Phase Semantics for Light Linear Logic (Extended

Electronic Notes in Theoretical Computer Science 6 (1997)

URL: http://www.elsevier.nl/locate/entcs/volume6.html

14 pages

Phase Semantics for Light Linear Logic (Extended Abstract) Max I. Kanovich 1 Department of Theoretical and Applied Linguistics Russian State University for the Humanities Miusskaya 6, 125267 Moscow, Russia Email: [email protected]

Mitsuhiro Okada 2

Email:

Department of Philosophy Keio University 2-15-45 Mita, Minato-ku Tokyo 108, Japan

[email protected]

Andre Scedrov 3 Department of Mathematics University of Pennsylvania Philadelphia, PA 19104-6395 USA Email: [email protected]

http://www.cis.upenn.edu/~scedrov

Abstract

Light linear logic 1] is a renement of the propositions-as-types paradigm to polynomialtime computation. A semantic setting for the underlying logical system is introduced here in terms of bred phase spaces. Strong completeness is established, with a purely semantic proof of cut elimination as a consequence. A number of mathematical examples of bred phase spaces are presented that illustrate subtleties of light linear logic.

c 1997 Published by Elsevier Science B. V.

Kanovich, Okada and Scedrov

1 Introduction

Typed lambda calculi have long been recognized as analogous to formal logical calculi of intuitionistic logic. In technical terms this correspondence is known as the Curry-Howard isomorphism or the propositions-as-types paradigm. Logic provides not only basic input/output specications (i.e., types or formulas), but also a setting for well-typed programs (i.e., terms or formal proofs), as well as a mode of execution of well-typed programs by means of term reduction or normalization 2]. The advent of linear logic 3] with its intrinsic ability to re ect computational resources has made it possible to rene the propositions-as-types paradigm to computational complexity specications. A bounded version of linear logic (BLL) was introduced in 4], in which the reuse of resources is bounded in advance, and in which any functional term of appropriate type encodes a polynomial-time algorithm. Conversely, any polynomial-time function arises in this way. A detailed comparison of this approach to various other logical characterizations of polynomial-time functions may be found in 4,1]. A major advantage of BLL is that the system itself is (locally) polynomial-time. The run-time normalization complexity is implicit in the system and does not need to be enforced explicitly in the syntax. From a strictly logical point of view, however, BLL still suers from the presence of explicit resource parameters (whose technical role is to indicate input/output size ratios.) In this sense BLL is not a purely logical system. This diculty is resolved in Girard's light linear logic (LLL) 1], which keeps all the advantages of BLL, but avoids mentioning the resources altogether. In LLL resources can be synthesized by purely logical means. The basic idea in 1] is to set up the structural rules and the logical rules for modalities more carefully than in linear logic so that the computational power of normalization can be well-controlled. In the course of setting up such wellcontrolled rules central points are to dispense with the principles AA and A A, but to retain the exponential isomorphism A B ' A B . LLL shares some of these and other technical features with the systems studied in 5,6]. A more subtle, but equally important point of LLL is to reject the principle A B A B . In order to compensate for this, LLL adds a self-dual modality x that satises AxA and xA A and xA xB x A B . Although syntax of LLL is well-understood thanks to Girard's careful analysis 1], semantics for LLL has remained an open question. Surprisingly, an aswer is suggested by another research direction, namely !

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Initial results obtained during the visit to University of Pennsylvania in Summer 1996, supported by ONR Grants N00014-92-J-1916 and N00014-96-1-0555. 2 Partially supported by the Grant-in-Aid for Scientic Research (Ministry of Education, Science and Culture, Japan) and the Ogata-Jyosei Grant (Keio University). Initial results obtained during the visit to University of Pennsylvania in September 1996, supported by ONR Grant N00014-96-1-0555. 3 Partially supported by NSF Grant CCR-94-00907, by ONR Grant N00014-92-J-1916, and by a Centennial Research Fellowship from the American Mathematical Society. 1

2


by the work of the rst author and T. Ito on extensions of linear logic with certain features of temporal logic 7]. Models of temporal logic 8] distinguish among semantic objects at dierent points in time (much as Kripke models distinguish among semantic objects in dierent worlds.) Temporal logic models also feature a semantic operator next such that next A at time t is A at time t . Our starting point is that not only does LLL modality x behave in many ways like the operator next (except for the self-dual nature of x), but that, for instance, the principle AA fails in such a stratied setting. Informally, consider a semantic setting for linear logic and repeat it, each time at a dierent level t. Let A t be the given denition of applied to At+1 . In this reading, for any t, A t yields xA t but there is no general reason why A t should yield At . In fact, a closer analysis reveals that the semantic intuition of levels or stages t is related to the syntactic notion of nesting depth of proof-boxes in LLL. (The basic idea described above may be modied slightly by means of explicit transitions between levels so that semantic denitions at a given level t refer to transitions to t rather than to other levels, see Section 2.) In this paper this analysis is applied in the context of phase semantics for linear logic 3,911]. We explain how A B AB can fail in nature. We also establish Strong Completeness for LLL (i.e., valid formulas are provable without the cut rule), and thus we obtain a purely semantic proof of Cut Elimination (i.e., provable formulas are also provable without the cut rule.) Similar analysis may also be carried out in other semantic settings such as coherence spaces, which will be discussed elsewhere. It would also be interesting to see if such semantic methods can also establish the stronger version of cut elimination that proof normalization reductions terminate. We would like to thank Vincent Danos for very informative conversations. + 1

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2 Fibred Phase Spaces

A phase space M ? is a commutative monoid M with a distinguished subset ? M , called bottom. For any subset M , dene ? def fx 2 M jx ?g fx 2 M j8y 2 xy 2 ?g. A subset M is closed i ?? . Writing for the neutral element of M , let 1 be the subset f g??, that is, ??. It is readily seen that 1 is a closed submomoid. A homomorphism of phase spaces, or simply a phase homomorphism, is a monoid homomorphism h M ! M 0 such that h ? ?0. A phase space induces a natural preorder on the underlying monoid compatible with monoid multiplication: (

)

=

=

=

1

1

:

(

)

x y ,def x 2 fyg??:

Note that a phase homomorphism is not required to be monotone in the induced preorder. More generally, a phase structure is a commutative monoid M with a clo3


sure operator on M , that is, a mapping Cl from subsets of M to subsets of M satisfying the following four properties for any M : (Cl1) Cl(), (Cl2) Cl(Cl()) = Cl(), (Cl3) ) Cl() Cl( ), (Cl4) Cl() Cl( ) Cl( ).

A subset M is said to be closed i Cl() = . One can again dene a preorder compatible with monoid multiplication: x y i Cl(fxg) Cl(fyg). A phase space is a special case where Cl() =def ??. For a given mapping g : M ! M 0 , let us consider its lower approximations , that is, mappings f : M ! M 0 such that for every a 2 M there exists b 2 M such that b a and f (a) g(b). In this case we also say that f is bounded by g. We are particularly interested in lower approximations that satisfy a certain continuity property. A mapping f : M ! M 0 has the intermediate value property i for every a b 2 M such that f (a) 2 10 and f (b) 2 10 , there exists c 2 M such that c a, c b, and f (a)f (b) = f (c). Note that the identity function has the intermediate value property with c = ab. However, in our applications, f will be bounded by 10, which will provide that f (a) 0 10 for all a 2 M . Example 2.1. Consider the reals with addition, where ? consists of the negative reals. In this phase space a b i a b. Any linear function h(x) = kx, with a positive k, is certainly a phase homomorphism. Let f be a continuous function such that f h and f 0.

6M 0

c a b f (x)

f (c) = f (a) + f (b)

h(x )

M -

f has the intermediate value property, both in the ordinary sense and as a mapping of phase spaces. The latter is a special case of the former on (1 minfa bg] because limx!1 f (x) = 1. Example 2.2. Let the phase structure M0 consist of the nonpositive integers, with a b =def minfa bg. Let Cl() =def fz 2 M0j9x 2 :z xg. The properties (Cl1)(Cl4) hold. Note that all elements of M0 are idempotent, that is, aa = a for all a. Also, 1 is the entire monoid. 4


Let M1 be the integers with addition. The properties (Cl1)(Cl4) again hold if Cl def fz 2 M1 j9x 2 :z xg. In this case, 1 1 , here meaning the integers . Let h0 M1 ! M0 be the constant function and let f0 M1 ! M0 be the function 0, if a > f0 a a , if a (

) =

= (

0]

0

:

0

:

0

( )=

1

0

f0 has the intermediate value property: if a b take c a, and if b < a then take c b. Then f0 a f0 b minff0 a f0 b g f0 c . 2 =

=

( )

( ) =

( )

( )

=

( )

A bred phase space is a family f Mn ?n hn fngn0, where for each integer n , Mn ?n is a phase space, hn Mn+1 ! Mn is a phase homomorphism, and fn Mn+1 ! Mn is a mapping with the intermediate value property such that fn is bounded by hn. A bred phase structure is dened similarly, but each hn is only required to be a monoid homomorphism. Given a bred phase structure, consider a family fngn0, where each n Mn is closed in Mn. One says that is closed. For any closed fngn0 and fngn0 one denes 1 , and in the natural way induced from the original denition in 3]: (

0

(

)

)

:

:

=

=

=

( (

( (

1)n = 1n

> n Mn

(

)

=

0)n = Cln ()

(

&

&

(

)n = )n = )n = )n =

n \ n Cln n n Cln n n n fz 2 Mnjz n n ng: (

)

(

)

x and are dened in the following way: x n Cln hn n+1 n Cln fn n+1 \ 1n \ Jn where Jn Mn is a submonoid of Mn such that every element of Jn is a weak idempotent, i.e., 8a 2 Jn a n a n a (after Y. Lafont.) !

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=

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=

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In a bred phase space one further denes: ? n ?n ? n n?n n n?n n n?n ?n x x ? ? n n f n?+1 \ 1n \ Jn ?n 5 (

(

(

.................. ............ .....

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=

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=

)

=(

)

=( (

(?

)

=(

(

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+1

)

)


Example 2.2., continued. Let h0 f0 M1 ! M0 be as in Example 2.2. For n , let Mn M1 and hn x fn x x for all x. Let J0 M0 , 1 1 , and let n be any closed subset of Mn, n 6 . Then 0 Cl f0 1 1 . Thus 0 1 . We show that 0 1 , that is, 0 is not a subset of 0. Indeed, 1 1 Cl 1 1 1 . Thus 0 Cl f0 1 1 Cl f0 1 1 . Also note that 1 0 Cl f0 1 1 , which does not in:

1

=

= (

(

))

(

) =

(

) =

=

= 1

)) = (

= (

2]

(!

3]

=

(

(

1]

(

((

(!

+

(!

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= (

2]

(!(

2]

(!(

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=

(!(

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) = (

2])) = (

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=

))

(

(

)) =

3]

)

=

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((

0])) = (

clude the neutral element of M0 .

1]

2

0

3 Fibred phase semantics In this Section we dene the bred phase semantics for propositional LLL. We shall extend our bred phase semantics to the second-order case (i.e., to the full LLL 1]) in Section 5. Let us recall basic elements of the syntax of propositional LLL from 1]. A propositional formula is dened in the same way as in linear logic, but one adds the new modalities x and x, namely, if A is a formula then xA and xA are formulas. As usual in linear logic, linear negation A? is used as an abbreviation in the sense of the de Morgan dual, except for atomic formulas p?. x and x are duals of each other 4 , i.e., xA ? def x A? , and xA ? def x A? . In addition to formulas, the syntax of LLL involves several punctuation marks that facilitate the management of contexts. Intuitively, if A and B are formulas, an expression A B is intended to represent AB , an expression A B is intended to represent A B , and an expression A is intended to represent A. These expressions are themselves not formulas. Formally, a block is either a multiset A1 A2 : : : An of formulas, where n , or an expression A , where A is a formula. A sequent is an expression ` , where is a multiset A1 A2 : : : Ak of blocks, where k . Note that sequents are allowed to be empty, but the blocks are not. We shall observe the following notation: Roman capitals for formulas, boldface Roman capitals for blocks, and Greek capitals for nite multisets of blocks mutually separated by semicolons. The inference rules of LLL are included in the Appendix. Given a bred phase space f Mn ?n hn fngn0, for each propositional formula A one associates a closed family A f A ngn0 in the obvious way by using the semantic operations described in the previous section, starting with any valuation, i.e., any assignment of closed families to propositional atoms. A is called the inner value of A. A valuation satis es a formula A i for each n, n 2 A n. A formula is valid i it is satised in any valuation (

)

=

(

)

(

)

=

(

)

.................. ............ .....

]

?

1

]

0

(

)

=

1

(

(

)

)

Contrary to 1], we do not assume x = x because this is not needed for the main features of LLL related to polynomial time. In particular, polynomial-time functions are naturally represented in an intuitionistic version of LLL 1], which, as a type system, is a renement of system F 12,2]. 4

6


in any bred phase space. These notions are readily extended to sequents by using the intended representation of punctuation marks. Lemma 3.1 In any bred phase structure, n x n. Lemma 3.2 Let and be closed families in a bred phase structure. Then n n. Proof. Since n n Cln n n , it suces to show n n n. Then by (Cl4), it suces to show (!

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=

((!

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fn n+1 \ n \ Jn fn n+1 \ n \ Jn Cln fn n+1 \ n+1 \ n \ Jn : Take an arbitrary element d from the left hand-side. d is of the form fn a fn b for some a 2 n+1, b 2 n+1. First, notice that fn a fn b 2 Jn. This is because fn a 2 Jn, fn b 2 Jn and Jn is a submonoid (of Mn). Also, fn a fn b 2 1n this is because fn a 2 1n, fn b 2 1n, and 1n is a submonoid of Mn . We need to show that fn a fn b fn c for some c 2 n+1. By the intermediate value property of fn, 9c 2 Mn+1c n+1 a c n+1 b and fn a fn b fn c . But c n+1 a 2 n+1 implies c 2 n+1 and c n+1 b 2 n+1 implies c 2 n+1 since n+1 and n+1 are Cln+1-closed. 2 (

(

)

1

)

(

(

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1

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(

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1

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( )

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Theorem 3.3 (Soundness) If a formula is provable in propositional LLL,

then it is valid. The soundness of ordinary phase semantics for linear logic is a special case when for every n, Mn = Mn+1 and hn and fn are the identity functions. There is also an important generalization of our soundness theorem to bred phase structures, where every valuation satises every formula provable in an intuitionistic version of propositional LLL, ILLL 1]. (A more detailed discussion of the syntax of ILLL is included in the appendix.) Viewed in this way, our Example 2.2 provides a natural mathematical setting in which the ILLL formulas !A!A!(AA) and !1 are not satised. Note that in Example 2.2, redening f0(0) to be 0 instead of 1 yields an example in which !1 is satised but !A!A!(A A) is not.

4 Strong Completeness The completeness theorem may be proved in the following strong form. Theorem 4.1 (Strong Completeness) If a propositional formula is valid, then it is provable in propositional LLL without the cut rule. This implies the cut-elimination theorem. Theorem 4.2 (Cut-Elimination) If a formula is provable in propositional LLL, then it is provable in propositional LLL without the cut rule. 7


Proof. If A is provable in LLL, A is valid by the Soundness Theorem. Then

by the Strong Completeness Theorem, A is cut-free provable in LLL. 2 Remark. Cut-Elimination fails if one adds to LLL the !-rule with the empty context, i.e., when n (see Appendix.) Indeed, let p be a propositional ? atom. The sequent ` p p ? is cut-free provable in LLL itself, the sequent ` is cut-free provable as an instance of the new rule, and hence the sequent ` p? p is provable by cut. But this sequent has no cut-free proofs. = 0 !(

&1) ? ?

!1

!(

&1) ?

We prove the Strong Completeness Theorem in the same manner as in Okada11]. For that purpose, we consider the commutative monoid M of nite multisets of blocks, with multiset union as the monoid operation (which we continue to indicate by semicolon concatenation). The empty set is the neutral element of M . Let us write `cf for ` is provable in propositional LLL without the cut rule. Given a sequent , dene the outer value k k as

k k f `cf

=

:

g:

Recall that the original denition of the canonical phase model for linear logic in 3] uses, in the present notation, k k f ` is provableg. Let ? M be the subset kk. Note that the outer value k k is closed since 2 k k?. Let J be the submonoid f A1 A2 : : : Ak Ai is a formula and k g. The M-contraction rule of LLL states precisely that every element of J is a weak idempotent. The following denition formalizes the intended meaning of punctuation marks. We assume a mapping that orders formulas and blocks according to some canonical ordering. The connectives and are associated to the left. is dened as With these conventions, given a sequent , the formula ?, (i) If , then (ii) If A1 : : : An, n , then A(1) : : : A(n), A, (iii) If A , then (iv) If A1 A2 : : : Ak where each Ai is a block and k , then A(1) : : : A(k) .

=

:

]

]

] :

0

................ ............ .....

=

................... ............ .....

=

................... ........... ........... ..... ....... ............. ....

1

................. ............ ....

]

................... ........... .....

Proposition 4.3 If `cf

.................. . .......... .....

=

=?

1

................... .......... .....

. That is, `cf Let us dene a mapping h M ! M as h h A1 : : : An x A1 : : : An n h A x A ,

then

....... ......... ............ .....

:

( )=

(

(

..... ...... ..... .... .......... .....

=

=

=

)=

(

]) =

?

)

...... .......... ....... ..... .. ....

1

8

....... ......... ............ .....

.

.................. ........... .....

=


h(A1 : : : Ak) = h(A1) : : : h(Ak) k 1 Proposition 4.4 h is a phase homomorphism. Consider the function f : M ! M dened as ( f () = A1 ] : : : An] if = A1 : : : An for n 1.

, otherwise. In particular, if = , then = ? by denition, and hence f () = ?]. The M-weakening rule of LLL implies that f () . In fact, it is clearly the case that f () 2 1 \ J for any sequent . We also have the following lemma. Lemma 4.5 f has the intermediate value property. Furthermore, f is bounded by h. Our canonical model is the bred phase space f(Mn ?n) hn fngn0, where Mn = M , ?n = ?, hn = h, and fn = f . We shall drop the indices for the rest of this section. Finally, we consider the valuation p = kpk for any atomic formula p. The following lemma is obtained in the manner similar to Okada 11]. Lemma 4.6 (Main Lemma) For any propositional formula A, A kAk. Strong Completeness follows from the Main Lemma. .................. ............. ....

]

......... ...... .. ............ ......

Proof of Strong Completeness. Assume that A is valid. Hence 1 2 A for any model, in particular for this canonical model. Therefore, 2 A in this model. On the other hand, A kAk. Hence 2 kAk. By denition, this means A is provable in LLL without the cut rule. 2 Let us also note another consequence of the Main Lemma. Corollary 4.7 For any propositional formula A, A 2 A?. Proof. By the Main Lemma, A kAk. It suces to show A A ?, namely, 8 2 A( A 2 ?). But A kAk means that for any 2 A, `cf A. Therefore A A ?. 2 The Main Lemma has another formulation, which will be essential for the second-order case in the next section. Lemma 4.8 For any propositional formula A, A? 2 A kAk. Proof. By the Main Lemma, A kAk for any A. By Corollary 4.7 for A?, A? 2 A?? = A . 2

5 Second-Order Completeness Girard 1] formulated LLL as a second-order propositional system. Let us adjust the underlying idea in Okada 11] to extend the bred phase semantics 9


to the second-order case so that the soundness, strong completeness and cutelimination theorems apply to the full LLL. A further extension to higher-order (nite-order) LLL may also be possible using a modied version of higher-order phase models introduced in Okada 11]. Let us write AfX g to indicate that X is a vector of propositional variables containing the free variables of A. Let AfB=X g or AfB g denote the formula obtained from AfX g by substituting the vector of formulas B for X . Let A f=X g or Afg denote the the result of the inner value construction starting with the vector of closed families as the value of the variable list X . In this section we use Form to denote the set of second-order formulas. Let f(Mn ?n fn hn)gn0 be a bred phase space. Consider an assignment that to any formula A (possibly with free propositional variables), associates a set < A > of closed families. Let < A >n denote f()n : 2< A >g. Then the second-order propositional quantiers may be interpreted as follows:

\

((

8XA))n =def

((

9XA))n =def Cln(

n 2n B2Form

Anfn=X g

n 2n B2Form

Anfn=X g):

A second-order bred phase model is a bred phase space f(Mn ?n fn hn)gn0 together with an assignment that associates a set < A > of closed families to any formula A, such that the following condition holds: For any formula AfX g, where X = X1 : : : Xk is a vector of second-order propositional variables, for any vector of formulas B = B1 : : : Bk , for any vector of closed families = 1 : : : k , whenever (j )n 2< Bj >n for all n 0 and all 1 j k, then it is the case that (A )nf()n=X g 2< AfB=X g >n for all n 0. A formula is closed i it has no free variables. A closed formula A is valid i in any second-order phase model, 1n 2 (A)n for all n 0. Theorem 5.1 (Soundness, Second-Order Version) If a closed formula is provable in LLL, then it is valid. The canonical phase space is dened as in the previous section, but `cf now means provable in second-order LLL without the cut rule. For any formula A, dene < A > as

< A >= f closed : A? 2 kAkg: The set < A > corresponds to the set of candidates of reducibility of type A in 12,2].

Lemma 5.2 (Main Lemma, Second-Order Version) For any formulas AfX g and B , and for any 2< B >, AfB=X g? 2 A f=X g kAfB=X gk. 10


In other words, the canonical phase space and the assignment < A > just dened form a second-order phase model. As before, we obtain strong completeness and hence cut-elimination. Theorem 5.3 (Strong Completeness, Second-Order Version) If a closed formula is valid, then it is provable in LLL without the cut rule. Theorem 5.4 (Cut-Elimination, Second-Order Version) If a formula is provable in LLL, then it is also provable in LLL without the cut rule. The methods and results again specialize to ELL and extend to the intuitionistic version.

References 1] Jean-Yves Girard. Light linear logic. Manuscript, 1995. Available by anonymous ftp from lmd.univ-mrs.fr as /pub/girard/LLL.ps.Z. 2] Jean-Yves Girard, Yves Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science 7. Cambridge University Press, 1988. 3] Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1102, 1987. 4] Jean-Yves Girard, Andre Scedrov, and Philip J. Scott. Bounded linear logic: A modular approach to polynomial time computability. Theoretical Computer Science, 97:166, 1992. 5] Vincent Danos, Jean-Baptiste Joinet, and Harold Schellinx. The structure of exponentials: Uncovering the dynamics of linear logic proofs. In G. Gottlob, A. Leitsch, and D. Mundici, editors, Proceedings of the Third Kurt Gdel Colloquium on Computational Logic and Proof Theory, pages 159171. SpringerVerlag LNCS 713, 1993. 6] Simone Martini and Andrea Masini. On the ne structure of the exponential rule. In J.-Y. Girard, Y. Lafont, and L. Regnier, editors, Advances in Linear Logic, London Mathematical Society Lecture Notes, Vol. 222, pages 197210. Cambridge University Press, 1995. 7] Max I. Kanovich and Takayasu Ito. Temporal linear logic specications for concurrent processes. In Proceedings of the Twelfth Annual IEEE Symposium on Logic in Computer Science, Warsaw, Poland, 1997. 8] Amir Pnueli. The temporal logic of programs. In Proc. 18-th Annual Symposium on the Foundations of Computer Science, pages 4657. 1977. 9] Yves Lafont. The undecidability of second order linear logic without exponentials. Journal of Symbolic Logic, 61:541548, 1996. 10] Yves Lafont and Andre Scedrov. The undecidability of second order multiplicative linear logic. Information and Computation, 125:4651, 1996.

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11] Mitsuhiro Okada. Phase semantics for higher order completeness, cut-elimination and normalization proofs (extended abstract). In J.-Y. Girard, M. Okada, and A. Scedrov, editors, ENTCS (Electronic Notes in Theoretical Computer Science) Vol.3: A Special Issue on Linear Logic 96, Tokyo Meeting. URL: http://www.elsevier.nl/locate/entcs/volume3.html, 1996. Full version submitted to Theoretical Computer Science, also available by anonymous ftp at http://abelard.flet.mita.keio.ac.jp/person/mitsu/paper.html or ftp://abelard.flet.mita.keio.ac.jp/pub/Papers/mitsu. 12] Jean-Yves Girard. Une extension de l'interprtation de Gdel a l'analyse, et son application a l'limination des coupures dans l'analyse et la thorie des types. In J.E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, pages 6392. North-Holland, 1971.

12


A LLL rules Let us recall LLL inference rules from 1]. The x rules have been modied since we are not assuming that x is self-dual. The exchange rules are omitted because we are dealing with multisets. Identity/Negation

`A A

` A ` A? (cut) `

? (identity)

Structure

` `

`

A

(M-weakening)

]

` A

` A B

(A-weakening)

` A B B A A (M-contraction) (A-contraction) ` A ` A B

]

]

]

Logic

`

1

` (false) ` ?

(one)

` AB (par) ` A B

` A `B (times) ` AB

` >

(true)

.................. ............. ...

(no rule for zero)

` A (left plus) ` AB ` B (right plus) ` AB

` A ` B (with) ` A B

&

` B1 : : : Bn A

` B1 : : : Bn A (where n )

]

] !

(of course)

1

13

` `

A (why not) A

?

]


` B1 j : : : jBk A1 : : : Am1 Am (neutral) ` B1 : : : Bk xA1 : : : xAm1 rAm (where k m , where r is either x or x, and where each j is either a semicolon or a comma, namely, B1j : : : jBk means that B1 : : : Bn are formulas

]

]

0

separated by commas or semicolons.) Note that the conclusion contains at most one principal x.

` AfB=X g (there is) ` 9XA

` A (for all: X is not free in ) ` 8XA

Intuitionistic propositional formulas are built from propositional atoms and the constant by the connectives , and the modalities x. Intuitionistic sequents are expressions of the form A1 A2 : : : Ak ` B , where B and the formulas in the blocks Ai are intuitionistic. Because of the position of blocks to the left of the `, the intended interpretation of the punctuation marks is dual to the one stated above, i.e., A B is intended to represent A B , A B is intended to represent A B , and A is intended to represent A. An intuitionistic sequent A1 A2 : : : Ak ` B may be interpreted in the language of LLL as the sequent ` A?1 A?2 : : : A?k B , where A?i denotes the block in which every formula in the block Ai is negated (where C D ? is C D?,) and the punctuation marks are left the same. The inference rules of ILLL are those that remain correct after this translation 1]. 1

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