Undecidable problem

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

Partial polymorphic type inference and higher-order unification

Abstract: We show that the problem of partial type inference in the nth-order polymorphic l-calculus is equivalent to nth-order unification. On the one hand, this means that partial type inference in polymorphic l-calculi of order 2 or higher is undecidable. On the other hand, higher-order unification is often tractable in practice, and our translation entails a very useful algorithm for partial type inference in the o-order polymorphic l-calculus. We present an implementation in lProlog in full.

Message Sequence Graphs and Decision Problems on Mazurkiewicz Traces

Abstract: Message sequence chEirts (MSC) are a graphical specification language widely used for designing communication protocols. Our starting point are two decision problems concerning the correctness and the consistency of a design based by MSC graphs. Both problems are shown to be undecidable, in general. Using a natural connectivity assumption from Mazurkiewicz trace theory we show both problems to be EXPSPACE-complete for locally synchronized graphs. The results are based on new complexity results for star-connected rational trace languages.

Probabilistic automata on finite words: decidable and undecidable problems

Abstract: This paper tackles three algorithmic problems for probabilistic automata on finite words: the Emptiness Problem, the Isolation Problem and the Value 1 Problem The Emptiness Problem asks, given some probability 0 ≤ λ ≤ 1, whether there exists a word accepted with probability greater than λ, and the Isolation Problem asks whether there exist words whose acceptance probability is arbitrarily close to λ Both these problems are known to be undecidable [11, 4, 3] About the Emptiness problem, we provide a new simple undecidability proof and prove that it is undecidable for automata with as few as two probabilistic transitions The Value 1 Problem is the special case of the Isolation Problem when λ = 1 or λ = 0 The decidability of the Value 1 Problem was an open question We show that the Value 1 Problem is undecidable Moreover, we introduce a new class of probabilistic automata, #-acyclic automata, for which the Value 1 Problem is decidable

Complexity of Two-Dimensional Patterns

Abstract: In dynamical systems such as cellular automata and iterated maps, it is often useful to look at a language or set of symbol sequences produced by the system. There are well-established classification schemes, such as the Chomsky hierarchy, with which we can measure the complexity of these sets of sequences, and thus the complexity of the systems which produce them. In this paper, we look at the first few levels of a hierarchy of complexity for two-or-more-dimensional patterns. We show that several definitions of “regular language” or “local rule” that are equivalent in d=1 lead to distinct classes in d≥2. We explore the closure properties and computational complexity of these classes, including undecidability and L, NL, and NP-completeness results. We apply these classes to cellular automata, in particular to their sets of fixed and periodic points, finite-time images, and limit sets. We show that it is undecidable whether a CA in d≥2 has a periodic point of a given period, and that certain “local lattice languages” are not finite-time images or limit sets of any CA. We also show that the entropy of a d-dimensional CA's finite-time image cannot decrease faster than t −d unless it maps every initial condition to a single homogeneous state.