Undecidable problem

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

Analysis of recursively parallel programs

Abstract: We propose a general formal model of isolated hierarchical parallel computations, and identify several fragments to match the concurrency constructs present in real-world programming languages such as Cilk and X10. By associating fundamental formal models (vector addition systems with recursive transitions) to each fragment, we provide a common platform for exposing the relative difficulties of algorithmic reasoning. For each case we measure the complexity of deciding state-reachability for finite-data recursive programs, and propose algorithms for the decidable cases. The complexities which include PTIME, NP, EXPSPACE, and 2EXPTIME contrast with undecidable state-reachability for recursive multi-threaded programs.

Numerical document queries

Abstract: A query against a database behind a site like Napster may search, e.g., for all users who have downloaded more jazz titles than pop music titles. In order to express such queries, we extend classical monadic second-order logic by Presburger predicates which pose numerical restrictions on the children (content) of an element node and provide a precise automata-theoretic characterization. While the existential fragment of the resulting logic is decidable, it turns out that satisfiability of the full logic is undecidable. Decidable satisfiability and a querying algorithm even with linear data complexity can be obtained if numerical constraints are only applied to those contents of elements where ordering is irrelevant. Finally, it is sketched how these techniques can be extended also to answer questions like, e.g., whether the total price of the jazz music downloaded so far exceeds a user's budget.

On the complexity of space bounded interactive proofs

Abstract: Two results on interactive proof systems with two-way probabilistic finite-state verifiers are proved. The first is a lower bound on the power of such proof systems if they are not required to halt with high probability on rejected inputs: it is shown that they can accept any recursively enumerable language. The second is an upper bound on the power of interactive proof systems that halt with high probability on all inputs. The proof method for the lower bound also shows that the emptiness problem for one-way probabilistic finite-state machines is undecidable. In the proof of the upper bound some results of independent interest on the rate of convergence of time-varying Markov chains to their halting states are obtained. >

Undecidable Verification Problems for Programs with Unreliable Channels

Abstract: We consider the class of finite-state systems communicating through unbounded butlossyFIFO channels (calledlossy channel systems) These systems have infinite state spaces due to the unboundedness of the channels In an earlier paper, we showed that the problems of checking reachability, safety properties, and eventuality properties are decidable for lossy channel systems In this paper, we show that the following problems are undecidable:?The model checking problem in propositional temporal logics such as propositional linear time temporal logic (PTL) and computation tree logic (CTL)?The problem of deciding eventuality properties with fair channels: do all computations eventually reach a given set of states if the unreliable channels satisfy fairness assumptions ?The results are obtained through reduction from a variant of the Post correspondence problem