Decision Problems For Convex Languages

In this paper we examine decision problems associated with various classes of convex languages, studied by Ang and Brzozowski (under the name "continuous languages"). We show that we can decide whether a given language L is prefix-, suffix-, factor-, or subword-convex in polynomial time if L is represented by a DFA, but that the problem is PSPACE-hard if L is represented by an NFA. In the case that a regular language is not convex, we prove tight upper bounds on the length of the shortest words demonstrating this fact, in terms of the number of states of an accepting DFA. Similar results are proved for some subclasses of convex languages: the prefix-, suffix-, factor-, and subword-closed languages, and the prefix-, suffix-, factor-, and subword-free languages.Comment: preliminary version. This version corrected one typo in Section 2.1.1, line

Generalizing input-driven languages: theoretical and practical benefits

Regular languages (RL) are the simplest family in Chomsky's hierarchy. Thanks to their simplicity they enjoy various nice algebraic and logic properties that have been successfully exploited in many application fields. Practically all of their related problems are decidable, so that they support automatic verification algorithms. Also, they can be recognized in real-time. Context-free languages (CFL) are another major family well-suited to formalize programming, natural, and many other classes of languages; their increased generative power w.r.t. RL, however, causes the loss of several closure properties and of the decidability of important problems; furthermore they need complex parsing algorithms. Thus, various subclasses thereof have been defined with different goals, spanning from efficient, deterministic parsing to closure properties, logic characterization and automatic verification techniques. Among CFL subclasses, so-called structured ones, i.e., those where the typical tree-structure is visible in the sentences, exhibit many of the algebraic and logic properties of RL, whereas deterministic CFL have been thoroughly exploited in compiler construction and other application fields. After surveying and comparing the main properties of those various language families, we go back to operator precedence languages (OPL), an old family through which R. Floyd pioneered deterministic parsing, and we show that they offer unexpected properties in two fields so far investigated in totally independent ways: they enable parsing parallelization in a more effective way than traditional sequential parsers, and exhibit the same algebraic and logic properties so far obtained only for less expressive language families

Local Algorithms for Sparse Spanning Graphs

We initiate the study of the problem of designing sublinear-time (local) algorithms that, given an edge (u,v) in a connected graph G=(V,E), decide whether (u,v) belongs to a sparse spanning graph G\u27 = (V,E\u27) of G. Namely, G\u27 should be connected and |E\u27| should be upper bounded by (1+epsilon)|V| for a given parameter epsilon > 0. To this end the algorithms may query the incidence relation of the graph G, and we seek algorithms whose query complexity and running time (per given edge (u,v)) is as small as possible. Such an algorithm may be randomized but (for a fixed choice of its random coins) its decision on different edges in the graph should be consistent with the same spanning graph G\u27 and independent of the order of queries. We first show that for general (bounded-degree) graphs, the query complexity of any such algorithm must be Omega(sqrt{|V|}). This lower bound holds for graphs that have high expansion. We then turn to design and analyze algorithms both for graphs with high expansion (obtaining a result that roughly matches the lower bound) and for graphs that are (strongly) non-expanding (obtaining results in which the complexity does not depend on |V|). The complexity of the problem for graphs that do not fall into these two categories is left as an open question

2008 Abstracts Collection -- IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science

This volume contains the proceedings of the 28th international conference on the Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2008), organized under the auspices of the Indian Association for Research in Computing Science (IARCS)

Discounted-Sum Automata with Multiple Discount Factors

Discounting the influence of future events is a key paradigm in economics and it is widely used in computer-science models, such as games, Markov decision processes (MDPs), reinforcement learning, and automata. While a single game or MDP may allow for several different discount factors, discounted-sum automata (NDAs) were only studied with respect to a single discount factor. For every integer ? ? ??{0,1}, as opposed to every ? ? ???, the class of NDAs with discount factor ? (?-NDAs) has good computational properties: it is closed under determinization and under the algebraic operations min, max, addition, and subtraction, and there are algorithms for its basic decision problems, such as automata equivalence and containment. We define and analyze discounted-sum automata in which each transition can have a different integral discount factor (integral NMDAs). We show that integral NMDAs with an arbitrary choice of discount factors are not closed under determinization and under algebraic operations. We then define and analyze a restricted class of integral NMDAs, which we call tidy NMDAs, in which the choice of discount factors depends on the prefix of the word read so far. Tidy NMDAs are as expressive as deterministic integral NMDAs with an arbitrary choice of discount factors, and some of their special cases are NMDAs in which the discount factor depends on the action (alphabet letter) or on the elapsed time. We show that for every function ? that defines the choice of discount factors, the class of ?-NMDAs enjoys all of the above good properties of integral NDAs, as well as the same complexities of the required decision problems. To this end, we also improve the previously known complexities of the decision problems of integral NDAs, and present tight bounds on the size blow-up involved in algebraic operations on them. All our results hold equally for automata on finite words and for automata on infinite words

Monadic Decomposability of Regular Relations

Monadic decomposibility - the ability to determine whether a formula in a given logical theory can be decomposed into a boolean combination of monadic formulas - is a powerful tool for devising a decision procedure for a given logical theory. In this paper, we revisit a classical decision problem in automata theory: given a regular (a.k.a. synchronized rational) relation, determine whether it is recognizable, i.e., it has a monadic decomposition (that is, a representation as a boolean combination of cartesian products of regular languages). Regular relations are expressive formalisms which, using an appropriate string encoding, can capture relations definable in Presburger Arithmetic. In fact, their expressive power coincide with relations definable in a universal automatic structure; equivalently, those definable by finite set interpretations in WS1S (Weak Second Order Theory of One Successor). Determining whether a regular relation admits a recognizable relation was known to be decidable (and in exponential time for binary relations), but its precise complexity still hitherto remains open. Our main contribution is to fully settle the complexity of this decision problem by developing new techniques employing infinite Ramsey theory. The complexity for DFA (resp. NFA) representations of regular relations is shown to be NLOGSPACE-complete (resp. PSPACE-complete)

Towards the true tree: Bioinformatic approaches in the phylogenetics and molecular evolution of the Endopterygota

In this thesis, I use bioinformatic approaches to address new and existing issues surrounding large-scale phylogenetic analysis. A phylogenetic analysis pipeline is developed to aid an investigation of the suitability of integrating Cytochrome Oxidase Subunit 1 (cox1) into phylogenetic supermatrices. In the first two chapters I assess the effect of varying cox1 sample size within a large variable phylogenetic context. As well as intuitive results on increased quality with greater taxon sampling, there are clear monophyly patters relating to local taxonomic sampling. Specifically, more monophyletic resampled taxa in cases when fewer consubfamilials are represented, with a tendency for these to remain unchanged in the degree of monophyly when rarefied. Sampling analyses are extended in chapter two using a mined Scarabaeoidea multilocus dataset, where taxa from given loci are used to improve existing matrices. Improvement in phylogenetic signal is best achieved by targeting cox1 to existing taxa, which suggests minimum parameters for cox1 adoption in large-scale phylogenetics. In chapter 3 I address recently-arisen issues related to phyloinformatic analysis of sequence-delineated matrices. There is ongoing work on setting species boundaries by sequence variation alone, but incongruence results in methodological issues upon integrating multiple loci delineated in this way. In the final chapter I assess the impact of heterogeneous substitution rates on large scale cox1 datasets. Although the number of heterogeneous sites in Coleoptera cox1 is substantial, their presence is found to be beneficial, as their removal negatively impacts the ability of the alignment to generate the 'known' topology. The homoplasy and heterogeneous characteristics of cox1 have not substantially impacted its utility, thus the cox1 datasets have potential to play a substantial role in the tree-of-life

Algebraic Aspects of Families of Fuzzy Languages

We study operations on fuzzy languages such as union, concatenation, Kleene $\star$, intersection with regular fuzzy languages, and several kinds of (iterated) fuzzy substitution. Then we consider families of fuzzy languages, closed under a fixed collection of these operations, which results in the concept of full Abstract Family of Fuzzy Languages or full AFFL. This algebraic structure is the fuzzy counterpart of the notion of full Abstract Family of Languages that has been encountered frequently in investigating families of crisp (i.e., non-fuzzy) languages. Some simpler and more complicated algebraic structures (such as full substitution-closed AFFL, full super-AFFL, full hyper-AFFL) will be considered as well.\ud In the second part of the paper we focus our attention to full AFFL's closed under iterated parallel fuzzy substitution, where the iterating process is prescribed by given crisp control languages. Proceeding inductively over the family of these control languages, yields an infinite sequence of full AFFL-structures with increasingly stronger closure properties