General motivations Model theory Recursion theory Lambda calculus - PowerPoint PPT Presentation

A N INTERACTIVE SEMANTICS FOR CLASSICAL PROOFS Michele Basaldella JAIST February 19, 2013

I NTRODUCTION

General motivations ◮ Model theory ◮ Recursion theory ◮ Lambda calculus ◮ Set theory ◮ Lattice theory ◮ Domain theory ◮ . . .

General motivations ◮ Model theory ◮ Recursion theory ◮ Lambda calculus ◮ Set theory ◮ Lattice theory ◮ Domain theory ◮ . . . ◮ Proof theory

General motivations ◮ Model theory ◮ Recursion theory ◮ Lambda calculus ◮ Set theory ◮ Lattice theory ◮ Domain theory ◮ . . . ◮ Proof theory We need a good theory of proofs .

Soundness and completeness theorem(s) ◮ Usual soundness and completeness theorems in logic state that F is provable if and only if F is true.

Soundness and completeness theorem(s) ◮ Usual soundness and completeness theorems in logic state that F is provable if and only if F is true. ◮ The aim of this talk is to show soundness and completeness theorems for proofs : roughly speaking, π is a proof of F if and only if **********.

Soundness and completeness theorem(s) ◮ Usual soundness and completeness theorems in logic state that F is provable if and only if F is true. ◮ The aim of this talk is to show soundness and completeness theorems for proofs : roughly speaking, π is a proof of F if and only if **********. ◮ I will use tools originally developed for the analysis of linear logic proofs in a different context.

Soundness and completeness theorem(s) ◮ Usual soundness and completeness theorems in logic state that F is provable if and only if F is true. ◮ The aim of this talk is to show soundness and completeness theorems for proofs : roughly speaking, π is a proof of F if and only if **********. ◮ I will use tools originally developed for the analysis of linear logic proofs in a different context. ◮ More specifically, the main inspiration is Girard’s ludics: ********** is a property determined by interaction .

Logic ◮ Logic = classical logic . ◮ Language = infinitary formulas . ◮ Proof–system = (a variant of) Tait’s calculus . Why this kind of logic? ◮ A purely logical approach to (first order, classical) arithmetic. ◮ All the relevant results also hold for the finitary restriction.

Logic ◮ Logic = classical logic . ◮ Language = infinitary formulas . ◮ Proof–system = (a variant of) Tait’s calculus . Why this kind of logic? ◮ A purely logical approach to (first order, classical) arithmetic. ◮ All the relevant results also hold for the finitary restriction. ◮ The delicate point is . . . Contraction rule.

Contraction Different “ degrees ” of contraction : ◮ Implicit contraction ⊢ Γ Γ Γ , A ⊢ Γ Γ , A Γ ⊢ Γ Γ Γ , B ⊢ Γ Γ Γ , C ⊢ Γ Γ Γ , A ∨ B ∨ C ⊢ Γ Γ Γ , A ∧ B ∧ C “No” contraction “No” contraction ⊢ Γ Γ Γ , B ∨ C , A ⊢ Γ Γ ⊢ Γ Γ ⊢ Γ Γ Γ , A Γ , B Γ , C ⊢ Γ Γ , A ∨ B ∨ C Γ ⊢ Γ Γ Γ , A ∧ B ∧ C Backtracking Backtracking ⊢ Γ Γ , A ∨ B ∨ C , A Γ ⊢ Γ Γ Γ , A ∧ B ∧ C , A ⊢ Γ Γ , A ∧ B ∧ C , B Γ ⊢ Γ Γ Γ , A ∧ B ∧ C , C ⊢ Γ Γ , A ∨ B ∨ C Γ ⊢ Γ Γ Γ , A ∧ B ∧ C Full contraction Full contraction

Main system ◮ Formulas : F , G , H , . . . generated in the usual way, using connectives ∨ , ∧ , ⊥ . . . . ◮ Sequents : Θ Θ Θ , Φ Φ Φ , . . . = finite non–empty sequences of formulas ⊢ F 0 , . . . , F n − 1 . ◮ Rules for deriving sequents. { Θ Θ a } a ∈ S Θ ( r ) Θ Θ Θ ◮ Derivations = well–founded trees labeled by sequent (which are “locally correct”). DEF � � System A = F , S , R , D

Auxiliary system ◮ Formulas : as in A ; ◮ Sequents ’ : Θ Θ Θ , Φ Φ Φ , . . . = unary sequences of formulas ⊢ ∗ F . ◮ Rules ’ for deriving sequents. { Θ Θ Θ a } a ∈ S ( r ) Θ Θ Θ ◮ Derivations ’ = well–founded trees labeled by sequent (which are “locally correct”). DEF � � System B = F , S’ , R’ , D’ ◮ Every sequent of B is derivable.

Interaction (I) ◮ Cut–elimination = an operation from trees labeled by sequents to trees labeled by sequents. ◮ Closed cuts = cuts of the form . . . . . . . π 0 . π n − 1 . π ⊢ ∗ F ⊥ ⊢ ∗ F ⊥ ⊢ F 0 , . . . , F n − 1 . . . n − 1 cut 0 where π is a derivation of ⊢ F 0 , . . . , F n − 1 in A , and π i is a derivation of ⊢ ∗ F ⊥ i in B , for each i < n . ◮ Cut elimination of closed cuts does not produce any cut–free sequent . . .

Interaction (II) ◮ . . . but the procedure of cut–elimination still makes sense: . . . . . . . π 0 . π 1 . π ⊢ ∗ F ⊥ ⊢ ∗ G ⊥ ⊢ F ∨ G , F ⊢ ∗ F ⊥ ∧ G ⊥ ⊢ F ∨ G cut reduces to . . . . . π 0 . π 1 . . . . ⊢ ∗ F ⊥ ⊢ ∗ G ⊥ . π 0 . π ⊢ ∗ F ⊥ ∧ G ⊥ ⊢ ∗ F ⊥ ⊢ F ∨ G , F cut

Interaction (II) ◮ . . . but the procedure of cut–elimination still makes sense: . . . . . . . π 0 . π 1 . π ⊢ ∗ F ⊥ ⊢ ∗ G ⊥ ⊢ F ∨ G , F ⊢ ∗ F ⊥ ∧ G ⊥ ⊢ F ∨ G cut reduces to . . . . . π 0 . π 1 . . . . ⊢ ∗ F ⊥ ⊢ ∗ G ⊥ . π 0 . π ⊢ ∗ F ⊥ ∧ G ⊥ ⊢ ∗ F ⊥ ⊢ F ∨ G , F cut ◮ We can study the properties of this procedure .

Generalization (I) ◮ We can also consider a more general version of closed cuts . . . . . . . π 0 . π n − 1 . π ⊢ F 0 , . . . , F n − 1 ⊢ ∗ G 0 . . . ⊢ ∗ G n − 1 cut where π is a derivation of ⊢ F 0 , . . . , F n − 1 in A and π i is a derivation of ⊢ ∗ G i in B , for each i < n . There are new situations to consider: ◮ Error : . . . π . . ⊢ F 1 ∨ F 2 , F 1 . π ′ ⊢ F 1 ∨ F 2 ⊢ ∗ G 1 ∨ G 2 cut reduces to an “ error .”

Generalization (II) ◮ Reduction : . . . . . . . . . π 1 . π 2 . π 3 . π ⊢ F 1 ∨ F 2 , F 1 ⊢ ∗ G 1 ⊢ ∗ G 2 ⊢ ∗ G 3 ⊢ F 1 ∨ F 2 ⊢ ∗ G 1 ∧ G 2 ∧ G 3 cut reduces to . . . . . . . π 1 . π 2 . π 3 . . . . ⊢ ∗ G 1 ⊢ ∗ G 2 ⊢ ∗ G 3 . π 1 . π ⊢ F 1 ∨ F 2 , F 1 ⊢ ∗ G 1 ∧ G 2 ∧ G 3 ⊢ ∗ G 1 cut

Generalization (II) ◮ Reduction : . . . . . . . . . π 1 . π 2 . π 3 . π ⊢ F 1 ∨ F 2 , F 1 ⊢ ∗ G 1 ⊢ ∗ G 2 ⊢ ∗ G 3 ⊢ F 1 ∨ F 2 ⊢ ∗ G 1 ∧ G 2 ∧ G 3 cut reduces to . . . . . . . π 1 . π 2 . π 3 . . . . ⊢ ∗ G 1 ⊢ ∗ G 2 ⊢ ∗ G 3 . π 1 . π ⊢ F 1 ∨ F 2 , F 1 ⊢ ∗ G 1 ∧ G 2 ∧ G 3 ⊢ ∗ G 1 cut ◮ We can study the properties of this procedure .

Generalization (+) ◮ Instead of considering derivations in A , we shall consider proof–terms , that we call tests T , U , V , . . . ◮ Intuition: : derivations in A Tests = Untyped lambda terms : derivations in minimal logic (natural deduction) ◮ A test does not contain all the information of a derivation. But we can consider closed cuts of the form T ⊢ ∗ G 0 . . . ⊢ ∗ G n − 1 cut and define a procedure of reduction ( interaction ).

T REES

Notation ◮ N ∗ = { s , t , u , . . . } = the set of finite sequences of natural numbers . ◮ Some sequences: = the empty sequence ; ( ) a = unary sequence; a 0 a 1 = binary sequence; a 0 a 1 · · · a k − 1 = k –ary sequence. ◮ st = the concatenation of s and t . ◮ In particular, if s is a k –ary sequence and a ∈ N , then sa is ( k + 1 ) –ary sequence. ⇒ there is u ∈ N ∗ such that t = su . DEF ◮ Prefix order : s ⊑ t ⇐

Trees ◮ A tree T is a non–empty subset of N ∗ such that if t ∈ T and s ⊑ t , then s ∈ T . ◮ Since T is non–empty, ( ) ∈ T . ( ) is called the root of T . ◮ An infinite branch in T is a infinite subset S ⊆ T of the form S = { ( ) , a 0 , a 0 a 1 , . . . , a 0 a 1 · · · a n − 1 , . . . } . ◮ A tree is said to be well–founded if it does not contain an infinite branch. ◮ A labeled tree is a pair L = ( T , ϕ ) consisting of a tree T and a function ϕ defined on T . ◮ ϕ is called the labeling function of L . The codomain of ϕ is called the set of labels . ◮ We write tree � � � � L and lab L for the underlying tree of L and its labeling function respectively, i.e., if L = ( T , ϕ ) , then � � � � tree L = T and lab L = ϕ .

S YSTEM A

System A System A is a variant of Tait’s calculus (1968). ◮ Finite sequences instead of finite sets. ◮ No propositional variables in this talk. ◮ Only subsets of natural numbers as index sets.

Formulas The formulas of our language are inductively defined as follows: if for some S ⊆ N , { G a } a ∈ S is a family of formulas, then � S G a and � S G a are formulas. Some terminology and notation: ◮ � S G a = disjunction ; ◮ � S G a = conjunction ; DEF ◮ 0 = � ∅ G a ; DEF ◮ 1 = � ∅ G a .

Negation and sequents The negation of a formula F , noted by F ⊥ , is the formula recursively defined as follows: � ⊥ DEF G a ⊥ � � ⊥ DEF G a ⊥ � � � � � � � S G a = � ; S G a = � . S S In particular, 0 ⊥ = 1 , and 1 ⊥ = 0 . The negation is involutive : F ⊥⊥ = F . Θ Φ Φ , . . . of A is a non–empty finite sequence A sequent Θ Θ , Φ ⊢ F 0 , . . . , F n − 1 of formulas ( n > 0).