International Journal of Foundations of Computer Science

Traces can be viewed as parallel processes and the "mean speedup" of a trace monoid has been introduced as a measure of "intrinsic parallelism" of the monoid. We study the problem of computing the mean speedup under two conditions: uniform distribution on the words of given length and uniform distribution on the traces of given height. In the first case, we give an approximability result showing a probabilistic fully polynomial time approximation scheme, while, in the second case, we prove that the problem is NP-hard to approximate within n^1-c for every ∊ > 0, unless NP = coR. A further consequence is the hardness of the problem of generating traces of a given height uniformly at random.

We consider fragments of first-order logic over finite words. In particular, we deal with first-order logic with a restricted number of variables and with the lower levels of the alternation hierarchy. We use the algebraic approach to show decidability of expressibility within these fragments. As a byproduct, we survey several characterizations of the respective fragments. We give complete proofs for all characterizations and we provide all necessary background. Some of the proofs seem to be new and simpler than those which can be found elsewhere. We also give a proof of Simon's theorem on factorization forests restricted to aperiodic monoids because this is simpler and sufficient for our purpose.

We consider the equivalence problem for labeled Markov chains (LMCs), where each state is labeled with an observation. Two LMCs are equivalent if every finite sequence of observations has the same probability of occurrence in the two LMCs. We show that equivalence can be decided in polynomial time, using a reduction to the equivalence problem for probabilistic automata, which is known to be solvable in polynomial time. We provide an alternative algorithm to solve the equivalence problem, which is based on a new definition of bisimulation for probabilistic automata. We also extend the technique to decide the equivalence of weighted probabilistic automata.

Then, we consider the equivalence problem for labeled Markov decision processes (LMDPs), which asks given two LMDPs whether for every scheduler (i.e. way of resolving the nondeterministic decisions) for each of the processes, there exists a scheduler for the other process such that the resulting LMCs are equivalent. The decidability of this problem remains open. We show that the schedulers can be restricted to be observation-based, but may require infinite memory.

Size (the number of states) of finite probabilistic automata with an isolated cut-point can be exponentially smaller than the size of any equivalent finite deterministic automaton. However, the proof is non-constructive. The result is presented in two versions. The first version depends on Artin's Conjecture (1927) in Number Theory. The second version does not depend on conjectures not proved but the numerical estimates are worse. In both versions the method of the proof does not allow an explicit description of the languages used. Since our finite probabilistic automata are reversible, these results imply a similar result for quantum finite automata.

We investigate the state complexity of union and intersection for finite languages. Note that the problem of obtaining the tight bounds for both operations was open. First we compute upper bounds using structural properties of minimal deterministic finite-state automata for finite languages. Then, we show that the upper bounds are tight if we have a variable sized alphabet that can depend on the size of input deterministic finite-state automata. In addition, we prove that the upper bounds are unreachable for any fixed sized alphabet.

Conjunctive grammars, introduced by Okhotin, extend context-free grammars by an additional operation of intersection in the body of any production of the grammar. Several theorems and algorithms for context-free grammars generalize to the conjunctive case. Okhotin posed nine open problems concerning those grammars. One of them was a question, whether a conjunctive grammars over a unary alphabet generate only regular languages. We give a negative answer, contrary to the conjectured positive one, by constructing a conjunctive grammar for the language {a⁴ⁿ : n ∈ ℕ}. We also generalize this result: for every set of natural numbers L we show that {aⁿ : n ∈ L} is a conjunctive unary language, whenever the set of representations in base-k system of elements of L is regular, for arbitrary k.

We show that for all integers n and α such that n ⩽ α ⩽ 2ⁿ, there exists a minimal nondeterministic finite automaton of n states with a four-letter input alphabet whose equivalent minimal deterministic finite automaton has exactly α states. It follows that in the case of a four-letter alphabet, there are no "magic numbers", i.e., the holes in the hierarchy. This improves a similar result obtained by Geffert for a growing alphabet of size n + 2.

As is well-known, Axel Thue constructed an infinite word over a 3-letter alphabet that contains no squares, that is, no nonempty subwords of the form xx. In this paper we consider a variation on this problem, where we try to avoid approximate squares, that is, subwords of the form xx' where |x| = |x'| and x and x' are "nearly" identical.

We develop a theory of regular aperiodic ω-languages in parallel with the theory around the Wagner hierarchy. In particular, we characterize the Wadge degrees of regular aperiodic ω-languages, find an effective version of the Wadge reducibility adequate for this class of languages and prove "aperiodic analogs" of the Büchi-Landweber determinacy theorem and of the Landweber's characterization of regular open and regular G_δ sets.

Bideterministic automata are deterministic automata with the property of their reversal automata also being deterministic. Bideterministic automata have previously been shown to be unique (up to an isomorphism) minimal NFAs with respect to the number of states. In this paper, we show that in addition to state minimality, bideterministic automata are also transition-minimal NFAs. However, as this transition minimality is not necessarily unique, we also present the necessary and sufficient conditions for a bideterministic automaton to be uniquely transition-minimal among NFAs. These results are obtained using the notion of the universal automaton. We also present some properties of the universal automaton. Furthermore, we show that bideterministic automata are transition-minimal ∊-NFAs.

The implication of multivalued dependencies (MVDs) in relational databases has originally and independently been defined in the context of some fixed finite universe by Delobel, Fagin, and Zaniolo. Biskup observed that the original axiomatisation for MVD implication does not reflect the fact that the complementation rule is merely a means to achieve database normalisation. He proposed two alternative ways to overcome this deficiency: i) an axiomatisation that does represent the role of the complementation rule adequately, and ii) a notion of MVD implication in which the underlying universe of attributes is left undetermined together with an axiomatisation of this notion.

In this paper we investigate multivalued dependencies with null values (NMVDs) as defined and axiomatised by Lien. We show that Lien's axiomatisation does not adequately reflect the role of the complementation rule, and extend Biskup's findings for MVDs in total database relations to NMVDs in partial database relations. Moreover, a correspondence between (minimal) axiomatisations in fixed universes that do reflect the property of complementation and (minimal) axiomatisations in undetermined universes is shown.

For a string w ∈ {0,1, 2,…, d-1}*, let val_d(w) denote the integer whose base d representation is the string w and let MSD_d(x) denote the most significant or the leading digit of a positive integer x when x is written as a base d integer. Hirvensalo and Karhumäki [9] studied the problem of computing the leading digit in the ternary representation of 2^x ans showed that this problem has a polynomial time algorithm. In [16], some applications are presented for the problem of computing the leading digit of A^B for given inputs A and B. In this paper, we study this problem from a formal language point of view. Formally, we consider the language L_b,d,j = {w|w ∈ {0,1, 2,…, d-1}*, 1 ≤ j ≤ 9, MSD_b(d^val_b(w))) = j} (and some related classes of languages) and address the question of whether this and some related languages are context-free. Standard pumping lemma proofs seem to be very difficult for these languages. We present a unified and simple combinatorial technique that shows that these languages are not unambiguous context-free languages. The Benford-Newcomb distribution plays a central role in our proofs.

We study tissue-like P systems which use string objects and communicate by introducing communication symbols in the strings. We prove that these systems are computationally complete and moreover, they are computationally efficient in the sense that NP-complete problems can be solved in this framework in polynomial time.