We report on an extension of Haskell with open type-level functions and equality constraints that unifies earlier work on GADTs, functional dependencies, and associated types. The contribution of the paper is that we identify and characterise the key technical challenge of entailment checking; and we give a novel, decidable, sound, and complete algorithm to solve it, together with some practically-important variants. Our system is implemented in GHC, and is already in active use.

We show that no algorithm exists for deciding whether a finite task for three or more processors is wait-free solvable in the asynchronous read-write shared-memory model. This impossibility result implies that there is no constructive (recursive) characterization of wait-free solvable tasks. It also applies to other shared-memory models of distributed computing, such as the comparison-based model. Key words: asynchronous distributed computation, task-solvability, wait-free computation, contractibility problem AMS subject classification: 68Q05, 68Q22 1 Introduction A fundamental area in the theory of distributed computation is the study of asynchronous wait-free shared-memory distributed algorithms. Characterizing the class of distributed tasks that can be solved, no matter how "inefficiently", is an important step towards a complexity theory for distributed computation. A breakthrough was the demonstation by Fisher, Lynch, and Paterson [FLP85] that certain simple tasks, such as cons...

We introduce normalised rewriting, a new rewrite relation. It generalises former notions of rewriting modulo E, dropping some conditions on E. For example, E can now be the theory of identity, idempotency, the theory of Abelian groups, the theory of commutative rings. We give a new completion algorithm for normalised rewriting. It contains as an instance the usual AC completion algorithm, but also the wellknown Buchberger's algorithm for computing standard bases of polynomial ideals. We investigate the particular case of completion of ground equations, In this case we prove by a uniform method that completion modulo E terminates, for some interesting E. As a consequence, we obtain the decidability of the word problem for some classes of equational theories. We give implementation results which shows the efficiency of normalised completion with respect to completion modulo AC. 1 Introduction Equational axioms are very common in most sciences, including computer science. Equations can ...

Abstract. Garside’s results and the existense of the greedy normal form for braids are shown to be true for the singular braid monoid. An analogue of the presentation of J. S. Birman, K. H. Ko and S. J. Lee for the braid group is also obtained for this monoid.

Abstract. Given a short exact sequence of groups with certain conditions, 1 → F → G → H → 1, we prove that G has solvable conjugacy problem if and only if the corresponding action subgroup A � Aut(F) is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form Z 2 ⋊ Fm, F2 ⋊ Fm, Fn ⋊ Z, and Z n ⋊A Fm with virtually solvable action group A � GLn(Z). Also, we give an easy way of constructing groups of the form Z 4 ⋊ Fn and F3 ⋊ Fn with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem. As an application, an alternative solution to the conjugacy problem in Aut(F2) is given. 1.

Dedicated to the memory of W. W. Boone We show in this article that the most usual finiteness conditions on a subgroup of a finitely generated group all have equivalent formulations in terms of formal language theory. This correspondence gives simple proofs of various theorems concerning intersections of subgroups and the preservation of finiteness conditions in a uniform manner. We then establish easily the theorems of Greibach and of Griffiths by considering free reductions of languages that describe the computations of pushdown automata in one case and of Turing machines in the other, thus making clear that they are essentially the same. 1.

Classical decision problems such as the word- and conjugacy problem are introduced and methods are given for solving them in certain cases. All the methods we present involve Van-Kampen diagrams as one of the most powerful tools when dealing with the classical decision problems. 1. Introduction In 1912 Max Dehn formulated in his article ,, Uber unendliche diskontinuierliche Gruppen" ("On infinite discontinuous groups") three fundamental problems for infinite groups given by finite presentations: the identity problem, the transformation problem, and the isomorphism problem. The following is a translation of Dehn's definition of the first two problems called in modern terms the word problem and the conjugacy problem: The identity problem (word problem): Let an arbitrary element of the group be given as a product of the generators. Find a method to decide in a finite number of steps whether or not this element equals the identity element. The transformation problem (conjugacy proble...

this paper. Our special interest is in groups which are presented to us as being generated by a small set of elements, be these permutations of vertices of a graph, matrices describing automorphisms of linear codes, or classes of homotopies of a knot described only by the relations that they satisfy (see [Hac 87]). Although the axioms that define the notion of a group are rather simple, and in spite of the abundance of knowledge about large classes of groups, one is frequently frustrated by the paucity of methods for dealing with groups described by a small set of generators and relations that hold between them. What is lacking in the standard texts of classical algebra and group theory is a counterpart of numerical methods in differential equations. Yet, such computational methods in group theory have been developed along the years under the influence of external problems as well as from within, especially by the needs of the classification of finite simple groups and the Burnside Groups problem. It should be emphasized that the present computational methods build on careful analysis of algorithmic aspects of known theories. Also, that the practical use of these algorithms became possible in a meaningful way only with the advent of computer technology. We will present in this article some of the known computational methods for investigating groups given by generators and relations, comment

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

Abstract. Most of the existing work in real number computation theory concentrates on complexity issues rather than computability aspects. Though some natural problems like deciding membership in the Mandelbrot set or in the set of rational numbers are known to be undecidable in the Blum-Shub-Smale (BSS) model of computation over the reals, there has not been much work on different degrees of undecidability. A typical question into this direction is the real version of Post’s classical problem: Are there some explicit undecidable problems below the real Halting Problem? In this paper we study three different topics related to such questions: First an extension of a positive answer to Post’s problem to the linear setting. We then analyze how additional real constants increase the power of a BSS machine. And finally a real variant of the classical word problem for groups is presented which we establish reducible to and from (that is, complete for) the BSS Halting problem. 1