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Linear Logic Pages Linear Logic Pages Yves Lafont 1999 (last correction in 2017) http://iml.univ-mrs.fr/~lafont/linear/ Sequent calculus Proofs Formulas- Sequents and rules- Basic equivalences and second order definability Alternative formulations: Two-sided sequent calculus- Hybrid sequent calculus Variations: Intuitionistic Linear Logic- More structural rules Reductions Theorem: Cut can be eliminated and identity can be restricted to the atomic case. Cut elimination (key cases)- Cut elimination (commutative cases)- Expansion of identities Corollaries: subformula property, strong and weak consistency, splitting property, disjunction property, and existence property. About provable formulas Translations Polarities- Intuitionistic Logic- Classical Logic Miscellaneous Invariants About exponential rules 2 Phase semantics and Decision problems MLL = Multiplicative Linear Logic MALL = Multiplicative Additive Linear Logic MELL = Multiplicative Exponential Linear Logic LL = Linear Logic WLL = Affine Linear Logic NP = non deterministic polynomial time PSPACE = polynomial space NEXPTIME = non deterministic exponential time Propositional case Phase spaces- Phase models- Syntactical model Theorem: If A is a propositional formula, the following properties are equivalent: 1. A is provable; 2. A holds in any phase model; 3. A holds in the syntactical model; 4. A is cut-free provable. Corollary: cut elimination (propositional case) Finite model property Finite models - Semilinear models MLL MALL MELL LL WLL finite model property yes yes no no yes semilinear model property yes yes ? no yes decidability yes yes ? no yes Provability problem MLL MALL MELL LL propositional case NP-complete PSPACE-complete ? undecidable first order case NP-complete NEXPTIME-complete undecidable undecidable second order case undecidable undecidable undecidable undecidable 3 Formulas Formulas are built from atoms and units, using (binary) connectives, modalities, and quantifiers: • An atom is a propositional (i.e. second order) variable α or its dual α?. More generally, it is an atomic predicate ? α(t1; : : : ; tn) or its dual α (t1; : : : ; tn), where t1; : : : ; tn are first order terms. • The multiplicative connectives are ⊗ (times, or multiplicative conjunction) and its dual (par, or multiplicative disjunction). The corresponding units are 1 (one) and ? (bottom). ` • The additive connectives are & (with, or additive conjunction) and its dual ⊕ (plus, or additive disjunction). The corresponding units are > (top) and 0 (zero). • The exponential modalities are ! (of course) and its dual ? (why not). • The quantifiers are 8 (for all, or universal quantifier) and its dual 9 (exists, or existential quantifier). A quantifier applies to a first order variable x or to a second order variable α. If A is an atom, A? stands for its dual. In particular, A?? = A. Linear negation is extended to all formulas by de Morgan equations: (A ⊗ B)? = A? B?; (A B)? = A? ⊗ B?; 1? = ?; ?? = 1; ` ` (A & B)? = A? ⊕ B?; (A ⊕ B)? = A? & B?; >? = 0; 0? = >; (!A)? = ?A?; (?A)? = !A?; (8ξ:A)? = 9ξ:A?; (9ξ:A)? = 8ξ:A?: ? Linear implication is defined by A ( B = A B. ` If x is a first order variable and t is a first order term, A[t=x] stands for the formula A where all free occurrences of x have been replaced by t (and bound variables have been renamed when necessary). Similarly, if α is a propositional variable and B is a formula, A[B/α] stands for the formula A where all free occurrences of α (respectively α?) have been replaced by B (respectively B?). 4 Sequents and rules Sequents are of the form ` Γ where Γ is a (possibly empty) sequence of formulas A1;:::;An. In practise, the sequent ` Γ is identified with the sequence Γ, and a sequent consisting of a single formula is identified with this formula. ? ? ? Γ stands for A1 ;:::;An (respectively !Γ for !A1;:::; !An and ?Γ for ?A1;:::; ?An), and if ∆ is another sequence of formulas, Γ ` ∆ stands for ` Γ?; ∆. A sequent is provable if it can be derived using the following rules: ` Γ; A; B; ∆ id ` A; Γ ` A?; ∆ x ? cut ` Γ; B; A; ∆ ` A; A ` Γ; ∆ ` A; Γ ` B; ∆ ` A; B; Γ ` Γ ⊗ 1 ? ` A ⊗ B; Γ; ∆ ` A B; Γ ` 1 ` ?; Γ ` ` ` A; Γ ` B; Γ ` A; Γ ` B; Γ > & ⊕1 ⊕2 ` A & B; Γ ` A ⊕ B; Γ ` A ⊕ B; Γ ` >; Γ ` A; ?Γ ` A; Γ ` ?A; ?A; Γ ` Γ ! ?d ?c ?w ` !A; ?Γ ` ?A; Γ ` ?A; Γ ` ?A; Γ ` A; Γ ` A[τ/ξ]; Γ 8 9 ` 8ξ:A; Γ ` 9ξ:A; Γ Note that exchange is the only structural rule. The rules for exponentials are respectively called promotion, dereliction, contraction, and weakening. In the 8-rule, ξ must have no free occurrence in Γ, but it is well understood that a bound variable can always be renamed. In the 9-rule, τ is a first order term (if ξ is a first order variable) or a formula (if ξ is a second order variable). By exchange, any permutation of Γ can be derived from Γ, so that in practise, sequents are considered as finite multisets, and exchange is implicit. Similarly, the following rules are derivable: ` Γ; ∆ ` ?Γ; ?Γ; ∆ ` ∆ ?D ?C ?W ` ?Γ; ∆ ` ?Γ; ∆ ` ?Γ; ∆ 5 Basic equivalences and second order definability We say that A and B are equivalent and we write A ≡ B if both A ` B and B ` A are provable. This amounts to say that, for any Γ, the sequent ` A; Γ is provable if and only if ` B; Γ is provable, or more generally, that ` Γ[A/α] is provable if and only if ` Γ[B/α] is provable. Here are some typical equivalences: A ⊗ (B ⊗ C) ≡ (A ⊗ B) ⊗ C; A ⊗ B ≡ B ⊗ A; A ⊗ 1 ≡ A A (B C) ≡ (A B) C; A B ≡ B A; A ? ≡ A; ` ` ` ` ` ` ` A &(B & C) ≡ (A & B)& C; A & B ≡ B & A; A ≡ A & A; A & 1 ≡ A; A ⊕ (B ⊕ C) ≡ (A ⊕ B) ⊕ C; A ⊕ B ≡ B ⊕ A; A ≡ A ⊕ A; A ⊕ 0 ≡ A; A ⊗ (B ⊕ C) ≡ (A ⊗ B) ⊕ (A ⊗ C);A ⊗ 0 ≡ 0; A (B & C) ≡ (A B)&(A C);A > ≡ >; ` ` ` ` !!A ≡ !A; !A ≡ !A ⊗ !A; !1 ≡ 1; !(A & B) ≡ !A ⊗ !B; !> ≡ 1; ??A ≡ ?A; ?A ≡ ?A ?A; ?? ≡ ?; ?(A ⊕ B) ≡ ?A ?B; ?0 ≡ ?; ` ` 8ξ:8ζ:A ≡ 8ζ:8ξ:A; 8ξ:(A & B) ≡ 8ξ:A & 8ξ:B; A ≡ 8ξ:A; A 8ξ:B ≡ 8ξ:(A B); ` ` 9ξ:9ζ:A ≡ 9ζ:9ξ:A; 9ξ:(A ⊕ B) ≡ 9ξ:A ⊕ 9ξ:B; A ≡ 9ξ:A; A ⊗ 9ξ:B ≡ 9ξ:(A ⊗ B): In the last two equivalences of the last two lines, the variable ξ must have no free occurrence in A. By definition of linear implication, we also get: A ( (B ( C) ≡ (A ⊗ B) ( C; A ( (B C) ≡ (A ( B) C; ` ` ? ? ? A ( B ≡ B ( A ; 1 ( A ≡ A; A ( ? ≡ A ; A ( (B & C) ≡ (A ( B)&(A ( C); (A ⊕ B) ( C ≡ (A ( C)&(B ( C); A ( > ≡ >; 0 ( A ≡ >; A ( 8ξ:B ≡ 8ξ:(A ( B); 9ξ:B ( A ≡ 8ξ:(B ( A): In the last two equivalences, the variable ξ must have no free occurrence in A. Note also the following equivalences: 1 ≡ 8α.α ( α; A ⊕ B ≡ 8α:!(A ( α) ( !(B ( α) ( α; 0 ≡ 8α.α; ? ? ≡ 9α.α ⊗ α ;A & B ≡ 9α:!(α ( A) ⊗ !(α ( B) ⊗ α; > ≡ 9α.α: In other words: • Multiplicative units are definable in terms of second order quantifiers and multiplicative connectives. • Additive connectives are definable in terms of second order quantifiers, multiplicative connectives, and exponen- tials. • Additive units are definable in terms of second order quantifiers. 6 Two-sided sequent calculus Formulas are built in the same way as in the one-sided calculus, except that linear negation and linear implication are primitive connectives. Sequents are of the form Γ ` ∆ where Γ and ∆ are finite sequences of formulas. The rules are the following: Γ ` ∆; A; B; Θ Γ; A; B; ∆ ` Θ Γ ` A; ∆ Θ;A ` Λ ` x x ` id cut Γ ` ∆; B; A; Θ Γ; B; A; ∆ ` Θ A ` A Γ; Θ ` ∆; Λ Γ;A ` ∆ Γ ` A; ∆ ? ? Γ;A ` B; ∆ Γ ` A; ∆ Θ;B ` Λ ? ` · ? · ` ` ( ( ` Γ ` A ; ∆ Γ;A ` ∆ Γ ` A ( B; ∆ Γ; Θ;A ( B ` ∆; Λ Γ ` A; ∆ Θ ` B; Λ Γ; A; B ` ∆ Γ ` ∆ ` ⊗ ⊗ ` ` 1 1 ` Γ; Θ ` A ⊗ B; ∆; Λ Γ;A ⊗ B ` ∆ ` 1 Γ; 1 ` ∆ Γ;A ` ∆ Θ;B ` Λ Γ ` A; B; ∆ Γ ` ∆ ` ` ? ` ` ? Γ; Θ;A B ` ∆; Λ Γ ` A B; ∆ ? ` Γ ` ?; ∆ ` ` ` ` Γ ` A; ∆ Γ ` B; ∆ Γ;A ` ∆ Γ;B ` ∆ ` & & ` & ` ` > Γ ` A & B; ∆ Γ;A & B ` ∆ 1 Γ;A & B ` ∆ 2 Γ ` >; ∆ Γ;A ` ∆ Γ;B ` ∆ Γ ` A; ∆ Γ ` B; ∆ ⊕ ` ` ⊕ ` ⊕ 0 ` Γ;A ⊕ B ` ∆ Γ ` A ⊕ B; ∆ 1 Γ ` A ⊕ B; ∆ 2 Γ; 0 ` ∆ !Γ ` A; ?∆ Γ;A ` ∆ Γ; !A; !A ` ∆ Γ ` ∆ ` ! !d ` !c ` !w ` !Γ ` !A; ?∆ Γ; !A ` ∆ Γ; !A ` ∆ Γ; !A ` ∆ !Γ;A ` ?∆ Γ ` A; ∆ Γ ` ?A; ?A; ∆ Γ ` ∆ ? ` ` ?d ` ?c ` ?w !Γ; ?A ` ?∆ Γ ` ?A; ∆ Γ ` ?A; ∆ Γ ` ?A; ∆ Γ ` A; ∆ Γ;A[τ/ξ] ` ∆ Γ;A ` ∆ Γ ` A[τ/ξ]; ∆ ` 8 8 ` 9 ` ` 9 Γ ` 8ξ:A; ∆ Γ; 8ξ:A ` ∆ Γ; 9ξ:A ` ∆ Γ ` 9ξ:A; ∆ In the ` 8-rule and in the 9 `-rule, ξ must have no free occurrence in Γ; ∆. De Morgan equations become equivalences, so that any formula A of the two-sided calculus is equivalent to a formula Ab of the one-sided calculus. A sequent Γ ` ∆ is provable in the two-sided calculus if and only if the sequent ` Γb?; ∆b is provable in the one-sided calculus. In particular, a sequent ` Γ is provable in the one-sided calculus if and only if it is provable in the two-sided calculus. Hybrid sequent calculus Formulas are the same as in the one-sided calculus. Sequents are of the form ` Θ ; Γ where Θ and Γ are finite sequences of formulas. We write ` Γ for ` ; Γ. The rules are the following (J.-M. Andreoli): 0 ` Θ;Γ; A; B; ∆ ` Θ; A; Θ ; A; Γ id ` Θ; A; Γ ` Θ; A?; ∆ x a ? cut ` Θ;Γ; B; A; ∆ ` Θ; A; Θ0 ;Γ ` Θ; A; A ` Θ;Γ; ∆ ` Θ; A; Γ ` Θ; B; ∆ ` Θ; A; B; Γ ` Θ;Γ ⊗ 1 ? ` Θ; A ⊗ B; Γ; ∆ ` Θ; A B; Γ ` Θ; 1 ` Θ; ?; Γ ` ` ` Θ; A; Γ ` Θ; B; Γ ` Θ; A; Γ ` Θ; B; Γ > & ⊕1 ⊕2 ` Θ; A & B; Γ ` Θ; A ⊕ B; Γ ` Θ; A ⊕ B; Γ ` Θ; >; Γ ` Θ; A ` Θ;A ;Γ ` Θ; A; Γ ` Θ; A[τ/ξ]; Γ ! ? 8 9 ` Θ;!A ` Θ;?A; Γ ` Θ; 8ξ:A; Γ ` Θ; 9ξ:A; Γ The second structural rule is called absorption.
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