Kanovich, M., Horn programming in linear logic is NP-complete,in:Proccedings of the 7th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz (1992), pp. 200--210.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... of propositional linear logic, the pspace completeness of multiplicative additive 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 ....

....to be decidable in np by the same method. The number of axiom nodes in a MLL1 proof is linear in the size of the conclusion, and each axiom node in the search has size at most quadratic in the size of the conclusion. This results in a tight bound since the propositional fragment MLL is np complete [8]. Theorem 3.5 MLL1 is np complete. 4 An Optimization We now present an optimization of LLV called LLO that exploits the permutabilities of linear logic. As is clear from the table in Appendix A, there are only a few pairs of impermutable rules in LL. A rule R1 is said to be impermutable below ....

M. Kanovich. Horn programming in linear logic is NPcomplete. In Proc. 7-th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, California, pages 200{ 210. IEEE Computer Society Press, Los Alamitos, California, June 1992.

....we can decide whether a sequent of the form (3.1) is provable in ITLL # . The temporal linear logic system in [24] does not hold completeness theorem for timed Petri nets because of including non logical axioms. Therefore, decidability of the logical system in [24] has not been argued yet. In [8], Kanovich researched several kinds of Horn fragment of linear logic, for example, the (#, Horn fragment, the Horn fragment, the # Horn fragment, and so on. Using similar method, we may obtain the decidabilities of various kinds of Horn fragment of temporal linear logic. This is one of our ....

M.I.Kanovich, "Horn programming in linear logic is NPcomplete, " Proc. 7th Annual Symposium on Logic in Computer Science, pp.200--210, June 1992.

....the empty multiset, H is the head of the clause, and G is the body of the clause. We will follow the common practice from the logic programming literature of writing the head of a clause first by reversing the sense of the implication: that is, the clause above is written #x[H # G] Kanovich [17, 18] introduced a very similar set of formulas he calls linear Horn clauses: these are essentially process clauses in contrapositive form (replace . ....

M. Kanovich. Horn programming in linear logic is NPcomplete. In Proceedings of the Seventh Annual IEEE Synposium on Logic in Computer Science, pages 200-- 210. IEEE Computer Society Press, June 1992.

....multiset. Process clauses are closed formulas of the form #x[G # H] where H is not and all free variables of G are free in H. These clause have been used in [Mil93] to encode a calculus similar to the # calculus. A nearly identical subset of linear logic has also been proposed by Kanovich [Kan92,Kan94]: if you write process clauses in their contrapositive form (replacing the connectives . with 1, and #, ....

Max Kanovich. Horn programming in linear logic is NP-complete. In Proceedings of the Seventh Annual IEEE Synposium on Logic in Computer Science, pages 200--210. IEEE Computer Society Press, June 1992.

....this language only when a compact representation of programs is needed. Otherwise we shall represent facts and rules as LL extralogical axioms. Thus (A B ( C) D E ( F : is represented as a set fA B C; D E F; g. 2.2. 1 Complexity of our fragment of LL Kanovich [45, 46, 48] proved that the derivability problem of the LL subset consisting only of tensor , modal storage operator , and linear implication ( is directly equivalent to Petri net reachability problem and thus is decidable. In [46, 47] he proved that standard Minsky machines [72] can be directly encoded ....

M. I. Kanovich. Horn Programming in Linear Logic is NP-complete. In Proceedings of Seventh Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, California, June 1992.

....that the multiplicative bang ( Gammaffi, fragment, MBLL (also called MELL) can encode Petri Net reachability. On the other hand it is known that MALL (the exponent free or multiplicative additive fragment of linear logic) is decidable (and PSPACE complete) and that MLL is NP complete [7] (see also below) It is thus of great interest to understand the boundary region (TBLL, PrTLL, MBLL) both to pin down as much as possible where undecidability happens as possibly to shed more light on the complexity of Petri Net reachability from a linear logic perspective. Here we investigate ....

Kanovich, M. [1992] "Horn Programming in Linear Logic is NP-Complete" in Proc. 7-th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, California, pp. 200-210, IEEE Computer Society Press, Los Alamitos, California.

....a marking M is reachable from a marking M 0 if and only if the following Horn sequent M # 0 , T r # #M # is provable in Linear Logic, where M # 0 , M # are corresponding formulas and T r # is a corresponding sequence of formulas. Remark. The Horn fragment of linear logic is NP complete. [5, 6] 3 Timed Petri Nets and Temporal Linear Logic 3.1 Timed Petri Nets We consider place timed Petri nets in the paper. Definition 2 (Timed Petri Net) A (place) Timed Petri Net (TPN) is a tuple (P l, T r, Ar, #) where: P l: Finite set of places T r: Finite set of transitions (disjoint with P ....

Max I. Kanovich. Horn programming in linear logic is NP-complete. In Seventh Annual Symposium on Logic in Computer Science, pages 200--210, Santa Cruz, California, June 1992. IEEE Computer Society Press.

....The Weakening Rule. In [Lincoln et al. 1990] it was demonstrated that the MLLW is NP complete, and in [Kanovich,1991] it was proved that the MLL is NP complete too. In order to nd the exact bound of NP completeness in Linear Logic, the Horn fragment of Linear Logic (HLL) has been introduced in [Kanovich,1992], i.e. the fragments containing only sequents of the form ; where contains only simple products, and contains simple products and implications of the form (X Y ) where X and Y are simple products. A simple product is a tensor product of a number of positive literals and constants) It is ....

....is correct: 1) the set f1; 2; kg can be partitioned into m disjoint sets S 1 ; S 2 ; Sm (three elements in each) such that P i2S j s i = b, for j = 1; m. In [Garey and Jonson,1979] it was demonstrated that the equipartition problem is NPcomplete. 2 Summary In [Kanovich,1992] it was shown that both HLL s with Weakening and without Weakening are NP complete, even if they contain occurrences of only two literals and do not contain constants at all. On the other hand, it was observed in [Kanovich,1992] that the one literal HLL s (both with Weakening and without ....

[Article contains additional citation context not shown here]

M.I.Kanovich. Horn Programming in Linear Logic is NP-complete. In Proc. 7-th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, June 1992, pp.200-210

....a marking M is reachable from a marking M 0 if and only if the following Horn sequent M # 0 , T r # #M # is provable in Linear Logic, where M # 0 , M # are corresponding formulas and T r # is a corresponding sequence of formulas. Remark 2.2. 1 The Horn fragment of linear logic is NP complete [14, 15]. For a Petri net in Fig. 2.1, the reachability that M is reachable from M 0 is presented as a sequent p 1# p 2# p 2 , p 1# p 2# p 2 # p 1# p 3# p 3 ) p 3 # p 2 ) # p 1# p 2# p 3 . Indeed, this sequent is provable in linear logic. 9 2.3 Phase Semantics for Linear ....

....where ITLL # is a subsystem of TLL which appears in Subsection 5.2.2. The temporal linear logic system in [33] does not satisfy the completeness theorem for timed Petri nets because it includes non logical axioms. Therefore, decidability of the logical system in [33] has not been established yet. [15] researched several kinds of Horn fragments of linear logic, for example, the (#, Horn fragment, the Horn fragment, the # Horn fragment, and so on. Using a similar method, we may obtain results concerning decidability of various kinds of Horn fragments of TLL. We plan to do research on this ....

Max I. Kanovich. Horn programming in linear logic is NP-complete. In Seventh Annual Symposium on Logic in Computer Science, pages 200--210, Santa Cruz, California, June 1992. IEEE Computer Society Press. 37

....we can decide whether a sequent of the form (3.1) is provable in ITLL # . The temporal linear logic system in [24] does not hold completeness theorem for timed Petri nets because of including non logical axioms. Therefore, decidability of the logical system in [24] has not been argued yet. In [8], Kanovich researched several kinds of Horn fragment of linear logic, for example, the (#, Horn fragment, the Horn fragment, the # Horn fragment, and so on. Using similar method, we may obtain the decidabilities of various kinds of Horn fragment of temporal linear logic. This is one of our ....

M.I.Kanovich, "Horn programming in linear logic is NPcomplete, " Proc. 7th Annual Symposium on Logic in Computer Science, pp.200--210, June 1992.

....Logic is also reduced to its normal fragment and the derivability in this fragment is characterized in terms of analogous games. 2 Normal fragment and its computational interpretation Now we give the de nition of the normal fragment. The normal fragment is an expansion of the Horn fragment [2, 3]. Let us recall some de nitions from [2] The research described in this publication was made possible in part by Grant No. NFQ000 from the International Science Foundation De nitions A simple product is a tensor product of literals and the constant 1. For example: 1, p, p q) A Horn ....

....and the derivability in this fragment is characterized in terms of analogous games. 2 Normal fragment and its computational interpretation Now we give the de nition of the normal fragment. The normal fragment is an expansion of the Horn fragment [2, 3] Let us recall some de nitions from [2]. The research described in this publication was made possible in part by Grant No. NFQ000 from the International Science Foundation De nitions A simple product is a tensor product of literals and the constant 1. For example: 1, p, p q) A Horn implication is an implication of the form: X ....

[Article contains additional citation context not shown here]

M.I.Kanovich. Horn Programming in Linear Logic is NP-complete. In Proc. 7-th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, June 1992, pp.200-210

....from modal logics to describe the needed fragments of FOL ( 19] An example of a substructural modal logic is the logic of categories ( 8] we mentioned above. To found our proposal we quote some complexity results of substructural logics. Horn programming in linear logic is NP complete [20]. Provability in the non modal fragment of Propositional Linear Logic (MALL) is PSPACE complete ( 21] Provability in the non modal fragment of (predicative) Linear Logic (MALL1) is NEXPTIME complete ( 22] 23] We note that one still needs to be careful. If one re introduces the ....

M. Kanovich, Horn programming in linear logic is NP-complete, in Proc. 7th annual IEEE symposium on Logic in Computer Science, Santa Cruz, California, pp. 200-210. 1992. (full paper appeared in Annals of Pure and Applied Logic).

....free) linear logic is undecidable. This is obtained by faithfully representing computations on counter machines by proof search in propositional linear logic, so that the representing search successfully ends in the axioms iff the counter machine computation terminates. On the other hand, Kanovich [22] shows that the multiplicative fragment alone is np complete. 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 ....

....result in [24] just discussed, this shows that mall1 is nexptime complete. In the case of the first order multiplicative fragment of linear logic, the same proof search procedure is in np when applied to formulas from this fragment. Thus it follows from earlier propositional np hardness results [22] (see above) that first order multiplicative linear logic is np complete. 4 Example: Simulating Validity Recall the nonmodal fragment of propositional linear logic, mall. This fragment contains multiplicative connectives Omega , ....

M. Kanovich. Horn programming in linear logic is NP-complete. In Proc. 7-th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, California, pages 200--210. IEEE Computer Society Press, Los Alamitos, California, June 1992. Full paper to appear in Annals of Pure and Applied Logic.

....and 8L . This takes the shape of three kinds of choice points in the parsing process: selecting the pertinent branch for every additive conjunction, identifying some node variables and instantiating feature variables in an appropriate manner. The NP completeness of the implicative fragment of ILL (Kanovich, 1992) shows that it is hopeless to nd a general parsing algorithm for IG that works in polynomial time in the worst cases. Experience has shown that, fortunately, these worst cases rarely occur in parsing natural languages. Nevertheless, the exibility of IG entails a combinatory explosion of the ....

M. Kanovich. 1992. Horn programming in linear logic is NP-complete. In LICS'92, Jun 92, Santa Cruz, California, pages 200-210.

....that the multiplicative bang ( Omega , Gammaffi, fragment, MBLL (also called MELL) can encode Petri Net reachability. On the other hand it is known that MALL (the exponent free or multiplicative additive fragment of linear logic) is decidable (and PSPACE complete) and that MLL is NP complete [7] (see also below) It is thus of great interest to understand the boundary region (TBLL, PrTLL, MBLL) both to pin down as much as possible where undecidability happens as possibly to shed more light on the complexity of Petri Net reachability from a linear logic perspective. Here we investigate ....

Kanovich, M. [1992] "Horn Programming in Linear Logic is NP-Complete" in Proc. 7-th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, California, pp. 200-210, IEEE Computer Society Press, Los Alamitos, California.

.... of propositional linear logic, the pspace completeness of multiplicative additive 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 ....

....to be decidable in np by the same method. The number of axiom nodes in a MLL1 proof is linear in the size of the conclusion, and each axiom node in the search has size at most quadratic in the size of the conclusion. This results in a tight bound since the propositional fragment MLL is np complete [8]. Theorem 3.5 MLL1 is np complete. 4 An Optimization We now present an optimization of LLV called LLO that exploits the permutabilities of linear logic. As is clear from the table in Appendix A, there are only a few pairs of impermutable rules in LL. A rule R1 is said to be impermutable below ....

M. Kanovich. Horn programming in linear logic is NPcomplete. In Proc. 7-th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, California, pages 200-- 210. IEEE Computer Society Press, Los Alamitos, California, June 1992.

....1998 and our anonymous referees for feedback and discussion. 1 2 Glue Compilation Deduction in linear logic is known to be intractable. Even the propositional tensor fragment which provides the scaffolding for the glue language approach [ Dalrymple et al. 1996 ] is known to be NP complete [ Kanovich, 1992 ] In practice, however, it has turned out that the glue analyses published in the literature are computationally feasible. The reason is that the meaning constructors (i.e. linear logic premises) employed in such analyses exhibit a simple and restricted combinatorial pattern. Gupta and ....

Kanovich, M. 1992. Horn programming in linear logic is NP-complete. In Seventh Annual IEEE Symposium on Logic in Computer Science. IEEE Computer Society Press. 200--210.

....and the standard computational complexity classes on the other. Lincoln et al. 72] show that the full propositional (i.e. quantifier free) linear logic is undecidable. However, even the fragment of propositional linear logic that does not allow modalities is unexpectedly expressive. Kanovich [61] shows that the multiplicative fragment is np complete. 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 ....

M. Kanovich. Horn programming in linear logic is NP-complete. In Proc. 7-th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, California, pages 200--210. IEEE Computer Society Press, Los Alamitos, California, June 1992.

....cut free multiplicative proof may be guessed and checked in polynomial time. The decision problem is np hard by reduction from 3 Partition, a problem which requires a perfect partitioning of groups of objects in much the same way that linear logic requires a complete accounting of propositions [17, 18]. Somewhat surprisingly, there is an alternate encoding of 3 Partition in multiplicative linear logic that does not use any propositions, that is, using only the constants 1 and and the connectives Omega and Gammaffi. Thus this multiplicative constant only fragment of linear logic is also ....

....in this fragment has been studied in [5] but although expspace hard [31] Petri net reachability is known to be decidable [20] It is not known how much more expressive this fragment of linear logic might be. The fourth fragment containing only the multiplicative connectives is NP Complete [17, 18]. Connectives Linear Logic In Fragment Complexity Omega Phi Undecidable [26] Omega Phi PSPACE Complete [26] Omega unknown Omega NP Complete [17] In summary, linear logic is an expressive logic with an intrinsic accounting of resources. Although a non classical logic, ....

M. Kanovich. Horn programming in linear logic is np-complete. In Proc. 7-th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz. IEEE Computer Society Press, Los Alamitos, California, June 1992.

....encode constraints on the form of a cutfree proof in the conclusion sequent. Linear logic is therefore expressive in a manner that intuitionistic and classical logic are not. The classification of the complexity and decidability of fragments of linear logic highlights some of this expressiveness [25, 21]. Our embedding of the implicational fragment of propositional intuitionistic logic in the imall fragment of linear logic provides an alternative proof for the pspace hardness of imall. More importantly, it provides insight into the use and elimination of the structural rules from iil through the ....

M. Kanovich. Horn programming in linear logic is NP-complete. In Proc. 7-th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, California, pages 200--210. IEEE Computer Society Press, Los Alamitos, California, June 1992.

....search in fragments of linear logic. Provability in full propositional (i.e. quantifier free) linear logic is undecidable [18] However, even the fragment 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 ....

M. Kanovich. Horn programming in linear logic is NP-complete. In Proc. 7-th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, California, pages 200--210. IEEE Computer Society Press, Los Alamitos, California, June 1992.

.... search procedures in various fragments of linear logic (we refer to [5, 7] for backgrounds on linear logic) Knowing about some results on complexity for them (for example, Multiplicative Linear Logic MLL is NP complete and Multiplicative and Additive Linear Logic MALL is PSPACE complete) [8, 9], they are based on various approaches as resolution [10] tableaux methods [2, 6] or canonical proofs [4, 5] Even if the later one allows to reduce some non determinism sources, through particular proof forms, it does not propose a good management of the additive connectives [4] Thus, to ....

M. I. Kanovich, Horn Programming in Linear Logic is NP-Complete, In 7 th Logic in Computer Science, IEEE Press, 200-210, 1992.

....4. 1) ffl First order multiplicative additive LL (no exponentials) is NEXPTIME hard [114] ffl Multiplicative additive LL (no exponentials and quantifiers) is PSPACE complete [112, 113] ffl Multiplicative LL and Horn LL are NP complete, which is proven by reduction to the 3 Partition problem [91, 92]. ffl Even severely limited parts of the fragments mentioned in the preceding item still remain NP complete [115] Severely limited means constant only (no propositional letters at all, but the multiplicative additive neutral elements are part of the language) or constant free, with only two ....

M. Kanovich. Horn programming in linear logic is NP-complete. In Logic in Computer Science (LICS'92), pages 200--210, Santa Cruz, CA, June 1992. IEEE Computer Society Press. Also University of Amsterdam ITLI Prepublication Series X-91-14.

....by Max Kanovich in electronic mail [10] Together with the earlier result [15] that the multiplicatives are in np, Kanovich s result showed that this decision problem is np complete. Kanovich later updated his argument to show that the Horn fragment of the multiplicatives is also np complete [11, 12], using a novel computational interpretation of this fragment of linear logic. This paper continues this trend by providing a proof that evaluating expressions in true, false, and, and or in multiplicative linear logic is np complete. That is, even without propositions, multiplicative linear logic ....

M. Kanovich. Horn programming in linear logic is np-complete. In Proc. 7-th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, pages 200--210. IEEE Computer Society Press, Los Alamitos, California, June 1992.

....Actually [4] characterized exactly the computational complexity of ALL Theorem 3. The decision problem for ALL is NC 1 Complete under many one, D log time reductions. The result stems on [2, 3] and is based on a game semantics which could be implemented on parallel machine. See also [12, 13] for complexity results about another fragments) However, it seems interesting to have a feasible sequential procedure easily implementable as a module of a general theorem prover, to speed up the proof search process. We now present a polynomial time algorithm to build a proof of E F if there ....

M. I. Kanovich, Horn Programming in Linear Logic is NP-Complete, In 7 th Symposium on Logic in Computer Science , IEEE Press, 200-210, 1992.

..... B using A GammaffiB (which may be read right to left as the definition of the connective Gammaffi) Using the connective Gammaffi one can define other more specific fragments such as the Horn fragment of MLL [25], but these results will be largely omitted here. In order to gain an intuition about provability, we will usually be speaking informally of a computational process searching for a proof of a formula from the bottom up in a sequent calculus. Thus given a conclusion sequent, we attempt to find its ....

.... (note that every connective is analyzed exactly once in any cut free MLL proof) The decision problem is np hard by reduction from 3 Partition, a problem which requires a perfect partitioning of groups of objects in much the same way that linear logic requires a complete accounting of propositions [23, 24, 25]. The proof of correctness of the encoding makes heavy use of the balanced property of MLL, which states that if a formula is provable in MLL, then the number of positive and negative occurrences of each literal are equal. This property can be used as a necessary condition to provability in MLL ....

M. Kanovich. Horn programming in linear logic is NP-complete. In Proc. 7-th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, California, pages 200--210. IEEE Computer Society Press, Los Alamitos, California, June 1992.

....application of these rules where the instantiation of the variables is stopped until the axioms application. We have proposed a simple example, for which we have knowledge about expressiveness and proof search, but we can apply this analysis method to different fragments of linear logic as in [15, 16, 17], considered as basis for linear logic programming. 8 Related and further work This work on proof normalization in linear logic presents similarities and differences with other works on various fragments of LL, mainly those focusing on extensions of logic programming. The study of permutability ....

....on CLL fragments which appears as a specific rule defined mainly from an analysis of the permutation properties as but with a bottom up proof direction. Finally, we cannot forget to mention the relationship with the fundamental results about complexity and decidability in LL. Kanovich s works [17] aim to develop a computational interpretation of the logic and to obtain efficient decision algorithms based on a bottom up approach. To do it, he considers the Horn fragment of LL from a computational and a logical point of view and then generalizes the approach by introduction of the additives ....

M.I. Kanovich. Horn programming in linear logic is NP-complete. In 7th IEEE Symposium on Logic in Computer Science, pages 200--210, Santa-Cruz, California, June 1992.

.... is decidable; it is a special case of propositional linear calculus with every formula preceded by ) The complexity results for fragments of Linear Logic are also not very encouraging; for example the multiplicative fragment Omega ; which corresponds to Horn clause programming, is np complete [49, 50]. During goal directed proof search inference rules are used backwards (the conclusion is given and we are searching for the premises) The specific proof theoretic properties of Linear Logic justify that one can apply initially only the four pure logic rules listed above until the goal is split ....

M. Kanovich. Horn programming in linear logic is NP-complete. In Logic in Computer Science (LICS'92), pages 200--210, Santa Cruz, CA, June 1992. IEEE Computer Society Press. Also University of Amsterdam ITLI Prepublication Series X-91-14.

....computational paradigm is also indicated by the complexity and undecidability of provability in fragments of linear logic. These results are consequences of direct, lockstep simulations of computations on generic machines. Provability of purely multiplicative propositional formulas is np complete [6]. In fact, the decision problem for constant only formulas in multiplicative propositional linear logic is also np complete [10] If the additives are also allowed, provability for propositional formulas is pspace complete [7] Finally, if the modalities are also allowed, full propositional (i.e. ....

M. Kanovich. Horn programming in linear logic is NP-complete. In Proc. 7-th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, California, pages 200--210. IEEE Computer Society Press, Los Alamitos, California, June 1992.

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Kanovich, M., Horn programming in linear logic is NP-complete,in:Proccedings of the 7th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz (1992), pp. 200--210.

No context found.

Max Kanovich. Horn programming in linear logic is NP-complete. In Proccedings of the 7th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, pages 200--210, Los Alamitos, California, 1992. IEEE Computer Society Press.

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M. Kanovich. Horn programming in linear logic is NP-complete. In Symposium on Logic in Computer Science (LICS), pages 200210. IEEE, 1992.

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M. Kanovich. Horn programming in linear logic is np-complete. In Proc. 7-th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, pages 200--210. IEEE Computer Society Press, Los Alamitos, California, June 1992.

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M. Kanovich. Horn programming in linear logic is NPcomplete. In Symposium on Logic in Computer Science (LICS), pp. 200--210. IEEE, 1992.

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M. Kanovich. Horn programming in linear logic is NP-complete. In Proc. 7-th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, California, pages 200--210. IEEE Computer Society Press, Los Alamitos, California, June 1992.

No context found.

Max Kanovich. Horn programming in linear logic is NP-complete. In Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science, pages 200-210. IEEE Computer Society Press, June 1992.

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Max I. Kanovich. Horn programming in linear logic is NP-complete. In Seventh Annual Symposium on Logic in Computer Science, pages 200--210, Santa Cruz, California, June 1992. IEEE Computer Society Press.

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Max I. Kanovich. Horn programming in linear logic is NP-complete. In Seventh Annual Symposium on Logic in Computer Science, pages 200--210, Santa Cruz, California, June 1992. IEEE Computer Society Press. 37

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Max I. Kanovich. 1992. Horn programming in linear logic is NP-complete. In Seventh Annual IEEE Symposium on Logic in Computer Science, pages 200--210, Los Alamitos, California.

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