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23G.Mints, S.Soloviev
CLASSICAL MULTIPLICATIVE LINEAR LOGIC □ INTUITIONISTIC MLL
Abstract. It is known how to present every deduction in the {!, I}free Classical Multiplicative Linear Logic as (the result of an obvious translation of) a deduction in the intuitionistic MLL. We extend the result to the language with I and give short proofs which do not use proof nets.
1. Introduction
One of the most important computational interpretations of logical proofs uses intuitionistic logic and Curry-Howard isomorphism between natural deduction and lambda terms. One of the goals of linear logic [2] was to provide an improved proof-theoretic model of computation which ensures uniqueness of the normal form of a derivation by means of a new formalism of proof nets, which works even for classical linear logic and provides a lot of symmetry. More traditional computational interpretation uses intuitionistic linear logic (cf. [3]) which admits a form of Curry-Howard isomorphism [7]. The results in the literature [10, 1] show how to present every deduction in the {!, I}-free Classical Multiplicative Linear Logic as (the result of an obvious translation of) a deduction in the intuitionistic MLL. We extend the result to the language with I and give short proofs which do not use proof nets.
Let us remind that the most important applications of linear logic in algebra depend on the language of MLL with the constant I, cf. [5, 6, 8].
Formulas of the !-free Classical Multiplicative Linear Logic CMLL are constructed from literals (propositional variables p, q, p', ..., constant I and their negations p, I) by the tensor product ® and par connective p. Derivable objects of CMLL are sequents, i.e. multisets of formulas. CMLL is axiomatized as follows.
Axioms p, p I,., I, p, p, I
Inference rules
® r, A A, B p r, A, B
r, A, A ® B r, ApB
Formulas of the Intuitionistic Multiplicative Linear Logic IMLL are constructed from propositional variables and constant I by linear
implication ô and tensor product ®. Derivable objects of IMLL are sequents T^A, where r is a multiset of formulas and A is a formula. IMLL is axiomatized as follows.
Axioms p^p I,--,I, p^p, ^ I Inference rules
r ^ A A ^ B r, A ^ A ® B
r, A ^ B r ^ Aô B
r ^ A A, B ^ C Aô B, r, A ^ C
A, B, r ^ C
A®B, r ^ C
We prove (Theorem 2 below) that every deduction in the classical MLL is (the result of an obvious translation of) a deduction in the intuitionistic MLL up to natural isomorphisms
and involution, i.e. interchanging p and p for some variables p. This suggests using ordinary lambda-terms to describe CMLL since CurryHoward isomorphism holds for usual typed lambda-terms and IMLL [7]. Our translation from CMLL into IMLl has an inverse * described in the section 3. Both of them completely preserve the structure of the derivation tree. This shows that every derivation in CMLL is essentially a derivation in IMLL. Moreover, one can fix the goal formula in an arbitrary way. This constitutes one of the differences with the negative translation of the traditional classical propositional logic into intuitionistic logic. The negative translation adds new negations with the corresponding antecedent and succedent rules and levels down important distinctions in the original formula.
This paper incorporates some suggestions of the referee of a previous version.
2. Reduction of CMLL to the balanced I-free fragment
A formula or sequent is balanced if each propositional variable occurs there exactly twice, once positively, once negatively. An instance of a formula or derivation is a result of substituting some propositional variables by formulas. I-instance is obtained when all these formulas are just I. The following well-known proposition (cf. [5]) provides a reduction to balanced sequents.
Lemma 1 Every derivation d in CMLL is an instance of a derivation of a balanced sequent.
Proof. Every occurrence of a propositional variable in the last sequent of d is traceable to a unique occurrence in a unique axiom of d. Replace
p®IQp
I □ I^I
(1)
occurrencies traceable to different occurrencies of axioms by distinct variables. □
Note. p^p&p is a counterexample for additive linear logic and the classical propositional calculus.
The next reduction eliminates multiplicative constants.
Lemma 2 Every derivation in CMLL is an I-instance of an I-free derivation of a balanced sequent up to transformations.
Proof. Consider given derivation d: r of a balanced sequent r in CMLL. Every occurrence of I in r is traceable to a unique occurrence of I in an axiom. Ifit comes from one of the first occurrences_of I in an axiom I, ..., I, p, p, replace these occurrences of I by qi, ..., qn and the last p by p®q1...®qn for distinct fresh variables q1, ..., qn (and make the same replacement for all occurrences traceable to these). If it is one of the last two occurrences in I, ..., I, I, I or in I, I, replace both occurrences by a fresh variable q. If it is an occurrence in an axiom I, replace it by q pp for a fresh q. □
3. Derivations of IMLL-sequents
Consider the standard translation of IMLL into CMLL:
(r^A)* := r, A
where
A ® B:= Ap B; Ao B:=A ® B; A:=A; ApB:= A ® B
and induced translation of derivations. Double negation over A is inserted to replace the linear implication Ao B by Ap B. As a warm-up consider the case when involution is not needed. Note that A below is allowed to contain arbitrary many formulas.
Theorem 1 If rA are multisets of formulas of IMLL and d: (r=^>A)* is a derivation in CMLL, then A consists of one formula and d^e * for some e: r=^>A in IMLL.
Proof. Induction on d. If d is an axiom p, p or I, p, p then A=p and e is p^p or I, p^p. If lastrule(d)=® then
Ti, A1', A f2, A2', B Ti, A1, A B, f2, A2 - or -
d: r, A', A ® B d: A ® B, r, A
In the first case A1 '=A2'^0 by the induction hypothesis (IH), and one has e: T^A®B. In the second case A1^0 and A 1^A consists of one formula, so that e: Ao B, T^A:
r ^ A r2 ^ B
r ^ A r2, B ^ A
or
e: r ^ A ® B e: AoB, r ^ A
Finally, if the lastrule(d)= p, one has:
r, A, A, B
d: r, A, ApB
since Ai^0, or
A, B, r, A
d: Ap B, r, A
r, A ^ B e: r ^ Ao B
A, B, r ^ A e: A®B, r ^ A
□
4. General case
Theorem 2 For every balanced I-free sequent of CMLL and its derivation d: U, C in CMLL there are formulas (intuitionistic translations) U1, C1 in IMLL, an involution i and a deduction e: U1 ^ C1 in IMLL (all depending of the choice of C) such that d^e*1.
Proof. Induction on d. The case d=p, p is obvious. Consider subcases depending of the lastrule(d) and a position of C. Let lastrule(d)=® and C is the principal formula:
f: r, A g: A, B d: r, A, A ® B
Then by IH there are fi: r^Ai, gi: A i ^Bi and involutions i', i'' such that ff l, g^gi 1 . Note that propositional variables in the premises are distinct, and define: i:=i'ui",
fi: ri^Ai gi: Ai^ Bi
e: ri, Ai ^ Ai ® Bi Remaining cases are similar.
r, A, C A, B ri, Ai ^ Ci Ai ^ Bi
d: r, A, A ® B, C e: r i, Ai, BiO Ai ^ Ci
r, A, B r i, Ai ^ Bi r i, Bi ^ Ai --- or -
d: r, ApB
e: ri ^ Aio Bi
e: r i ^ Bio Ai
In this case one can choose arbitrarily between two possible "witchings" of ApB.
r, A, B, C ri, Ai, Bi ^ Ci
d: r, ApB, C e: r, Ai ® Bi ^ Ci
□
Conclusion
Let us sum up.
For every derivation d: C in CMLL one has a derivation Int(d):^Ci in IMLL and an involution i such that d^(Int(d)) id and CDC^for some substitution d.
Moreover, for I-free balanced formula C the derivation Int(d) depends only of C.
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