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 Γ ⇒ G cut

• 1. Construct a proof-search tree TF for a formula F by

bottom-up applications of cut-free Gentzen-style rules R.

• 2. If all branches of TF terminate in axioms, then this tree is a

cut-free derivation of F.

• 3. Otherwise, by K¨
• nig’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¨

• del 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 Γ ⇒ ∆, α ∨ β ⇒ ∨ α, Γ ⇒ 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: S1 ∗ . . . ∗ Sn Si 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 Si to Sj. 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 Γ ⇒ ϕ&ψ ⇒ & ϕ, ψ, Γ ⇒ γ ϕ&ψ, Γ ⇒ γ & ⇒ ϕ, Γ ⇒ γ ψ, Γ ⇒ γ ϕ ∨ ψ, Γ ⇒ γ ⇒ ∨ Γ ⇒ ϕ Γ ⇒ ϕ ∨ ψ ∨ ⇒ Γ ⇒ ψ Γ ⇒ ϕ ∨ ψ

Γ ⇒ ϕ ψ, Γ ⇒ γ ϕ → ψ, Γ ⇒ γ →⇒ ϕ, Γ ⇒ ψ Γ ⇒ ϕ → ψ ⇒→ Γ ⇒ ϕ(t) Γ ⇒ ∃xϕ(x) ⇒ ∃ ϕ(y), Γ ⇒ θ ∃xϕ(x), Γ ⇒ θ ∃ ⇒ ϕ(t), Γ ⇒ θ ∀xϕ(x), Γ ⇒ θ ∀ ⇒ Γ ⇒ ϕ(y) Γ ⇒ ∀xϕ(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: S1 ∗ . . . ∗ Sn The number i is the place of the component Si 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 i0 = i, i1, . . . in = j such that rikik+1 for all k < n.

The most important configuration: T[{α}] := T ∗α, Γ ⇒ γ∗T1∗i : α, Γ1 ⇒ γ1∗. . .∗Tn∗i : α, Γn ⇒ γn∗Tn+1 (1) where i is the place of the component α, Γ ⇒ γ, all occurrences of i : α in antecedents are shown. Si 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&β}] & ⇒ T[{α}] T[{β}] T[{α ∨ β}] ∨ → T[{α[x/t]}] T[{∀xα}] ∀ ⇒ T[{α[x/b]}] T[{∃xα}] ∀ ⇒ 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 ⇒ ∀xϕ ∗ T ′ ∗ i : Γ ⇒ ϕ[x/y] T ∗ Γ ⇒ ∀xϕ ∗ T ′ ⇒ ∀ T ∗ Γ ⇒ ∃xϕ ∗ T ′ ∗ i : Γ ⇒ ϕ[x/t] T ∗ Γ ⇒ ∃xϕ ∗ T ′ ⇒ ∃ In ∀ ⇒, ⇒ ∃ applied to Si, the term t contains only variables present in components Sj 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, Γ1 ⇒ α∗. . .∗i : α → β, Γn ⇒ 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 Sj

• f the conclusion containing explicitly shown α → β and new

component Sj′ added for Sj, 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 S1 ∗ S2 ∗ . . . ∗ Sn in G* can be pruned into a derivation of one of the sequents Si 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* T1 . . . Tn 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 = Si

1 ∗ Si 2 ∗ . . . Si ni

S∞

k

=

• i

Si

ka ⇒

• i

Si

ks

Definition

W := {j : Sj occurs in T i for some i} D(j) := the set of all free variables and constants in all sequents S∞

l

for l such that Rlk. 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

• f the components is derivable:

d : α, α → β ⇒ α ∗ ⇒ β α, β, ⇒ γ ∗ β ⇒ β α, β, ⇒ γ ∗ ⇒ β trans 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.