Results 1  10
of
134
Decision Problems for Propositional Linear Logic
, 1990
"... Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. We show that unlike most other propositional (quantifierfree) logics, full propositional linear logic is undecidable. Further, we prove that without the modal storage operator, ..."
Abstract

Cited by 111 (19 self)
 Add to MetaCart
Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. We show that unlike most other propositional (quantifierfree) logics, full propositional linear logic is undecidable. Further, we prove that without the modal storage operator, which indicates unboundedness of resources, the decision problem becomes pspacecomplete. We also establish membership in np for the multiplicative fragment, npcompleteness for the multiplicative fragment extended with unrestricted weakening, and undecidability for certain fragments of noncommutative propositional linear logic. 1 Introduction Linear logic, introduced by Girard [14, 18, 17], is a refinement of classical logic which may be derived from a Gentzenstyle sequent calculus axiomatization of classical logic in three steps. The resulting sequent system Lincoln@CS.Stanford.EDU Department of Computer Science, Stanford University, Stanford, CA 94305, and the Computer Science Labo...
Typability and Type Checking in System F Are Equivalent and Undecidable
 ANNALS OF PURE AND APPLIED LOGIC
, 1998
"... Girard and Reynolds independently invented System F (a.k.a. the secondorder polymorphically typed lambda calculus) to handle problems in logic and computer programming language design, respectively. Viewing F in the Curry style, which associates types with untyped lambda terms, raises the questions ..."
Abstract

Cited by 70 (5 self)
 Add to MetaCart
(Show Context)
Girard and Reynolds independently invented System F (a.k.a. the secondorder polymorphically typed lambda calculus) to handle problems in logic and computer programming language design, respectively. Viewing F in the Curry style, which associates types with untyped lambda terms, raises the questions of typability and type checking. Typability asks for a term whether there exists some type it can be given. Type checking asks, for a particular term and type, whether the term can be given that type. The decidability of these problems has been settled for restrictions and extensions of F and related systems and complexity lowerbounds have been determined for typability in F, but this report is the first to resolve whether these problems are decidable for System F. This report proves that type checking in F is undecidable, by a reduction from semiunification, and that typability in F is undecidable, by a reduction from type checking. Because there is an easy reduction from typability to type checking, the two problems are equivalent. The reduction from type checking to typability uses a novel method to construct lambda terms that simulate arbitrarily chosen type environments. All the results also hold for the lambdaIotacalculus.
Computation with finite stochastic chemical reaction networks
 Natural Computing
, 2008
"... Abstract. A highly desired part of the synthetic biology toolbox is an embedded chemical microcontroller, capable of autonomously following a logic program specified by a set of instructions, and interacting with its cellular environment. Strategies for incorporating logic in aqueous chemistry have ..."
Abstract

Cited by 49 (14 self)
 Add to MetaCart
(Show Context)
Abstract. A highly desired part of the synthetic biology toolbox is an embedded chemical microcontroller, capable of autonomously following a logic program specified by a set of instructions, and interacting with its cellular environment. Strategies for incorporating logic in aqueous chemistry have focused primarily on implementing components, such as logic gates, that are composed into larger circuits, with each logic gate in the circuit corresponding to one or more molecular species. With this paradigm, designing and producing new molecular species is necessary to perform larger computations. An alternative approach begins by noticing that chemical systems on the small scale are fundamentally discrete and stochastic. In particular, the exact molecular counts of each molecular species present, is an intrinsically available form of information. This might appear to be a very weak form of information, perhaps quite difficult for computations to utilize. Indeed, it has been shown that errorfree Turing universal computation is impossible in this setting. Nevertheless, we show a design of a chemical computer that achieves fast and reliable Turinguniversal computation using molecular counts. Our scheme uses only a small number of different molecular species to do computation of arbitrary complexity. The total probability of error of the computation can be made arbitrarily small (but not zero) by adjusting the initial molecular counts of certain species. While physical implementations would be difficult, these results demonstrate that molecular counts can be a useful form of information for small molecular systems such as those operating within cellular environments. Key words. stochastic chemical kinetics; molecular counts; Turinguniversal computation; probabilistic computation 1. Introduction. Many
Decision Problems For Patterns
 Journal of Computer and System Sciences
, 1995
"... We settle an open problem, the inclusion problem for pattern languages [1, 2]. This is the first known case where inclusion is undecidable for generative devices having a trivially decidable equivalence problem. The study of patterns goes back to the seminal work of Thue [16] and is important also, ..."
Abstract

Cited by 33 (3 self)
 Add to MetaCart
(Show Context)
We settle an open problem, the inclusion problem for pattern languages [1, 2]. This is the first known case where inclusion is undecidable for generative devices having a trivially decidable equivalence problem. The study of patterns goes back to the seminal work of Thue [16] and is important also, for instance, in recent work concerning inductive inference and learning. Our results concern both erasing and nonerasing patterns. Categories and Subject Descriptors: F.4.3 [Mathematical Logic and Formal Languages ]: Formal Languages  Decision problems, Algebraic language theory; F.4.1 [Mathe matical Logic and Formal Languages]: Mathematical Logic  Computability theory. General Terms: Theory, Formal Languages Additional Key Words and Phrases: Patterns, Inclusion problems, Equivalence problems, Descriptive patterns, Unavoidable patterns 1 Introduction. The main result Instead of an exhaustive definition for a language, [7], it is sometimes better to give more leeway in the defi...
Decidability and complexity results for timed automata and semilinear hybrid automata
 In: Hybrid Systems: Computation and Control, 2000, LNCS
"... Abstract. We define a new class of hybrid automata for which reachability is decidable—a proper superclass of the initialized rectangular hybrid automata—by taking parallel compositions of simple components. Attempting to generalize, we encounter timed automata with algebraic constants. We show tha ..."
Abstract

Cited by 32 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We define a new class of hybrid automata for which reachability is decidable—a proper superclass of the initialized rectangular hybrid automata—by taking parallel compositions of simple components. Attempting to generalize, we encounter timed automata with algebraic constants. We show that reachability is undecidable for these algebraic timed automata by simulating twocounter Minsky machines. Modifying the construction to apply to parametric timed automata, we reprove the undecidability of the emptiness problem, and then distinguish the dense and discretetime cases with a new result. The algorithmic complexity— both classical and parametric—of oneclock parametric timed automata is also examined. We finish with a table of computabilitytheoretic complexity results, including that the existence of a Zeno run is Σ11complete for semilinear hybrid automata; it is too complex to be expressed in firstorder arithmetic. 1
Deciding Provability of Linear Logic Formulas
 Advances in Linear Logic
, 1994
"... Introduction There are many interesting fragments of linear logic worthy of study in their own right, most described by the connectives which they employ. Full linear logic includes all the logical connectives, which come in three dual pairs: the exponentials ! and ?, the additives & and \Phi, ..."
Abstract

Cited by 30 (0 self)
 Add to MetaCart
(Show Context)
Introduction There are many interesting fragments of linear logic worthy of study in their own right, most described by the connectives which they employ. Full linear logic includes all the logical connectives, which come in three dual pairs: the exponentials ! and ?, the additives & and \Phi, and the multiplicatives\Omega and . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ . SRI International Computer Science Laboratory, Menlo Park CA 94025 USA. Work supported under NSF Grant CCR9224858. lincoln@csl.sri.com http://www.csl.sri.com/lincoln/lincoln.html Patrick Lincoln For the most part we will consider fragments of linear logic built up using these connectives in any combination. For example, full linear logic formulas may employ any connective, while multiplic
Decision Problems for Pushdown Threads
 Programming Research Group, University of Amsterdam
, 2005
"... Abstract. Threads as contained in a thread algebra emerge from the behavioral abstraction from programs in an appropriate program algebra. Threads may make use of services such as stacks, and a thread using a single stack is called a pushdown thread. Equivalence of pushdown threads is shown decidabl ..."
Abstract

Cited by 29 (26 self)
 Add to MetaCart
(Show Context)
Abstract. Threads as contained in a thread algebra emerge from the behavioral abstraction from programs in an appropriate program algebra. Threads may make use of services such as stacks, and a thread using a single stack is called a pushdown thread. Equivalence of pushdown threads is shown decidable whereas pushdown thread inclusion is undecidable. This is again an example of a borderline crossing where the equivalence problem is decidable, whereas the inclusion problem is not. 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 ..."
Abstract

Cited by 25 (6 self)
 Add to MetaCart
(Show Context)
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
Linear Logic
, 1992
"... this paper we will restrict attention to propositional linear logic. The sequent calculus notation, due to Gentzen [10], uses roman letters for propositions, and greek letters for sequences of formulas. A sequent is composed of two sequences of formulas separated by a `, or turnstile symbol. One may ..."
Abstract

Cited by 23 (1 self)
 Add to MetaCart
(Show Context)
this paper we will restrict attention to propositional linear logic. The sequent calculus notation, due to Gentzen [10], uses roman letters for propositions, and greek letters for sequences of formulas. A sequent is composed of two sequences of formulas separated by a `, or turnstile symbol. One may read the sequent \Delta ` \Gamma as asserting that the multiplicative conjunction of the formulas in \Delta together imply the multiplicative disjunction of the formulas in \Gamma. A sequent calculus proof rule consists of a set of hypothesis sequents, displayed above a horizontal line, and a single conclusion sequent, displayed below the line, as below: Hypothesis1 Hypothesis2 Conclusion 4 Connections to Other Logics
On Showing Lower Bounds for ExternalMemory Computational Geometry Problems
"... . In this paper we consider lower bounds for externalmemory computational geometry problems. We find that it is not quite clear which model of computation to use when considering such problems. As an attempt of providing a model, we define the external memory Turing machine model, and we derive low ..."
Abstract

Cited by 22 (4 self)
 Add to MetaCart
. In this paper we consider lower bounds for externalmemory computational geometry problems. We find that it is not quite clear which model of computation to use when considering such problems. As an attempt of providing a model, we define the external memory Turing machine model, and we derive lower bounds for a number of problems, including the element distinctness problem, in this model. For these lower bounds we make the standard assumption that records are indivisible. Waiving the indivisibility assumption we show how to beat the lower bound for element distinctness. As an alternative model, we briefly discuss an externalmemory version of the algebraic computation tree. 1. Introduction The Input/Output (or just I/O) communication between fast internal memory and slower external storage is the bottleneck in many largescale computations. The significance of this bottleneck is increasing as internal computation gets faster, and as parallel computation gains popularity. Currently,...