Kripke Models, Proof Search and Cut-elimination for LJ Grigori Mints - PowerPoint PPT Presentation

Kripke Models, Proof Search and Cut-elimination for LJ Grigori Mints Stanford University/SRI

Abstract Existing Sch¨ utte-style completeness proofs for intuitionistic predicate logic with respect to Kripke models provide cut-elimination only for some semantic tableau formulations. Beth models extend this to multiple-succedent Gentzen calculus, but simple translation back to familiar one-succedent Gentzen calculus LJ introduces cuts. We present a short (non-effective) proof of completeness for Kripke models and cut-elimination for LJ.

Sch¨ utte’s Schema 1960 Γ ⇒ α α, Γ ⇒ G cut Γ ⇒ G 1. Construct a proof-search tree T F for a formula F by bottom-up applications of cut-free Gentzen-style rules R . 2. If all branches of T F terminate in axioms, then this tree is a cut-free derivation of F . 3. Otherwise, by K¨ onig’s Lemma there is an infinite branch B of this tree. Turn B into a model M refuting F : M �| = F . This argument proves completeness: if F is not derivable by R then F is not valid (= not true in all models).

History Classical Predicate Logic: G¨ odel 1930, Second Order Logic: Tait, 1966, Higher Order Logic: Prawitz ans Takahashi 1967 Intuitionistic Logic: Beth 1956

A disconect in Beth’s proof The rules R use multiple-succedent sequents. Γ ⇒ ∆ , α, β α, Γ ⇒ Γ ⇒ ∆ , α ∨ β ⇒ ∨ Γ ⇒ ∆ , ¬ α ⇒ ¬ This is not important for completeness theorem: multiple-succedent rules are directly derivable in LJ using cut . Also, Beth-style proof provides completeness for Beth models, but not for much more popular Kripke models.

Kripke’s proof Completeness for Kripke models: Kripke, 1965 by Sch¨ utte’s schema. Semantic tableau: S 1 ∗ . . . ∗ S n S i are multiple-succedent sequents; a relation of accessibility: rij , 1 ≤ i < j ≤ n semantic tableau T ⇓ a characteristic formula χ ( T ) The translation of every tableau rule is derivable in multiple-succedent sequent formulation plus cut . A Transfer rule: If rij then transfer every formula α from the antecedent of S i to S j . to ensure monotonicity: Rww ′ implies w | = α ⇒ w ′ | = α.

No simple transformation of tableau derivation into a cut-free sequent derivation is known. Present paper: a proof-search procedure for intuitionistic predicate logic that directly provides cut-elimination for LJ. It uses semantic tableaux and applies antecedent inference rules in parallel to all accessible sequents when the rule is applied to a parent sequent. This device provides also some economy in proof search.

Why proofs by one-succedent sequents are desirable? 1. Connection to natural deduction (Prawitz translation). 2. Extracting programs from intuitionistic proofs.

system G ≈ LJ Axioms: ϕ, Γ ⇒ ϕ ⊥ , Γ ⇒ P for atomic formulas P Inference rules: Γ ⇒ ϕ Γ ⇒ ψ ϕ, ψ, Γ ⇒ γ ⇒ & ϕ & ψ, Γ ⇒ γ & ⇒ Γ ⇒ ϕ & ψ ϕ, Γ ⇒ γ ψ, Γ ⇒ γ Γ ⇒ ϕ Γ ⇒ ψ ⇒ ∨ Γ ⇒ ϕ ∨ ψ ∨ ⇒ ϕ ∨ ψ, Γ ⇒ γ Γ ⇒ ϕ ∨ ψ

Γ ⇒ ϕ ψ, Γ ⇒ γ ϕ, Γ ⇒ ψ →⇒ Γ ⇒ ϕ → ψ ⇒→ ϕ → ψ, Γ ⇒ γ Γ ⇒ ϕ ( t ) ϕ ( y ) , Γ ⇒ θ Γ ⇒ ∃ x ϕ ( x ) ⇒ ∃ ∃ x ϕ ( x ) , Γ ⇒ θ ∃ ⇒ ϕ ( t ) , Γ ⇒ θ Γ ⇒ ϕ ( y ) ∀ x ϕ ( x ) , Γ ⇒ θ ∀ ⇒ Γ ⇒ ∀ x ϕ ( x ) ⇒ ∀ with standard provisos in quantifier rules.

Tableau System G* for Predicate Logic Marked formula : i : α . The notation i : Γ means that i is attached to each formula of the set Γ. Sequent : Γ → α , Γ is a finite set of formulas and marked formulas. A tableau: S 1 ∗ . . . ∗ S n The number i is the place of the component S i in T . The length | T | = n of a tableau T . A binary immediate accessibility relation r on { 1 , . . . , n } . rij → i < j ; R : the reflexive transitive closure of r : Rij iff i = j or there are i 0 = i , i 1 , . . . i n = j such that ri k i k +1 for all k < n .

The most important configuration: T [ { α } ] := T ∗ α, Γ ⇒ γ ∗ T 1 ∗ i : α, Γ 1 ⇒ γ 1 ∗ . . . ∗ T n ∗ i : α, Γ n ⇒ γ n ∗ T n +1 (1) where i is the place of the component α, Γ ⇒ γ , all occurrences of i : α in antecedents are shown. S i is the parent component; the i : α are subordinate .

The rules of G* Axioms of G*: T ∗ ϕ, Γ ⇒ ϕ ∗ T ′ T ∗ ⊥ , Γ ⇒ ϕ ∗ T ′ Antecedent rules T [ { α, β } ] T [ { α } ] T [ { β } ] ∨ → T [ { α & β } ] & ⇒ T [ { α ∨ β } ] T [ { α [ x / t ] } ] T [ { α [ x / b ] } ] ∀ ⇒ ∀ ⇒ T [ {∀ x α } ] T [ {∃ x α } ] In all antecedent rules except →⇒ the relation r for a premise of a rule is the same as in the conclusion.

Succedent rules T ∗ Γ ⇒ ϕ → ψ ∗ T ′ ∗ ϕ, i : Γ ⇒ ψ ⇒→ T ∗ Γ ⇒ ϕ → ψ ∗ T ′ T ∗ Γ ⇒ ϕ & ψ ∗ T ′ ∗ i : Γ ⇒ ϕ T ∗ Γ ⇒ ϕ & ψ ∗ T ′ ∗ i : Γ ⇒ ψ ⇒ & T ∗ Γ ⇒ ϕ & ψ ∗ T ′ T ∗ Γ ⇒ ϕ ∨ ψ ∗ T ′ ∗ i : Γ ⇒ ϕ ∗ i : Γ ⇒ ψ ⇒ ∨ Γ ⇒ ϕ ∨ ψ T ∗ Γ ⇒ ∀ x ϕ ∗ T ′ ∗ i : Γ ⇒ ϕ [ x / y ] ⇒ ∀ T ∗ Γ ⇒ ∀ x ϕ ∗ T ′ T ∗ Γ ⇒ ∃ x ϕ ∗ T ′ ∗ i : Γ ⇒ ϕ [ x / t ] ⇒ ∃ T ∗ Γ ⇒ ∃ x ϕ ∗ T ′ In ∀ ⇒ , ⇒ ∃ applied to S i , the term t contains only variables present in components S j with Rji . The relation r for the premise extends the relation for the conclusion by the pair ( i , n + 1) when one component is added, and by pairs ( i , n + 1) , ( i , n + 2) when two components are added (as in → ∨ ). Here n is the length of the conclusion.

The rule →⇒ T [ α → β ; α ] T [ β ] →⇒ T [ α → β ] where T [ α → β ; α ] := T [ α → β ] ∗ α → β, Γ ⇒ α ∗ i : α → β, Γ 1 ⇒ α ∗ . . . ∗ i : α → β, Γ n ⇒ α In the premise T [ β ] the relation r is the same as in the conclusion. In T [ α → β ; α ] the relation r is extended: for each component S j of the conclusion containing explicitly shown α → β and new component S j ′ added for S j , add the pair rjj ′ .

Theorem System G* is equivalent to G: A sequent is derivable in G* iff it is derivable in G. Proof. Each of the inclusions is proved separately. Lemma (Pruning) Any derivation of a tableau S 1 ∗ S 2 ∗ . . . ∗ S n in G* can be pruned into a derivation of one of the sequents S i in G by deleting some components or whole tableaux. Lemma System G is contained in G*. Proof. Add redundant sequents.

Proof-Search in Predicate Logic; Completeness A proof-search procedure for G* consists of tree extension steps . bottom-up applications of one of the rules of G* T 1 . . . T n T Inference rules are applied (bottom-up) during proof-search in a fair way: every possible application of a rule to every component in every tableau is made, except in closed tableaux. Let B be a branch of the proof-search tree: T 0 , T 1 , . . . (2) T k = S i 1 ∗ S i 2 ∗ . . . S i n i � � S i S i S ∞ = ka ⇒ k ks i i

Definition W := { j : S j occurs in T i for some i } D ( j ) := the set of all free variables and constants in all sequents S ∞ for l such that Rlk. l M = ( W , R ∞ , D , | =) where j | = α iff α ∈ S ∞ ja for atomic formulas α . The relation R ∞ is the union of relations R in tableaux T i . Lemma M is a Kripke model.

Lemma If B is a branch of the proof search tree then α ∈ S ∞ ja implies j | = α ; α = S ∞ js implies j �| = α. Theorem G* is complete.

Note. The Pruning lemma is false in the presence of the Transfer rule since there are tableaux derivable in G*+transfer where none of the components is derivable: α, β, ⇒ γ ∗ β ⇒ β trans α, α → β ⇒ α ∗ ⇒ β α, β, ⇒ γ ∗ ⇒ β d : α, α → β ⇒ γ ∗ ⇒ β

Such a tableau is actually encountered in a proof-search tree of a sequent: d α, α → β ⇒ β ∨ γ ∗ α, α → β ⇒ β α, α → β ⇒ β ∨ γ up to structural rules.