International Journal of Foundations of Computer Science

We study a novel representation of semiautomata, which is motivated by the method of trace-assertion specifications of software modules. Each state of the semiautomaton is represented by an arbitrary word leading to that state, the canonical word. The transitions of the semiautomaton give rise to a right congruence, the state-equivalence, on the set of input words of the semiautomaton: two words are state-equivalent if and only if they lead to the same state. We present a simple algorithm for finding a set of generators for state-equivalence. Directly from this set of generators, we construct a confluent prefix-rewriting system which permits us to transform any word to its canonical representative. In general, the rewriting system may allow infinite derivations. To address this issue, we impose the condition of prefix-continuity on the set of canonical words. A set is prefix-continuous if, whenever a word w and a prefix u of w are in the set, then all the prefixes of w longer than u are also in the set. Prefix-continuous sets include prefix-free and prefix-closed sets as special cases. We prove that the rewriting system is Noetherian if and only if the set of canonical words is prefix-continuous. Furthermore, if the set of canonical words is prefix-continuous, then the set of rewriting rules is irredundant. We show that each prefix-continuous canonical set corresponds to a spanning forest of the semiautomaton.

Cover automata were introduced a few years ago for designing a compact representation of finite languages. Our aim is to extend this notion to cover transducers for functions with finite domain. Given two alphabets Σ and Ω, and a function α : Σ* → Ω* of order l (the maximal length of a word in the domain of α), a cover transducer for α is any subsequential transducer that realizes the function α when its input is restricted to the set of words of Σ* having a length not greater than l. We study the problem of reducing the number of states of a cover transducer. We report experimental results, from an implementation using WFSC (Weighted Finite State Compiler), a Xerox tool for handling weighted finite state automata and transducers.

In the standard definition of a P system, a computation step consists of a parallel application of a "maximal" set of nondeterministically chosen rules. Referring to this system as a parallel P system, we consider in this paper a sequential P system, in which each step consists of an application of a single nondeterministically chosen rule. We show the following:

1. For 1-membrane purely catalytic systems (pure CS's), the sequential version is strictly weaker than the parallel version in that the former defines (i.e., generates) exactly the semilinear sets, whereas the latter is known to define nonrecursive sets.

2. For 1-membrane communicating P systems (CPS's), the sequential version can only define a proper subclass of the semilinear sets, whereas the parallel version is known to define nonrecursive sets.

3. Adding a new type of rule of the form: ab → a_xb_yc_comed_come to the CPS (a natural generalization of the rule ab → a_xb_yc_come in the original model), where x, y ∈ {here, out}, to the sequential 1-membrane CPS makes it equivalent to a vector addition system.

4. Sequential 1-membrane symport/antiport systems (SA's) are equivalent to vector addition systems, contrasting the known result that the parallel versions can define nonrecursive sets.

5. Sequential 1-membrane SA's whose rules have radius 1, (1,1), (1,2) (i.e., of the form (a, out), (a, in), (a, out; b, in), (a, out; bc, in)) generate exactly the semilinear sets. However, if the rules have radius 1, (1,1), (2,1) (i.e., of the form (ab, out; c, in)), the SA's can only generate a proper subclass of the semilinear sets.

In this paper we study some properties of cyclic unary regular languages. We find a connection between the uniqueness of the minimal NFA for certain cyclic unary regular languages and a Diophantine equation studied by Sylvester.

We also obtain some results on the radius of unary languages. We show that the nondeterministic radius of a cyclic unary regular language L is not necessarily obtained by any of the minimal NFAs for L. We give a class of examples which demonstrates that the nondeterministic radius of a regular language cannot necessarily even be approximated by the minimal radius of its minimal NFAs.

The decidability of the shuffle decomposition problem for regular languages is a long standing open question. We consider decompositions of regular languages with respect to shuffle along a regular set of trajectories and obtain positive decidability results for restricted classes of trajectories. Also we consider decompositions of unary regular languages. Finally, we establish in the spirit of the Dassow-Hinz undecidability result an undecidability result for regular languages shuffled along a fixed linear context-free set of trajectories.

Informally put, the semifeasible sets are the class of sets having a polynomial-time algorithm that, given as input any two strings of which at least one belongs to the set, will choose one that does belong to the set. We provide a tutorial overview of the advice complexity of the semifeasible sets. No previous familiarity with either the semifeasible sets or advice complexity will be assumed, and when we include proofs we will try to make the material as accessible as possible via providing intuitive, informal presentations.

Karp and Lipton introduced advice complexity about a quarter of a century ago.¹⁸ Advice complexity asks, for a given power of interpreter, how many bits of "help" suffice to accept a given set. Thus, this is a notion that contains aspects both of descriptional/informational complexity and of computational complexity. We will see that for some powers of interpreter the (worst-case) complexity of the semifeasible sets is known right down to the bit (and beyond), but that for the most central power of interpreter—deterministic polynomial time—the complexity is currently known only to be at least linear and at most quadratic.

While overviewing the advice complexity of the semifeasible sets, we will stress also the issue of whether the functions at the core of semifeasibility—so-called selector functions—can without cost be chosen to possess such algebraic properties as commutativity and associativity. We will see that this is relevant, in ways both potential and actual, to the study of the advice complexity of the semifeasible sets.

Gemmating P systems were introduced as a theoretical model based on the biological idea of the gemmation of mobile membranes. In the general model of extended gemmating P systems, strings are modified either by evolution rules in the membranes or while sending them to another membrane. We here consider the restricted variant of extended gemmating P systems with pre-dynamic rules where strings are only modified at the ends while sending them from one membrane to another one. In a series of papers the number of membranes being sufficient for obtaining computational completeness has steadily been decreased. In this paper we now prove the optimal result, i.e., gemmating P systems only using pre-dynamic rules are already computationally complete with three membranes, even in the non-extended case and with the minimal weight of rules possible. Moreover, we also show that for gemmating tissue P systems two cells suffice, and if we allow the environment to be fully involved in the communication of strings, even one cell together with the environment can manage the task to generate any recursively enumerable language.

If the machines of some type A have enough resources to (i) solve problems that no machine of type B can solve, and (ii) simulate any unary two-way deterministic finite automaton that has access to a linearly-bounded counter, then typically no recursive function can upper bound the increase in the size of description when a machine of type A is replaced by an equivalent machine of type B.

Non-recursive trade-offs between different representations of languages reveal a basic phenomenon. The gain in economy of description can be arbitrary. The purpose of this paper is to survey the main aspects and results regarding different representations of languages whose relative succinctness is not recursively bounded. Basic properties of descriptional systems and reasonable size measures are addressed, and the unified fundamental proof schemes emerging from the literature are presented. A comprehensive overview of results is given. Finally, some new results are shown. In particular, it is proved that between each two levels of the infinite one-way k-head finite automata hierarchies there is a non-recursive trade-off. Moreover, non-recursive trade-offs are shown between nondeterministic 2-head and deterministic k-head automata.

In this paper, we study the tradeoffs in descriptional complexity of NFA (nondeterministic finite automata) of various amounts of ambiguity. We say that two classes of NFA are separated if one class can be exponentially more succinct in descriptional sizes than the other. New results are given for separating DFA (deterministic finite automata) from UFA (unambiguous finite automata), UFA from MDFA (DFA with multiple initial states) and UFA from FNA (finitely ambiguous NFA). We present a family of regular languages that we conjecture to be a good candidate for separating FNA from LNA (linearly ambiguous NFA).

Language equations with all Boolean operations and concatenation and a particular order on the set of solutions are proved to be equal in expressive power to the first-order Peano arithmetic. In particular, it is shown that the class of sets representable using k variables (for every k ≥ 2) is exactly the k-th level of the arithmetical hierarchy, i.e., the sets definable by recursive predicates with k alternating quantifiers. The property of having an extremal solution is shown to be nonrepresentable in first-order arithmetic.

As is well known, each minimal NFA for a regular language L is isomorphic to a subautomaton of the so-called universal automaton for L. We explore and compare various conditions on sets of states of which are related to the fact that induced subautomata of accept the whole language L. The methods of several previous works on minimalizations of NFA can be modified so that they fit in our approach. We also propose a new algorithm which is easy to implement.

Metalinear CD grammar systems are context-free CD grammar systems where each component consists of metalinear productions. The maximal number of nonterminals in all starting productions is referred to as the width of a CD grammar system. It is shown that between the class of CD grammar systems of width m + 1 and of width m there are savings concerning the size of the grammar system not bounded by any recursive function. This is called a non-recursive trade-off. Furthermore, it is proven that there are non-recursive trade-offs between the class of metalinear CD grammar systems of width m and the class of (2m - 1)-linear context-free grammars. In addition, some decidability results are presented.

Iwama et al. showed that there exists an n-state binary nondeterministic finite automaton such that its equivalent minimal deterministic finite automaton has exactly 2ⁿ - α states, for all n ≥ 7 and 5 ≤ α ≤ 2n-2, subject to certain coprimality conditions. We investigate the same question for both unary and binary symmetric difference nondeterministic finite automata. In the binary case, we show that for any n ≥ 4, there is an n-state symmetric difference nondeterministic finite automaton for which the equivalent minimal deterministic finite automaton has 2^{n - 1} + 2^{k - 1} - 1 states, for 2 < k ≤ n - 1. In the unary case, we consider a large practical subclass of unary symmetric difference nondeterministic finite automata: for all n ≥ 2, we argue that there are many values of α such that there is no n-state unary symmetric difference nondeterministic finite automaton with an equivalent minimal deterministic finite automaton with 2ⁿ - α states, where 0 < α < 2^{n - 1}. For each n ≥ 2, we quantify such values of α precisely.

The problem of negative design of DNA languages is addressed, that is, properties and construction methods of large sets of words that prevent undesired bonds when used in DNA computations. We recall a few existing formalizations of the problem and then define the property of sim-bond-freedom, where sim is a similarity relation between words. We show that this property is decidable for context-free languages and polynomial-time decidable for regular languages. The maximality of this property also turns out to be decidable for regular languages and polynomial-time decidable for an important case of the Hamming similarity. Then we consider various construction methods for Hamming bond-free languages, including the recently introduced method of templates, and obtain a complete structural characterization of all maximal Hamming bond-free languages. This result is applicable to the θ-k-code property introduced by Jonoska and Mahalingam.