Lincoln, P. and T. C. Winkler, Constant-only multiplicative linear logic is NP-complete, 135(1), 1994 pp. 155--169.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....linear logic has found applications in logic programming [14, 19] modeling concurrent computation [11] planning [18] and other areas. Its expressiveness, however, results in a high complexity. Propositional linear logic is undecidable. The multiplicative fragment (MLL) is already NP complete [16]. The complexity of the multiplicative exponential fragment (MELL) is still unknown. Consequently, proof search in linear logic is di#cult to automate. Girard s sequent calculus [12] although covering all of linear logic, contains too many redundancies to be useful for e# cient proof search. ....

P. Lincoln and T. Winkler. Constant-only multiplicative linear logic is NP-complete. TCS, 135:155-169, 1994.

....linear logic without exponentials but extended with second order propositional quantifiers. The decision problem for MELL is still open, but almost all the other problems have been solved: ffl The NP completeness of MLL has been obtained by Kanovich [K1] Moreover, Lincoln and Winkler [LW] have established that MLL without proper atoms is already NP complete. 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 ....

Lincoln, P., and Winkler, T. (1994), Constant-Only Multiplicative Linear Logic is NPComplete, Theoretical Computer Science 135, 155--169.

....change. It has found applications in logic programming [14,20] planing [19] modeling concurrent computation [11] and other areas. Its expressiveness, however results in a high complexity. Validity is undecidable for propositional linear logic. The multiplicative fragment is already NP complete [16]. The complexity of the multiplicative exponential fragment (MELL) is still unknown. Consequently, proof search in linear logic is dicult to automate. Various calculi have been developed for linear logic. Beginning with the sequent calculus and proof nets by Girard [12] several optimizations have ....

P. Lincoln and T. Winkler. Constant-only multiplicative linear logic is NP-complete. TCS, 135:155-169, 1994.

....In fact, in a noticeable contrast to the standard np completeness of the satisfiability of propositional formulas in classical logic, in linear logic even the decision problem for constant only multiplicative propositional formulas is np complete. This result, obtained by Lincoln and Winkler [27], shows that a simple minded, efficient, truth table style characterization of provability in the multiplicative fragment of linear logic would imply that p = np. Lincoln et al. 23] study the fragment of propositional linear logic that consists only of the multiplicative and the additive ....

P. Lincoln and T. Winkler. Constant-Only Multiplicative Linear Logic is NP-Complete. Manuscript, September 1992. Available using anonymous ftp from host ftp.csl.sri.com and the file pub/lincoln/comult-npc.dvi.

....linear logic has found applications in logic programming [14,19] modeling concurrent computation [11] planning [18] and other areas. Its expressiveness, however, results in a high complexity. Propositional linear logic is undecidable. The multiplicative fragment (MLL) is already NP complete [16]. The complexity of the multiplicative exponential fragment (MELL) is still unknown. Consequently, proof search in linear logic is dicult to automate. Girard s sequent calculus [12] although covering all of linear logic, contains too many redundancies to be useful for ef cient proof search. ....

P. Lincoln and T. Winkler. Constant-only multiplicative linear logic is NP-complete. TCS, 135:155-169, 1994.

.... of a multiplicative formula A using only multiplicative units as atoms, such a choice would lead to identify all proofs of A ; Hence the correctness problem for such proof nets contains the decision problem for the constant only multiplicative fragment, which is known to be NP complete, see [LW92]. But the general shape of the known criterions is coNP 8 , hence the existence of a criterion of the same shape is very unlikely. Our solution corresponds to a version of proof nets in which weakened formulas are attached to a formula of the net. ffl The geometry of interaction of ....

Lincoln, P. & Winkler, T. : Constant-only Multiplicative Linear Logic is NP-complete, Theoretical Computer Science 135,1994,pp.155-159.

....change. It has found applications in logic programming [14,20] planing [19] modeling concurrent computation [11] and other areas. Its expressiveness, however results in a high complexity. Validity is undecidable for propositional linear logic. The multiplicative fragment is already NP complete [16]. The complexity of the multiplicative exponential fragment (MELL) is still unknown. Consequently, proof search in linear logic is difficult to automate. Various calculi have been developed for linear logic. Beginning with the sequent calculus and proof nets by Girard [12] several optimizations ....

P. Lincoln and T. Winkler. Constant-only multiplicative linear logic is NP-complete. TCS, 135:155--169, 1994.

....of linear logic rules may be found in, e.g. 45, 72, 89] Let us mention that the nonmodal fragment of linear logic was anticipated by a calculus proposed by Lambek [66, 67] motivated by linguistic considerations of syntax of natural languages. A remarkable result of Lincoln and Winkler [75] shows that there is no simple minded truth table characterization of provability even for the multiplicative fragment of linear logic, unless p = np (see below) In this sense semantics of linear logic is necessarily involved. Basic linear algebra constructions on finite dimensional vector spaces ....

....Furthermore, in a marked contrast to the standard np completeness of the satisfiability of propositional formulas in classical logic, in linear logic even the decision problem for constant only multiplicative propositional formulas is np complete. This result, obtained by Lincoln and Winkler [75], shows that a simple minded, efficient, truth table style characterization of provability in the multiplicative fragment of linear logic would imply that p = np. Beyond the multiplicatives, Lincoln et al. 72] show pspace completeness of the nonmodal propositional fragment (i.e. the ....

P. Lincoln and T. Winkler. Constant-Only Multiplicative Linear Logic is NP-Complete. Manuscript, September 1992. Available using anonymous ftp from host ftp.csl.sri.com and the file pub/lincoln/comult-npc.dvi.

....in exactly one way; if it is some negative atom, the same is true because of the linearly balanced assumption. Since G is acyclic, every backtracking path must stop at some node, which is the root. It is well known that the problem of deciding the provability of a given MLL formula is NP complete [9]. Here we shall consider provability decision problems restricted to linearly balanced formulas: ProofNet : Given an MLL proof structure, is it a proof net EssNet : Given an IMLL essential net, is it correct In this paper we give a linear time algorithm for EssNet (Algorithm A in Figure 4) ....

P. Lincoln and T. Winkler. Constant-only multiplicative linear logic is NPcomplete. Theoretical Computer Science, 135:155-159, 1994.

....the same is true because of the linearly balanced assumption. Since G is acyclic, every backtracking path must stop at some node, which is the root. An Algorithm for Verifying Essential Nets 5 It is well known that the problem of deciding the provability of a given MLL formula is NPcomplete [10]. Here we shall consider provability decision problems restricted to linearly balanced formulas: ProofNet : Given an MLL proof structure, is it a proof net EssNet : Given an IMLL essential net, is it correct In this paper we give a linear time algorithm for EssNet (Algorithm A in Figure 4) By ....

P. Lincoln and T. Winkler. Constant-only multiplicative linear logic is NP-complete. Theoretical Computer Science, 135:155-159, 1994.

....linear logic without exponentials but extended with second order propositional quantifiers. The decision problem for MELL is still open, but almost all the other problems have been solved: ffl The NP completeness of MLL has been obtained by Kanovich [K1] Moreover, Lincoln and Winkler [LW] have established that MLL without proper atoms is already NP complete. 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 ....

Lincoln, P., and Winkler, T. (1994), Constant-Only Multiplicative Linear Logic is NPComplete, Theoretical Computer Science 135, 155--169.

....confluent modulo the permuting conversions represented by these rewirings. The necessity of this more complex view of coherence is supported by results which show that the addition of unit rules to the multiplicative system greatly adds to the computational complexity of provability in the system (Lincoln and Winkler 1994). In this paper, we will need the rewiring system again to handle the weakening rules of the exponential fragment as well as the units themselves. It would be a mistake to imagine that the proof nets (or circuits) we draw in this paper are merely pictures. There is now a fairly deep formalism ....

Lincoln, P. and Winkler, I. (1994) Constant-only multiplicative linear logic is NP-complete.

....linear logic with exponentials, i. e, unbounded uses of these rules, is undecidable [23] though classical propositional logic is only NP complete [8] Bounding the uses of these rules regains decidability, but even the small multiplicative propositional fragment without variables is NPcomplete [25]. The decidability is maintained when lifting the multiplicativeadditive fragment to first order logic, although the problem is NEXPTIME hard, thus provably intractable [24] Finally, a complementary approach to our work is to compare the minimal lengths of proofs that different calculi allow for, ....

P. Lincoln and T. Winkler. Constant-only multiplicative linear logic is NP-complete. Available through anonymous ftp from ftp.csl.sri.com, file /pub/lincoln/comultnpc.dvi, 1992.

....linear logic has found applications in logic programming [14,19] modeling concurrent computation [11] planning [18] and other areas. Its expressiveness, however, results in a high complexity. Propositional linear logic is undecidable. The multiplicative fragment (MLL) is already NP complete [16]. The complexity of the multiplicative exponential fragment (MELL) is still unknown. Consequently, proof search in linear logic is difficult to automate. Girard s sequent calculus [12] although covering all of linear logic, contains too many redundancies to be useful for efficient proof search. ....

P. Lincoln and T. Winkler. Constant-only multiplicative linear logic is NP-complete. TCS, 135:155-169, 1994.

....multiplicative fragment of this system, the syntax tree T of a formula A is a proof structure (without weakening boxes) with conclusion A. Therefore, deciding whether A is provable or not corresponds to verifying correctness of the proof structure T. This fragment of linear logic is NP complete [LW94]. Hence, there is no chance to get any polynomial parsing technique for proof nets without weakening boxes. The previous considerations on the constant only multiplicative case give an informal argument against any trivial generalization of DR correctness to the case of proof nets without ....

P. Lincoln and T. Winkler. Constant-only multiplicative linear logic is NP-complete. Theoretical Computer Science, 135:155--169, 1994.

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P. Lincoln and T. Winkler. Constant-Only Multiplicative Linear Logic is NP-Complete. Theoretical Computer Science, 135:155--169, 1994. Available using anonymous ftp from host ftp.csl.sri.com and the file pub/lincoln/comultnpc. dvi.

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P. Lincoln and T. Winkler. Constant-Only Multiplicative Linear Logic is NP-Complete. Manuscript to be submitted to Proc. MFPS 8, Oxford, 1992.

.... linear logic (MALL) and the np completeness of multiplicative linear logic (MLL) with unrestricted weakening [10] Using similar techniques, Kanovich showed that multiplicative linear logic is np complete [8] Lincoln and Winkler demonstrated np completeness of MLL using only the constants [13]. Lincoln and Scedrov had previously shown this fragment of linear logic to be nexptime hard [11] By extending this proof search paradigm, we are able to show here that the rst order version of MALL is decidable in nexptime. We also observe that rstorder MLL is np complete and hence of the same ....

P. Lincoln and T. Winkler. Constant-Only Multiplicative Linear Logic is NP-Complete. To Appear in TCS, 1993.

....propositional linear logic are considered here: the multiplicative additive fragment, mall, and the multiplicative fragment extended with additive constants, mll . mall is pspace complete [18] The np completeness of mll follows from the np completeness of the pure multiplicative fragment, mll [16, 19]. These are global hardness properties in that they provide lower bounds on proponent s optimal strategy. In chess and in many other intricate games, however, choosing the best next move often seems just as hard as developing a complete winning strategy. In other words, these games are locally ....

P. Lincoln and T. Winkler. Constant-Only Multiplicative Linear Logic is NP-Complete. Theoretical Computer Science, 135:155--169, 1994. 7

.... linear logic (MALL) and the np completeness of multiplicative linear logic (MLL) with unrestricted weakening [10] Using similar techniques, Kanovich showed that multiplicative linear logic is np complete [8] Lincoln and Winkler demonstrated np completeness of MLL using only the constants [13]. Lincoln and Scedrov had previously shown this fragment of linear logic to be nexptime hard [11] By extending this proof search paradigm, we are able to show here that the first order version of MALL is decidable in nexptime. We also observe that firstorder MLL is np complete and hence of the ....

P. Lincoln and T. Winkler. Constant-Only Multiplicative Linear Logic is NP-Complete. To Appear in TCS, 1993.

.... of linear logic that does not allow modalities is strikingly expressive: provability of multiplicative propositional formulas is np complete [17] In fact, the decision problem for constant only formulas in multiplica 1 INTRODUCTION 3 tive propositional linear logic is also np complete [20]. If the additives are also allowed, provability for propositional formulas is pspace complete [18] and in fact this fragment allows a structural embedding of a cut free proof system for implicational propositional intuitionistic logic [19] The decidability of second order propositional linear ....

P. Lincoln and T. Winkler. Constant-Only Multiplicative Linear Logic is NP-Complete. Manuscript to be submitted to Proc. MFPS 8, Oxford, 1992.

....of mell, and the decidability of various fragments of propositional linear logic without exponentials but extended with second order propositional quantifiers. The decision problem for mell is still open. The np completeness of mll has been obtained by Kanovich [14] Lincoln and Winkler [18] have established that the provability of multiplicative propositional sentences built on constants, not literals, is already np complete. mall1 with function symbols is nexptime complete: the hardness has been obtained by Lincoln and Scedrov [17] and the membership, and hence completeness, by ....

P. Lincoln and T. Winkler. Constant-Only Multiplicative Linear Logic is NP-Complete. Theoretical Computer Science, 135:155--169, 1994. Available using anonymous ftp from host ftp.csl.sri.com and the file pub/lincoln/comultnpc. dvi.

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Lincoln, P. and T. C. Winkler, Constant-only multiplicative linear logic is NP-complete, 135(1), 1994 pp. 155--169.

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Patrick Lincoln and Timothy C. Winkler. Constant-only multiplicative linear logic is NP-complete. In Theoretical Computer Science, volume 135(1), pages 155--169. 1994.

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P. Lincoln and T. Winkler. Constant-Only Multiplicative Linear Logic is NP-Complete. Theoretical Computer Science, 135(1):155--169, Dec. 1994.

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