Dedicated, by her co-authors, to the memory of Evelyn Nelson who died after the paper was submitted Dedicated by Saharon Shelah to his friend Alan Mekler In the literature two notions of the word problem for a variety occur. A variety has a decidable word problem if every finitely presented algebra in the variety has a decidable word problem. It has a uniformly decidable word problem if there is an algorithm which given a finite presentation produces an algorithm for solving the word problem of the algebra so presented. A variety is given with finitely many axioms having a decidable, but not uniformly decidable, word problem. Other related examples are given as well. The following two options occur in the literature for what is meant by the solvability of the word problem for a variety V: (1) there is an algorithm which, given a finite presentation 9 * in finitely many generators and relations, solves the word problem for 9 relative to the

Using 1947 work of Post showing that the word problem for semigroups is unsolvable, we explicitly exhibit an algebraic characterization of the bits of the halting probability Ω. Our proof closely follows a 1978 formulation of Post’s work by M. Davis. The proof is selfcontained and not very complicated.

Abstract. Realizability theory can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between constructive mathematics and programming by translating specifications in constructive logic into annotated interface code in Objective Caml. The system supports a rich input language allowing descriptions of complex mathematical structures. RZ does not extract code from proofs, but allows any implementation method, from handwritten code to code extracted from proofs by other tools. 1

We prove that the word problem for every monoid presented by a fixed 2-homogeneous semi-Thue system can be solved in log-space, which generalizes a result of Lipton and Zalcstein for free groups. The uniform word problem for the class of all 2-homogeneous semi-Thue systems is shown to be complete for symmetric log-space.

In previously studied cases in varieties of algebras, the word and isomorphism problems have had the same solution. For abelian groups and loops, for example, both are soluble [15, Section 3.3; 9; 10]; for groups, semigroups and lattice-ordered groups, for example, both are insoluble [7, 19, 1, 18, 12, 11]. In contrast, Miller [17, p. 77], shows that there are recursive classes of finitely presented groups with uniformly soluble word problem but with insoluble isomorphism problem. Our contribution is to provide a variety of algebras defined by a finite number of laws where this dichotomy also occurs. Specifically, we have the following. THEOREM A. (1) The class of finitely presented abelian lattice-ordered groups has uniformly soluble word problem. (2) The isomorphism problem for the class of finitely presented lattice-ordered groups is insoluble. In other words, there is a single algorithm which when given an arbitrary finitely presented abelian lattice-ordered group (from any recursive listing of them all) and a word in its generators determines whether or not the word is 0, yet there is no algorithm which can always distinguish non-isomorphic finitely presented abelian lattice-ordered groups. Part (1) of Theorem A was already known. Part (2) of the theorem is established using the ideas of Baker and Beynon to show that with each compact polyhedron we can associate—in an effective manner—a finitely generated one relator abelian lattice-ordered group so that two compact polyhedra are piecewise linear homeomorphic if and only if the associated finitely presented abelian lattice-ordered groups are isomorphic. Since Markov [16] has proved that the piecewise linear homeomorphism problem for compact polyhedra is insoluble, Theorem A follows. Actually, the duality and the full strength of Markov's superb theorem give a very strong dichotomy, as follows. THEOREM B. The isomorphism problem for m generator 1 relator abelian latticeordered groups is insoluble whenever m ^ 10.

Abstract. We give a decision procedure for the satisfiability of finite sets of ground equations and disequations in the constructor theory: the terms used may contain both uninterpreted and constructor function symbols. Constructor function symbols are by definition injective and terms built with distinct constructors are themselves distinct. This corresponds to properties of (co-)inductive type constructors in inductive type theory. We do this in a framework where function symbols can be partially applied and equations between functions are allowed. We describe our algorithm as an extension of congruence-closure and give correctness, completeness and termination arguments. We then proceed to discuss its limits and extension possibilities by describing its implementation in the Coq proof assistant. Among problems in equational reasoning, a crucial one is the word problem: does a set of equations entail another one? In 1947, Post and Markov [15, 7] showed that this is undecidable. What is decidable is whether an equation

RZ is a tool which translates axiomatizations of mathematical structures to program specifications using the realizability interpretation of logic. This helps programmers correctly implement data structures for computable mathematics. RZ does not prescribe a particular method of implementation, but allows programmers to write efficient code by hand, or to extract trusted code from formal proofs, if they so desire. We used this methodology to axiomatize real numbers and implemented the specification computed by RZ. The axiomatization is the standard domaintheoretic construction of reals as the maximal elements of the interval domain, while the implementation closely follows current state-of-the-art implementations of exact real arithmetic. Our results shows not only that the theory and practice of computable mathematics can coexist, but also that they work together harmoniously.

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We describe a novel approach to verification of software systems centered around an underlying database. Instead of applying general-purpose techniques with only partial guarantees of success, it identifies restricted but reasonably expressive classes of applications and properties for which sound and complete verification can be performed in a fully automatic way. This leverages the emergence of high-level specification tools for database-centered applications that not only allow fast prototyping and improved programmer productivity but, as a side effect, provide convenient targets for automatic verification. We present theoretical and practical results on verification of database-driven systems. The results are quite encouraging and suggest that, unlike arbitrary software systems, significant classes of databasedriven systems may be amenable to automatic verification. This relies on a novel marriage of database and model checking techniques, of relevance to both the database and the computer aided verification communities. 1.

In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathematical denitions and inferences in such a way that they are unshakable." Hilbert's program relied explicitly on formal systems (equivalently, computational systems) to provide certainty in mathematics. The concepts of computation and formal system were not dened in his time, but Descartes' method may be understood as seeking certainty in essentially the same way. In this article, I explain formal systems as concrete artifacts, and investigate the way in which they provide a high level of certainty| arguably the highest level achievable by rational discourse. The rich understanding of formal systems achieved by mathematical logic and computer science in this century illuminates the nature of programs,...