Undecidable problem

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

The Complexity of Nash Equilibria in Simple Stochastic Multiplayer Games

Abstract: We analyse the computational complexity of finding Nash equilibria in simple stochastic multiplayer games. We show that restricting the search space to equilibria whose payoffs fall into a certain interval may lead to undecidability. In particular, we prove that the following problem is undecidable: Given a game $\mathcal G$, does there exist a pure-strategy Nash equilibrium of $\mathcal G$ where player 0 wins with probability 1. Moreover, this problem remains undecidable if it is restricted to strategies with (unbounded) finite memory. However, if mixed strategies are allowed, decidability remains an open problem. One way to obtain a provably decidable variant of the problem is to restrict the strategies to be positional or stationary. For the complexity of these two problems, we obtain a common lower bound of NP and upper bounds of NP and PSpace respectively.

Conditional planning in the discrete belief space

Abstract: Probabilistic planning with observability restrictions, as formalized for example as partially observable Markov decision processes (POMDP), has a wide range of applications, but it is computationally extremely difficult. For POMDPs, the most general decision problems about existence of policies satisfying certain properties are undecidable. We consider a computationally easier form of planning that ignores exact probabilities, and give an algorithm for a class of planning problems with partial observability. We show that the basic backup step in the algorithm is NP-complete. Then we proceed to give an algorithm for the backup step, and demonstrate how it can be used as a basis of an efficient algorithm for constructing plans.

Satisfiability and Query Answering in Description Logics with Global and Local Cardinality Constraints

Abstract: We introduce and investigate the expressive description logic (DL) ALCSCC++, in which the global and local cardinality constraints introduced in previous papers can be mixed. On the one hand, we prove that this does not increase the complexity of satisfiability checking and other standard inference problems. On the other hand, the satisfiability problem becomes undecidable if inverse roles are added to the languages. In addition, even without inverse roles, conjunctive query entailment in this DL turns out to be undecidable. We prove that decidability of querying can be regained if global and local constraints are not mixed and the global constraints are appropriately restricted. The latter result is based on a locally-acyclic model construction, and it reduces query entailment to ABox consistency in the restricted setting, i.e., to ABox consistency w.r.t. restricted cardinality constraints in ALCSCC, for which we can show an ExpTime upper bound.

Tradeoffs in metaprogramming

Abstract: The design of metaprogramming languages requires appreciation of the tradeoffs that exist between important language characteristics such as safety properties, expressive power, and succinctness. Unfortunately, such tradeoffs are little understood, a situation we try to correct by embarking on a study of metaprogramming language tradeoffs using tools from computability theory. Safety properties of metaprograms are in general undecidable; for example, the property that a metaprogram always halts and produces a type-correct instance is Π02-complete. Although such safety properties are undecidable, they may sometimes be captured by a restricted language, a notion we adapt from complexity theory. We give some sufficient conditions and negative results on when languages capturing properties can exist: there can be no languages capturing total correctness for metaprograms, and no 'functional' safety properties above Σ03 can be captured. We prove that translating a metaprogram from a general-purpose to a restricted metaprogramming language capturing a property is tantamount to proving that property for the metaprogram.