International Journal of Foundations of Computer Science

Let 2N be the class of families of problems solvable by families of two-way nondeterministic finite automata of polynomial size. We characterize 2N in terms of families of formulas of transitive-closure logic. These formulas apply the transitive-closure operator on a quantifier-free disjunctive normal form of first-order logic with successor and constants, where (i) apart from two special variables, all others are equated to constants in every clause, and (ii) no clause simultaneously relates these two special variables and refers to fixed input cells. We prove that automata with polynomially many states are as powerful as formulas with polynomially many clauses and polynomially large constants. This can be seen as a refinement of Immerman’s theorem that nondeterministic logarithmic space matches positive transitive-closure logic ($\mathrm{\text{NL}}=\mathrm{\text{FO}}+\mathrm{\text{posTC}}$).

We characterize the complexity of the PATS problem for patterns of fixed height and color count in variants of the model where seed glues are either chosen or fixed and identical (so-called non-uniform and uniform variants). We prove that both variants are NP-complete for patterns of height 2 or more and admit $O(n)$-time algorithms for patterns of height 1. We also prove that if the height and number of colors in the pattern is fixed, the non-uniform variant admits a $O(n)$-time algorithm while the uniform variant remains NP-complete. The NP-completeness results use a new reduction from a constrained version of a problem on finite state transducers.

It is one of the most famous open problems to determine the minimum amount of states required by a deterministic finite automaton to distinguish a pair of strings, which was stated by Christian Choffrut more than thirty years ago.

We investigate the same question for different automata models and we obtain new upper and lower bounds for some of them including alternating, ultrametric, quantum, and affine finite automata.

Recently, a method to decide the NL-complete problem of whether the language accepted by a given deterministic finite automaton (DFA) can also be accepted by some reversible deterministic finite automaton (REV-DFA) has been derived. Here, we show that the corresponding problem for nondeterministic finite automata (NFA) is PSPACE-complete. The recent DFA method essentially works by minimizing the DFA and inspecting it for a forbidden pattern. We here study the degree of irreversibility for a regular language, the minimal number of such forbidden patterns necessary in any DFA accepting the language, and show that the degree induces a strict infinite hierarchy of language families. We examine how the degree of irreversibility behaves under the usual language operations union, intersection, complement, concatenation, and Kleene star, showing tight bounds (some asymptotically) on the degree.

We show how to train a weighted regular tree grammar such that it best approximates a weighted linear context-free tree grammar concerning the Kullback–Leibler divergence between both grammars. Furthermore, we construct a regular tree grammar that approximates the tree language induced by a context-free tree grammar.

Regular expressions are often ambiguous. We present a novel method based on Brzozowski’s derivatives to aid the user in diagnosing ambiguous regular expressions. We introduce a derivative-based finite state transducer to generate parse trees and minimal counter-examples. The transducer can be easily customized to either follow the POSIX or Greedy disambiguation policy and based on a finite set of examples it is possible to examine if there are any differences between POSIX and Greedy.

This paper presents a practical algorithm for the uniform membership problem of labeled multidigraphs of tree-width at most 2 for spanning tree automata. Though the theoretical existence of a linear-time algorithm for the membership problem for graphs of bounded tree-width was shown in the previous study, the implementation of the linear-time algorithm is expected to be impractical because it requires the construction of a finite-state automaton whose size is super-exponential in the size of the tree automaton and the tree-width of graphs. In addition, the tree automaton itself should be a part of the input in practical situations.

We introduce and investigate stack transducers, which are one-way stack automata with an output tape. A one-way stack automaton is a classical pushdown automaton with the additional ability to move the stack head inside the stack without altering the contents. For stack transducers, we distinguish between a digging and a non-digging mode. In digging mode, the stack transducer can write on the output tape when its stack head is inside the stack, whereas in non-digging mode, the stack transducer is only allowed to emit symbols when its stack head is at the top of the stack. These stack transducers have a motivation from natural-language interface applications, as they capture long-distance dependencies in syntactic, semantic, and discourse structures. We study the computational capacity for deterministic digging and non-digging stack transducers, as well as for their non-erasing and checking versions. We finally show that even for the strongest variant of stack transducers the stack languages are regular.

In a number of fields, it is necessary to compare a witness string with a distribution. One possibility is to compute the probability of the string for that distribution. Another, giving a more global view, is to compute the expected edit distance from a string randomly drawn to the witness string. This number is often used to measure the performance of a prediction, the goal then being to return the median string, or the string with smallest expected distance. To be able to measure this, computing the distance between a hypothesis and that distribution is necessary. This paper proposes two solutions for computing this value, when the distribution is defined with a probabilistic finite state automaton. The first is exact but has a cost which can be exponential in the length of the input string, whereas the second is a fully polynomial-time randomized schema.

We study the two-dimensional pattern matching implemented using the deterministic two-dimensional on-line tessellation automaton. This restricted two-dimensional cellular automaton is able to simulate the Baker–Bird algorithm, which was proposed as the first algorithm for the two-dimensional pattern matching. We explore capabilities of this automaton to carry out the matching task against an arbitrary set of equal-sized patterns. To measure amount of resources needed to accomplish it, we introduce the pattern complexity of a picture language. We show that this complexity ranges from a constant to exponential one. All of these are illustrated by giving examples of two-dimensional on-line tessellation automata matching sets of patterns, describing general techniques of how to construct them and proving lower bounds on the pattern complexity of some picture languages as well as operations over them.