Abstract. We consider membership problems for rational subsets of the semigroup of [Formula: see text] matrices over [Formula: see text]. For a semigroup [Formula: see text], the rational subsets [Formula: see text] are defined as the sets accepted by nondeterministic finite automatons whose transitions are labeled by elements of [Formula: see text]. In general, it is undecidable on inputs [Formula: see text] and [Formula: see text] whether [Formula: see text] belongs to [Formula: see text]. Therefore, we restrict our attention to the family [Formula: see text] of flat rational subsets of [Formula: see text] over [Formula: see text], where [Formula: see text] is a subsemigroup of [Formula: see text]. It consists of finite unions of the form [Formula: see text], where [Formula: see text] and [Formula: see text]. Assuming that the membership for [Formula: see text] is decidable, we prove various results when the membership for [Formula: see text] is decidable. If [Formula: see text] is a subgroup of a group [Formula: see text], then we provide a rather general condition when [Formula: see text] is an (effective) relative Boolean algebra. This leads to one of our main results that the emptiness problem for Boolean combinations of sets in [Formula: see text] is decidable. It is possible that such a strong decidability result cannot be pushed any further for groups sitting between [Formula: see text] and [Formula: see text]. To support this possibility, we prove the following dichotomy: If [Formula: see text] is a finitely generated group such that [Formula: see text], then either [Formula: see text] or [Formula: see text] contains an extension of the Baumslag–Solitar group [Formula: see text] of infinite index. It is open whether the membership for rational subsets is decidable in the latter case. For singular matrices, we will show that the membership problem for [Formula: see text] is decidable in doubly exponential time, where [Formula: see text] is the monoid generated by [Formula: see text].

We study the emptiness and $\lambda$-reachability problems for unary and binary Probabilistic Finite Automata (PFA) and characterise the complexity of these problems in terms of the degree of ambiguity of the automaton and the size of its alphabet. Our main result is that emptiness and $\lambda$-reachability are solvable in EXPTIME for polynomially ambiguous unary PFA and if, in addition, the transition matrix is binary, we show they are in NP. In contrast to the Skolem-hardness of the $\lambda$-reachability and emptiness problems for exponentially ambiguous unary PFA, we show that these problems are NP-hard even for finitely ambiguous unary PFA. For binary polynomially ambiguous PFA with fixed and commuting transition matrices, we prove NP-hardness of the $\lambda$-reachability (dimension 9), nonstrict emptiness (dimension 37) and strict emptiness (dimension 40) problems.

(Multi-type) branching processes are a natural and well-studied model for generating random infinite trees. Branching processes feature both nondeterministic and probabilistic branching, generalizing both transition systems and Markov chains (but not generally Markov decision processes). We study the complexity of model checking branching processes against linear-time omega-regular specifications: is it the case almost surely that every branch of a tree randomly generated by the branching process satisfies the omega-regular specification? The main result is that for LTL specifications this problem is in PSPACE, subsuming classical results for transition systems and Markov chains, respectively. The underlying general model-checking algorithm is based on the automata-theoretic approach, using unambiguous Büchi automata.

We study the emptiness and λ-reachability problems for unary and binary Probabilistic Finite Automata (PFA) and characterise the complexity of these problems in terms of the degree of ambiguity of the automaton and the size of its alphabet. Our main result is that emptiness and λ-reachability are solvable in EXPTIME for polynomially ambiguous unary PFA and if, in addition, the transition matrix is over {0, 1}, we show they are in NP. In contrast to the Skolem-hardness of the λ-reachability and emptiness problems for exponentially ambiguous unary PFA, we show that these problems are NP-hard even for finitely ambiguous unary PFA. For binary polynomially ambiguous PFA with commuting transition matrices, we prove NP-hardness of the λ-reachability (dimension 9), nonstrict emptiness (dimension 37) and strict emptiness (dimension 40) problems.

We consider the following variant of the Mortality Problem: given $k\times k$ matrices $A_1, A_2, \dots,A_{t}$, does there exist nonnegative integers $m_1, m_2, \dots,m_t$ such that the product $A_1^{m_1} A_2^{m_2} \cdots A_{t}^{m_{t}}$ is equal to the zero matrix? It is known that this problem is decidable when $t \leq 2$ for matrices over algebraic numbers but becomes undecidable for sufficiently large $t$ and $k$ even for integral matrices. In this paper, we prove the first decidability results for $t>2$. We show as one of our central results that for $t=3$ this problem in any dimension is Turing equivalent to the well-known Skolem problem for linear recurrence sequences. Our proof relies on the Primary Decomposition Theorem for matrices that was not used to show decidability results in matrix semigroups before. As a corollary we obtain that the above problem is decidable for $t=3$ and $k \leq 3$ for matrices over algebraic numbers and for $t=3$ and $k=4$ for matrices over real algebraic numbers. Another consequence is that the set of triples $(m_1,m_2,m_3)$ for which the equation $A_1^{m_1} A_2^{m_2} A_3^{m_3}$ equals the zero matrix is equal to a finite union of direct products of semilinear sets. For $t=4$ we show that the solution set can be non-semilinear, and thus it seems unlikely that there is a direct connection to the Skolem problem. However we prove that the problem is still decidable for upper-triangular $2 \times 2$ rational matrices by employing powerful tools from transcendence theory such as Baker's theorem and S-unit equations.

We show the surprising result that the cutpoint isolation problem is decidable for probabilistic finite automata where input words are taken from a letter-bounded context-free language. A context-free language ℒ is letter-bounded when ℒ ⊆ a₁^* a₂^* ⋯ a_𝓁^* for some finite 𝓁 > 0 where each letter is distinct. A cutpoint is isolated when it cannot be approached arbitrarily closely. The decidability of this problem is in marked contrast to the situation for the (strict) emptiness problem for PFA which is undecidable under the even more severe restrictions of PFA with polynomial ambiguity, commutative matrices and input over a letter-bounded language as well as to the injectivity problem which is undecidable for PFA over letter-bounded languages. We provide a constructive nondeterministic algorithm to solve the cutpoint isolation problem, which holds even when the PFA is exponentially ambiguous. We also show that the problem is at least NP-hard and use our decision procedure to solve several related problems.

This work relates numerical problems on matrices over the rationals to symbolic algorithms on words and finite automata. Using exact algebraic algorithms and symbolic computation, we prove new decidability results for 2 × 2 matrices over Q. Namely, we introduce a notion of flat rational sets: if M is a monoid and N ≤ M is its submonoid, then flat rational sets of M relative to N are finite unions of the form L0g1 L1 ··· gtLt where all Lis are rational subsets of N and gi ∈ M. We give quite general sufficient conditions under which flat rational sets form an effective relative Boolean algebra. As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of GL(2, Q) over GL(2, Z) is decidable.

We consider the Membership and the Half-Space Reachability problems for matrices in dimensions two and three. Our first main result is that the Membership Problem is decidable for finitely generated sub-semigroups of the Heisenberg group over rational numbers. Furthermore, we prove two decidability results for the Half-Space Reachability Problem. Namely, we show that this problem is decidable for sub-semigroups of GL(2,Z) and of the Heisenberg group over rational numbers.

We consider the membership problem for matrix semigroups, which is the problem to decide whether a matrix belongs to a given finitely generated matrix semigroup. In general, the decidability and complexity of this problem for two-dimensional matrix semigroups remains open. Recently there was a significant progress with this open problem by showing that the membership is decidable for 2x2 nonsingular integer matrices. In this paper we focus on the membership for singular integer matrices and prove that this problem is decidable for 2x2 integer matrices whose determinants are equal to 0, 1, -1 (i.e. for matrices from GL(2,Z) and any singular matrices). Our algorithm relies on a translation of numerical problems on matrices into combinatorial problems on words and conversion of the membership problem into decision problem on regular languages.

The main result of this paper is the decidability of the membership problem for 2 × 2 nonsingular integer matrices. Namely, we will construct the first algorithm that for any nonsingular 2 × 2 integer matrices M1,…, Mn and M decides whether M belongs to the semigroup generated by {M1,…, Mn}. Our algorithm relies on a translation of numerical problems on matrices into combinatorial problems on words. It also makes use of some algebraic properties of well-known subgroups of GL(2, Z) and various new techniques and constructions that help to convert matrix equations into the emptiness problem for intersection of regular languages.

The decision problems on matrices were intensively studied for many decades as matrix products play an essential role in the representation of various computational processes. However, many computational problems for matrix semigroups are inherently difficult to solve even for problems in low dimensions and most matrix semigroup problems become undecidable in general starting from dimension three or four. This paper solves two open problems about the decidability of the vector reachability problem over a finitely generated semigroup of matrices from SL(2, Z) and the point to point reachability (over rational numbers) for fractional linear transformations, where associated matrices are from SL(2, Z). The approach to solving reachability problems is based on the characterization of reachability paths between points which is followed by the translation of numerical problems on matrices into computational and combinatorial problems on words and formal languages. We also give a geometric interpretation of reachability paths and extend the decidability results to matrix products represented by arbitrary labelled directed graphs. Finally, we will use this technique to prove that a special case of the scalar reachability problem is decidable.

Abstract Let E be a computably enumerable (c.e.) equivalence relation on the set ω of natural numbers. We say that the quotient set $\omega /E$ (or equivalently, the relation E ) realizes a linearly ordered set ${\cal L}$ if there exists a c.e. relation ⊴ respecting E such that the induced structure ( $\omega /E$ ; ⊴) is isomorphic to ${\cal L}$ . Thus, one can consider the class of all linearly ordered sets that are realized by $\omega /E$ ; formally, ${\cal K}\left( E \right) = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$ . In this paper we study the relationship between computability-theoretic properties of E and algebraic properties of linearly ordered sets realized by E . One can also define the following pre-order $ \le _{lo} $ on the class of all c.e. equivalence relations: $E_1 \le _{lo} E_2 $ if every linear order realized by E 1 is also realized by E 2 . Following the tradition of computability theory, the lo -degrees are the classes of equivalence relations induced by the pre-order $ \le _{lo} $ . We study the partially ordered set of lo -degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among the lo -degrees.

This paper studies labeled sample compression for multi-label concept classes. For a specific extension of the notion of VC-dimension to multi-label classes, we prove that every maximum multi-label class of dimension d has a sample compression scheme in which every sample is compressed to a subset of size at most d. We further show that every multi-label class of dimension 1 has a sample compression scheme using only sets of size at most 1. As opposed to the binary case, the latter result is not immediately implied by the former, since there are multi-label concept classes of dimension 1 that are not contained in maximum classes of dimension 1.

Abstract We study the following open question in computable model theory: does there exist a structure of computable dimension two which is the prime model of its first-order theory? We construct an example of such a structure by coding a certain family of c.e. sets with exactly two one-to-one computable enumerations into a directed graph. We also show that there are examples of such structures in the classes of undirected graphs, partial orders, lattices, and integral domains.

Abstract In this paper we answer the following well-known open question in computable model theory. Does there exist a computable not ℵ0-categorical saturated structure with a unique computable isomor-phism type? Our answer is affirmative and uses a construction based on Kolmogorov complexity. With a variation of this construction, we also provide an example of an ℵ1-categorical but not ℵ0-categorical saturated -structure with a unique computable isomorphism type. In addition, using the construction we give an example of an ℵ1-categorical but not ℵ0-categorical theory whose only non-computable model is the prime one.