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Kripke-Style Semantics for Normal Systems Arnon Avron and Ori Lahav Tel Aviv University LATD 2010

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Outline “Normal” Gentzen systems Generalized non-deterministic Kripke-style semantics General soundness and completeness theorem Analycity of normal Gentzen systems Extensions to hypersequents Application: non-deterministic connectives in Gödel logic 2

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Normal Gentzen Systems Fully-structural propositional sequential systems Normal derivation rule: ◦ The rule: s 1 s 2 … s m / c ◦ Its application: 3 (s 1 )+context 1 … (s m )+context m (c)+context

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Normal Rules Example 4 ,φ ,ψ ,φψ,φ ,ψ ,φψ p 1 p 2 / p 1 p 2 ,φ ψ φψ,φ ψ φψ

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Normal Rules Example 5 ,φ ,ψ ,φψ,φ ,ψ ,φψ p 1 p 2 / p 1 p 2 ,φ ψ φψ,φ ψ φψ Each premise is associated with a context restriction. p 1 p 2 / p 1 p 2 wff wff wff

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Many-Sided Sequents Classical sequent: ( , are finite sets of formulas) n-sided sequent: 1 ; 2 ; … ; n Notation ◦ Let I be a finite set of signs. ◦ A signed formula: i ψ (i I and ψ is a formula). ◦ A sequent is a finite set of signed formulas. Negative interpretation [Baaz, Fermüller, Zach `93] {i 1 ψ 1, …, i n ψ n } is true iff i k is not the truth-value assigned to ψ k for some i k n Classical sequent notation I = {t,f} is{ t ψ | ψ } { f ψ | ψ } 6

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Normal Rules A normal rule: ◦ Premises: sequents s 1 … s m ◦ Context restrictions: corresponding sets of signed formulas 1 … m ◦ A Conclusion: a sequent c Its application: 7 (s 1 ) (s’ 1 ) … (s m ) (s’ m ) (c) s’

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Examples S4’s Box I = {t,f} p 1, / p 1 ={t φ | φ wff } {f φ | φ wff } 3-Valued Lukasiewicz's Implication [Zach `93] I = {f,I,t} f p 1, , I p 1, I p 2, , t p 2, / t p 1 p 2 ={ i φ | i I, φ wff } (i.e. no context restriction) 8 ,ψ ,ψ ,ψ ,ψ , φ ; ; Σ ; , φ, ψ ; Σ ; ; Σ, ψ ; ; Σ, φ ψ

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Normal Gentzen Systems Axioms Weakenings Cut Finite set of normal Gentzen rules 9 iφ, jφiφ, jφ s s, i φ s, i 1 φ … s, i n φ s for every i j in I for every i I where I ={i 1,…,i n }

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Examples of Normal Gentzen Systems Normal Gentzen systems are suitable for: ◦ Classical logic ◦ Intuitionistic logic ◦ Dual-intuitionistic logic ◦ Bi-intuitionistic logic ◦ S4 and S5 ◦ n-valued Lukasiewicz logic ◦ … 10

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Kripke Semantics Kripke Frame ◦ Set of worlds W ◦ Legal set of relations R 1,R 2,… W W ◦ Legal valuation v : W x wff I A sequent s is true in a W iff i φ s. v(a, φ ) i A sequent is R-true in a W iff it is true in every b W s.t. aRb 11

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G-Legal Kripke Frames A normal system G induces a set of G-legal Kripke frames: ◦ A preorder R for every context-restriction in G. If is the set of all signed formulas, R is the identity relation. ◦ Generalized persistence condition If i φ , v(a, φ )=i implies v(b, φ )=i for every b s.t. aR b. ◦ The valuation should respect the logical rules of G Respect a rule s 1, 1 ,…, s m, m / c : If 1 i m. σ (s i ) is R i -true in a W, then σ (c) is true in a 12

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Rules Semantics = R wff ◦ v(a, ) is free when v(a, )=f and (v(b, )=f or v(b, )=t for every b a) Example: Primal Intuitionistic Implication (Gurevich ‘09) 13 if v(a, )=t then v(b, )=t for every b a Persistence v(a, )=t if v(b, )=t for every b a Respect rule #1 v(a, )=f if v(a, )=t and v(a, )=f Respect rule #2 p 2, wff / p 1 p 2 1 , , , p 1, wff wff , p 2 , wff wff / p 1 p 2 2

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Strong Soundness and Completeness For every normal system G, C ├ G s iff every G-legal frame which is a model of C is also a model of s. Applications: ◦ Several known completeness theorems are immediately obtained as special cases. ◦ Compactness theorem. ◦ Basis for semantic proofs of analycity. ◦ Semantic decision procedure for analytic systems. 14

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Analycity A normal system is (strongly) analytic iff it has the subformula property, i.e. C ├ G s implies that there exists a proof of s from C in G that contains only subformulas of C and s. Analycity implies: decidability, consistency and conservativity. 15

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Semantic Characterization of Analycity Def Given a set E of formulas, an E-Semiframe is a Kripke frame, except for: v : W E I instead of v : W wff I. Notation Given a set E of formulas, C ├ G E s iff there exists a proof of s from C in G containing only formulas from E. Thm C ├ G E s iff every G-legal E-semiframe which is a model of C is also a model of s. Cor G is analytic iff for every C and s, if every G-legal frame which is a model of C is also a model of s, then this is also true with respect to sub[C,s]-semiframes. 16

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Semantic Proofs of Analycity To prove that G is analytic, it suffices to prove: Every G-legal semiframe can beextended to a G-legal frame. This can be easily done in many important cases: ◦ Normal systems for classical logic, intuitionistic logic, dual and bi-intuitionistic logics, S4, S5, n-valued Lukasiewicz's logic… ◦ The family of coherent canonical systems. 17

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Normal Hypersequent Systems This approach can be extended to hypersequent systems (with internal contraction). Generalized communication rule: Examples: S4.3Gödel logic 18 G | s 1 (s 2 ) G | s 2 (s 1 ) G | s 1 | s 2 R is linear

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Application: Adding Non-Deterministic Connectives to Gödel Logic General Motivations: ◦ Uncertainty or ambiguity on the level of the connectives ◦ Gurevich’s Motivation: proof-search complexity Examples ◦ v( ) [v( ), v( ) v( )] ◦ v( ) [min{v( ),v( )}, max{v( ),v( )}] Remains decidable 19 G | G | G | , G | , G | , G | , G | , G | , G | , G | , G | ,

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