Mathematical Foundations Of Computer Science Research Articles

Effective Projections on Group Shifts to Decide Properties of Group Cellular Automata

Many decision problems concerning cellular automata are known to be decidable in the case of algebraic cellular automata, that is, when the state set has an algebraic structure and the automaton acts as a morphism. The most studied cases include finite fields, finite commutative rings and finite commutative groups. In this paper, we provide methods to generalize these results to the broader case of group cellular automata, that is, the case where the state set is a finite (possibly non-commutative) finite group. The configuration space is not even necessarily the full shift but a subshift — called a group shift — that is a subgroup of the full shift on [Formula: see text], for any number [Formula: see text] of dimensions. We show, in particular, that injectivity, surjectivity, equicontinuity, sensitivity and nilpotency are decidable for group cellular automata, and non-transitivity is semi-decidable. Injectivity always implies surjectivity, and jointly periodic points are dense in the limit set. The Moore direction of the Garden-of-Eden theorem holds for all group cellular automata, while the Myhill direction fails in some cases. The proofs are based on effective projection operations on group shifts that are, in particular, applied on the set of valid space-time diagrams of group cellular automata. This allows one to effectively construct the traces and the limit sets of group cellular automata. A preliminary version of this work was presented at the conference Mathematical Foundations of Computer Science 2020.

On Information Orders on Metric Spaces

Information orders play a central role in the mathematical foundations of Computer Science. Concretely, they are a suitable tool to describe processes in which the information increases successively in each step of the computation. In order to provide numerical quantifications of the amount of information in the aforementioned processes, S.G. Matthews introduced the notions of partial metric and Scott-like topology. The success of partial metrics is given mainly by two facts. On the one hand, they can induce the so-called specialization partial order, which is able to encode the existing order structure in many examples of spaces that arise in a natural way in Computer Science. On the other hand, their associated topology is Scott-like when the partial metric space is complete and, thus, it is able to describe the aforementioned increasing information processes in such a way that the supremum of the sequence always exists and captures the amount of information, measured by the partial metric; it also contains no information other than that which may be derived from the members of the sequence. R. Heckmann showed that the method to induce the partial order associated with a partial metric could be retrieved as a particular case of a celebrated method for generating partial orders through metrics and non-negative real-valued functions. Motivated by this fact, we explore this general method from an information orders theory viewpoint. Specifically, we show that such a method captures the essence of information orders in such a way that the function under consideration is able to quantify the amount of information and, in addition, its measurement can be used to distinguish maximal elements. Moreover, we show that this method for endowing a metric space with a partial order can also be applied to partial metric spaces in order to generate new partial orders different from the specialization one. Furthermore, we show that given a complete metric space and an inf-continuous function, the partially ordered set induced by this general method enjoys rich properties. Concretely, we will show not only its order-completeness but the directed-completeness and, in addition, that the topology induced by the metric is Scott-like. Therefore, such a mathematical structure could be used for developing metric-based tools for modeling increasing information processes in Computer Science. As a particular case of our new results, we retrieve, for a complete partial metric space, the above-explained celebrated fact about the Scott-like character of the associated topology and, in addition, that the induced partial ordered set is directed-complete and not only order-complete.

Connected Subgraph Defense Games

We study a security game over a network played between a defender and kattackers. Every attacker chooses, probabilistically, a node of the network to damage. The defender chooses, probabilistically as well, a connected induced subgraph of the network of lambda nodes to scan and clean. Each attacker wishes to maximize the probability of escaping her cleaning by the defender. On the other hand, the goal of the defender is to maximize the expected number of attackers that she catches. This game is a generalization of the model from the seminal paper of Mavronicolas et al. Mavronicolas et al. (in: International symposium on mathematical foundations of computer science, MFCS, pp 717–728, 2006). We are interested in Nash equilibria of this game, as well as in characterizing defense-optimal networks which allow for the best equilibrium defense ratio; this is the ratio of k over the expected number of attackers that the defender catches in equilibrium. We provide a characterization of the Nash equilibria of this game and defense-optimal networks. The equilibrium characterizations allow us to show that even if the attackers are centrally controlled the equilibria of the game remain the same. In addition, we give an algorithm for computing Nash equilibria. Our algorithm requires exponential time in the worst case, but it is polynomial-time for lambda constantly close to 1 or n. For the special case of tree-networks, we further refine our characterization which allows us to derive a polynomial-time algorithm for deciding whether a tree is defense-optimal and if this is the case it computes a defense-optimal Nash equilibrium. On the other hand, we prove that it is {mathtt {NP}}-hard to find a best-defense strategy if the tree is not defense-optimal. We complement this negative result with a polynomial-time constant-approximation algorithm that computes solutions that are close to optimal ones for general graphs. Finally, we provide asymptotically (almost) tight bounds for the Price of Defense for any lambda ; this is the worst equilibrium defense ratio over all graphs.

Ideen zu einer Kritik ‚algorithmischer‘ Rationalität

Abstract A critique of algorithmic rationalisation offers at best some initial reasons and preliminary ideas. Critique is understood as a reflection on validity. It is limited to an “epistemological investigation” of the limits of the calculable or of what appears “knowable” in the mode of the algorithmic. The argumentation aims at the mathematical foundations of computer science and goes back to the so-called “foundational crisis of mathematics” at the beginning of the 20th century with the attempt to formalise concepts such as calculability, decidability and provability. The Gödel theorems and Turing’s halting problem prove to be essential for any critical approach to “algorithmic rationalisation”. Both, however, do not provide unambiguous results, at best they run towards what later became known as “Gödel’s disjunction”. The chosen path here, however, suggests the opposite way, insofar as, on the one hand, the topos of creativity appear constitutive for what can be regarded as cognitive “algorithmic rationalisation” and which encounters systematic difficulties in the evaluation of non-trivial results. On the other hand, the investigations lead to a comparison between the “mediality” of formally generated structures, which have to distinguish between object-and metalanguages, and the “volatile” differentiality of human thought, which calls for syntactically non-simulatable sense structures.

Depth-4 Lower Bounds, Determinantal Complexity: A Unified Approach

Tavenas (Proceedings of mathematical foundations of computer science (MFCS), 2013) has recently proved that any $$n^{O(1)}$$ -variate and degree n polynomial in $$\mathsf {VP}$$ can be computed by a depth-4 $$\Sigma \Pi \Sigma \Pi $$ circuit of size $$2^{O(\sqrt{n}\log n)}$$ . So, to prove $$\mathsf {VP}\ne \mathsf {VNP}$$ it is sufficient to show that an explicit polynomial in $$\mathsf {VNP}$$ of degree n requires $$2^{\omega (\sqrt{n}\log n)}$$ size depth-4 circuits. Soon after Tavenas’ result, for two different explicit polynomials, depth-4 circuit-size lower bounds of $$2^{\Omega (\sqrt{n}\log n)}$$ have been proved (see Kayal et al. in Proceedings of symposium on theory of computing, ACM, 2014b. http://doi.acm.org/10.1145/2591796.2591847 ; Fournier et al. in Proceedings of symposium on theory of computing, ACM, 2014). In particular, using a combinatorial design Kayal et al. (2014b) construct an explicit polynomial in $$\mathsf {VNP}$$ that requires depth-4 circuits of size $$2^{\Omega (\sqrt{n}\log n)}$$ and Fournier et al. (Proceedings of symposium on theory of computing, ACM, 2014) show that the iterated matrix multiplication polynomial (which is in $$\mathsf {VP}$$ ) also requires $$2^{\Omega (\sqrt{n}\log n)}$$ size depth-4 circuits. In this paper, we identify a simple combinatorial property such that any polynomial f that satisfies this property would achieve a similar depth-4 circuit-size lower bound. In particular, it does not matter whether f is in $$\mathsf {VP}$$ or in $$\mathsf {VNP}$$ . As a result, we get a simple unified lower-bound analysis for the above-mentioned polynomials. Another goal of this paper is to compare our current knowledge of the depth-4 circuit-size lower bounds and the determinantal complexity lower bounds. Currently, the best known determinantal complexity lower bound is $$\Omega (n^2)$$ for permanent of a $$n\times n$$ matrix (which is a $$n^2$$ -variate and degree n polynomial) due to Cai et al. (Proceedings of symposium on theory of computing, ACM, 2008). We prove that the determinantal complexity of the iterated matrix multiplication polynomial is $$\Omega (dn)$$ where d is the number of matrices and n is the dimension of the matrices. In particular, our result settles the determinantal complexity of the iterated matrix multiplication polynomial to $$\Theta (dn)$$ . To the best of our knowledge, a $$\Theta (n)$$ bound for the determinantal complexity for the iterated matrix multiplication polynomial was known only for any constant $$d>1$$ , due to Jansen (Theory Comput Syst 49(2):343–354, 2011).

Features of development and implementation of courses of the educational field «Mathematics» in distance education system

Effective application of distance technologies, even at the level of individual courses in most universities is faced with a number of organizational, methodological problems. The final design of the regulatory framework the problem of implementation of these technologies in the field of theoretical research and experimental work in the practical implementation of the plane. The authors consider a wide range of problems arising from the mass introduction of e-learning in educational practice, their classification on various grounds. When the authors of the work are based on practical experience in the development of the course Mathematical Foundations of Computer Science for students training areas Teacher Education profiles Informatics and Computer Science and Foreign Language on the basis of a learning management system Moodle. Also studied a number of aspects related to the characteristics of the studied subject area (in Mathematics example of educational area). We formulate the necessary conditions for effective use of e-learning technologies. These conditions, from the point of view of the authors, speakers, among others participated in the implementation of the university administration, the formation of a uniform by the level of the basic mathematical training a group of students, the formation of a thesaurus (semantic web) course in the classroom working with students, as well as a phased approach to solving complex text assignments, requiring the construction of a formal mathematical model in the subject area. We offer some simple criteria for evaluating the effectiveness of (increased performance and students' motivation, optimization of time working teachers, reducing the number of chronic debtors in the discipline).

Embedding signed graphs in the line

Signed graphs are graphs with an assignment of a positive or a negative sign to each edge. These graphs are helpful to represent different types of networks. For instance, they have been used in social networks, where a positive sign in an edge represents friendship between the two endpoints of that edge, while a negative sign represents enmity. Given a signed graph, an important question is how to embed such a graph in a metric space so that in the embedding every vertex is closer to its positive neighbors than to its negative ones. This problem is known as Sitting Arrangement (SA) problem and it was introduced by Kermarrec et al. (Proceedings of the 36th International Symposium on Mathematical Foundations of Computer Science (MFCS), pp. 388---399, 2011). Cygan et al. (Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science (MFCS), 2012) proved that the decision version of SA problem is NP-Complete when the signed graph has to be embedded into the Euclidean line. In this work, we study the minimization version of SA (MinSA) problem in the Euclidean line. We relate MinSA problem to the well known quadratic assignment (QA) problem. We establish such a relation by proving that local minimums in MinSA problem are equivalent to local minimums in a particular case of QA problem. In this document, we design two heuristics based on the combinatorial structure of MinSA problem. We experimentally compare their performances against heuristics designed for QA problem. This comparison favors the proposed heuristics.

On Enumerating Monomials and Other Combinatorial Structures by Polynomial Interpolation

We study the problem of generating the monomials of a black box polynomial in the context of enumeration complexity. We present three new randomized algorithms for restricted classes of polynomials with a polynomial or incremental delay, and the same global running time as the classical ones. We introduce TotalBPP, IncBPP and DelayBPP, which are probabilistic counterparts of the most common classes for enumeration problems. Our interpolation algorithms are applied to algebraic representations of several combinatorial enumeration problems, which are so proved to belong to the introduced complexity classes. In particular, the spanning hypertrees of a 3-uniform hypergraph can be enumerated with a polynomial delay. Finally, we study polynomials given by circuits and prove that we can derandomize the interpolation algorithms on classes of bounded-depth circuits. We also prove the hardness of some problems on polynomials of low degree and small circuit complexity, which suggests that our good interpolation algorithm for multilinear polynomials cannot be generalized to degree 2 polynomials. This article is an improved and extended version of Strozecki (Mathematical Foundations of Computer Science, pp. 629---640, 2010) and of the third chapter of the author's Ph.D. Thesis (Strozecki, Ph.D. Thesis, 2010).

An [formula omitted] algorithm for hyper-minimizing a (minimized) deterministic automaton

We improve a recent result [A. Badr, Hyper-minimization in O ( n 2 ) , Internat. J. Found. Comput. Sci. 20 (4) (2009) 735–746] for hyper-minimized finite automata. Namely, we present an O ( n log n ) algorithm that computes for a given deterministic finite automaton (dfa) an almost-equivalent dfa that is as small as possible—such an automaton is called hyper-minimal. Here two finite automata are almost-equivalent if and only if the symmetric difference of their languages is finite. In other words, two almost-equivalent automata disagree on acceptance on finitely many inputs. In this way, we solve an open problem stated in [A. Badr, V. Geffert, I. Shipman, Hyper-minimizing minimized deterministic finite state automata, RAIRO Theor. Inf. Appl. 43 (1) (2009) 69–94] and by Badr. Moreover, we show that minimization linearly reduces to hyper-minimization, which shows that the time-bound O ( n log n ) is optimal for hyper-minimization. Independently, similar results were obtained in [P. Gawrychowski, A. Jeż, Hyper-minimisation made efficient, in: Proc. 34th Int. Symp. Mathematical Foundations of Computer Science, in: LNCS, vol. 5734, Springer, 2009, pp. 356–368].

Progresses in the analysis of stochastic 2D cellular automata: A study of asynchronous 2D minority

Cellular automata are often used to model systems in physics, social sciences, biology that are inherently asynchronous. Over the past 20 years, studies have demonstrated that the behavior of cellular automata drastically changes under asynchronous updates. Still, the few mathematical analyses of asynchronism focus on 1D probabilistic cellular automata, either on single examples or on specific classes. As for other classic dynamical systems in physics, extending known methods from 1D to 2D systems is a long lasting challenging problem. In this paper, we address the problem of analyzing an apparently simple 2D asynchronous cellular automaton: 2D Minority where each cell, when fired, updates to the minority state of its neighborhood. Our simulations reveal that in spite of its simplicity, the minority rule exhibits a quite complex response to asynchronism. By focusing on the fully asynchronous regime, we are however able to describe completely the asymptotic behavior of this dynamics as long as the initial configuration satisfies some natural constraints. Besides these technical results, we have strong reasons to believe that our techniques relying on defining an energy function from the transition table of the automaton may be extended to the wider class of threshold automata. 2 2 An abstract version of this paper has been published in [D. Regnault, N. Schabanel, É. Thierry, Progresses in the analysis of stochastic 2D cellular automata: A study of asynchronous 2D minority, in: LNCS Proc. of 32nd Symp. of Mathematical Foundations of Computer Science, MFCS, vol. 4708, Springer, 2007, pp. 320–332].

Complexity of DNF minimization and isomorphism testing for monotone formulas

We investigate the complexity of finding prime implicants and minimum equivalent DNFs for Boolean formulas, and of testing equivalence and isomorphism of monotone formulas. For DNF related problems, the complexity of the monotone case differs strongly from the arbitrary case. We show that it is DP-complete to check whether a monomial is a prime implicant for an arbitrary formula, but the equivalent problem for monotone formulas is in L. We show PP-completeness of checking if the minimum size of a DNF for a monotone formula is at most k, and for k in unary, we show that the complexity of the problem drops to coNP. In Christopher Umans [Christopher Umans, The minimum equivalent DNF problem and shortest implicants, Journal of Computer and System Sciences 63 (4) (2001) 597–611] a similar problem for arbitrary formulas was shown to be ∑ 2 p -complete. We show that calculating the minimum equivalent DNF for a monotone formula is possible in output-polynomial time if and only if P = NP. Finally, we disprove a conjecture from Steffen Reith [Steffen Reith, On the complexity of some equivalence problems for propositional calculi, in: Proceedings of the 28th International Symposium on Mathematical Foundations of Computer Science (MFCS), vol. 2747, Lecture Notes in Computer Science, Springer, 2003, pp. 632–641] by showing that checking whether two formulas are isomorphic has the same complexity for arbitrary formulas as for monotone formulas.

Monotonicity of non-deterministic graph searching

In graph searching, a team of searchers are aiming at capturing a fugitive moving in a graph. In the initial variant, called invisible graph searching, the searchers do not know the position of the fugitive until they catch it. In another variant, the searchers permanently know the position of the fugitive, i.e. the fugitive is visible. This latter variant is called visible graph searching. A search strategy that catches any fugitive in such a way that the part of the graph reachable by the fugitive never grows is called monotone. A priori, monotone strategies may require more searchers than general strategies to catch any fugitive. This is however not the case for visible and invisible graph searching. Two important consequences of the monotonicity of visible and invisible graph searching are: (1) the decision problem corresponding to the computation of the smallest number of searchers required to clear a graph is in NP, and (2) computing optimal search strategies is simplified by taking into account that there exist some that never backtrack. Fomin et al. [F.V. Fomin, P. Fraigniaud, N. Nisse, Nondeterministic graph searching: From pathwidth to treewidth, in: Proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science, MFCS’05, 2005, pp. 364–375] introduced an important graph searching variant, called non-deterministic graph searching, that unifies visible and invisible graph searching. In this variant, the fugitive is invisible, and the searchers can query an oracle that permanently knows the current position of the fugitive. The question of the monotonicity of non-deterministic graph searching was however left open. In this paper, we prove that non-deterministic graph searching is monotone. In particular, this result is a unified proof of monotonicity for visible and invisible graph searching. As a consequence, the decision problem corresponding to non-deterministic graph searching belongs to NP. Moreover, the exact algorithms designed by Fomin et al. do compute optimal non-deterministic search strategies.

Efficient implementation of algorithms for approximate exponentiation

We present efficient implementations of algorithms for the following fundamental problem: Given as input three positive integers x, y and j, compute the leading j digits of x y . A special case of this problem ( k = 2 and j = 1 ) was recently studied by Hirvensalo and Karhumaki [M.Hirvensalo, J.Karhumaki, Computing partial information out of uncomputable one—The first digit of 2 n to base 3 as an example, in: Mathematical Foundations of Computer Science, in: Lecture Notes in Computer Science, Springer-Verlag, Inc., 2002] for which they presented a polynomial time algorithm. Specifically an algorithm of bit complexity O ( n 2 log 3 n log log n ) where n = | y | is the number of digits in y. But their algorithm is not efficient in practice. For example, finding the leading digit of 2 y where y is a 500 digit positive integer takes several hours. Hirvensalo and Karhumaki's algorithm is based on computing a rational approximation to ln 2 (and a few other constants) to a high-degree of precision. Our approach is fundamentally different from theirs: we use a modified addition chain algorithm in which the multiplication is truncated to varying number of digits at various steps, followed by a self-tester that validates the truncation. Our algorithm runs several orders of magnitude faster. For example, on an input in which x and y have a few thousand digits, our program computes the leading 1000 digits in under 3 minutes. Since the approximate exponentiation has many application [B.Ravikumar, A Las Vegas randomized approximation algorithm for approximate exponentiation and its applications, in preparation] we hope that our algorithm will stimulate further research on this problem.

Pseudorandom Bits for Constant‐Depth Circuits with Few Arbitrary Symmetric Gates

We exhibit an explicitly computable pseudorandom generator stretching l bits into $m(l) = l^{\Omega(\log l)}$ bits that look random to constant‐depth circuits of size $m(l)$ with $\log m(l)$ arbitrary symmetric gates (e.g., PARITY, MAJORITY). This improves on a generator by Luby, Velickovic, and Wigderson [Proceedings of the Second Israel Symposium on Theory of Computing Systems, 1993, pp. 18–24] that achieves the same stretch but fools only circuits of depth 2 with one arbitrary symmetric gate at the top. Our generator fools a strictly richer class of circuits than Nisan’s generator for constant‐depth circuits (but Nisan’s generator has a much bigger stretch) [Combinatorica, 11 (1991), pp. 63–70]. In particular, we conclude that every function computable by uniform $\poly(n)$‐size probabilistic constant‐depth circuits with $O(\log n)$ arbitrary symmetric gates is in $\mathit{TIME}(2^{n^{o(1)}})$. This seems to be the richest probabilistic circuit class known to admit a subexponential derandomization. Our generator is obtained by constructing an explicit function $f : \zo^n \to \zo$ that is very hard on average for constant‐depth circuits of size $s(n) = n^{\Omega(\log n)}$ with $\log s(n)$ arbitrary symmetric gates, and plugging it into the Nisan–Wigderson pseudorandom generator construction [J. Comput. System Sci., 49 (1994), pp. 149–167]. The proof of the average‐case hardness of this function is a modification of arguments by Razborov and Wigderson [Inform. Process. Lett., 45 (1993), pp. 303–307] and Hansen and Miltersen [Proceedings of the 29th International Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Comput. Sci. 3153, Springer‐Verlag, Berlin, 2004, pp. 334–345] and combines Håstad’s switching lemma [Computational Limitations of Small‐Depth Circuits, MIT Press, Cambridge, MA, 1987] with a multiparty communication complexity lower bound by Babai, Nisan, and Szegedy [J. Comput. System Sci., 45 (1992), pp. 204–232].

The Dual Complexity Space as the Dual of a Normed Cone

In [M. Schellekens, The Smyth completion: A common foundation for denotational semantics and complexity analysis, in: Proc. MFPS 11, Electronic Notes in Theoretical Computer Science, vol. 1 (1995), 23 pages] M. Schellekens introduced the complexity (quasi-metric) space as a part of the research in Theoretical Computer Science and Topology, with applications to the complexity analysis of algorithms. Later on, S. Romaguera and M. Schellekens ([S. Romaguera, M. Schellekens, Quasi-metric properties of complexity spaces, Topology Appl. 98 (1999), 311–322]) introduced the so-called dual complexity (quasi-metric) space and established several quasi-metric properties of the complexity space via the analysis of th e dual. These authors also proved in [S. Romaguera, M. Schellekens, Duality and quasi-normability for complexity spaces, Appl. Gen. Topology 3 (2002), 91–112] that actually the dual complexity space C∗ can be modeled as a norm-weightable cone whose induced quasi-metric is Smyth complete. This fact suggests the existence of deep connections between a general theory of (dual) complexity spaces and Asymmetric Functional Analysis. These connections have been recently explored in [L.M. Garcá-Raffi, S. Romaguera, E.A. Sánchez-Pérez, Sequence spaces and asymmetric norms in the theory of compuational complexity, Math. Comput. Model 36 (2002), 1–11], [L.M. Garcá-Raffi, S. Romaguera, E.A. Sánchez-Pérez, The supremum asymmetric norm on sequence spaces: a general framework to measure complexity distances, in: Proceedings of the Second Irish Conference on the Mathematical Foundations of Computer Science and Information Technology (MFCSIT 2002), Galway, Ireland, July 2002; Electronic Notes in Theoret. Comput. Sci. 74 (2003), URL: http://www.elsevier.nl/locate/entcs/volume74.htm 12 pages] and [M. O'Keefe, S. Romaguera, M. Schellekens, Norm-weightable Riesz spaces and the dual complexity space, in: Proceedings of the Second Irish Conference on the Math ematical Foundations of Computer Science and Information Technology (MFCSIT 2002), Galway, Ireland, July 2002; Electronic Notes in Theoret. Comput. Sci. 74 (2003), URL: http://www.elsevier.nl/locate/entcs/volume74.htm 17 pages]. In particular, it was proved in [L.M. Garcá-Raffi, S. Romaguera, E.A. Sánchez-Pérez, Sequence spaces and asymmetric norms in the theory of compuational complexity, Math. Comput. Model 36 (2002), 1–11] that the so-called dual p-complexity space Cp∗, with 1⩽p<∞, is isometrically isomorphic to the positive cone of the classical Banach space lp. The space C1∗ is exactly the dual complexity space, and thus it is isometrically isomorphic to the positive cone of the Banach space l1 of all absolutely summable real sequences. Here, we continue the analysis of the structure of the dual complexity space C∗. We show that it is the dual space of the positive c one of the Banach space c0 of all real sequences converging to zero, and that its dual space is the positive cone of the Banach space l∞ of all bounded real sequences. Furthermore, the dual space of Cp∗, 1<p<∞, is Cq∗ where 1/p+1/q=1. These results extend to this setting well-known theorems of the classical theory of Functional Analysis.

Improving a fixed parameter tractability time bound for the shadow problem

Consider a forest of k trees and n nodes together with a (partial) function σ mapping leaves of the trees to non-root nodes of other trees. Define the shadow of a leaf ℓ to be the subtree rooted at σ(ℓ). The shadow problem asks whether there is a set S of leaves exactly one from each tree such that none of these leaves lies in the shadow of another leaf in S. This graph theoretical problem as shown in Franco et al. (Discrete Appl. Math. 96 (1999) 89) is equivalent to the falsifiability problem for pure implicational Boolean formulas over n variables with k occurences of the constant false as introduced in: Heusch J. Wiedermann, P. Hajek (Eds.), Proceedings of the Twentieth International Symposium on Mathematical Foundations of Computer Science (MFCS’95), Prague, Czech Republic, Lecture Notes in Computer Science, Vol. 969, Springer, Berlin, 1995, pp. 221–226, where its NP-completeness is shown for arbitrary values of k and a time bound of O( n k ) for fixed k was obtained. In Franco et al. (1999) this bound is improved to O( n 2 k k ) showing the problem's fixed parameter tractability (Congr. Numer. 87 (1992) 161). In this paper the bound O( n 33 k ) is achieved by dynamic programming techniques thus significantly improving the fixed parameter part.

Preface: Volume 74

This volume contains the Proceedings of the Second Irish Conference on the Mathematical Foundations of Computer Science and Information Technology (MFCSIT 2002). The conference was a two-day event and took place in the National University of Ireland, Galway, (NUI, Galway) on18th and 19st July, 2002.MFCSIT2002 is the second of a series of conferences to be held in Ireland whose main focus is the mathematical and foundational issues arising out of computer science and information technology. The first of these conferences, MFCSIT2000, was held at NUI, Cork on 20/21 July2000. The Proceedings were published as Volume 40 of the Electronic Notes in Theoretical Computer Science (ENTCS), (T. Hurley, M. Airchinnigh, M. Schellekens and A. Seda (eds.))The following extract from the introduction to the first conference is well worth repeating: “It is now widely recognised that there has been a very significant increase in Ireland over the last few years in activities relating to the production of software and hardware by many computer manufacturers, including some of the world's best-known names in the computer industry. However, it was felt by a number of people˙˙ that a need had emerged for a forum which could provide the means for discussion ˙˙ of related theoretical questions, and so help maintain a balance between theory and practice. The MFCSIT conferences are intended to provide such a forum and hence to help maintain this balance.”It is my pleasure to record a debt of gratitude to a considerable number of people who made the meeting the success it was, both on my own behalf and on behalf of my fellow editors: Sharon Flynn, Mícheál Mac an Airchinnigh, Niall Madden, Michael McGettrick, Michel Schellekens, and Anthony Seda.We were fortunate in having seven internationally well-known speakers who gave key-note addresses: Steve Brookes, Matthew Hennessey, Derek Holt, Peter Jeavons, Bill Lawvere, David MacKay and Mark Sofroniou. The speakers brilliantly set the themes for this conference with their fascinating and stimulating talks. We are indeed grateful to these for their participation and talks - many of the participants expressed appreciation of the quality and depth of these key-note addresses.We had two special sessions, one on Coding Theory organised by Pat Fitzpatrick, and one on Topology in Computer Science organised by Michel Schellekens and Anthony Seda. Our sincere thanks go to the organisers of these special sessions, and to the invited speakers for those sessions: Achim Jung, Mike Smyth, Dieter Spreen, David McKay, Edgar Martinez-Moro and Eimear Byrne.On the editing side, we are extremely grateful to all those who refereed the proceedings. We are particularly indebted to Niall Madden who bore the brunt of the editing and preparation of the proceedings; his expertise was invaluable.A conference cannot exist, of course, without participants, so our thanks to those who attended. (Altogether there were 57 registered participants plus a number of occasional visitors.) We are grateful also to those who submitted papers, to those who chaired sessions, and to Jimmie Browne, Registrar and Deputy-President of NUI, Galway, who opened the conference.Finally, we wish to acknowledge the following sponsors, without whose support the conference could not have taken place: Science Foundation IrelandThe Registrar's Office, NUI, GalwayThe Mathematics Department, NUI, GalwayThe Information Technology Department, NUI, GalwayThe School of Mathematics, Applied Mathematics and Statistics, NUI, CorkThe Computer Science Department, NUI, CorkSepember 19, 2003Ted Hurley

Speeding up the detection of evolutive tandem repeats

We recently introduced evolutive tandem repeats with jump (using Hamming distance) (Proc. MFCS’02: the 27th Internat. Symp. Mathematical Foundations of Computer Science, Warszawa, Otwock, Poland, August 2002, Lecture Notes in Computer Science, Vol. 2420, Springer, Berlin, pp. 292–304) which consist in a series of almost contiguous copies having the following property: the Hamming distance between two consecutive copies is always smaller than a given parameter e. In this article, we present a significative improvement that speeds up the detection of evolutive tandem repeats. It is based on the progressive computation of distances between candidate copies participating to the evolutive tandem repeat. It leads to a new algorithm, still quadratic in the worst case, but much more efficient on average, authorizing larger sequences to be processed.

Randomness, Computability, and Density

We study effectively given positive reals (more specifically, computably enumerable reals) under a measure of relative randomness introduced by Solovay [manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, NY, 1975] and studied by Calude, Hertling, Khoussainov, and Wang [Theoret. Comput. Sci., 255 (2001), pp. 125--149], Calude [Theoret. Comput. Sci., 271 (2002), pp. 3--14], Kucera and Slaman [SIAM J. Comput., 31 (2002), pp. 199--211], and Downey, Hirschfeldt, and LaForte [Mathematical Foundations of Computer Science 2001, Springer-Verlag, Berlin, 2001, pp. 316--327], among others. This measure is called domination or Solovay reducibility and is defined by saying that $\alpha$ dominates $\beta$ if there are a constant c and a partial computable function $\varphi$ such that for all positive rationals $q < \alpha$ we have $\varphi(q)\!\downarrow < \beta$ and $\beta- \varphi(q) \leqslant c(\alpha- q)$. The intuition is that an approximating sequence for $\alpha$generates one for $\beta$ whose rate of convergence is not much slower than that of the original sequence. It is not hard to show that if $\alpha$ dominates $\beta$, then the initial segment complexity of $\alpha$ is at least that of $\beta$. In this paper we are concerned with structural properties of the degree structure generated by Solovay reducibility. We answer a natural question in this area of investigation by proving the density of the Solovay degrees. We also provide a new characterization of the random computably enumerable reals in terms of splittings in the Solovay degrees. Specifically, we show that the Solovay degrees of computably enumerable reals are dense, that any incomplete Solovay degree splits over any lesser degree, and that the join of any two incomplete Solovay degrees is incomplete, so that the complete Solovay degree does not split at all. The methodology is of some technical interest, since it includes a priority argument in which the injuries are themselves controlled by randomness considerations.

Preface: Volume 40

This volume contains the Proceedings of the First Irish Conference on the Mathematical Foundations of Computer Science and Information Technology (MFCSIT2000). The conference was a two-day event and took place in the National University of Ireland, Cork, on 20th and 21st July, 2000.MFCSIT2000 was the first conference to be held in Ireland whose main focus was the mathematical and foundational issues arising out of computer science and information technology. It is now widely recognised that there has been a very significant increase in Ireland over the last few years in activities relating to the production of software and hardware by many computer manufacturers, including some of the world's best-known names in the computer industry. However, it was felt by a number of people, including the guest editors of this volume, that a need had emerged for a forum which could provide the means for discussion, within Ireland, of related theoretical questions, and so help maintain a balance between theory and practice. The MFCSIT conferences are intended to provide such a forum and hence to help maintain this balance.It is my pleasure to record a debt of gratitude to a considerable number of people who made the meeting the success it was, both on my own behalf and on behalf of my fellow editors: Ted Hurley, Mícheál Mac an Airchinnigh, and Michel Schellekens. First, we were fortunate in having five internationally-known speakers who gave key-note addresses: Jaco de Bakker, Abbas Edalat, Dick Hamlet, Giorgio Levi, and Dana Scott. Not only did they give fascinating and stimulating talks, but they also represented the main themes of the conference amongst the many possible themes which legitimately fall within the wide scope of its title. We were fortunate also in having a Special Session on Security and Concurrency as part of the conference. This session was undoubtedly a very considerable success in its own right, and our thanks are due to Matthew Hennessy and Mike Mislove who so ably organized it.Thanks are also due to Simon Foley and Michel Schellekens who greatly helped me as members of the local organizing committee, to student helpers Eleanor Clifford and Daithi O'Crualaoich, and special thanks to Pascal Hitzler for sterling work in preparing websites, processing abstracts and for undertaking many other tasks. On the editing side, sincere thanks to all those who refereed the proceedings, and to Professor Mike Mislove for much assistance in preparing final versions of papers. A conference cannot exist, of course, without participants, so our thanks to those who attended, to those who submitted papers and to those who chaired sessions. Thanks too to Michael O'Sullivan, NUI-Cork's Vice-President for Planning, Communications and Development, for opening the meeting.Finally, we wish to acknowledge the support of our sponsors, as follows.The Arts Faculty, NUI, CorkThe Department of Computer Science, NUI, CorkThe Department of Mathematics, NUI, GalwayThe School of Mathematics, Applied Mathematics and Statistics, NUI, CorkEMC Ireland, CorkEnterprise IrelandLogic Programming Associates, LondonThe National Software Directorate, DublinPFH Computers, Cork