In Floyd-Hoare logic, programs are dynamic while assertions are static (hold at states). In action logic the two notions become one, with programs viewed as on-the-fly assertions whose truth is evaluated along intervals instead of at states. Action logic is an equational theory ACT conservatively extending the equational theory REG of regular expressions with operations preimplication a!b (had a then b) and postimplication b/a (b if-ever a). Unlike REG, ACT is finitely based, makes a reflexive transitive closure, and has an equivalent Hilbert system. The crucial axiom is that of pure induction, (a!a) = a!a. This work was supported by the National Science Foundation under grant number CCR-8814921. 1 Introduction Many logics of action have been proposed, most of them in the past two decades. Here we define action logic, ACT, a new yet simple juxtaposition of old ideas, and show off some of its attractive aspects. The language of action logic is that of equational regular expressio...

The varieties RA of relation algebras and DA of dynamic algebras are similar with regard to definitional capacity, admitting essentially the same equational definitions of converse and star. They differ with regard to completeness and decidability. The RA definitions that are incomplete with respect to representable relation algebras, when expressed in their DA form are complete with respect to representable dynamic algebras. Moreover, whereas the theory of RA is undecidable, that of DA is decidable in exponential time. These results follow from representability of the free intensional dynamic algebras. Dept. of Computer Science, Stanford, CA 94305. This paper is based on a talk given at the conference Algebra and Computer Science, Ames, Iowa, June 2-4, 1988. It will appear in the proceedings of that conference, to be published by SpringerVerlag in the Lecture Notes in Computer Science series. This work was supported by the National Science Foundation under grant number CCR-8814921 ...

this paper, we study 2-dimensional analogies of this idea: semigroup diagrams, monoid pictures, annular diagrams, cylindric pictures and braided pictures. While the groups of linear diagrams are all free, we get a large class of groups which are representable by 2-dimensional semigroup diagrams. Semigroup diagrams, are well-known geometrical objects used in the study of Thue systems (=semigroup presentations). They were first formally introduced by Kashintsev [16], see also Remmers [29], Stallings [34] or Higgins [13]. The role of semigroup diagrams in the study of semigroups is similar to the role of van Kampen diagrams in the study of groups (see [22] or [26])

INTRODUCTION Non-commutative logic is a unication of : | commutative linear logic (Girard 1987) and | cyclic linear logic (Girard 1989; Yetter 1990), a classical conservative extension of the Lambek calculus (Lambek 1958). In a previous paper with Abrusci (Abrusci and Ruet 1999) we presented the multiplicative fragment of non-commutative logic, with proof nets and a sequent calculus based on the structure of order varieties, and a sequentialization theorem. Here we consider full propositional non-commutative logic. Non-commutative logic. Let us rst review the basic ideas. Consider the purely noncommutative fragment of linear logic, obtained by removing the exchange rule entirely : ` ; ; ; , ` ; ; ; y This work has been partly carried out at LIENS-CNRS, Ecole Normale Superieure (Paris), at McGill University

Using a result of B.H. Neumann we extend Eilenberg’s Equality Theorem to a general result which implies that the multiplicity equivalence problem of two (nondeterministic) multitape finite automata is decidable. As a corollary we solve a long standing open problem in automata theory, namely, the equivalence problem for multitape deterministic finite automata. The main theorem states that there is a finite test set for the multiplicity equivalence of finite automata over conservative monoids embeddable in a fully ordered group. 1

The essential postulates of classical thermodynamics are formulated, from which the second law is deduced as the principle of increase of entropy in irreversible adiabatic processes that take one equilibrium state to another. The entropy constructed here is defined only for equilibrium states and no attempt is made to define it otherwise. Statistical mechanics does not enter these considerations. One of the main concepts that makes everything work is the comparison principle (which, in essence, states that given any two states of the same chemical composition at least one is adiabatically accessible from the other) and we show that it can be derived from some assumptions about the pressure and thermal equilibrium. Temperature is derived from entropy, but at the start not even the concept of ‘hotness’ is assumed. Our formulation offers a certain clarity and rigor that goes beyond most textbook discussions of the second law.

This chapter presents a review of various interpretations of the fuzzy membership function together with ways of obtaining a membership function. We emphasize that different interpretations of the membership function call for different elicitation methods. We try to make this distinction clear using techniques from measurement theory.

Two different but related measurement problems are considered within the fuzzy set theory. The first problem is the membership measurement and the second is property ranking. These two measurement problems are combined and two axiomatizations of fuzzy set theory are obtained. In the first one, the indifference is transitive but in the second one this drawback is removed by utilizing interval orders.

Abstract. A natural topology on the space of left orderings of an arbitrary semi-group is introduced. It is proved that this space is compact and that for free abelian groups it is homeomorphic to the Cantor set. An application of this result is a new proof of the existence of universal Gröbner bases. 1. Orderings for semi-groups Given a semi-group G (ie. a set with an associative binary operation), a linear order, <, on G is a left order if a < b implies ca < cb, for any c. Similarly, a linear order,<, is a right order if a < b implies ac < bc, for any c ∈ G. The sets of all left and right orderings of G are denoted by LO(G) and RO(G) respectively. If G is a group then there is a 1-1 correspondence between these two sets which associates with any left ordering, <l, a right ordering, <r, such that a <r b if and only if b −1 <l a −1. For more about ordering of groups see [4, 7, 8]. Let Ua,b ⊂ LO(G) denote the set of all left orderings, <, for which a < b. We can put a topology on LO(G) in one of the following two ways. Definition 1.1. LO(G) has the smallest topology for which all the sets Ua,b are open. Any open set in this topology is a union of sets of the form

this paper. We let (AP