Kai Salomaa

The state complexities of some basic operations on regular languages

Abstract We consider the state complexities of some basic operations on regular languages. We show that the number of states that is sufficient and necessary in the worst case for a deterministic finite automaton (DFA) to accept the catenation of an m-state DFA language and an n-state DFA language is exactly m2n − 2n − 1, for m, n ⩾ 1. The result of 2n − 1 + 2n − 2 states is obtained for the star of an n-state DFA language, n1. State complexities for other basic operations and for regular languages over a one-letter alphabet are also studied.

State Complexity of Basic Operations on Finite Languages

The state complexity of basic operations on regular languages has been studied in [9,10,11]. Here we focus on finite languages. We show that the catenation of two finite languages accepted by an m- state and an n-state DFA, respectively, with m > n is accepted by a DFA of (m - n + 3)2n-2 - 1 states in the two-letter alphabet case, and this bound is shown to be reachable. We also show that the tight upperbounds for the number of states of a DFA that accepts the star of an n-state finite language is 2n-3 + 2n-4 in the two-letter alphabet case. The same bound for reversal is 3 ? 2p-1 - 1 when n is even and 2p - 1 when n is odd. Results for alphabets of an arbitrary size are also obtained. These upper-bounds for finite languages are strictly lower than the corresponding ones for general regular languages.

A SHARPENING OF THE PARIKH MAPPING

In this paper we introduce a sharpening of the Parikh map- ping and investigate its basic properties. The new mapping is based on square matrices of a certain form. The classical Parikh vector appears in such a matrix as the second diagonal. However, the matrix prod- uct gives more information about a word than the Parikh vector. We characterize the matrix products and establish also an interesting in- terconnection between mirror images of words and inverses of matrices. Mathematics Subject Classification. 68Q45, 68Q70.

Decision Problems for Patterns

We settle an open problem, the inclusion problem for pattern languages. This is the first known case where inclusion is undecidable for generative devices having a trivially decidable equivalence problem. The study of patterns goes back to the seminal work of Thue and is important also, for instance, in recent work concerning inductive inference and learning. Our results concern both erasing and nonerasing patterns.

State complexity of combined operations

The state complexity of combined operations is studied. We show that the state complexity of a combined operation can be very different from the composition of the state complexities of the participating individual operations. However, the estimate through individual nondeterministic state complexities for each of the combined operations being considered is very similar to the actual state complexity. Several open problems related to state complexity are also proposed.

Deterministic tree pushdown automata and monadic tree rewriting systems

Abstract We show that the deterministic tree pushdown automata of J. H. Gallier and R. V. Book ( Theoret. Comput. Sci. 37 (1985), 123–150) are strictly more powerful than the corresponding automata of K. M. Schimpf (Ph. D. dissertation, University of Pennsylvania, 1982). In fact, even one of the additional features of the former automata, the capability to delete or to duplicate subtrees of the tree stack increases the recognition power. Also we show that finite unions of congruence classes of canonical monadic tree rewriting systems can be recognized by deterministic tree pushdown automata without the additional acceptance conditions used in op. cit . For right-linear monadic tree rewriting systems the same is true for unions of congruence classes over regular tree languages.

Pattern languages with and without erasing

The paper deals with the problems related to finding a pattern common to all words in a given set. We restrict our attention to patterns expressible by the use of variables ranging over words. Two essentially different cases result, depending on whether or not the empty word belongs to the range. We investigate equivalence and inclusion problems, patterns descriptive for a set, as well as some complexity issues. The inclusion problem between two pattern languages turns out to be of fundamental theoretical importance because many problems in the classical combinatorics of words can be reduced to it.

Metric Lexical Analysis

We study automata-theoretic properties of distances and quasi-distances between words. We show that every additive distance is finite. We also show that every additive quasi-distance is regularitypreserving, that is, the neighborhood of any radius of a regular language with respect to an additive quasi-distance is regular. As an application we present a simple algorithm that constructs a metric (fault-tolerant) lexical analyzer for any given lexical analyzer and desired radius (fault-tolerance index).

Nondeterministic State Complexity of Basic Operations for Prefix-Free Regular Languages

We investigate the nondeterministic state complexity of basic operations for prefix-free regular languages. The nondeterministic state complexity of an operation is the number of states that are necessary and sufficient in the worst-case for a minimal nondeterministic finite-state automaton that accepts the language obtained from the operation. We establish the precise state complexity of catenation, union, intersection, Kleene star, reversal and complementation for prefix-free regular languages.

Inclusion is Undecidable for Pattern Languages

The inclusion problem for (nonerasing) pattern languages was raised by Angluin [1] in 1980. It has been open ever since. In this paper, we settle this open problem and show that inclusion is undecidable for (both erasing and nonerasing) pattern languages. In addition, we show that a special case of the inclusion problem, i.e., the inclusion problem for terminal-free erasing pattern languages, is decidable.