We show that the problem of checking careful synchronizability of partial finite automata is PSPACE-complete. Also the problems of checking D 1-, D 2-, and D 3-directability of nondeterministic finite automata are PSPACE-complete; moreover, the restrictions of all these problems to automata with two input letters remain PSPACE-complete.

Computational Complexity of Certain Problems Related to Carefully Synchronizing Words for Partial Automata and Directing Words for Nondeterministic Automata

We show that the problem of checking careful synchronizability of partial finite automata and the problem of finding the shortest carefully synchronizing word are PSPACE-complete. We show that the problem of checking D1, D2 and D3-directability of nondeterministic finite automata and the problem of finding the shortest D1, D2 and D3-directing word are PSPACE-complete. The restrictions of these problems to 2-letter automata remain PSPACE-complete.

Complexity of problems concerning carefully synchronizing words for PFA and directing words for NFA

We consider the careful synchronization of partial automata with only one undefined transition and the generalized synchronization of nondeterministic automata with only one ambiguous transition. For each of the two cases we prove that the problem of checking whether or not a given automaton is synchronizable is PSPACE-complete. The restrictions of these problems to 2-letter automata are also PSPACE-complete.

Synchronization of automata with one undefined or ambiguous transition

A set of nonnegative matrices $\mathcal{M}=\{M_1, M_2, \ldots, M_k\}$ is called primitive if there exist indices $i_1, i_2, \ldots, i_m$ such that $M_{i_1} M_{i_2} \ldots M_{i_m}$ is positive (i.e. has all its entries $>0$). The length of the shortest such product is called the exponent of $\mathcal{M}$. The concept of primitive sets of matrices comes up in a number of problems within control theory, non-homogeneous Markov chains, automata theory etc. Recently, connections between synchronizing automata and primitive sets of matrices were established. In the present paper, we significantly strengthen these links by providing equivalence results, both in terms of combinatorial characterization, and computational aspects. We study the maximal exponent among all primitive sets of $n \times n$ matrices, which we denote by $\exp(n)$. We prove that $\lim_{n\rightarrow\infty} \tfrac{\log \exp(n)}{n} = \tfrac{\log 3}{3}$, and moreover, we establish that this bound leads to a resolution of the Cerný problem for carefully synchronizing automata. We also study the set of matrices with no zero rows and columns, denoted by $\mathcal{NZ}$, due to its intriguing connections to the Cerný conjecture and the recent generalization of Perron-Frobenius theory for this class. We characterize computational complexity of different problems related to the exponent of $\mathcal{NZ}$ matrix sets, and present a quadratic bound on the exponents of sets belonging to a special subclass. Namely, we show that the exponent of a set of matrices having total support is bounded by $2n^2 -5n +5$.

Primitive sets of nonnegative matrices and synchronizing automata

A set of nonnegative matrices $\mathcal{M}=\{M_1, M_2, \ldots, M_k\}$ is called primitive if there exist possibly equal indices $i_1, i_2, \ldots, i_m$ such that $M_{i_1} M_{i_2} \cdots M_{i_m}$ is entrywise positive. The length of the shortest such product is called the exponent of $\mathcal{M}$. Recently, connections between synchronizing automata and primitive sets of matrices were established. In the present paper, we strengthen these links by providing equivalence results, both in terms of combinatorial characterization and computational complexity. We pay special attention to the set of matrices without zero rows and columns, denoted by $\mathscr{NZ}$, due to its intriguing connections to the Cerný conjecture. We rely on synchronizing automata theory to derive a number of results about primitive sets of matrices. Making use of an asymptotic estimate by Rystsov [Cybernetics, 16 (1980), pp. 194--198], we show that the maximal exponent $\exp(n)$ of primitive sets of $n \times n$ matrices satisfy $\lim_...

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Primitive Sets of Nonnegative Matrices and Synchronizing Automata

Improved upper bounds on synchronizing nondeterministic automata

In this paper we give two families of P systems with active membranes that can solve the satisfiability problem of propositional formulas in linear time in the number of propositional variables occurring in the input formula. These solutions do not use polarizations of the membranes or non-elementary membrane division but use separation rules with relabeling. The first solution is a uniform one, but it is not polynomially uniform. The second solution, which is based on the first one, is a polynomially semi-uniform solution.

A new approach for solving SAT by p systems with active membranes

In this paper we solve the SAT problem (the satisfiability problem of propositional formulas in conjunctive normal form) by two polynomially uniform families of P systems with active membranes. The novelty of these solutions is that these P systems can solve the SAT problem in linear time in the number of propositional variables occurring in the input. This means that the number of computation steps is independent form the number of clauses of the input. To achieve this efficiency our systems employ only the standard rules of P systems with active membranes plus membrane creation rules. Moreover, in the first solution the P systems do not use the polarizations of the membranes but use such membrane division rules which can change the labels of the involved membranes. In the second solution the P systems do not employ membrane label changing but use the polarizations of the membranes instead.

Solving SAT by P Systems with Active Membranes in Linear Time in the Number of Variables

We prove that every single-tape deterministic Turing machine working in $t(n)$ time, for some function $t:\mathbb {N}\rightarrow \mathbb {N}$, can be simulated by a uniform family of polarizationless P systems with active membranes. Moreover, this is done without significant slowdown in the working time. Furthermore, if $\log t(n)$ is space constructible, then the members of the uniform family can be constructed by a family machine that uses $O(\log t(n))$ space.

Simulating Turing Machines with Polarizationless P Systems with Active Membranes

In Membrane Computing, the solution of a decision problem $X$ belonging to the complexity class P via a polynomially uniform family of recognizer P systems is trivial, since the polynomial encoding of the input can involve the solution of the problem. The design of such solution has one membrane, two objects, two rules and one computation step. Stricto sensu, it is a solution in the framework of Membrane Computing, but it does not use Membrane Computing strategies. In this paper, we present three designs of uniform families of P systems that solve the decision problem STCON by using Membrane Computing strategies (pure Membrane Computing techniques): P systems with membrane creation, P systems with active membranes with dissolution and without polarizations and P systems with active membranes without dissolution and with polarizations. Since STCON is NL-complete, such designs are constructive proofs of the inclusion of NL in $\mathbf{PMC}_\mathcal{MC}$, $\mathbf{PMC}_{\mathcal{AM}^0_{+d}}$ and $\mathbf{PMC}_{\mathcal{AM}^+_{-d}}$.

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Papers

Improved upper bounds on synchronizing nondeterministic automata

A new approach for solving SAT by p systems with active membranes

Solving SAT by P Systems with Active Membranes in Linear Time in the Number of Variables

Simulating Turing Machines with Polarizationless P Systems with Active Membranes

Solving the ST-Connectivity Problem with Pure Membrane Computing Techniques