Undecidable problem

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

Dealing with communication for dynamic multithreaded recursive programs.

Abstract: This paper presents a new contribution to the model-checking of multithreaded programs with recursive procedure calls, result passing between recursive procedures, dynamic creation of parallel processes, and synchronisation between parallel threads. To represent such programs accurately, we define the model SPAD that can be seen as the extension with synchronisation of the class PAD (the subclass of the rewrite systems PRS where parallel composition is not allowed in the lefthand sides of the rules). We consider in this paper the reachability problem of this model, which is undecidable. As in [BET03a,BET04], we reduce this problem to the computation of abstractions of the sets of execution paths of the program, and we propose a generic technique that can compute different abstractions (of different precisions and different costs) of these sets.

The Universal Fragment of Presburger Arithmetic with Unary Uninterpreted Predicates is Undecidable

Abstract: The first-order theory of addition over the natural numbers, known as Presburger arithmetic , is decidable in double exponential time. Adding an uninterpreted unary predicate to the language leads to an undecidable theory. We sharpen the known boundary between decidable and undecidable in that we show that the purely universal fragment of the extended theory is already undecidable. Our proof is based on a reduction of the halting problem for two-counter machines to unsatisfiability of sentences in the extended language of Presburger arithmetic that does not use existential quantification. On the other hand, we argue that a single ∀∃ quantifier alternation turns the set of satisfiable sentences of the extended language into a Σ(1,1)-complete set. Some of the mentioned results can be transfered to the realm of linear arithmetic over the ordered real numbers. This concerns the undecidability of the purely universal fragment and the Σ(1,1)-hardness for sentences with at least one quantifier alternation. Finally, we discuss the relevance of our results to verification. In particular, we derive undecidability results for quantified fragments of separation logic, the theory of arrays, and combinations of the theory of equality over uninterpreted functions with restricted forms of integer arithmetic. In certain cases our results even imply the absence of sound and complete deductive calculi.

Undecidability of infinite post correspondence problem for instances of Size 9

Abstract: In the infinite Post Correspondence Problem an instance (h,g) consists of two morphisms h and g , and the problem is to determine whether or not there exists an infinite word ω such that h(ω) = g(ω) . This problem was shown to be undecidable by Ruohonen (1985) in general. Recently Blondel and Canterini (Theory Comput. Syst. 36 (2003) 231–245) showed that this problem is undecidable for domain alphabets of size 105. Here we give a proof that the infinite Post Correspondence Problem is undecidable for instances where the morphisms have domains of 9 letters. The proof uses a recent result of Matiyasevich and Senizergues and a modification of a result of Claus.

Quantifying over boolean announcements.

Abstract: Various extensions of public announcement logic have been proposed with quantification over announcements. The best-known extension is called arbitrary public announcement logic, APAL. It contains a primitive language construct Box phi intuitively expressing that 'after every public announcement of a formula, formula phi is true.' The logic APAL is undecidable and it has an infinitary axiomatization. Now consider restricting the APAL quantification to public announcements of boolean formulas only, such that Box phi intuitively expresses that 'after every public announcement of a boolean formula, formula phi is true.' This logic can therefore called boolean arbitrary public announcement logic, BAPAL. The logic BAPAL is the subject of this work. It is decidable and it has a finitary axiomatization. These results may be considered of interest, as for various applications quantification over booleans is sufficient in formal specifications.