Undecidable problem

About: Undecidable problem is a research topic. Over the lifetime, 3135 publications have been published within this topic receiving 71238 citations.

Analyses of unsatisfiability for equational logic programming

Abstract: The problem of unifying pairs of terms with respect to an equational theory (as well as detecting the unsatisfiability of a system of equations) is, in general, undecidable. In this work, we define a framework based on abstract interpretation for the (static) analysis of the unsatisfiability of equation sets. The main idea behind the method is to abstract the process of semantic unification of equation sets based on narrowing. The method consists of building an abstract narrower for equational theories, and executing the sets of equations to be detected for unsatisfiability in the approximated narrower. As an instance of our framework, we define a new analysis whose accuracy is enhanced by some simple loop-checking technique. This analysis can also be actively used for pruning the search tree of an incremental equational constraint solver, and can be integrated with other methods in the literature. Standard methods are shown to be an instance of our framework. To the best of our knowledge, this is the first framework proposed for approximating equational unification.

The varieties of arboreal experience

Abstract: Publisher Summary This chapter discusses the varieties of arboreal experience. The Paris–Harrington Theorem is a variant of the Finite Ramsey Theorem, which, though provable in ordinary set theory (and so a theorem of ordinary mathematical practice), is not provable in formal number theory. This independence being of a character different from that of Godel's undecidable sentence or that associated with independence results in set theory, the result has been much touted by logicians seeking social acceptance by the rest of the mathematical world as something closely akin to the coming of the millennium. This is unfortunate as the result really is remarkable philosophically because of this new character of independence and mathematically for reasons requiring a new paragraph. The difference between the usual Finite Ramsey Theorem and the Paris–Harrington Theorem is surprisingly consequential, concerning computationally hairy combinatorial problems with unfeasible bounds; but the former's unfeasible bounds are readily intelligible, while the latter's are not. The relation between Kruskal's Theorem and ordinals is fairly close. In fact, Kruskal's Theorem is most memorably stated, if not in terms of ordinals and well-ordering, at least in terms of well-quasi-ordering.

Captive Cellular Automata

Abstract: We introduce a natural class of cellular automata characterised by a property of the local transition law without any assumption on the states set. We investigate some algebraic properties of the class and show that it contains intrinsically universal cellular automata. In addition we show that Rice’s theorem for limit sets is no longer true for that class, although infinitely many properties of limit sets are still undecidable.

Set Constraints in Some Equational Theories

Abstract: Set constraints are relations between sets of ground terms over a given alphabet. They give a natural formalism for many problems in program analysis, type inference, order-sorted unification, and constraint logic programming. In this paper we start studies of set constraints in the environment given by equational specifications. We show that in the case of associativity (i.e., in free monoids) as well as in the case of associativity and commutativity (i.e., in commutative monoids) the problem of consistency of systems of set constraints is undecidable; in linear nonerasing shallow theories the consistency of systems of positive set constraints is NEXPTIME-complete and in linear shallow theories the problem for positive and negative set constraints is decidable.

Effectiveness of data dependence analysis

Abstract: Data dependence testing is the basic step in detecting loop level parallelism in numerical programs. The problem is undecidable in the general case. Therefore, work has been concentrated on a simplified problem, affine memory disambiguation. In this simpler domain, array references and loops bounds are assumed to be linear integer functions of loop variables. Dataflow information is ignored. For this domain, we have shown that in practice the problem can be solved accurately and efficiently.(1) This paper studies empirically the effectiveness of this domain restriction, how many real references are affine and flow insensitive. We use Larus's llpp system(2) to find all the data dependences dynamically. We compare these to the results given by our affine memory disambiguation system. This system is exact for all the cases we see in practice. We show that while the affine approximation is reasonable, memory disambiguation is not a sufficient approximation for data dependence analysis. We propose extensions to improve the analysis.