H. Schellinx. Some Syntactical Observations on Linear Logic. Journal of Logic and Computation, 1(4):537--559, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....embedding of intuitionistic logic into linear logic, for formulae, sequents and for proofs. The translation, known as the Girard embedding (the ( 0 embedding of [Gir87] was claimed to be correct and faithful. A detailed proof wasn t provided in that paper this was supplied by Schellinx in [Sch91]. Sch94] TS96] show how the Girard embedding of proofs induces a sequent calculus for implicational intuitionistic logic this Gentzen system is known as IU. We know of no satisfactory semantic justification for the form of IU. In this chapter we discuss IU and the Girard embedding and give a ....

....of Gamma; P oe Q ) R is (oe L ) then in the translation we would have: Gamma g ) P g Gamma g ) P g (P ) Gamma g ; Q g ) R g Gamma g ; P g GammaffiQ g ) R g ( Gammaffi L ) We know that the embedding is correct for provability. The following theorem can be found in [Sch91]: Theorem 5.1 IL Gamma ) P iff CLL Gamma g ) P g . PROOF: The proof can be found in [Sch91] Xi In [Gir87] a translation of proofs is also given, showing how to interpret a (natural deduction) proof into a linear sequent calculus proof. It is translations of proofs that we are really ....

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H. Schellinx. Some Syntactical Observations on Linear Logic. Journal of Logic and Computation, 1(4):537--559, 1991.

....it is needed. Proposition 1 Let B be a first order formula over the logical constants Omega ; Gammaffi; 8; 1; The sequent 1 Gamma B is provable in L if and only if B is provable in linear logic. For this sublogic, provability in classical and intuitionistic linear logic coincide [30]. Two formulas B and C are equivalent, written B j C, if the sequents B Gamma C and C Gamma B are provable in L. We note the following equivalences: 1 j ; B C) j B Omega C, B Gammaffi (C D) j (B Gammaffi C) B Gammaffi D) B Gammaffi (8x:C) j 8x(B Gammaffi C) provided x ....

Harold Schellinx. Some syntactical observations on linear logic. Technical Report ML-90-08, ITLI Prepublication Series for Mathematical Logic and Foundations, University of Amsterdam, 1990. 11

....sequent over Gammaffi, Phi, and , only involves intuitionistic sequents. Thus, such a proof yields an intuitionistic proof if we erase and replace the additive connectives by the standard connectives oe, A more complex argument is needed in order to handle 0 (see Schellinx [11]) Classical logic can also be represented in linear logic. 3 Representing Classical Logic into Linear Logic Given a classical sequent A 1 ; Am Gamma B 1 ; B n , we will consider that the occurrences of B 1 ; B n are positive, and that the occurrences of A 1 ; Am ....

H. Schellinx. Some syntactical observations on linear logic. Logic and Computation (1990). to appear.

....and . Thanks to its simmetry, the inference systems of LL can be given with two equivalent formulations: one with one sided sequents Delta, the other with two sided sequents Gamma Delta (see Appendix A) Here Delta and Gamma are multisets of formulas. Intuitionistic linear logic (ILL) (Schellinx 1991) is usually defined by restricting the right hand side of sequents to an empty or a singleton multiset. However, similarly to the classical case, there exist multi conclusion formulations of ILL with special restriction on some of the rules, e.g. full intuitionistic linear logic (FILL) Hyland ....

H. Schellinx. Some syntactical observations on linear logic. Journal of Logic and Computation, 1(4):537--559, 1991.

....(it is the principal formula) and the proof is divided into two parts, following the usual pattern. But the other succedent formulae, that are included in the set , are erased. Such rules are similar to the so called special rules of multi conclusion sequent calculus for intuitionistic logic [43, 48]. Assuming that we would like to deal with negations instead of just considering formulae through De Morgan laws, we should also add the following rules: A; A L :A :A R Let us consider G (E (A B) A C) B D) E F to illustrate the use of this 27 :A A: ....

H. Schellinx. Some syntactical observations on linear logic. Journal of Logic and Computation, 1(4):537559, 1991.

....can be obtained from classical sequent calculus of CL (see appendix 1. by restricting sequents to a single conclusion (see appendix 2. It is important to recall that there exists a multi conclusion sequent calculus for IL (see appendix 3. that is well adapted to study efficient proof search [13, 31, 32]. Such a calculus is in fact a restriction of classical 4 LABELLED DEDUCTION sequent calculus including some particular rules called special rules. It allows to naturally consider the intuitionistic provability like a perturbation of classical provability. Thus, we can consider the following ....

H. Schellinx. Some syntactical observations on linear logic. Journal of Logic and Computation, 1(4):537--559, 1991.

....that the subproof above each ( oe) and ( 8) rule constitutes a proof of its right side formula. Thus the transformed proof instantiates only LMM inference figures. Overall, the method recalls Schellinx s proof theoretic justification of the embedding of intuitionistic logic into linear logic (Schellinx, 1991). In considering just a single application P of ( oe) or ( 8) there is a simple characterization of which other rule applications permute below P. Lemma 5 (permutability) Let D be an LMP derivation containing an application P of ( oe) or ( 8) which introduces the eigenvariable ff in the ....

Schellinx, H. (1991). Some syntactical observations on linear logic. Journal of Logic and Computation, 1(4):537--559.

.... The resulting logical framework and in particular its proof theoretical formalization revealed strong connections with many computational models such as the Petri nets [15, 21] and the calculus [4] Proof theoretic investigations yielded both Intuitionistic and Classical formulations of LL [43] with interesting results on cut elimination for sublogics based on fragments of connectives and on permutability of their rules [2, 18, 31] For people not acquainted with these subject, in the following we shall describe the main features of LL under a proof theoretical perspective. Some ....

H. Schellinx. Some Syntactical Observations on Linear Logic. Journal of Logic and Computation, 1(4):537-- 559, 1991.

....to arbitrarily duplicate and reduce formulas on both sides of sequents. The resulting logic revealed strong connections with many computational models such as Petri nets [16] and Milner s calculus [4] Proof theoretic investigations yielded both Intuitionistic and Classical formulations of LL [40] with interesting results on cut elimination for fragments of connectives and on permutability of rules [2, 19] For people not acquainted with these subjects, in the following we shall describe the main features of LL under a proof theoretical perspective. Some preliminary notions on ....

H. Schellinx. Some Syntactical Observations on Linear Logic. Journal of Logic and Computation, 1(4):537-- 559, 1991.

.... as conversion (computation) has been extended to this new setting yielding definitions of more complex (and more refined) term calculi, e.g. 2, 51] Proof theoretic investigations allowed to study, as in the Classical case, both Intuitionistic and Classical formulations of Linear Logic [68]; results were obtained on cut elimination with different fragments of connectives and on a systematic study of the permutability of their rules [9, 33, 51] Proof search paradigms, based on the interpretation of formulas as programs and proofs as computations, have been investigated with a ....

H. Schellinx. Some Syntactical Observations on Linear Logic. Journal of Logic and Computation, 1(4):537--559, 1991.

....sequent over Gammaffi, Phi, and , only involves intuitionistic sequents. Thus, such a proof yields an intuitionistic proof if we erase and replace the additive connectives by the standard connectives oe, A more complex argument is needed in order to handle 0 (see Schellinx [11]) Classical logic can also be represented in linear logic. 3 Representing Classical Logic into Linear Logic Given a classical sequent A 1 ; Am Gamma B 1 ; B n , we will consider that the occurrences of B 1 ; B n are positive, and that the occurrences of A 1 ; Am ....

H. Schellinx. Some syntactical observations on linear logic. Logic and Computation, 1990. to appear.

.... area of study is directly relevant to the logic programming use of linear logic, where linear logic sequents are taken to be logic programs which execute by performing proof search [20, 5, 4, 19] This area of research is also directly relevant to the construction of linear logic theorem provers [40, 44, 35, 6, 11, 10, 41]. The results here also lead into the study of semantics of linear logic, pointing to deep connections between various fragments of linear logic and familiar structures from computer science [12, 29, 30] In particular, work has progressed in attempting to find viewpoints where the proof theory of ....

H. Schellinx. Some syntactical observations on linear logic. Journal of Logic and Computation, 1:537--559, 1991.

....(2) implies (1) can be proved by a careful analysis of cut free proofs in ILL. Finally, the equivalence between (2) and (3) comes from the fact that, in the absence of additive units and of second order quantifiers, classical linear logic is a conservative extension of intuitionistic linear logic [Sx]. This encoding uses exponentials and additives in an essential way. Exponentials are really necessary because MALL is decidable [LMSS] but the decidability of MELL is still an open problem. 2. Encoding two counter machines in MALL2 Now we want to adapt the above encoding to MALL2. This means ....

H. Schellinx. Some syntactical observations on linear logic. Journal of Logic and Computation 1 (4), 537--559. 1991. THE UNDECIDABILITY OF LL2 WITHOUT EXPONENTIALS 7

....having to add indices. It is used however merely as a notational device. 2 Observe that this is far less obvious in case we consider Girard s translation, defined on the full language, as an embedding of intuitionistic logic into classical linear logic, though also in that case it holds. See Schellinx(1991). Note that this implies that Girard s translation does not give us a proper translation of sequent calculus derivations: the inductive procedure makes it necessary to introduce cuts in the linear derivation. If we define the skeleton sk( of a linear derivation to be the (classical or ....

Schellinx, H. (1991). Some syntactical observations on linear logic. Journal of Logic and Computation, 1(4):537--559.

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H. Schellinx. Some syntactical observations on linear logic. Journal of Logic and Computation, 1:537--559, 1991.

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