5,286 research outputs found

    Highly Undecidable Problems For Infinite Computations

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    We show that many classical decision problems about 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are Π21\Pi_2^1-complete, hence located at the second level of the analytical hierarchy, and "highly undecidable". In particular, the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, and the unambiguity problem are all Π21\Pi_2^1-complete for context-free omega-languages or for infinitary rational relations. Topological and arithmetical properties of 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are also highly undecidable. These very surprising results provide the first examples of highly undecidable problems about the behaviour of very simple finite machines like 1-counter automata or 2-tape automata.Comment: to appear in RAIRO-Theoretical Informatics and Application

    Synchronizing weighted automata

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    We introduce two generalizations of synchronizability to automata with transitions weighted in an arbitrary semiring K=(K,+,*,0,1). (or equivalently, to finite sets of matrices in K^nxn.) Let us call a matrix A location-synchronizing if there exists a column in A consisting of nonzero entries such that all the other columns of A are filled by zeros. If additionally all the entries of this designated column are the same, we call A synchronizing. Note that these notions coincide for stochastic matrices and also in the Boolean semiring. A set M of matrices in K^nxn is called (location-)synchronizing if M generates a matrix subsemigroup containing a (location-)synchronizing matrix. The K-(location-)synchronizability problem is the following: given a finite set M of nxn matrices with entries in K, is it (location-)synchronizing? Both problems are PSPACE-hard for any nontrivial semiring. We give sufficient conditions for the semiring K when the problems are PSPACE-complete and show several undecidability results as well, e.g. synchronizability is undecidable if 1 has infinite order in (K,+,0) or when the free semigroup on two generators can be embedded into (K,*,1).Comment: In Proceedings AFL 2014, arXiv:1405.527

    On equivalence, languages equivalence and minimization of multi-letter and multi-letter measure-many quantum automata

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    We first show that given a k1k_1-letter quantum finite automata A1\mathcal{A}_1 and a k2k_2-letter quantum finite automata A2\mathcal{A}_2 over the same input alphabet ÎŁ\Sigma, they are equivalent if and only if they are (n12+n22−1)∣Σ∣k−1+k(n_1^2+n_2^2-1)|\Sigma|^{k-1}+k-equivalent where n1n_1, i=1,2i=1,2, are the numbers of state in Ai\mathcal{A}_i respectively, and k=max⁥{k1,k2}k=\max\{k_1,k_2\}. By applying a method, due to the author, used to deal with the equivalence problem of {\it measure many one-way quantum finite automata}, we also show that a k1k_1-letter measure many quantum finite automaton A1\mathcal{A}_1 and a k2k_2-letter measure many quantum finite automaton A2\mathcal{A}_2 are equivalent if and only if they are (n12+n22−1)∣Σ∣k−1+k(n_1^2+n_2^2-1)|\Sigma|^{k-1}+k-equivalent where nin_i, i=1,2i=1,2, are the numbers of state in Ai\mathcal{A}_i respectively, and k=max⁥{k1,k2}k=\max\{k_1,k_2\}. Next, we study the language equivalence problem of those two kinds of quantum finite automata. We show that for kk-letter quantum finite automata, the non-strict cut-point language equivalence problem is undecidable, i.e., it is undecidable whether L≄λ(A1)=L≄λ(A2)L_{\geq\lambda}(\mathcal{A}_1)=L_{\geq\lambda}(\mathcal{A}_2) where 0<λ≀10<\lambda\leq 1 and Ai\mathcal{A}_i are kik_i-letter quantum finite automata. Further, we show that both strict and non-strict cut-point language equivalence problem for kk-letter measure many quantum finite automata are undecidable. The direct consequences of the above outcomes are summarized in the paper. Finally, we comment on existing proofs about the minimization problem of one way quantum finite automata not only because we have been showing great interest in this kind of problem, which is very important in classical automata theory, but also due to that the problem itself, personally, is a challenge. This problem actually remains open.Comment: 30 pages, conclusion section correcte

    A Survey on Continuous Time Computations

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    We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

    Conjugacy of one-dimensional one-sided cellular automata is undecidable

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    Two cellular automata are strongly conjugate if there exists a shift-commuting conjugacy between them. We prove that the following two sets of pairs (F,G)(F,G) of one-dimensional one-sided cellular automata over a full shift are recursively inseparable: (i) pairs where FF has strictly larger topological entropy than GG, and (ii) pairs that are strongly conjugate and have zero topological entropy. Because there is no factor map from a lower entropy system to a higher entropy one, and there is no embedding of a higher entropy system into a lower entropy system, we also get as corollaries that the following decision problems are undecidable: Given two one-dimensional one-sided cellular automata FF and GG over a full shift: Are FF and GG conjugate? Is FF a factor of GG? Is FF a subsystem of GG? All of these are undecidable in both strong and weak variants (whether the homomorphism is required to commute with the shift or not, respectively). It also immediately follows that these results hold for one-dimensional two-sided cellular automata.Comment: 12 pages, 2 figures, accepted for SOFSEM 201

    Decidable and undecidable problems about quantum automata

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    We study the following decision problem: is the language recognized by a quantum finite automaton empty or non-empty? We prove that this problem is decidable or undecidable depending on whether recognition is defined by strict or non-strict thresholds. This result is in contrast with the corresponding situation for probabilistic finite automata for which it is known that strict and non-strict thresholds both lead to undecidable problems.Comment: 10 page

    Computational Processes and Incompleteness

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    We introduce a formal definition of Wolfram's notion of computational process based on cellular automata, a physics-like model of computation. There is a natural classification of these processes into decidable, intermediate and complete. It is shown that in the context of standard finite injury priority arguments one cannot establish the existence of an intermediate computational process

    Multi-Head Finite Automata: Characterizations, Concepts and Open Problems

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    Multi-head finite automata were introduced in (Rabin, 1964) and (Rosenberg, 1966). Since that time, a vast literature on computational and descriptional complexity issues on multi-head finite automata documenting the importance of these devices has been developed. Although multi-head finite automata are a simple concept, their computational behavior can be already very complex and leads to undecidable or even non-semi-decidable problems on these devices such as, for example, emptiness, finiteness, universality, equivalence, etc. These strong negative results trigger the study of subclasses and alternative characterizations of multi-head finite automata for a better understanding of the nature of non-recursive trade-offs and, thus, the borderline between decidable and undecidable problems. In the present paper, we tour a fragment of this literature
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