Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages

A language $L$ over an alphabet $\Sigma$ is suffix-convex if, for any words $x,y,z\in\Sigma^*$, whenever $z$ and $xyz$ are in $L$, then so is $yz$. Suffix-convex languages include three special cases: left-ideal, suffix-closed, and suffix-free languages. We examine complexity properties of these three special classes of suffix-convex regular languages. In particular, we study the quotient/state complexity of boolean operations, product (concatenation), star, and reversal on these languages, as well as the size of their syntactic semigroups, and the quotient complexity of their atoms.Comment: 20 pages, 11 figures, 1 table. arXiv admin note: text overlap with arXiv:1605.0669

Large Aperiodic Semigroups

The syntactic complexity of a regular language is the size of its syntactic semigroup. This semigroup is isomorphic to the transition semigroup of the minimal deterministic finite automaton accepting the language, that is, to the semigroup generated by transformations induced by non-empty words on the set of states of the automaton. In this paper we search for the largest syntactic semigroup of a star-free language having $n$ left quotients; equivalently, we look for the largest transition semigroup of an aperiodic finite automaton with $n$ states. We introduce two new aperiodic transition semigroups. The first is generated by transformations that change only one state; we call such transformations and resulting semigroups unitary. In particular, we study complete unitary semigroups which have a special structure, and we show that each maximal unitary semigroup is complete. For $n \ge 4$ there exists a complete unitary semigroup that is larger than any aperiodic semigroup known to date. We then present even larger aperiodic semigroups, generated by transformations that map a non-empty subset of states to a single state; we call such transformations and semigroups semiconstant. In particular, we examine semiconstant tree semigroups which have a structure based on full binary trees. The semiconstant tree semigroups are at present the best candidates for largest aperiodic semigroups. We also prove that $2^n-1$ is an upper bound on the state complexity of reversal of star-free languages, and resolve an open problem about a special case of state complexity of concatenation of star-free languages.Comment: 22 pages, 1 figure, 2 table

Most Complex Non-Returning Regular Languages

A regular language $L$ is non-returning if in the minimal deterministic finite automaton accepting it there are no transitions into the initial state. Eom, Han and Jir\'askov\'a derived upper bounds on the state complexity of boolean operations and Kleene star, and proved that these bounds are tight using two different binary witnesses. They derived upper bounds for concatenation and reversal using three different ternary witnesses. These five witnesses use a total of six different transformations. We show that for each $n\ge 4$ there exists a ternary witness of state complexity $n$ that meets the bound for reversal and that at least three letters are needed to meet this bound. Moreover, the restrictions of this witness to binary alphabets meet the bounds for product, star, and boolean operations. We also derive tight upper bounds on the state complexity of binary operations that take arguments with different alphabets. We prove that the maximal syntactic semigroup of a non-returning language has $(n-1)^n$ elements and requires at least $\binom{n}{2}$ generators. We find the maximal state complexities of atoms of non-returning languages. Finally, we show that there exists a most complex non-returning language that meets the bounds for all these complexity measures.Comment: 22 pages, 6 figure

Symmetric Groups and Quotient Complexity of Boolean Operations

The quotient complexity of a regular language L is the number of left quotients of L, which is the same as the state complexity of L. Suppose that L and L' are binary regular languages with quotient complexities m and n, and that the transition semigroups of the minimal deterministic automata accepting L and L' are the symmetric groups S_m and S_n of degrees m and n, respectively. Denote by o any binary boolean operation that is not a constant and not a function of one argument only. For m,n >= 2 with (m,n) not in {(2,2),(3,4),(4,3),(4,4)} we prove that the quotient complexity of LoL' is mn if and only either (a) m is not equal to n or (b) m=n and the bases (ordered pairs of generators) of S_m and S_n are not conjugate. For (m,n)\in {(2,2),(3,4),(4,3),(4,4)} we give examples to show that this need not hold. In proving these results we generalize the notion of uniform minimality to direct products of automata. We also establish a non-trivial connection between complexity of boolean operations and group theory

Checking Whether an Automaton Is Monotonic Is NP-complete

An automaton is monotonic if its states can be arranged in a linear order that is preserved by the action of every letter. We prove that the problem of deciding whether a given automaton is monotonic is NP-complete. The same result is obtained for oriented automata, whose states can be arranged in a cyclic order. Moreover, both problems remain hard under the restriction to binary input alphabets.Comment: 13 pages, 4 figures. CIAA 2015. The final publication is available at http://link.springer.com/chapter/10.1007/978-3-319-22360-5_2

Firms’ Proactiveness During the Crisis: Evidence from European Data

none3This paper contributes to the literature on the entrepreneurial behavior of firms during the economic crisis by investigating the determinants of proactive behavior on a large sample of European companies during the 2008–2009 financial crisis. We explore various dimensions of proactive behavior, including: investments in innovation, expanding product offer, undergoing quality certification, investing in tangible assets and avoiding layoff. Our findings show a surprising heterogeneity of determinants in the case of different proactivity measures, especially when considering the impact of public policies which support entrepreneurship. We also provide some evidence supporting the organizational learning hypothesis with regard to proactiveness, as we show that the previous crisis experience matters in the case of the adoption of proactive or reactive strategy by a firm.openJan Brzozowski; Marco Cucculelli; Valentina PeruzziBrzozowski, Jan Pawel; Cucculelli, Marco; Peruzzi, Valentin

Testing the Equivalence of Regular Languages

The minimal deterministic finite automaton is generally used to determine regular languages equality. Antimirov and Mosses proposed a rewrite system for deciding regular expressions equivalence of which Almeida et al. presented an improved variant. Hopcroft and Karp proposed an almost linear algorithm for testing the equivalence of two deterministic finite automata that avoids minimisation. In this paper we improve the best-case running time, present an extension of this algorithm to non-deterministic finite automata, and establish a relationship between this algorithm and the one proposed in Almeida et al. We also present some experimental comparative results. All these algorithms are closely related with the recent coalgebraic approach to automata proposed by Rutten

A Computational Interpretation of Context-Free Expressions

We phrase parsing with context-free expressions as a type inhabitation problem where values are parse trees and types are context-free expressions. We first show how containment among context-free and regular expressions can be reduced to a reachability problem by using a canonical representation of states. The proofs-as-programs principle yields a computational interpretation of the reachability problem in terms of a coercion that transforms the parse tree for a context-free expression into a parse tree for a regular expression. It also yields a partial coercion from regular parse trees to context-free ones. The partial coercion from the trivial language of all words to a context-free expression corresponds to a predictive parser for the expression

The crystal structure of superoxide dismutase from Plasmodium falciparum

Background: Superoxide dismutases (SODs) are important enzymes in defence against oxidative stress. In Plasmodium falciparum, they may be expected to have special significance since part of the parasite life cycle is spent in red blood cells where the formation of reactive oxygen species is likely to be promoted by the products of haemoglobin breakdown. Thus, inhibitors of P. falciparum SODs have potential as anti-malarial compounds. As a step towards their development we have determined the crystal structure of the parasite's cytosolic iron superoxide dismutase. Results: The cytosolic iron superoxide dismutase from P. falciparum (PfFeSOD) has been overexpressed in E. coli in a catalytically active form. Its crystal structure has been solved by molecular replacement and refined against data extending to 2.5 angstrom resolution. The structure reveals a two-domain organisation and an iron centre in which the metal is coordinated by three histidines, an aspartate and a solvent molecule. Consistent with ultracentrifugation analysis the enzyme is a dimer in which a hydrogen bonding lattice links the two active centres. Conclusion: The tertiary structure of PfFeSOD is very similar to those of a number of other iron-and manganese-dependent superoxide dismutases, moreover the active site residues are conserved suggesting a common mechanism of action. Comparison of the dimer interfaces of PfFeSOD with the human manganese-dependent superoxide dismutase reveals a number of differences, which may underpin the design of parasite-selective superoxide dismutase inhibitors

Quotient Complexity of Regular Languages

The past research on the state complexity of operations on regular languages is examined, and a new approach based on an old method (derivatives of regular expressions) is presented. Since state complexity is a property of a language, it is appropriate to define it in formal-language terms as the number of distinct quotients of the language, and to call it "quotient complexity". The problem of finding the quotient complexity of a language f(K,L) is considered, where K and L are regular languages and f is a regular operation, for example, union or concatenation. Since quotients can be represented by derivatives, one can find a formula for the typical quotient of f(K,L) in terms of the quotients of K and L. To obtain an upper bound on the number of quotients of f(K,L) all one has to do is count how many such quotients are possible, and this makes automaton constructions unnecessary. The advantages of this point of view are illustrated by many examples. Moreover, new general observations are presented to help in the estimation of the upper bounds on quotient complexity of regular operations