Stability and Complexity of Minimising Probabilistic Automata

We consider the state-minimisation problem for weighted and probabilistic automata. We provide a numerically stable polynomial-time minimisation algorithm for weighted automata, with guaranteed bounds on the numerical error when run with floating-point arithmetic. Our algorithm can also be used for "lossy" minimisation with bounded error. We show an application in image compression. In the second part of the paper we study the complexity of the minimisation problem for probabilistic automata. We prove that the problem is NP-hard and in PSPACE, improving a recent EXPTIME-result.Comment: This is the full version of an ICALP'14 pape

Minimisation of Multiplicity Tree Automata

We consider the problem of minimising the number of states in a multiplicity tree automaton over the field of rational numbers. We give a minimisation algorithm that runs in polynomial time assuming unit-cost arithmetic. We also show that a polynomial bound in the standard Turing model would require a breakthrough in the complexity of polynomial identity testing by proving that the latter problem is logspace equivalent to the decision version of minimisation. The developed techniques also improve the state of the art in multiplicity word automata: we give an NC algorithm for minimising multiplicity word automata. Finally, we consider the minimal consistency problem: does there exist an automaton with $n$ states that is consistent with a given finite sample of weight-labelled words or trees? We show that this decision problem is complete for the existential theory of the rationals, both for words and for trees of a fixed alphabet rank.Comment: Paper to be published in Logical Methods in Computer Science. Minor editing changes from previous versio

Finite-automaton aperiodicity is PSPACE-complete

AbstractIn this paper, we solve an open problem raised by Stern (1985) — “Is finite-automaton aperiodicity PSPACE-complete?” — by providing an affirmative answer. We also characterize the exact complexity of two other problems considered by Stern: (1) dot-depth-one language recognition and (2) piecewise testable language recognition. We show that these two problems are logspace- complete for NL (the class of languages accepted by nondeterministic logspace-bounded Turing machines

DFA Minimization Algorithms in Map-Reduce

Map-Reduce has been a highly popular parallel-distributed programming model. In this thesis, we study the problem of minimizing Deterministic Finite State Automata (DFA). We focus our attention on two well-known (serial) algorithms, namely the algorithms of Moore (1956) and of Hopcroft (1971). The central cost-parameter in Map-Reduce is that of communication cost i.e., the amount of data that has to be communicated between the processes. Using techniques from Communication Complexity we derive an O(kn log{n}) lower bound and O(kn^3 log{n}) upper bound for the problem, where n is the number of states in the DFA to be minimized,and k is the size of its alphabet. We then develop Map-Reduce versions of both Moore's and Hopcroft's algorithms, and show that their communication cost is O(kn^2 (log {n} + log {k})). Both methods have been implemented and tested on large DFA, with 131,072 states. The experiments verify our theoretical analysis, and also reveal that Hopcroft's algorithm -- considered superior in the sequential framework -- is very sensitive to skew in the topology of the graph of the DFA, whereas Moore's algorithm handles skew without major efficiency loss

The parallel complexity of finite-state automata problems

AbstractThe goal of this paper is to study the exact complexity of several important problems concerning finite-state automata and to classify the degrees of ambiguity of nondeterministic finite-state automata. Our results are as follows: (1) Minimization of deterministic finite automata is NC1-complete for NL. (2) Testing whether the degree of ambiguity of a nondeterministic finite automaton is exponential, or polynomial, or bounded is NC1-complete for NL. (3) Checking whether a given nondeterministic finite automaton is unambiguous or k-ambiguous is NC1-complete for NL, where k is some fixed constant. (4) The bounded nonuniversality problem for nondeterministic finite automata (which is the problem of deciding whether L(M) ⋔ Σ≤n ≠ Σ≤n for a given nondeterministic finite automaton M and a unary integer n) is log-space complete for NP. (5) The bounded nonuniversality problem for unambiguous finite automata is in DET (the class of problems NC1-reducible to computing the determinants of integer matrices), and for deterministic finite automata, it is NC1-complete for NL. (6) The inequivalence problems for unambiguous and k-ambiguous finite automata are both in DET, where k is some fixed constant