Undecidable problem

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

On the Complexity of Constant Propagation

Abstract: Constant propagation (CP) is one of the most widely used optimizations in practice (cf. [9]). Intuitively, it addresses the problem of statically detecting whether an expression always evaluates to a unique constant at run-time. Unfortunately, as proved by different authors [4, 16], CP is in general undecidable even if the interpretation of branches is completely ignored. On the other hand, it is certainly decidable in more restricted settings, like on loop-free programs (cf. [7]). In this paper, we explore the complexity of CP for a three-dimensional taxonomy. We present an almost complete complexity classification, leaving only two upper bounds open.

Finite and infinite regular thue systems

Abstract: This dissertation studies certain classes of Thue system, concentrating on the Church-Rosser property. The following new results are obtained about infinite regular Thue systems S: (1) if S is Church-Rosser, the word problem is solvable in linear time; (2) if S is monadic Church-Rosser, it defines a nontrivial boolean algebra of DCFLs; (3) if S is monadic Church-Rosser and so is another system T, equivalence of S and T is decidable; (4) if S is monadic, it is decidable if S is Church-Rosser; (5) if S is not monadic it is undecidable if S is Church-Rosser. The following new results are obtained about finite Thue systems S: (1) it is undecidable if there exists another finite Thue system T which is equivalent to S and is Church-Rosser (respectively: almost confluent, preperfect); (2) it is undecidable if S generates a Church-Rosser congruence. Some of these results generalise results about finite Thue systems, and some answer what had previously been open questions. The results are obtained using the theories of automata and formal languages, of Turing machines, and of finitely presented groups.

Axiomatizable theories with few axiomatizable extensions

Abstract: In this paper we prove two theorems. They answer questions raised by Myhill in 1956. (We recall the well-known fact that Myhill's invention of the maximal set in 1956 [2] stemmed from his attempt to prove I below.) I. There exists an axiomatizable, essentially undecidable theory in standard formalization such that all axiomatizable extensions of are finite extensions. II. There exists an axiomatizable but undecidable theory in standard formalization such that (a) has a consistent, complete, decidable extension , (b) If is an axiomatizable extension of then either (i) is a finite extension of , or (ii) is a finite extension of .

On Forward Closure and the Finite Variant Property

Abstract: Equational unification is an important research area with many applications, such as cryptographic protocol analysis. Unification modulo a convergent term rewrite system is undecidable, even with just a single rule. To identify decidable (and tractable) cases, two paradigms have been developed — Basic Syntactic Mutation [14] and the Finite Variant Property [6]. Inspired by the Basic Syntactic Mutation approach, we investigate the notion of forward closure along with suitable redundancy constraints. We show that a convergent term rewriting system R has a finite forward closure if and only if R has the finite variant property. We also show the undecidability of the finiteness of forward closure, therefore determining if a system has the finite variant property is undecidable.