Lincoln, P., Scedrov, A. and Shankar, N. (1995), Decision Problems for Second Order Linear Logic, in "Proc. 10-th Annual IEEE Symposium on Logic in Computer Science, San Diego, California," 476--485.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....notations: and stands for the left and right implications, and stands for the tensor product. See Table 3 for the inference rules) It is clear that LC N ILL ILL LL LLW. Main results. Lincoln, Scedrov, and Shankar showed the undecidability of IMLL2 and IMALL2 by embedding of LJ2 [LSS]. Lafont has proved the undecidability of MALL2 [Laf1] Then Lafont and Scedrov proved that MLL2 is undecidable too [LS] Emms showed embedding of LJ2 into N IMLL2. Kanovich demonstrated the undecidability of N MLL2, cyclic LL and second order Lambek Calculus (LC2) Kan2] On the other hand, ....

P. Lincoln, A. Scedrov, N. Shankar. Decision Problem for Second Order Linear Logic. 10-th Annual IEEE Symposium on Logic in Computer Science, San Diego, California, IEEE Computer Society Press. 1995.

....rise recently to many papers and results. P. Lincoln, A. Scedrov and N. Shankar have shown that the multiplicative fragment of second order intuitionistic linear logic (IMLL2) is undecidable by encoding second order intuitionistic propositional logic known to be undecidable into IMLL2 [10]. M. Emms has extended this strategy to prove the undecidability of the second order Lambek Calculus (L2) 3] which can be viewed as the multiplicative fragment of intuitionistic non commutative linear logic. The undecidability of MALL2 has been shown by Y. Lafont [6] and the undecidability of ....

....the antecedent of a sequent, it respects a certain form and if it constitutes the succedent of a sequent, it respects another form. Example 3. 1 The formula C = 8X(X ( X Omega X) is added to the antecedent of sequents to simulate contraction in second order intuitionistic logic inside IMALL2 [10]. The negative translation of C in MALL2 is the formula 9X(X Omega (X OX ) which is not a member of MALL2 because it does not verify Condition 1. This observation is logical because contraction causes the undecidability of second order intuitionistic logic. Dropping commutativity from ....

P. Lincoln, A. Scedrov, and N. Shankar. Decision problems for second order linear logic. In D. Kozen, editor, Tenth Annual IEEE Symposium on Logic in Computer Science, pages 476--485, San Diego, California, June 1995. 23

....to proceed by induction on the proof size. In the case of the generalized Lambek calculi we have studied in x4.1, Cut Elimination went hand in hand with decidability. For L2 this is not the case. Undecidability of L2 has been established in [Emms 95] via a generalization of a recent result by [Lincoln e.a. 95] These authors show that second order intuitionistic propositional logic (LJ2) a system which is known to be undecidable can be embedded into the multiplicative fragment of second order intuitionistic Linear Logic (IMLL2) i.e. LP2. The key idea of the embedding is to reintroduce the ....

....as shown in [Moortgat 95] This illustrates a second type of control that can be logically implemented in terms of the unary vocabulary: a procedural dynamic form of control rather than the structural static control of x4.2.3. The 3; 2 # annotation is a variation on the lock and key method of [Lincoln e.a. 95] one forces a particular execution strategy for successful proof search by decorating formulae with the 2 # ( lock ) and 3 ( key ) control operators. For the selection of the active formula, one uses the distributivity principes K1;K2, in combination with the base residuation logic for 3; 2 ....

Lincoln, P., A. Scedrov and N. Shankar (1995) `Decision problems for second-order Linear Logic'. Proceedings of the Tenth Annual IEEE Symposium on Logic in Computer Science.

....rise recently to many papers and results. P. Lincoln, A. Scedrov and N. Shankar have shown that the multiplicative fragment of second order intuitionistic linear logic (IMLL2) is undecidable by encoding second order intuitionistic propositional logic known to be undecidable into IMLL2 [10]. M. Emms has extended this strategy to prove the undecidability of second order Lambek Calculus (L2) 4] which can be viewed as the multiplicative fragment of intuitionistic non commutative linear logic. The undecidability of MALL2 has been shown by Y. Lafont [7] and the undecidability of the ....

Patrick Lincoln, Andre Scedrov, and Natarajan Shankar. Decision problems for second order linear logic. In D. Kozen, editor, Tenth Annual IEEE Symposium on Logic in Computer Science, pages 476--485, San Diego, California, June 1995.

....expressive power of exponentials by means of second order quantification. We may first replace Theta by Theta 1, so that we automatically get dereliction and weakening. We must add an explicit formula for contraction. Unfortunately, the formula = 8ff: ff Gammaffi ff Omega ff) used in [LSS] is too strong: For instance, it implies a Gammaffi a Omega a which does not correspond to any transition of the machine. Instead, we shall use = 8ff: ff 1 Gammaffi ff Omega ff) Now we can state the central result of this paper: Theorem 3. For any (i; p; q) 2 [0; m] Theta N Theta N, ....

P. Lincoln, A. Scedrov & N. Shankar. Decision Problems for Second Order Linear Logic. 10-th Annual IEEE Symposium on Logic in Computer Science, San Diego, California, IEEE Computer Society Press. 1995.

.... ffl MALL1 with function symbols is NEXPTIME complete: the hardness has been obtained by Lincoln and Scedrov [LSv] and the membership, and hence completeness, by Lincoln and Shankar [LSr] ffl The undecidability of MLL12 and MALL12 with function symbols has been proved by Amiot [A] Lincoln et al. [LSS] have recently showed the undecidability of IMLL2 and IMALL2, and the undecidability of MLL12 and MALL12 without function symbols. Lafont [Lt] then proved the undecidability of MALL2. Here we use a refinement of the methods from [LSS] and [Lt] to establish the undecidability of MLL2. E mail ....

....function symbols has been proved by Amiot [A] Lincoln et al. LSS] have recently showed the undecidability of IMLL2 and IMALL2, and the undecidability of MLL12 and MALL12 without function symbols. Lafont [Lt] then proved the undecidability of MALL2. Here we use a refinement of the methods from [LSS] and [Lt] to establish the undecidability of MLL2. E mail address: lafont lmd.univ mrs.fr E mail address: andre cis.upenn.edu. WWW: http: www.cis.upenn.edu andre. Partially supported by NSF Grant CCR 94 00907, by ONR Grant N00014 92 J 1916, and by CNRS. Scedrov is an American Mathematical ....

[Article contains additional citation context not shown here]

Lincoln, P., Scedrov, A. and Shankar, N. (1995), Decision Problems for Second Order Linear Logic, in "Proc. 10-th Annual IEEE Symposium on Logic in Computer Science, San Diego, California," 476--485.

.... ffl MALL1 with function symbols is NEXPTIME complete: the hardness has been obtained by Lincoln and Scedrov [LSv] and the membership, and hence completeness, by Lincoln and Shankar [LSr] ffl The undecidability of MLL12 and MALL12 with function symbols has been proved by Amiot [A] Lincoln et al. [LSS] have recently showed the undecidability of IMLL2 and IMALL2, and the undecidability of MLL12 and MALL12 without function symbols. Lafont [Lt] then proved the undecidability of MALL2. Here we use a refinement of the methods from [LSS] and [Lt] to establish the undecidability of MLL2. E mail ....

....function symbols has been proved by Amiot [A] Lincoln et al. LSS] have recently showed the undecidability of IMLL2 and IMALL2, and the undecidability of MLL12 and MALL12 without function symbols. Lafont [Lt] then proved the undecidability of MALL2. Here we use a refinement of the methods from [LSS] and [Lt] to establish the undecidability of MLL2. E mail address: lafont lmd.univ mrs.fr y E mail address: scedrov cis.upenn.edu. WWW: http: www.cis.upenn.edu andre. Partially supported by NSF Grant CCR 94 00907, by ONR Grant N00014 92 J 1916, and by CNRS. Scedrov is an American ....

[Article contains additional citation context not shown here]

Lincoln, P., Scedrov, A. and Shankar, N. (1995), Decision Problems for Second Order Linear Logic, in "Proc. 10-th Annual IEEE Symposium on Logic in Computer Science, San Diego, California," 476--485.

.... ffl MALL1 with function symbols is NEXPTIME complete: the hardness has been obtained by Lincoln and Scedrov [LSv] and the membership, and hence completeness, by Lincoln and Shankar [LSr] ffl The undecidability of MLL12 and MALL12 with function symbols has been proved by Amiot [A] Lincoln et al. [LSS] have recently showed the undecidability of IMLL2 and IMALL2, and the undecidability of MLL12 and MALL12 without function symbols. Lafont [Lt] then proved the undecidability of MALL2. Here we use a refinement of the methods from [LSS] and [Lt] to establish the undecidability of MLL2. E mail ....

....function symbols has been proved by Amiot [A] Lincoln et al. LSS] have recently showed the undecidability of IMLL2 and IMALL2, and the undecidability of MLL12 and MALL12 without function symbols. Lafont [Lt] then proved the undecidability of MALL2. Here we use a refinement of the methods from [LSS] and [Lt] to establish the undecidability of MLL2. E mail address: lafont lmd.univ mrs.fr y E mail address: andre cis.upenn.edu. WWW: http: www.cis.upenn.edu andre. Partially supported by NSF Grant CCR 94 00907, by ONR Grant N00014 92 J 1916, and by CNRS. Scedrov is an American Mathematical ....

[Article contains additional citation context not shown here]

Lincoln, P., Scedrov, A. and Shankar, N. (1995), Decision Problems for Second Order Linear Logic, in "Proc. 10-th Annual IEEE Symposium on Logic in Computer Science, San Diego, California," 476--485.

.... Linear Logic Amiot has shown that MLL (and MALL) with first and second order quantifiers and appropriate function symbols is undecidable [1, 2] In recent work, pure second order intuitionistic MLL (IMLL2) has been shown to be undecidable, through the encoding of second order intuitionistic logic [34]. The key point is that it is possible to encode contraction and weakening using second order formulas. C Delta = 8X:X Gammaffi(X Omega X) W Delta = 8X:X Gammaffi1 [ Sigma LJ2 Delta] Delta =C;C;C;W; Sigma] IMLL2 [ Delta] C encodes contraction and W encodes weakening. A ....

P. Lincoln, A. Scedrov, and N. Shankar. Decision problems for second order linear logic. Email Message 278 to Linear Mailing List, October 1994.

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Lincoln, P., A. Scedrov and N. Shankar (1995) `Decision problems for secondorder Linear Logic'. Proceedings of the Tenth Annual IEEE Symposium on Logic in Computer Science.

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