The variety of Kleene algebras with conversion is not finitely based

Given an arbitrary set A, one obtains the full Kleene algebra of binary relations over A by considering the operations of union, composition, reflexive-transitive closure, conversion, and the empty set and the identity relation as constants. Such algebras generate the variety of Kleene algebras (with conversion). As a result of a general analysis of identities satisfied by varieties having an involution operation, we prove that the variety of Kleene algebras with conversion has no finite equational axiomatization. In our argument we make use of the fact that the variety of Kleene algebras without conversion is not finitely based and that, relatively to this variety, the variety of Kleene algebras with conversion is finitely axiomatized.     

Bi-catenation and shuffle product of languages

The shuffle product of two words consists of all words obtained by inserting one word into another word sparsely. The shuffle product of two languages is the union of all the shuffle products of two words taken one from each of these two languages. The bi-catenation of two languages A andB is the set . A non-empty word which is not a power of any other word is called a primitive word. A language is a prefix code if no word in this language is a prefix of any other word in this language. This paper is devoted to the investigation of the elementary properties of bi-catenation and shuffle product of languages. The families of prefix codes, disjunctive languages and languages consisting of primitive words with respective to these two operations are studied. We characterize languages of which the bi-catenation or the shuffle product with any non-empty word are prefix codes. We also derive that for any bifix code A, both and , , are disjunctive languages, where Q is the set of all primitive words over an alphabet X with more than one letter and . For the shuffle product case, surprisingly is a regular language, where a is a letter of the alphabet X. Received: 22 September 1997 / 7 January 1998     

List partitions

By definition, apartition of a list is a division of that list into nonempty contiguous segments. Many programming and operations research problems can be specified in terms of list partitions, and we present a hierarchy of theorems for deriving programs from such specifications. Throughout, reasoning is conducted in an equational style using the calculus for program synthesis developed by Bird and Meertens.Supported by a BP research studentship.     

Equational Elements in Additive Algebras

Let K and Γ be a semiring and a finite ranked alphabet, respectively. We consider K -Γ -algebras with additive carriers and fully multilinear operations. The free objects in this category are the K -Γ -algebras of formal power series on trees with coefficients in K . We study equational elements in such structures. More precisely we establish the following classical result of Mezei and Wright: the equational elements of are the recognizable formal power series, whereas each equational element of an additive K -Γ -algebra is the image via the unique morphism of a recognizable formal power series. As a consequence we get closure of the family Eq( ) under recognizable substitution. The following version of Kleene's theorem is also established: the recognizable series on trees form the smallest family including polynomials and closed under OI substitution and star. A yield situation is a triple ( ) formed by an additive algebra (resp. semialgebra ) and a fully linear function transforming all operations in into products in . It is shown that for each algebraic element m of a semialgebra , there exist a yield situation ( ) and an equational element q ∈ such that m = Y(q) , thus generalizing corresponding well-known results in mathematical language theory. Several applications are given. Received March 1997, and in revised form July 1997, and in final form February 1998.     

Unavoidable Sets of Words of Uniform Length

A set of words X over a finite alphabet A is said to be unavoidable if all but finitely many words in A* have a factor in X. We examine the problem of calculating the cardinality of minimal unavoidable sets of words of uniform length; we correct an error in [8], state a conjecture offering a formula for the minimum size of these so called n-good sets for all values of n, and show that the conjecture is correct in an infinite number of cases.     

The equational theory of prebisimilarity over basic CCS with divergence

This paper studies the equational theory of prebisimilarity, a bisimulation-based preorder introduced by Hennessy and Milner in the early 1980s, over basic CCS with the divergent process Ω. It is well known that prebisimilarity affords a finite ground-complete axiomatization over this language; this study proves that this ground-complete axiomatization is also complete in the presence of an infinite set of actions. Moreover, in sharp contrast to this positive result, it is shown that prebisimilarity is not finitely based over basic CCS with the divergent process Ω when the set of actions is finite and nonempty.     

E-generalization using grammars

We extend the notion of anti-unification to cover equational theories and present a method based on regular tree grammars to compute a finite representation of E-generalization sets. We present a framework to combine Inductive Logic Programming and E-generalization that includes an extension of Plotkin's lgg theorem to the equational case. We demonstrate the potential power of E-generalization by three example applications: computation of suggestions for auxiliary lemmas in equational inductive proofs, computation of construction laws for given term sequences, and learning of screen editor command sequences.     

A spatial equational logic for the applied π-calculus

Spatial logics have been proposed to reason locally and modularly on algebraic models of distributed systems. In this paper we define the spatial equational logic A π L whose models are processes of the applied π-calculus. This extension of the π-calculus allows term manipulation and records communications as aliases in a frame, thus augmenting the predefined underlying equational theory. Our logic allows one to reason locally either on frames or on processes, thanks to static and dynamic spatial operators. We study the logical equivalences induced by various relevant fragments of A π L, and show in particular that the whole logic induces a coarser equivalence than structural congruence. We give characteristic formulae for some of these equivalences and for static equivalence. Going further into the exploration of A π L’s expressivity, we also show that it can eliminate standard term quantification.     

Reductions between disjoint NP-Pairs

Disjoint NP-pairs are pairs (A, B) of nonempty, disjoint sets in NP. We prove that all of the following assertions are equivalent: There is a many-one complete disjoint NP-pair; there is a strongly many-one complete disjoint NP-pair; there is a Turing complete disjoint NP-pair such that all reductions are smart reductions; there is a complete disjoint NP-pair for one-to-one, invertible reductions; the class of all disjoint NP-pairs is uniformly enumerable. Let A, B, C, and D be nonempty sets belonging to NP. A smart reduction between the disjoint NP-pairs (A, B) and (C, D) is a Turing reduction with the additional property that if the input belongs to A B, then all queries belong to C D. We prove under the reasonable assumption that UP ∩ co-UP has a P-bi-immune set that there exist disjoint NP-pairs (A, B) and (C, D) such that (A, B) is truth-table reducible to (C, D), but there is no smart reduction between them. This paper contains several additional separations of reductions between disjoint NP-pairs. We exhibit an oracle relative to which DistNP has a truth-table-complete disjoint NP-pair, but has no many-one-complete disjoint NP-pair.     

The variety generated by semi-Heyting chains

The purpose of this paper was to investigate the structure of semi-Heyting chains and the variety CSH{{\mathcal{CSH}}} generated by them. We determine the number of non-isomorphic n-element semi-Heyting chains. As a contribution to the study of the lattice of subvarieties of CSH,{{\mathcal{CSH}}}, we investigate the inclusion relation between semi-Heyting chains. Finally, we provide equational bases for CSH{{\mathcal{CSH}}} and for the subvarieties of CSH{{\mathcal{CSH}}} introduced in [5].     

Recognizability,hypergraph operations,and logical types

We study several algebras of graphs and hypergraphs and the corresponding notions of equational sets and recognizable sets. We generalize and unify several existing results which compare the associated equational and recognizable sets. The basic algebra on relational structures is based on disjoint union and quantifier-free definable operations. We expand it to an equivalent one by adding operations definable with “few quantifiers”, i.e., operations that take into account local information about elements or tuples. We also consider monadic second-order transductions and we prove that the inverse image of a recognizable set under such a transduction is recognizable.     

Relations rationnelles infinitaires

In this paper, we build a theory of infinitary rational relations, which is an extension of the theory of finitary rational relations, i. e. sets ofK-vectors of finite words which are recognized by finite automata withK tapes, and at the same time an extension of the theory of infinitary rational languages, i.e., sets of finite and infinite words which are recognized by finite automata (the condition of recognizability of an infinite word is that its reading by the automaton must go through a state, wich belongs to a designated subset, infinitly time). Our main result is a theorem similar to the Kleene theorem about rational languages of finite words: it is proved that the family of relations recognized by finite automata withK tapes is the family of relations obtained from the finite finitary relations with a finite sequence of operations of: union, product, finite star, and infinite star. Then the closure properties of this family of relations, are studied.      

An Equational Axiomatization of Dynamic Negation and Relational Composition

We consider algebras on binary relations with two main operators: relational composition and dynamic negation. Relational composition has its standard interpretation, while dynamic negation is an operator familiar to students of Dynamic Predicate Logic (DPL) (Groenendijk and Stokhof, 1991): given a relation R its dynamic negation R is a test that contains precisely those pairs (s,s) for which s is not in the domain of R. These two operators comprise precisely the propositional part of DPL.This paper contains a finite equational axiomatization for these dynamic relation algebras. The completenessresult uses techniques from modal logic. We also lookat the variety generated by the class of dynamic relation algebras and note that there exist nonrepresentable algebras in this variety, ones which cannot be construedas spaces of relations. These results are also proved for an extension to a signature containing atomic tests and union.     

Algebraic Structures of Rough Sets in Representative Approximation Spaces

In this paper a generalized notion of an approximation space is considered. By an approximation space we mean an ordered pair (U, ), where U is a finite nonempty set and is a covering of U. According to connections between rough sets and concepts we define two types of approximation operations. Hence we obtain two families of rough sets. We show that these families form lattices in special types of representative approximation spaces. The operations on rough sets defined in the above lattices are analogous to classical operations on sets.     

Fewest repetitions versus maximal-exponent powers in infinite binary words

A square is the concatenation of a nonempty word with itself. A word has period p if its letters at distance p match. The exponent of a nonempty word is the quotient of its length over its smallest period. In this article we give some new results on the trade-off between the number of squares and the number of maximal-exponent powers in infinite binary words, in the three cases where the maximal exponent is 7/3, 5/2, and 3. These are the only threshold values related to the question.     

Shuffle languages are in P

In this paper we show that shuffle languages are contained in one-way-NSPACE(log n) thus in P. We consider the class of shuffle languages which emerges from the class of finite languages through regular operations (union, concatenation, Kleene star) and shuffle operations (shuffle and shuffle closure). For every shuffle expression E we construct a shuffle automaton which accepts the language generated by E and we show that the automaton can be simulated by a one-way nondeterministic Turing machine in logarithmic space.     

On stabilizers of infinite words

The stabilizer of an infinite word over a finite alphabet Σ is the monoid of morphisms over Σ that fix . In this paper we study various problems related to stabilizers and their generators. We show that over a binary alphabet, there exist stabilizers with at least n generators for all n. Over a ternary alphabet, the monoid of morphisms generating a given infinite word by iteration can be infinitely generated, even when the word is generated by iterating an invertible primitive morphism. Stabilizers of strict epistandard words are cyclic when non-trivial, while stabilizers of ultimately strict epistandard words are always non-trivial. For this latter family of words, we give a characterization of stabilizer elements.     

Fuzzy implication can be arbitrarily complicated: A theorem

In fuzzy logic, there are several methods of representing implication in terms of &, ∨, and ¬; in particular, explicit representations define a class of S implications, implicit representations define a class of R implications. Some reasonable implication operations have been proposed, such as Yager's a^b, that are difficult to represent as S or R implications. For such operations, a new class of representations has recently been proposed, called A implications, for which the relationship between implications and the basic operations &, ∨, and ¬ is even more complicated. A natural question is: Is this complexity really necessary? In other words, is it true that A operations cannot be described as S or R operations, or they can, but we simply have not found these representations? In this paper we show that yes, the complexity is necessary, because there are operations that cannot be represented in a simpler form. © 1998 John Wiley & Sons, Inc.