Undecidable problem

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

Fast left-linear semi-unification

Abstract: Semi-unification is a generalization of both unification and matching with applications in proof theory, term rewriting systems, polymorphic type inference, and natural language processing. It is the problem of solving a set of term inequalities M1 ≤ N1, ..., M k <- N k , where ≤ is interpreted as the subsumption preordering on (first-order) terms. Whereas the general problem has recently been shown to be undecidable, several special cases are decidable.

Parameterized Verification of Communicating Automata under Context Bounds

Abstract: We study the verification problem for parameterized communicating automata (PCA), in which processes synchronize via message passing. A given PCA can be run on any topology of bounded degree (such as pipelines, rings, or ranked trees), and communication may take place between any two processes that are adjacent in the topology. Parameterized verification asks if there is a topology from a given topology class that allows for an accepting run of the given PCA. In general, this problem is undecidable even for synchronous communication and simple pipeline topologies. We therefore consider context-bounded verification, which restricts the behavior of each single process. For several variants of context bounds, we show that parameterized verification over pipelines, rings, and ranked trees is decidable. More precisely, it is PSPACE-complete for pipelines and rings, and EXPTIME-complete for ranked trees. Our approach is automata-theoretic. We build a finite (tree, respectively) automaton that identifies those topologies that allow for an accepting run of the given PCA. The verification problem then reduces to checking nonemptiness of that automaton.

Eliminating reflection from type theory

Abstract: Type theories with equality reflection, such as extensional type theory (ETT), are convenient theories in which to formalise mathematics, as they make it possible to consider provably equal terms as convertible. Although type-checking is undecidable in this context, variants of ETT have been implemented, for example in NuPRL and more recently in Andromeda. The actual objects that can be checked are not proof-terms, but derivations of proof-terms. This suggests that any derivation of ETT can be translated into a typecheckable proof term of intensional type theory (ITT). However, this result, investigated categorically by Hofmann in 1995, and 10 years later more syntactically by Oury, has never given rise to an effective translation. In this paper, we provide the first effective syntactical translation from ETT to ITT with uniqueness of identity proofs and functional extensionality. This translation has been defined and proven correct in Coq and yields an executable plugin that translates a derivation in ETT into an actual Coq typing judgment. Additionally, we show how this result is extended in the context of homotopy type theory to a two-level type theory.

On begins, meets and before

Abstract: Interval temporal logics (ITLs) are logics for reasoning about temporal statements expressed over intervals, i.e., periods of time. The most famous temporal logic for intervals studied so far is probably Halpern and Shoham's HS, which is the logic of the thirteen Allen's interval relations. Unfortunately, HS and most of its fragments are undecidable. This discouraged the research in this area until recently, when a number non-trivial decidable ITLs have been discovered. This paper is a contribution towards the complete classification of all different fragments of HS. We consider here different combinations of the interval relations begins (B), meets (A), later (L) and their inverses , and . We know from previous work that the combination is decidable only when finite domains are considered (and undecidable elsewhere), and that is decidable over the natural numbers. In the present paper we show that, over strongly discrete linear models (e.g. finite orders, the naturals, the integers), decidability of can be further extended to capture the language , which lies strictly in between and . The logic turns out to be maximal w.r.t decidability over the considered classes, and its satisfiability problem is EXPSPACE-complete. In this paper we also provide an optimal non-deterministic decision procedure, and we show that the language is powerful enough to polynomially encode metric constraints on the length of the current interval.

Solving Partial-Information Stochastic Parity Games

Abstract: We study one-sided partial-information 2-player concurrent stochastic games with parity objectives. In such a game, one of the players has only partial visibility of the state of the game, while the other player has complete knowledge. In general, such games are known to be undecidable, even for the case of a single player (POMDP). These undecidability results depend crucially on player strategies that exploit an infinite amount of memory. However, in many applications of games, one is usually more interested in finding a finite-memory strategy. We consider the problem of whether the player with partial information has a finite-memory winning strategy when the player with complete information is allowed to use an arbitrary amount of memory. We show that this problem is decidable.