The expressibility of languages and relations by word equations

Abstract

Classically, several properties and relations of words, such as "being a power of the same word", can be expressed by using word equations. This paper is devoted to a general study of the expressive power of word equations. As main results we prove theorems which allow us to show that certain properties of words are not expressible as components of solutions of word equations. In particular, "the primitiveness" and "the equal length" are such properties, as well as being "any word over a proper subalphabet". knowledge, no attempt to synthesis or of a systematization of this topic has been done. This was emphasized also in a recent survey [Choffrut and Karhumäki 1997], where some results of the field were collected.Classical relations on words that are characterized as solution sets of word equations are for instance, "two words X and Y are powers of the same word" if and only if they constitute a solution of the equation XY ϭ YX, and "two words X and Y are conjugates" if and only if they constitute a solution of the equation XZ ϭ ZY. In the first case, we need no extra variables, while in the second case an additional variable is needed, see Example 4. As above, we identify names of variables and particular solutions of an equation.Motivated by above, we say that a property of words-either a language L ʕ ⌺* or a k-ary relation ʕ (⌺*) k -is expressible by a word equation, if there exists an equation e with t Ն k variables over ⌺ such that: -L coincides with the values of a fixed component of all solutions of e, or -coincides with the values of k fixed components of all solutions of e.Obviously, languages are k-ary relations with k ϭ 1, but, due to the importance of this particular case, we have chosen to define those separately. We allow e to contain constants from ⌺. An important feature here is also that t can be larger than k, that is, additional variables are allowed. This increases essentially the expressive power of equations, and in particular makes it much easier to express certain properties by equations.As an illustration we recall the following: The union of solution sets of two equations can be expressed as a solution set of one equation, as was shown in Büchi and Senger [1986/1987] using 4 additional variables, and later improved to require only 2 additional ones by S. Grigorieff (personal communications), cf. also Choffrut and Karhumäki. Similarly, the inequality, that is the set of t-tuples of words which does not satisfy a given equation e with t variables, can be expressed as a union of the solution sets of finitely many equations, each of them using 3 extra variables, cf. for example, Choffrut and Karhumäki [1997], and consequently the inequality is expressible by one equation if additional variables are allowed.This way of expressing relations on words using word equations is very natural and resembles the way of expressing enumerable relations on integers by diophantine equations. However, the expressive power of our method is weaker. Namely, while diophantine equations can express all recursi...

Satisfiability of word equations with constants is in PSPACE

“…I would like to thank Prof. Filippo Mignosi (University of Palermo, Italy) for many discussions during my visit in Palermo in 1997. The discussions were about synchronizing factorizations introduced in Karhumaeki et al [2000]. I used in this paper two of his ideas.…”

“…I used in this paper two of his ideas. The first one was to use several words instead of one in the definition of F Q -factorization, see Karhumaeki et al [2000]. The second idea was to use context in the definition of a synchronizing factorization.…”

Complete Characterization of Zero-expressible Functions

“…The only existing tools to decide whether a language or relation is not expressible were introduced in [2,5,6]. There is almost nothing known about general case when the formula need not to be existential.…”

“…The class of expressible languages is incomparable with most classes occuring in the litterature like D0L languages, regular languages or context-free languages [5]. We consider another natural class of relations -the class of relations defined by length constraints only.…”

“…Since the relation {(a n , b n ) : n ≥ 0} being an intersection of an expressible relation a * × b * and a relation {(x, y) : |x| = |y|} is not expressible [5], the classes of expressible relations and the relations defined by length constraints are incomparable. We consider the intersection of these two classes.…”