Decision Problems for Subclasses of Rational Relations over Finite and Infinite Words
نویسندگان
چکیده
We consider decision problems for relations over finite and infinite words defined by finite automata. We prove that the equivalence problem for binary deterministic rational relations over infinite words is undecidable in contrast to the case of finite words, where the problem is decidable. Furthermore, we show that it is decidable in doubly exponential time for an automatic relation over infinite words whether it is a recognizable relation. We also revisit this problem in the context of finite words and improve the complexity of the decision procedure to single exponential time. The procedure is based on a polynomial time regularity test for deterministic visibly pushdown automata, which is a result of independent interest.
منابع مشابه
Rational Chebyshev Collocation approach in the solution of the axisymmetric stagnation flow on a circular cylinder
In this paper, a spectral collocation approach based on the rational Chebyshev functions for solving the axisymmetric stagnation point flow on an infinite stationary circular cylinder is suggested. The Navier-Stokes equations which govern the flow, are changed to a boundary value problem with a semi-infinite domain and a third-order nonlinear ordinary differential equation by applying proper si...
متن کاملHighly Undecidable Problems For Infinite Computations
We show that many classical decision problems about 1counter ω-languages, context free ω-languages, or infinitary rational relations, are Π12-complete, hence located at the second level of the analytical hierarchy, and “highly undecidable”. In particular, the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, and t...
متن کاملMulti-sequential Word Relations
Rational relations are binary relations of finite words that are realised by non-deterministic finite state transducers (NFT). A particular kind of rational relations is the sequential functions. Sequential functions are the functions that can be realised by input-deterministic transducers. Some rational functions are not sequential. However, based on a property on transducers called the twinni...
متن کاملThree Applications to Rational Relations of the High Undecidability of the Infinite Post Correspondence Problem in a Regular ω-Language
It was noticed by Harel in [Har86] that “one can define Σ 1 -complete versions of the well-known Post Correspondence Problem”. We first give a complete proof of this result, showing that the infinite Post Correspondence Problem in a regular ω-language is Σ 1 -complete, hence located beyond the arithmetical hierarchy and highly undecidable. We infer from this result that it is Π 1 -complete to d...
متن کاملThe Theory of Rational Relations on Transfinite Strings
In order to explain the purpose of this paper, we recall briefly how the theory of “rationality” developed in the last fourty years. The theory of sets of finite strings recognized by finite automata, also known as regular or rational sets was developed in the fifties. It rapidly extended in two directions. Indeed, by solving the decidability problem of the second order monadic theory of one su...
متن کامل