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The Undecidability of Second Order Multiplicative Linear Logic
, 1996
"... The multiplicative fragment of second order propositional linear logic is shown to be undecidable. Introduction Decision problems for propositional (quantifierfree) linear logic were first studied by Lincoln et al. [LMSS]. In referring to linear logic fragments, let M stand for multiplicatives, A ..."
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The multiplicative fragment of second order propositional linear logic is shown to be undecidable. Introduction Decision problems for propositional (quantifierfree) linear logic were first studied by Lincoln et al. [LMSS]. In referring to linear logic fragments, let M stand for multiplicatives, A for additives, E for exponentials (or modalities), 1 for first order quantifiers, 2 for second order propositional quantifiers, and I for "intuitionistic" version. In [LMSS] it was shown that full propositional linear logic is undecidable and that MALL is PSPACEcomplete. The main problems left open in [LMSS] were the NPcompleteness of MLL, the decidability 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, but almost all the other problems have been solved: ffl The NPcompleteness of MLL has been obtained by Kanovich [K1]. Moreover, Linco...
Typability and Type Checking in the SecondOrder lambdaCalculus Are Equivalent and Undecidable
, 1993
"... We consider the problems of typability and type checking in the Girard/Reynolds secondorder polymorphic typedcalculus, for which we use the short name "System F" and which we use in the "Curry style" where types are assigned to pureterms. These problems have been considere ..."
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We consider the problems of typability and type checking in the Girard/Reynolds secondorder polymorphic typedcalculus, for which we use the short name "System F" and which we use in the "Curry style" where types are assigned to pureterms. These problems have been considered and proven to be decidable or undecidable for various restrictions and extensions of System F and other related systems, and lowerbound complexity results for System F have been achieved, but they have remained "embarrassing open problems" 3 for System F itself. We first prove that type checking in System F is undecidable by a reduction from semiunification. We then prove typability in System F is undecidable by a reduction from type checking. Since the reverse reduction is already known, this implies the two problems are equivalent. The second reduction uses a novel method of constructingterms such that in all type derivations, specific bound variables must always be assigned a specific type. Using this technique, we can require that specif subterms must be typable using a specific, fixed type assignment in order for the entire term to be typable at all. Any desired type assignment maybe simulated. We develop this method, which we call \constants for free", for both the K and I calculi.
The Undecidability Of Second Order Linear Logic Without Exponentials
 Journal of Symbolic Logic
, 1995
"... . Recently, Lincoln, Scedrov and Shankar showed that the multiplicative fragment of second order intuitionistic linear logic is undecidable, using an encoding of second order intuitionistic logic. Their argument applies to the multiplicativeadditive fragment, but it does not work in the classical c ..."
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. Recently, Lincoln, Scedrov and Shankar showed that the multiplicative fragment of second order intuitionistic linear logic is undecidable, using an encoding of second order intuitionistic logic. Their argument applies to the multiplicativeadditive fragment, but it does not work in the classical case, because second order classical logic is decidable. Here we show that the multiplicativeadditive fragment of second order classical linear logic is also undecidable, using an encoding of twocounter machines originally due to Kanovich. The faithfulness of this encoding is proved by means of the phase semantics. In this paper, we write LL for the full propositional fragment of linear logic, MLL for the multiplicative fragment, MALL for the multiplicativeadditive fragment, and MELL for the multiplicativeexponential fragment. Similarly, we write ILL, IMLL, etc. for the fragments of intuitionistic linear logic, LL2, MLL2, etc. for the second order fragments of linear logic, and ILL2, IML...
Counter Machines and Verification Problems
"... We study various generalizations of reversalbounded multicounter machines and show that they have decidable emptiness, inniteness, disjointness, containment, and equivalence problems. The extensions include allowing the machines to perform linearrelation tests among the counters and parameterized ..."
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Cited by 11 (2 self)
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We study various generalizations of reversalbounded multicounter machines and show that they have decidable emptiness, inniteness, disjointness, containment, and equivalence problems. The extensions include allowing the machines to perform linearrelation tests among the counters and parameterized constants (e.g., \Is 3x 5y 2D 1 +9D 2 < 12?", where x; y are counters, and D 1 ; D 2 are parameterized constants). We believe that these machines are the most powerful machines known to date for which these decision problems are decidable. Decidability results for such machines are useful in the analysis of reachability problems and the verication /debugging of safety properties in innitestate transition systems. For example, we show that (binary, forward, and backward) reachability and safety are solvable for these machines. Key words: Counter machines, automated verication, innitestate systems, reachability 1 A short version [15] of this paper appeared in the Proceedings of the 25th International Symposium on Mathematical Foundations of Computer Science (MFCS 2000), Lecture Notes in Computer Science 1893 (Springer, Berlin, 2000) 426435. 2 The work by Oscar H. Ibarra and Jianwen Su has been supported in part by NSF grants IRI9700370 and IIS9817432. 3 The work by Zhe Dang and Richard A. Kemmerer has been supported in part by the Defense Advanced Research Projects Agency (DARPA) and Rome Laboratory, Air Force Material Command, USAF, under agreement number F306029710207. 4 The work by Tevk Bultan has been supported in part by NSF grant CCR9970976 and NSF CAREER award CCR9984822. Preprint submitted to Elsevier Science 17 April 2001 1
On P systems operating in sequential mode
 International Journal of Foundations of Computer Science
, 2004
"... 1. For 1membrane catalytic systems (CS's), the sequential version is strictlyweaker than the parallel version in that the former defines (i.e. generates) exactly the semilinear sets, whereas the latter is known to define nonrecursivesets. 2. For 1membrane communicating P systems (CPS's), ..."
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1. For 1membrane catalytic systems (CS's), the sequential version is strictlyweaker than the parallel version in that the former defines (i.e. generates) exactly the semilinear sets, whereas the latter is known to define nonrecursivesets. 2. For 1membrane communicating P systems (CPS's), the sequential versioncan only define a proper subclass of the semilinear sets, whereas the parallel version is known to define nonrecursive sets.3. Adding a new type of rule of the form: ab! axbyccomedcome to the CPS(a natural generalization of the rule ab! axbyccome in the original model),where x; y 2 fhere; outg, to the sequential 1membrane CPS makes itequivalent to a vector addition system.
On the Decidability of Modelchecking for P Systems
 JOURNAL OF AUTOMATA, LANGUAGES AND COMBINATORICS
"... Membrane computing is a branch of molecular computing that aims to develop models and paradigms that are biologically motivated. It identifies an unconventional computing model, namely a P system, from natural phenomena of cell evolutions and chemical reactions. Because of the nature of maximal para ..."
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Membrane computing is a branch of molecular computing that aims to develop models and paradigms that are biologically motivated. It identifies an unconventional computing model, namely a P system, from natural phenomena of cell evolutions and chemical reactions. Because of the nature of maximal parallelism inherent in the model, P systems have a great potential for implementing massively concurrent systems in an efficient way that would allow us to solve currently intractable problems. In this paper, we look at various models of P systems and investigate their modelchecking problems. We identify what is decidable (or undecidable) about modelchecking these systems under extended logic formalisms of CTL. We also report on some experiments on whether existing conservative (symbolic) modelchecking techniques can be practically applied to handle P systems with a reasonable size.
When Ambients Cannot be Opened
 In Proceedings of FoSSaCS 2003
, 2003
"... We investigate expressiveness of a fragment of the ambient calculus, a formalism for describing distributed and mobile computations. More precisely, we study expressiveness of the pure and public ambient calculus from which the has been removed, in terms of the reachability problem of the reduct ..."
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We investigate expressiveness of a fragment of the ambient calculus, a formalism for describing distributed and mobile computations. More precisely, we study expressiveness of the pure and public ambient calculus from which the has been removed, in terms of the reachability problem of the reduction relation. Surprisingly, we show that even for this very restricted fragment, the reachability problem is not decidable. At a second step, for a slightly weaker reduction relation, we prove that reachability can be decided by reducing this problem to markings reachability for Petri nets. Finally, we show that the nameconvergence problem as well as the modelchecking problem turn out to be undecidable for both the original and the weaker reduction relation. 1
Programmability of Chemical Reaction Networks
"... Summary. Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a formal model that considers a set of chemical reactions acting on a finite number of molecules in a wellstirred solution according to standard c ..."
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Summary. Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a formal model that considers a set of chemical reactions acting on a finite number of molecules in a wellstirred solution according to standard chemical kinetics equations. SCRNs have been widely used for describing naturally occurring (bio)chemical systems, and with the advent of synthetic biology they become a promising language for the design of artificial biochemical circuits. Our interest here is the computational power of SCRNs and how they relate to more conventional models of computation. We survey known connections and give new connections between SCRNs and
Twoway equational tree automata for AClike theories: Decidability and closure properties
 In Proc. 14th International Conference on Rewriting Techniques and Applications (RTA), volume 2706 of Lecture Notes in Computer Science
, 2003
"... Abstract. We study twoway tree automata modulo equational theories. We deal with the theories of Abelian groups), idempotent commutative monoids (¡£¢¨¤£©), and the theory of exclusiveor), as well as some variants including the theory of commutative monoids). We show that the oneway automata for a ..."
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Abstract. We study twoway tree automata modulo equational theories. We deal with the theories of Abelian groups), idempotent commutative monoids (¡£¢¨¤£©), and the theory of exclusiveor), as well as some variants including the theory of commutative monoids). We show that the oneway automata for all these theories are closed under union and intersection, and emptiness is decidable. For twoway automata the situation is more complex. In all these theories except¡£¢¨¤£©, we show that twoway automata can be effectively reduced to oneway automata, provided some care is taken in the definition