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Abstract. We survey results, both positive and negative, on regularity of maximal solutions of systems of implicit language equations and inequalities. These results concern inequalities with constant right-hand sides, one-sided linear inequalities, inequalities with restrictions on constants, and commutation equations and inequalities. In addition, we present some of these results in a generalized form in order to underline common principles. 1.

. The article is mainly concerned with the Kruskal tree theorem and the following observation: there is a duality at the level of binary relations between well and noetherian orders. The first step here is to extend Kruskal theorem from orders to binary relations so that the duality applies. Then, we describe the theorem obtained by duality and show that it corresponds to a theorem by Ferreira and Zantema which subsumes Dershowitz's seminal results on recursive path orderings. 1 Introduction 1.1 A duality between well and noetherian relations This paper investigates a duality between the two well-established concepts of well and noetherian order. An order ¯ on a set X is well when in every sequence (x i ) i2N of elements of X: 9(i; j) 2 N 2 ; i ! j and x i ¯ x j (1) An order OE is noetherian on X when it induces no infinite descending chain ::: OE x 2 OE x 1 . In other words, letting OE ? denote the complement 1 of OE, an order is noetherian when for every sequence (x i ) i2...

A modal reduction principle of the form [i1]... [in]p ⇒ [j1]... [jn ′]p can be viewed as a production rule i1 ·... · in → j1 ·... · jn ′ in a formal grammar. We study the extensions of the multimodal logic Km with m independent K modal connectives by finite addition of axiom schemes of the above form such that the associated finite set of production rules forms a regular grammar. We show that given a regular grammar G and a modal formula φ, deciding whether the formula is satisfiable in the extension of Km with axiom schemes from G can be done in deterministic exponential-time in the size of G and φ, and this problem is complete for this complexity class. Such an extension of Km is called a regular grammar logic. The proof of the exponential-time upper bound is extended to PDL-like extensions of Km and to global logical consequence and global satisfiability problems. Using an equational characterization of contextfree languages, we show that by replacing the regular grammars by linear ones, the above problem becomes undecidable. The last part of the paper presents non-trivial classes of exponential time complete regular grammar logics.

The notion of an unavoidable set of words appears frequently in the fields of mathematics and theoretical computer science, in particular with its connection to the study of combinatorics on words. The theory of unavoidable sets has seen extensive study over the past twenty years. In this paper we extend the definition of unavoidable sets of words to unavoidable sets of partial words. Partial words, or finite sequences that may contain a number of “do not know” symbols or holes, appear in natural ways in several areas of current interest such as molecular biology, data communication, DNA computing, etc. We demonstrate the utility of the notion of unavoidability on partial words by making use of it to identify several new classes of unavoidable sets of full words. Along the way we begin work on classifying the unavoidable sets of partial words of small cardinality. We pose a conjecture, and show that affirmative proof of this conjecture gives a sufficient condition for classifying all the unavoidable sets of partial words of size two. Lastly we give a result which makes the conjecture easy to verify for a significant number of cases.

Abstract. A not well-known result [9, Theorem 4.4] in formal language theory is that the Higman-Haines sets for any language are regular, but it is easily seen that these sets cannot be effectively computed in general. Here the Higman-Haines sets are the languages of all scattered subwords of a given language and the sets of all words that contain some word of a given language as a scattered subword. Recently, the exact level of unsolvability of Higman-Haines sets was studied in [10]. We focus on language families whose Higman-Haines sets are effectively constructible. In particular, we study the size of Higman-Haines sets for the lower classes of the Chomsky hierarchy, namely for the families of regular, linear contextfree, and context-free languages, and prove upper and lower bounds on the size of these sets. 1

This paper considers the problem of finding avoiding words given sets of partial words of the form {a ⋄ m1 a ⋄ m2... ⋄ mk a, b⋄ n1 b⋄ n2 nl b} where a and b are letters in the alphabet and ⋄ is a hole, or showing the set is unavoidable. Such a set is avoidable if and only if there exists a two-sided infinite full word with no factor compatible with a member of the set, and this word is called the avoiding word. For the case of k = 1, l = 1, we identify a strict minimum period for an avoiding word. For the case k = 1, l = 2, we refine a conjecture which was identified by Blanchet-Sadri et al. that, if proven, suffices to classify all two element sets [4]. We also discover a collection of unavoidable sets for the case k = 1, l = 3, and correct a previous result, which claimed no such sets exist. Finally, given our refined conjecture, we attempt to classify the remaining sets by identifying common avoiding words and exhibiting exactly which sets these words can avoid. In the process, we introduce a new technique to reduce a set if the period of the avoiding word is known. Using this technique, we are able to classify a significant number of sets previously unclassified.

In Ref. [2], Ehrenfeucht et al. showed that a set L of finite words is regular if and only if L is -closed under some monotone Well-Quasi-Order (WQO) over finite words. We extend this result to regular ω-languages. That is, (1) an ω-language L is regular if and only if L is -closed under a periodic extension of some monotone WQO over finite words, and (2) an ω-language L is regular if and only if L is -closed under a WQO over ω-words which is a continuous extension of some monotone WQO over finite words.

This paper proposed the sufficient and necessary condition for the regularity of ω-languages in terms of a Well-Quasi-Order (WQO). That is, an ω-language L is regular if and only if L is -closed and convergenceclosed under some monotonic WQO .

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