New problems complete for nondeterministic log space

Jones, Neil D.; Lien, Yao-Nan; Laaser, William T.

doi:10.1007/bf01683259

Cited by 124 publications

(61 citation statements)

References 6 publications

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“…On the other hand, Jones, Lien and Laaser [8] showed that GenSubSg is complete for NLOGSPACE. Now let A, F, h be an instance of Gen-Clo and let A be the algebra A, F .…”

Section: Proof Clearly a Triple A F H Is A Member Of Gen-clo If Amentioning

confidence: 99%

Computational Complexity of Term-Equivalence

Bergman

Juedes

Slutzki

1999

Int. J. Algebra Comput.

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Abstract. Two algebraic structures with the same universe are called term-equivalent if they have the same clone of term operations. We show that the problem of determining whether two finite algebras of finite similarity type are term-equivalent is complete for deterministic exponential time.When are two algebraic structures (presumably of different similarity types) considered to be "the same", for all intents and purposes? This is the notion of 'term-equivalence', a cornerstone of universal algebra.For example, the two-element Boolean algebra is generally thought of as the structure B = {0, 1}, ∧, ∨, ¬ , with the basic operations being conjunction, disjunction and negation. However, it is sometimes convenient to use instead the algebra B = {0, 1}, | whose only basic operation is the Sheffer stroke. Algebraically speaking, these two structures behave in exactly the same way. It is easy to transform B into B and back via the definitionsInformally, two algebras A and B are called term-equivalent if each of the basic operations of A can be built from the basic operations of B, and vice-versa. For a precise formulation, see Definition 3.3.Because of its fundamental role, it is natural to wonder about the computational complexity of determining term-equivalence. Specifically, given two finite algebras A and B of finite similarity type, are A and B termequivalent? There is a straightforward deterministic algorithm for solving this problem (Theorem 3.6), and its running time is exponential in the size of the input. In this paper we prove that this is the best possible: the termequivalence problem is complete for EXPTIME (Theorem 3.8). It follows from the Hierarchy Theorem of Hartmanis and Stearns [5] that the class EXPTIME is strictly larger than PTIME (the class of problems solvable

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“…On the other hand, Jones, Lien and Laaser [8] showed that GenSubSg is complete for NLOGSPACE. Now let A, F, h be an instance of Gen-Clo and let A be the algebra A, F .…”

Section: Proof Clearly a Triple A F H Is A Member Of Gen-clo If Amentioning

confidence: 99%

Computational Complexity of Term-Equivalence

Bergman

Juedes

Slutzki

1999

Int. J. Algebra Comput.

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“…In the following we investigate the computational complexity of the minimization problem for biautomata, and show that it is NL-complete, too. For proving NL-hardness we give a reduction from the following variant of the graph reachability problem [16,17] which is NL-complete, too.…”

Section: Computational Complexity Of (Hyper)-minimizationmentioning

confidence: 99%

Minimal and Hyper-Minimal Biautomata

Holzer¹,

Jakobi²

2014

Developments in Language Theory

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Abstract. We compare deterministic finite automata (DFAs) and biautomata under the following two aspects: structural similarities between minimal and hyper-minimal automata, and computational complexity of the minimization and hyper-minimization problem. Concerning classical minimality, the known results such as isomorphism between minimal DFAs, and NL-completeness of the DFA minimization problem carry over to the biautomaton case. But surprisingly this is not the case for hyper-minimization: the similarity between almostequivalent hyper-minimal biautomata is not as strong as it is between almost-equivalent hyper-minimal DFAs. Moreover, while hyper-minimization is NL-complete for DFAs, we prove that this problem turns out to be computationally intractable, i.e., NP-complete, for biautomata.

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“…Proof of (a) in Theorem 3 The Nlogspace-hardness of the reachability problem for SHA follows from the complexity of reachability problem for finite graphs [13]. On the other hand, the Nlogspace-hardness of the schedulability problem for SHA follows from the complexity of nonemptiness problem for Büchi automata (Proposition 10.12 of [17]).…”

Section: C1 Proofs From Sectionmentioning

confidence: 99%

Weak Singular Hybrid Automata

Krishna

Mathur

Trivedi

2014

Lecture Notes in Computer Science

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Abstract. The framework of Hybrid automata-introduced by Alur, Courcourbetis, Henzinger, and Ho-provides a formal modeling and analysis environment to analyze the interaction between the discrete and the continuous parts of hybrid systems. Hybrid automata can be considered as generalizations of finite state automata augmented with a finite set of real-valued variables whose dynamics in each state is governed by a system of ordinary differential equations. Moreover, the discrete transitions of hybrid automata are guarded by constraints over the values of these real-valued variables, and enable discontinuous jumps in the evolution of these variables. Singular hybrid automata are a subclass of hybrid automata where dynamics is specified by state-dependent constant vectors. Henzinger, Kopke, Puri, and Varaiya showed that for even very restricted subclasses of singular hybrid automata, the fundamental verification questions, like reachability and schedulability, are undecidable. Recently, Alur, Wojtczak, and Trivedi studied an interesting class of hybrid systems, called constant-rate multi-mode systems, where schedulability and reachability analysis can be performed in polynomial time. Inspired by the definition of constant-rate multi-mode systems, in this paper we introduce weak singular hybrid automata (WSHA), a previously unexplored subclass of singular hybrid automata, and show the decidability (and the exact complexity) of various verification questions for this class including reachability (NP-Complete) and LTL modelchecking (Pspace-Complete). We further show that extending WSHA with a single unrestricted clock or with unrestricted variable updates lead to undecidability of reachability problem.

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New problems complete for nondeterministic log space

Cited by 124 publications

References 6 publications

Computational Complexity of Term-Equivalence

Computational Complexity of Term-Equivalence

Minimal and Hyper-Minimal Biautomata

Weak Singular Hybrid Automata

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