent consists of single-conclusion sequents only
  • 20. HG Single-Conclusion Hypersequent SystemStructural Rules H ϕ⇒ϕ (EW ) H|Γ⇒E H|Γ⇒E H|Γ⇒ (IW ⇒) (⇒ IW ) H | Γ, ψ ⇒ E H|Γ⇒ψ
  • 21. HG Single-Conclusion Hypersequent SystemStructural Rules H ϕ⇒ϕ (EW ) H|Γ⇒E H|Γ⇒E H|Γ⇒ (IW ⇒) (⇒ IW ) H | Γ, ψ ⇒ E H|Γ⇒ψ H1 | Γ1 ⇒ ϕ H2 | Γ2 , ϕ ⇒ E (cut) H1 | H2 | Γ1 , Γ2 ⇒ E
  • 22. HG Single-Conclusion Hypersequent SystemStructural Rules H ϕ⇒ϕ (EW ) H|Γ⇒E H|Γ⇒E H|Γ⇒ (IW ⇒) (⇒ IW ) H | Γ, ψ ⇒ E H|Γ⇒ψ H1 | Γ1 ⇒ ϕ H2 | Γ2 , ϕ ⇒ E (cut) H1 | H2 | Γ1 , Γ2 ⇒ E H1 | Γ1 , Γ1 ⇒ E1 H2 | Γ2 , Γ2 ⇒ E2 (com) H1 | H2 | Γ1 , Γ2 ⇒ E1 | Γ2 , Γ1 ⇒ E2
  • 23. HG Single-Conclusion Hypersequent SystemLogical Rules H1 | Γ1 , ψ1 ⇒ E H2 | Γ2 , ψ2 ⇒ E (∨ ⇒) H1 | H2 | Γ1 , Γ2 , ψ1 ∨ ψ2 ⇒ E H | Γ ⇒ ψ1 H | Γ ⇒ ψ2 (⇒ ∨1 ) (⇒ ∨2 ) H | Γ ⇒ ψ1 ∨ ψ2 H | Γ ⇒ ψ1 ∨ ψ2
  • 24. HG Single-Conclusion Hypersequent SystemLogical Rules H1 | Γ1 , ψ1 ⇒ E H2 | Γ2 , ψ2 ⇒ E (∨ ⇒) H1 | H2 | Γ1 , Γ2 , ψ1 ∨ ψ2 ⇒ E H | Γ ⇒ ψ1 H | Γ ⇒ ψ2 (⇒ ∨1 ) (⇒ ∨2 ) H | Γ ⇒ ψ1 ∨ ψ2 H | Γ ⇒ ψ1 ∨ ψ2 H1 | Γ1 ⇒ ψ1 H2 | Γ2 , ψ2 ⇒ E (⊃⇒) H1 | H2 | Γ1 , Γ2 , ψ1 ⊃ ψ2 ⇒ E H | Γ, ψ1 ⇒ ψ2 (⇒⊃) H | Γ ⇒ ψ1 ⊃ ψ2
  • 25. HIF Single-Conclusion Hypersequent System H | Γ, ϕ{t/x} ⇒ E (∀ ⇒) H | Γ, ∀xϕ ⇒ E H | Γ ⇒ ϕ{y /x} (⇒ ∀) H | Γ ⇒ ∀xϕ where y doesn’t occur free in any component of the conclusion.
  • 26. HIF Single-Conclusion Hypersequent System H | Γ, ϕ{t/x} ⇒ E (∀ ⇒) H | Γ, ∀xϕ ⇒ E H | Γ ⇒ ϕ{y /x} (⇒ ∀) H | Γ ⇒ ∀xϕ where y doesn’t occur free in any component of the conclusion. similar rules for ∃
  • 27. Cut-Admissibility H1 | Γ1 ⇒ ϕ H2 | Γ2 , ϕ ⇒ E (cut) H1 | H2 | Γ1 , Γ2 ⇒ E
  • 28. Cut-Admissibility H1 | Γ1 ⇒ ϕ H2 | Γ2 , ϕ ⇒ E (cut) H1 | H2 | Γ1 , Γ2 ⇒ E One of the most important properties of a (hyper)sequent calculus, provides the key for proof-search Traditional syntactic cut-admissibility proofs are notoriously prone to errors, especially (but certainly not only) in the case of hypersequent systems the first proof of cut-elimination for HIF was erroneous A semantic proof is usually more reliable and easier to check A semantic proof usually provides also a proof of completeness as well as strong cut-admissibility
  • 29. MCG Multiple-Conclusion Hypersequent SystemStructural Rules H ϕ⇒ϕ (EW ) H|Γ⇒∆ H|Γ⇒∆ H|Γ⇒∆ (IW ⇒) (⇒ IW ) H | Γ, ψ ⇒ ∆ H | Γ ⇒ ∆, ψ H1 | Γ1 ⇒ ∆1 , ϕ H2 | Γ2 , ϕ ⇒ ∆2 (cut) H1 | H2 | Γ1 , Γ2 ⇒ ∆1 , ∆2 H1 | Γ1 , Γ1 ⇒ ∆1 H2 | Γ2 , Γ2 ⇒ ∆2 (com) H1 | H2 | Γ1 , Γ2 ⇒ ∆1 | Γ2 , Γ1 ⇒ ∆2
  • 30. MCG Multiple-Conclusion Hypersequent SystemStructural Rules H ϕ⇒ϕ (EW ) H|Γ⇒∆ H|Γ⇒∆ H|Γ⇒∆ (IW ⇒) (⇒ IW ) H | Γ, ψ ⇒ ∆ H | Γ ⇒ ∆, ψ H1 | Γ1 ⇒ ∆1 , ϕ H2 | Γ2 , ϕ ⇒ ∆2 (cut) H1 | H2 | Γ1 , Γ2 ⇒ ∆1 , ∆2 H1 | Γ1 , Γ1 ⇒ ∆1 H2 | Γ2 , Γ2 ⇒ ∆2 (com) H1 | H2 | Γ1 , Γ2 ⇒ ∆1 | Γ2 , Γ1 ⇒ ∆2 H | Γ ⇒ ∆1 , ∆2 (split) H | Γ ⇒ ∆1 | Γ ⇒ ∆2
  • 31. MCG Multiple-Conclusion Hypersequent SystemLogical Rules H1 | Γ1 ⇒ ∆1 , ψ1 H2 | Γ2 , ψ2 ⇒ ∆2 H | Γ, ψ1 ⇒ ψ2 (⊃⇒) (⇒⊃) H1 | H2 | Γ1 , Γ2 , ψ1 ⊃ ψ2 ⇒ ∆1 , ∆2 H | Γ ⇒ ψ1 ⊃ ψ2 H | Γ, ϕ{t/x} ⇒ ∆ H | Γ ⇒ ∆, ϕ{y /x} (∀ ⇒) (⇒ ∀) H | Γ, ∀xϕ ⇒ ∆ H | Γ ⇒ ∆, ∀xϕ
  • 32. Results MCG is strongly sound and complete with respect to the Kripke semantics of the standard first-order Gödel logic
  • 33. Results MCG is strongly sound and complete with respect to the Kripke semantics of the standard first-order Gödel logic MCG admits strong cut-admissibility:
  • 34. Results MCG is strongly sound and complete with respect to the Kripke semantics of the standard first-order Gödel logic MCG admits strong cut-admissibility: for every set H of hypersequents closed under substitution and a hypersequent H: H H iff there exists a proof of H from H in which the cut-formula of every application of the cut rule is in frm[H]
  • 35. Results MCG is strongly sound and complete with respect to the Kripke semantics of the standard first-order Gödel logic MCG admits strong cut-admissibility: for every set H of hypersequents closed under substitution and a hypersequent H: H H iff there exists a proof of H from H in which the cut-formula of every application of the cut rule is in frm[H] As a corollary, we obtain the same results for HIF, the original single-conclusion calculus
  • 36. Thank you!