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-sed sequent calculus - Hybr sequent calculus Variations: Intuitionistic Linear Logic - More structural rules Reductions Theorem: Cut can be eliminated and entity can be restricted to the atomic case. Cut elimination (key cases) - Cut elimination (commutative cases) - Expansion of entities 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 decability yes yes? no yes Provability problem MLL MALL MELL LL propositional case NP-complete PSPACE-complete? undecable first order case NP-complete NEXPTIME-complete undecable undecable second order case undecable undecable undecable undecable 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(t 1,..., t n ) or its dual α (t 1,..., t n ), where t 1,..., t n 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 (for all, or universal quantifier) and its dual (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, ( ξ.a) = ξ.a, ( ξ.a) = ξ.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 A 1,..., A n. In practise, the sequent Γ is entified with the sequence Γ, and a sequent consisting of a single formula is entified with this formula. Γ stands for A 1,..., A n (respectively!γ for!a 1,...,!A n and?γ for?a 1,...,?A n ), 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, x Γ, B, A, A, A A, cut Γ, B, A B, Γ, A, B, Γ A ` B, Γ ` 1 1 Γ, Γ B, Γ & A & B, Γ A B, Γ 1 B, Γ A B, Γ 2, Γ A,?Γ!!A,?Γ?d?A, Γ?A,?A, Γ?c?A, Γ Γ?A, Γ?w ξ.a, Γ A[τ/ξ], Γ ξ.a, Γ Note that exchange is the only structural rule. The rules for exponentials are respectively called promotion, dereliction, contraction, and weakening. In the -rule, ξ must have no free occurrence in Γ, but it is well understood that a bound variable can always be renamed. In the -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 consered as finite multisets, and exchange is implicit. Similarly, the following rules are derivable: Γ,?D?Γ,?Γ,?C?W 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 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, ξ. ζ.a ζ. ξ.a, ξ.(a & B) ξ.a & ξ.b, A ξ.a, A ` ξ.b ξ.(a ` B), ξ. ζ.a ζ. ξ.a, ξ.(a B) ξ.a ξ.b, A ξ.a, A ξ.b ξ.(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 ξ.b ξ.(a B), ξ.b A ξ.(b A). In the last two equivalences, the variable ξ must have no free occurrence in A. Note also the following equivalences: In other words: 1 α.α α, A B α.!(a α)!(b α) α, 0 α.α, α.α α, A & B α.!(α A)!(α B) α, α.α. 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 exponentials. Additive units are definable in terms of second order quantifiers. 6
Two-sed sequent calculus Formulas are built in the same way as in the one-sed 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 x Γ, B, A, Θ Γ, A, B, Θ x Γ, B, A, Θ A A Γ A, Θ, A Λ cut Γ, Θ, Λ Γ, A Γ A, Γ A, Γ, A Γ, A B, Γ A B, Γ A, Θ, B Λ Γ, Θ, A B, Λ Γ A, Θ B, Λ Γ, Θ A B,, Λ Γ, A, B Γ, A B 1 1 Γ Γ, 1 1 Γ, A Θ, B Λ ` Γ, Θ, A ` B, Λ Γ A, B, Γ A ` B, ` Γ Γ, Γ A, Γ B, & Γ A & B, Γ, A Γ, A & B & 1 Γ, B Γ, A & B & 2 Γ, Γ, A Γ, B Γ, A B Γ A, Γ A B, 1 Γ B, Γ A B, 2 Γ, 0 0!Γ A,?!!Γ!A,? Γ, A!d Γ,!A Γ,!A,!A!c Γ,!A Γ Γ,!A!w!Γ, A?!Γ,?A?? Γ A,?d Γ?A, Γ?A,?A,?c Γ?A, Γ?w Γ?A, Γ A, Γ ξ.a, Γ, A[τ/ξ] Γ, ξ.a Γ, A Γ, ξ.a Γ A[τ/ξ], Γ ξ.a, In the -rule and in the -rule, ξ must have no free occurrence in Γ,. De Morgan equations become equivalences, so that any formula A of the two-sed calculus is equivalent to a formula  of the one-sed calculus. A sequent Γ is provable in the two-sed calculus if and only if the sequent Γ, is provable in the one-sed calculus. In particular, a sequent Γ is provable in the one-sed calculus if and only if it is provable in the two-sed calculus. Hybr sequent calculus Formulas are the same as in the one-sed 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): Θ ; Γ, A, B, x Θ ; Γ, B, A, Θ, A, Θ ; A, Γ a Θ, A, Θ ; Γ Θ ; A, A Θ ; A, Γ Θ ; A, cut Θ ; Γ, Θ ; A, Γ Θ ; B, Θ ; A B, Γ, Θ ; A, B, Γ Θ ; A ` B, Γ ` Θ ; 1 1 Θ ; Γ Θ ;, Γ Θ ; A, Γ Θ ; B, Γ & Θ ; A & B, Γ Θ ; A, Γ Θ ; A B, Γ 1 Θ ; B, Γ Θ ; A B, Γ 2 Θ ;, Γ Θ ; A! Θ ;!A Θ, A ; Γ? Θ ;?A, Γ Θ ; A, Γ Θ ; ξ.a, Γ Θ ; A[τ/ξ], Γ Θ ; ξ.a, Γ The second structural rule is called absorption. In the -rule, ξ must have no free occurrence in Θ, Γ. A sequent Θ ; Γ is provable in the hybr calculus if and only if?θ, Γ is provable in the one-sed calculus. In particular, Γ is provable in the one-sed calculus if and only if it is provable in the hybr calculus. Note that, in practise, Θ can be consered as a finite set of formulas. 7
Intuitionistic Linear Logic Formulas are built as in (Classical) Linear Logic, except that there is no negation, no why not, and par is replaced by linear implication. Sequents are of the form Γ A where Γ is a finite sequence of formulas and A is a formula. The rules are the following: Γ, A, B, C x Γ, B, A, C A A Γ A, A C cut Γ, C Γ, A B Γ A B Γ A, B C Γ,, A B C Γ A B Γ, A B Γ, A, B C Γ, A B C 1 1 Γ C Γ, 1 C 1 Γ A Γ B & Γ A & B Γ, A C Γ, A & B C & 1 Γ, B C Γ, A & B C & 2 Γ Γ, A C Γ, B C Γ, A B C Γ A Γ A B 1 Γ B Γ A B 2 Γ, 0 C 0!Γ A!!Γ!A Γ, A C!d Γ,!A C Γ,!A,!A C!c Γ,!A C Γ C Γ,!A C!w Γ A Γ ξ.a Γ, A[τ/ξ] C Γ, ξ.a C Γ, A C Γ, ξ.a C Γ A[τ/ξ] Γ ξ.a In the -rule and in the -rule, ξ must have no free occurrence in Γ (respectively in Γ and C). If A is provable in Intuitionistic Linear Logic, then it is obviously provable in Linear Logic. The converse holds if A contains no additive unit (, 0) and no second order quantifier, but not in general: Linear Logic is not a conservative extension of Intuitionistic Linear Logic (H. Schellinx). For instance, the formula ((α β) 0) α is provable in Linear Logic, but not in Intuitionistic Linear Logic. More structural rules Linear Logic does not allow (unrestricted) contraction and weakening: A, A, Γ c Γ w If we add them, we essentially get Classical Logic with modalities. In that case, we have A B A&B, A`B A B, 1, and 0. Furthermore, the -rule becomes redundant, as well as the?c-rule and the?w-rule. If we only add (unrestricted) weakening, we get Affine Linear Logic. In that case, we have 1, 0, and the following sequents become provable: A B A & B, A B A ` B. Furthermore, the -rule becomes redundant, as well as the?w-rule. If we only add (unrestricted) contraction, we get Contractive Linear Logic. In that case, the following sequents become provable: A & B A B, A ` B A B. Furthermore, the?c-rule becomes redundant. Contractive Linear Logic must not be confused with Relevant Logic, which has an artificial distributivity law, and therefore, no good property of cut elimination. 8
Cut elimination (key cases) A cut with an entity is eliminated as follows: A, A cut A cut between two matching logical rules is eliminated as follows: B, A, B, Θ ` A B, Γ, A ` B, Θ cut Γ,, Θ B, A, B, Θ cut A,, Θ cut Γ,, Θ 1 Γ 1, Γ cut Γ Γ B, Γ A, & 1 A & B, Γ A B, cut Γ, B, Γ B, & 2 A & B, Γ A B, cut Γ, A, cut Γ, B, Γ B, cut Γ, A,?Γ A,!?d!A,?Γ?A, cut A,?Γ A, cut A,?Γ?A,?A,!?c!A,?Γ?A, cut A,?Γ!!A,?Γ A,?Γ!!A,?Γ?Γ,?Γ,?C?A,?A, cut?a,?γ, cut A,?Γ!?w!A,?Γ?A, cut?w ξ.a, Γ A [τ/ξ], ξ.a, cut Γ, A[τ/ξ], Γ A [τ/ξ], cut Γ, 9
Cut elimination (commutative cases) Commutation with most logical rules is straigthforward. Here are some typical examples: A, B, C, Γ ` A, B ` C, Γ A, cut B ` C, Γ, A, B, Γ C, Θ A, B C, Γ, Θ A, cut B C, Γ,, Θ A, B, C, Γ A, cut B, C, Γ, ` B ` C, Γ, A, B, Γ A, cut B, Γ, C, Θ B C, Γ,, Θ A, B, Γ A, C, Γ & A, B & C, Γ B & C, Γ, A, cut A, B, Γ A, A, C, Γ A, cut cut B, Γ, C, Γ, & B & C, Γ, A,, Γ A, cut, Γ,, Γ, In the case of the promotion rule, the commutation is the following:?a, B,?Γ A,?!!?A,!B,?Γ!A,? cut!b,?γ,??a, B,?Γ A,?!!A,? cut B,?Γ,?!!B,?Γ,? Expansion of entities Identities are expanded into atomic ones as follows: A B, A ` B A, A B, B A B, A, B ` A B, A ` B 1, 1 1 1, A & B, A B A, A 1 A, A B A & B, A B B, B 2 B, A B &, 0, 0!A,?A A, A?d A,?A!!A,?A ξ.a, ξ.a A, A A, ξ.a ξ.a, ξ.a 10
About provable formulas In a cut-free proof of a formula A, the last rule must be a logical rule. Therefore: is not provable (strong consistency); 0 is not provable (weak consistency); if A B is provable, then A and B are provable (splitting property); if A B is provable, then A or B is provable (disjunction property); if ξ.a is provable, then A[τ/ξ] is provable for some τ (existence property). To sum up: 1, are provable, but not, 0, α, α ; A B, as well as A & B, is provable if and only if A and B are provable; A ` B is provable if and only if the sequent A, B is provable; A B is provable if and only if A or B is provable;!a, as well as ξ.a, is provable if and only if A is provable;?a is provable whenever A is provable; ξ.a is provable if and only if A[τ/ξ] is provable for some τ. It may happen that: A and B are provable, but not A ` B (take A = B = 1); A ` B is provable but neither A nor B is provable (take A = α and B = α );?A is provable but not A (take A = α α ). Note also that: the empty sequent is not provable, so that A and A cannot be both provable; the sequent A 1,..., A n is provable if and only if the formula A 1` ` A n is provable. Polarities We say that a formula A is positive (respectively negative) if A!A (respectively A?A). Obviously, A is positive if and only if A is negative. Furthermore: 1 and 0 are positive, as well as!a for any A; if A and B are positive, so are A B and A B; if A is positive, so is ξ.a. By duality, we get: and are negative, as well as?a for any A; if A and B are negative, so are A ` B and A & B; if A is negative, so is ξ.a. We say that A is regular if A?!A. We have: if A is positive, then?a is regular; is regular, as well as?!a for any A; if A and B are regular, so is A ` B. 11
Intuitionistic Logic Intuitionistic formulas are built from atoms (α, or more generally, α(t 1,..., t n )) and units (, ) using binary connectives (,, ) and quantifiers (, ). Sequents are of the form Γ C where Γ is a sequence of formulas and C is a formula. The rules are the following: Γ, A, B, C x Γ, B, A, C Γ, A, A C c Γ, A C Γ C Γ, A C w A A Γ A, A C cut Γ, C Γ, A B Γ A B Γ A, B C Γ,, A B C Γ A Γ B Γ A B Γ, A C Γ, A B C 1 Γ, B C Γ, A B C 2 Γ Γ, A C Γ, B C Γ, A B C Γ A Γ A B 1 Γ B Γ A B 2 Γ, C Γ A Γ ξ.a Γ, A[τ/ξ] C Γ, ξ.a C Γ, A C Γ, ξ.a C Γ A[τ/ξ] Γ ξ.a In the -rule (respectively in the -rule), ξ must have no free occurrence in Γ (respectively in Γ, C). Alternatively, the rules for and can be formulated as follows (multiplicative version): Γ A B Γ, A B Γ, A, B C Γ, A B C The translation A A from Intuitionistic Logic into Linear Logic is defined by α = α, (A B) =!A B, (A B) = A & B, (A B) =!A!B, =, = 0, ( ξ.a) = ξ.a, ( ξ.a) = ξ.!a. A sequent Γ C is provable in Intuitionistic Logic if and only if its translation!γ C is provable in Linear Logic. The only if direction is easy. Conversely, it is clear that Γ C is provable in Intuitionistic Logic whenever its translation!γ C is provable in Intuitionistic Linear Logic: It suffices to conser the obvious translation A A from Intuitionistic Linear Logic into Intuitionistic Logic, defined by α = α, (A B) = A B, (A B) = (A & B) = A B, (A B) = A B, 1 = =, 0 =, (!A) = A, ( ξ.a) = ξ.a, ( ξ.a) = ξ.a. But since Linear Logic is not a conservative extension of Intuitionistic Linear Logic, it is more difficult to show that Γ C is provable in Intuitionistic Logic whenever its translation is provable in Linear Logic (H. Shellinx). The above translation is sometimes called the call-by-name translation. There is also a call-by-value translation, defined by α =!α, (A B) =!(A B ), (A B) = A B, (A B) = A B, = 1, = 0, ( ξ.a) =! ξ.a, ( ξ.a) = ξ.a. In that case, a proof of Γ C is translated into a proof of Γ C, using the fact that A is positive for any A. 12
Classical Logic Formulas are built from atoms (α, α, or more generally, α(t 1,..., t n ), α(t 1,..., t n )) and units (, ) using binary connectives (, ) and quantifiers (, ). Negation is defined on atoms and extended to all formulas by De Morgan equations. Implication is defined by A B = A B. Sequents are of the form Γ where Γ is a sequence of formulas. The rules are the following: Γ, A, B, x Γ, B, A, A, A, Γ c Γ w A, A A, cut Γ, B, Γ A B, Γ A B, Γ 1 B, Γ A B, Γ 2, Γ ξ.a, Γ A[τ/ξ], Γ ξ.a, Γ In the -rule, ξ must have no free occurrence in Γ. Alternatively, the rules for,, and can be formulated as follows (multiplicative version): B, A B, Γ, A, B, Γ A B, Γ A translation A A of (cut-free) Classical Logic into Linear Logic is given by α = α, ( α) = α (A B) =?A?B, (A B) = A B, = 1, = 0, ( ξ.a) = ξ.?a, ( ξ.a) = ξ.a. A variant is given by (A B) =?A &?B and =. In both cases, a sequent Γ is provable in Classical Logic if and only if its translation?γ is provable in Linear Logic. The only if direction is easy: It suffices to conser cut-free proofs. Conversely, it is clear that Γ is provable in Classical Logic whenever its translation?γ is provable in Linear Logic: It suffices to conser the obvious translation A A from Linear Logic into Classical Logic, defined by α = α, (α ) = α, (A B) = (A & B) = A B, (A ` B) = (A B) = A B, 1 = =, = 0 =, (!A) = (?A) = A, ( ξ.a) = ξ.a, ( ξ.a) = ξ.a. An alternative translation of (cut-free) Classical Logic into Linear Logic is given by α =?α, ( α) =?α (A B) = A & B, (A B) = A ` B, =, =, ( ξ.a) = ξ.a, ( ξ.a) =? ξ.a. In that case, a cut-free proof of Γ is translated into a proof of Γ, using the fact that A is negative for any A. A translation of (full) Classical Logic into Linear Logic is given by α =?!α, ( α) =?!?α, (A B) =?(!A!B ), (A B) = A ` B, =?1, =, ( ξ.a) =?! ξ.a, ( ξ.a) =? ξ.!a. In that case, a proof of Γ is translated into a proof of Γ, using the fact that A is regular and that ( A)?A for any A. 13
Invariants Here, we conser propositional formulas in the multiplicative fragment of Linear Logic, built from a set P of propositional variables. If Γ is a sequent, [Γ] stands for the number of occurrences of in Γ. Similarly, we define [Γ]`, [Γ] 1, [Γ], [Γ] α, and [Γ] α. Note that the following equation holds for any sequent of length n: [Γ] + [Γ]` + n = [Γ] 1 + [Γ] + α P([Γ] α + [Γ] α ). Furthermore, a provable sequent satisfies the following equations: [Γ] α = [Γ] α, [Γ]` + n = [Γ] + α P[Γ] α + 1, [Γ]` + [Γ] 1 + n = [Γ] + [Γ] + 2. This is easily checked by induction on proofs. Note that the last equation is a consequence of the previous ones. In the case of a provable formula, we get: [A] α = [A] α, [A]` = [A] + α P[A] α, [A]` + [A] 1 = [A] + [A] + 1. Those conditions are not sufficient: Take for instance A = (1 ` 1), or A = (α ` α) (α ` α ). About exponential rules If A is a formula, A (n) stands for the sequent A,..., A of length n. The following rules are derivable (weak promotion, digging, absorption, and multiplexing):!a,?γ??a, Γ?A, Γ?A, A, Γ?A, Γ A (n), Γ?A, Γ Furthermore, promotion is derivable from weak promotion and digging. Multiplexing is invertible in certain circumstances: A sequent?a, Γ with no occurrence of &,!, or second order is provable if and only if A (n), Γ is provable for some n. This is easily checked by induction on cut-free proofs. To see that this does not hold in general, take for instance A = α and Γ = α & 1, or Γ =!α. Similarly, a sequent?a, Γ with no occurrence of! or second order is provable if and only if (A ) (n), Γ is provable for some n. If A is a formula,! n A stands for the formula (A&1) (A&1) (n times) and? n A for the formula (A )` `(A ) (n times). The latter result can be generalized a follows: Conser a provable sequent with p occurrences of!, q occurrences of?, and no second order. Then, for all m 1,..., m p N, there are n 1,..., n q N such that the sequent obtained by replacing the p occurrences of! by! m1,...,! mp and the q occurrences of? by? n1,...,? nq is provable (approximation theorem). Note that the following rule is not admissible: C A C C C C 1 C!A For instance, if A = C = α!(α α α)!(α 1), the three premisses are provable but not the conclusion. 14
Phase spaces If M is a (multiplicative) mono and X, Y M, we write XY for the set {xy x X and y Y } and X Y for the set {z M xz Y for all x X}. A phase space is a pair (M, M ) where M is a commutative (multiplicative) mono and M M. If X M, we write X for X M. It is easy to prove the following properties: X Y if and only if XY M, XX M, if X Y then Y X, X X, X = X, (X Y ) = (XY ) = X Y, (X Y ) = (X Y ) = X Y. A fact is an X M such that X = X, or equivalently, X = Y for some Y M. For instance, M = {1} is a fact, as well as 1 M = ( M ) = {1}, M = M =, 0 M = ( M ) = M =. Note also that: if X M and Y is a fact, then X Y = (XY ) is a fact; if (X i ) i I is a family of facts, then i I X i = ( i I X i ) is a fact; if X M, then X is the smallest fact containing X. We write x y if {y} {x}, or equivalently, {x} {y}. We write x y if x y and y x, or equivalently, {x} = {y}. We define the following operations on facts: X ` Y = X Y = (X Y ), X Y = (X ` Y ) = (X Y ) = (XY ), X & Y = X Y = (X Y ), X Y = (X & Y ) = (X Y ) = (X Y ),?X = (X I M ),!X = (?X ) = (X I M ), where I M = {x 1 M x = x 2 }. In the last two definitions, I M may be replaced by any submono K M of J M = {x 1 M x x 2 } = {x M x x 2 and x 1}. Such a K M is called an exponential structure (Y. Lafont). For instance: K M = {1} (trivial structure), K M = I M (standard structure), K M = J M (full structure). Anyway, the notion of topolinear space is definitively obsolete. 15
Phase models Here we conser propositional formulas built from a set P of propositional variables. A phase model is a phase space (possibly with an exponential structure) together with a fact α M for each α P. The interpretation, which is already defined for units and propositional variables is extended to all formulas in the obvious way: (α ) M = (α M ), (A ` B) M = A M ` B M, (A B) M = A M B M, (A & B) M = A M & B M, (A B) M = A M B M, (?A) M =?A M, (!A) M =!A M. Since A M is always a fact, we also get (A ) M = (A M ) and (A B) M = (A M ) ` B M = A M B M. We say that A holds in the model if 1 A M. Note that the formula A B holds if and only if A M B M. More generally, if Γ is a sequence of formulas A 1,..., A n, we write Γ M for (A 1` ` A n ) M, and we say that the sequent Γ holds in the model if 1 Γ M, or equivalently, (A M 1 ) (A M n ) M. By induction on proofs, it is easy to see that if a sequent is provable, then it holds in any phase model (soundness). Here are some basic examples: If M =, then M and M are the only facts, and we get a Boolean model. In that case, soundness means that any provable formula is classically val. Let α P. If M = Z with addition, M = {0}, α M = {1}, and β M = {0} for each β P \ {α}, then for any multiplicative formula A, the set A M consists of the single element [A] α [A] α. In that case, soundness means that [A] α = [A] α whenever A is provable. If M = Z with addition, M = {1}, and α M = {1} for each α P, then for any multiplicative formula A, the set A M consists of the single element [A] + α P [A] α [A]`. In that case, soundness means that [A]` = [A] + α P [A] α whenever A is provable. A logical congruence on a phase model M is a congruence such that M is closed for : if x M and x y, then y M. For instance, = is the finest logical congruence and is the coarsest one. If is a logical congruence, then all facts are closed for, and the canonical projection π : M M/ induces a structure of phase model on the quotient mono M/ : M/ = π( M ), α M/ = π(α M ) for each α P, K M/ = π(k M ). It is easy to see that A M/ = π(a M ) for any formula A. In particular, A holds in M if and only if it holds in M/. Note that if K M is the standard exponential structure on M, then π(k M ) is not necessarily the standard exponential structure on M/. 16
Syntactical model The syntactical model is defined as follows: M is the free commutative mono generated by all formulas. In other words, M is the set of all sequents, consered as finite multisets, with multiset union; M is the set of all cut-free provable sequents, and α M = {α} for each α P. In other words, α M = {Γ M α, Γ is cut-free provable}; K M is the set of all sequents of the form?γ. By induction on formulas, A M {A} for any A (M. Okada). In particular, if A holds in the syntactical model, then the empty sequent belongs to {A}, which means that A is (cut-free) provable (completeness). Note the following points: By cut elimination, M is the set of all provable sequents. However, it is essential to conser cut-free proofs if one wants to use this model for proving cut elimination. In fact, A M = {A} for any A, but in the cut-free version of the syntactical model, only the inclusion can be proved directly, and this is enough for proving completeness and cut elimination. M can be replaced by the free commutative mono generated by all subformulas of A. Therefore, completeness holds for phase models whose underlying mono is finitely generated. M can also be replaced by M/, where is the smallest congruence such that Γ, Γ Γ for any Γ K M, so that π(k M ) = I M/. Therefore, completeness holds for phase models with the standard exponential structure. Completeness does not hold for phase models with the trivial exponential structure: For instance, α (!α 0) holds in any such phase model, but it is not provable. Finite models We say that that a model M is finite if it has finitely many facts. In that case, M/ is a finite mono and satisfies the same formulas as M. We say that a fragment of Linear Logic satisfies the finite model property if any formula of this fragment that hold in all finite models is provable. It is easy to see that such a fragment is decable. The multiplicative additive fragment satisfies the finite model property (Y. Lafont). To see that, it suffices to conser a finite version of the syntactical model. For instance: M is the free commutative mono generated by all subformulas of A; M = {Γ M Γ is cut-free provable or Γ > A }, where Γ = [Γ]` + n for any sequent Γ of length n, and α M = {α} for each α P. The multiplicative exponential fragment does not satisfy the finite model property: For instance, the formula!α!(α β)!(α β 1) β holds in any finite model, but it is not provable. We say that that a model M is affine if the set M is an eal of M, that is M M M. This amonts to say that M = 0 M, or equivalently, 1 M = M. In that case, any fact is an eal. There is a completeness theorem for Affine Linear Logic with respect to affine models, and Affine Linear Logic satisfies a finite model property. This comes from the fact that if M is a finitely generated free commutative mono, then all its eals are finitely generated. Consequently, Affine Linear Logic is decable (A. Kopylov). 17