The Calculus of Relations as a Foundation for Mathematics

A variable-free, equational logic $\mathcal{L}^\timesA variable-free, equational logic based on the calculus of relations (a theory of binary relations developed by De Morgan, Peirce, and Schr?der during the period 1864–1895) is shown to provide an adequate framework for the development of all of mathematics. The expressive and deductive powers of are equivalent to those of a system of first-order logic with just three variables. Therefore, three-variable first-order logic also provides an adequate framework for mathematics. Finally, it is shown that a variant of may be viewed as a subsystem of sentential logic. Hence, there are subsystems of sentential logic that are adequate to the task of formalizing mathematics. This paper is an expanded version of a talk given by the author at the Special Session on Automated Reasoning in Mathematics and Logic, held March 8–10, 2002, at the Georgia Institute of Technology, during the Joint Southeastern Section MAA/Southeast Regional AMS Meeting. The session was organized by Johan G. F. Belinfante.     

Tuples of Disjoint $\mathsf{NP}$ -Sets

Disjoint -pairs are a well studied complexity-theoretic concept with important applications in cryptography and propositional proof complexity. In this paper we introduce a natural generalization of the notion of disjoint -pairs to disjoint k-tuples of -sets for k≥2. We define subclasses of the class of all disjoint k-tuples of -sets. These subclasses are associated with a propositional proof system and possess complete tuples which are defined from the proof system. In our main result we show that complete disjoint -pairs exist if and only if complete disjoint k-tuples of -sets exist for all k≥2. Further, this is equivalent to the existence of a propositional proof system in which the disjointness of all k-tuples is shortly provable. We also show that a strengthening of this conditions characterizes the existence of optimal proof systems. An extended abstract of this paper appeared in the proceedings of the conference CSR 2006 (Lecture Notes in Computer Science 3967, 80–91, 2006). Supported by DFG grant KO 1053/5-1.     

Representing and Building Models for Decidable Subclasses of Equational Clausal Logic

We develop new techniques for the automated construction of models for subclasses of equational clausal logic. The models are represented by some specific class of rewrite rules. We show that the evaluation of arbitrary first-order formulae in these interpretations is a decidable problem. As an example of an application, we consider the class , a decidable subclass of first-order clausal logic without equality. In the equational case, is undecidable, but it is known to be decidable if all the equational literals are ground. We extend this result to a class of clause sets possibly containing nonground equational literals. The algorithms for extracting models from positively disconnected saturated sets proposed by Fermüller and Leitsch are extended in order to handle the full ordering restrictions of the resolution/paramodulation calculus.     

An Efficient Approach to Solving Random k-sat Problems

Proving that a propositional formula is contradictory or unsatisfiable is a fundamental task in automated reasoning. This task is coNP-complete. Efficient algorithms are therefore needed when formulae are hard to solve. Random sat formulae provide a test-bed for algorithms because experiments that have become widely popular show clearly that these formulae are consistently difficult for any known algorithm. Moreover, the experiments show a critical value of the ratio of the number of clauses to the number of variables around which the formulae are the hardest on average. This critical value also corresponds to a ‘phase transition’ from solvability to unsolvability. The question of whether the formulae located around or above this critical value can efficiently be proved unsatisfiable on average (or even for a.e. formula) remains up to now one of the most challenging questions bearing on the design of new and more efficient algorithms. New insights into this question could indirectly benefit the solving of formulae coming from real-world problems, through a better understanding of some of the causes of problem hardness. In this paper we present a solving heuristic that we have developed, devoted essentially to proving the unsatisfiability of random sat formulae and inspired by recent work in statistical physics. Results of experiments with this heuristic and its evaluation in two recent sat competitions have shown a substantial jump in the efficiency of solving hard, unsatisfiable random sat formulae.     

Feasible Operations on Proofs: The Logic of Proofs for Bounded Arithmetic

This paper presents a semantics for the logic of proofs in which all the operations on proofs are realized by feasibly computable functions. More precisely, we will show that the completeness of for the semantics of proofs of Peano Arithmetic extends to the semantics of proofs in Buss’ bounded arithmetic . In view of applications in epistemology of in particular and justification logics in general this result shows that explicit knowledge in the propositional framework can be made computationally feasible. This research supported by CUNY Community College Collaborative Incentive Research Grant 91639-0001 “Mathematical Foundations of Knowledge Representation”.     

Formal Correctness of a Quadratic Unification Algorithm

We present a case study using ACL2 to verify a nontrivial algorithm that uses efficient data structures. The algorithm receives as input two first-order terms, and it returns a most general unifier of these terms if they are unifiable, failure otherwise. The verified implementation stores terms as directed acyclic graphs by means of a pointer structure. Its time complexity is and its space complexity , and it can be executed in ACL2 at a speed comparable to a similar C implementation. We report the main issues encountered to achieve this formally verified implementation. This work has been supported by project TIN2004-03884 (Ministerio de Educación y Ciencia, Spain) and FEDER funds.     

The Expected Size of the Rule k Dominating Set

Dai, Li, and Wu proposed Rule k, a localized approximation algorithm that attempts to find a small connected dominating set in a graph. In this paper we consider the "average-case" performance of two closely related versions of Rule k for the model of random unit disk graphs constructed from n random points in an square. We show that if and then for both versions of Rule k, the expected size of the Rule k dominating set is as It follows that, for in a suitable range, the expected size of the Rule k dominating sets are within a constant factor of the optimum.     

Probabilistic Length-Reducing Two-Pushdown Automata

Hardness of a separation of nondeterminism, randomization and determinism for polynomial time computations has motivated the analysis of this issue for restricted models of computation. Following this line of research, we consider randomized length-reducing two-pushdown automata ( ), a natural extension of pushdown automata ( ). Our main results are as follows. We show that deterministic s are weaker than Las Vegas s which in turn are weaker than Monte Carlo s. Moreover, bounded two-sided error s are stronger than Monte Carlo s and they are able to recognize some languages which cannot be recognized nondeterministically. Finally, we prove that amplification is impossible for Las Vegas and Monte Carlo automata. Partially supported by MNiSW grant number N206 024 31/3826, 2006-2008. An extended abstract of this paper appeared in the MFCS06 Proceedings (Lecture Notes in Computer Science, vol. 4162, pp. 561–572, 2006).     

Efficient Algorithms for k Maximum Sums

We study the problem of computing the k maximum sum subsequences. Given a sequence of real numbers and an integer parameter k, the problem involves finding the k largest values of for The problem for fixed k = 1, also known as the maximum sum subsequence problem, has received much attention in the literature and is linear-time solvable. Recently, Bae and Takaoka presented a -time algorithm for the k maximum sum subsequences problem. In this paper we design an efficient algorithm that solves the above problem in time in the worst case. Our algorithm is optimal for and improves over the previously best known result for any value of the user-defined parameter k < 1. Moreover, our results are also extended to the multi-dimensional versions of the k maximum sum subsequences problem; resulting in fast algorithms as well.     

Convex Drawings of 3-Connected Plane Graphs

We use Schnyder woods of 3-connected planar graphs to produce convex straight-line drawings on a grid of size The parameter depends on the Schnyder wood used for the drawing. This parameter is in the range The algorithm is a refinement of the face-counting algorithm; thus, in particular, the size of the grid is at most The above bound on the grid size simultaneously matches or improves all previously known bounds for convex drawings, in particular Schnyder's and the recent Zhang and He bound for triangulations and the Chrobak and Kant bound for 3-connected planar graphs. The algorithm takes linear time. The drawing algorithm has been implemented and tested. The expected grid size for the drawing of a random triangulation is close to For a random 3-connected plane graph, tests show that the expected size of the drawing is      

A semantically complete extension sequence of the system $$\mathcal{L}^* $$

In this paper, the method of well-combined semantics and syntax proposed by Pavelka is applied to the research of the propositional calculus formal system . The partial constant values are taken as formulas, formulas are fuzzified in two manners of semantics and syntax, and inferring processes are fuzzified. A sequence of new extensions { } of the system is proposed, and the completeness of is proved.     

Exponential Lower Bounds for the Running Time of DPLL Algorithms on Satisfiable Formulas

DPLL (for Davis, Putnam, Logemann, and Loveland) algorithms form the largest family of contemporary algorithms for SAT (the propositional satisfiability problem) and are widely used in applications. The recursion trees of DPLL algorithm executions on unsatisfiable formulas are equivalent to treelike resolution proofs. Therefore, lower bounds for treelike resolution (known since the 1960s) apply to them. However, these lower bounds say nothing about the behavior of such algorithms on satisfiable formulas. Proving exponential lower bounds for them in the most general setting is impossible without proving P ≠ NP; therefore, to prove lower bounds, one has to restrict the power of branching heuristics. In this paper, we give exponential lower bounds for two families of DPLL algorithms: generalized myopic algorithms, which read up to n ^1−ε of clauses at each step and see the remaining part of the formula without negations, and drunk algorithms, which choose a variable using any complicated rule and then pick its value at random. ^★Extended abstract of this paper appeared in Proceedings of ICALP 2004, LNCS 3142, Springer, 2004, pp. 84–96. ^†Supported by CCR grant CCR-0324906. ^‡Supported in part by Russian Science Support Foundation, RAS program of fundamental research “Research in principal areas of contemporary mathematics,” and INTAS grant 04-77-7173. ^§Supported in part by INTAS grant 04-77-7173.     

An $\mathsf{EDF}$ -based restricted-migration scheduling algorithm for multiprocessor soft real-time systems

There has been much recent interest in the use of the earliest-deadline-first ( ) algorithm for scheduling soft real-time sporadic task systems on identical multiprocessors. In hard real-time systems, a significant disparity exists between -based schemes and Pfair scheduling: on M processors, the worst-case schedulable utilization for all known variants is approximately M/2, whereas it is M for optimal Pfair algorithms. This is unfortunate because -based algorithms entail lower scheduling and task-migration overheads. However, such a disparity in schedulability can be alleviated by easing the requirement that all deadlines be met, which may be sufficient for soft real-time systems. In particular, in recent work, we have shown that if task migrations are not restricted, then (i.e. , global ) can ensure bounded tardiness for a sporadic task system with no restrictions on total utilization. Unrestricted task migrations in global may be unappealing for some systems, but if migrations are forbidden entirely, then bounded tardiness cannot be guaranteed. In this paper, we address the issue of striking a balance between task migrations and system utilization by proposing an algorithm called , which is based upon and treads a middle path, by restricting, but not eliminating, task migrations. Specifically, under , the ability to migrate is required for at most M−1 tasks, and it is sufficient that every such task migrate between two processors and at job boundaries only. , like global , can ensure bounded tardiness to a sporadic task system as long as the available processing capacity is not exceeded, but, unlike global , may require that per-task utilizations be capped. The required cap is quite liberal, hence, should enable a wide range of soft real-time applications to be scheduled with no constraints on total utilization.

Some Relations between Approximation Problems and PCPs over the Real Numbers

In [10] it was recently shown that that is the existence of transparent long proofs for was established. The latter denotes the class of real number decision problems verifiable in polynomial time as introduced by Blum et al. [6]. The present paper is devoted to the question what impact a potential full real number theorem would have on approximation issues in the BSS model of computation. We study two natural optimization problems in the BSS model. The first, denoted by MAX-QPS, is related to polynomial systems; the other, MAX-q-CAP, deals with algebraic circuits. Our main results combine the PCP framework over with approximation issues for these two problems. We also give a negative approximation result for a variant of the MAX-QPS problem.     

On the Complexity of 2-Monotone Restarting Automata

The -automaton is the weakest form of the nondeterministic version of the restarting automaton that was introduced by Jančar et al. to model the so-called analysis by reduction. Here it is shown that the class ℒ(R) of languages that are accepted by -automata is incomparable under set inclusion to the class of Church-Rosser languages and to the class of growing context-sensitive languages. In fact this already holds for the class of languages that are accepted by 2-monotone -automata. In addition, we prove that already the latter class contains -complete languages, showing that already the 2-monotone -automaton has a surprisingly large expressive power. The results of this paper have been announced at DLT 2004 in Auckland, New Zealand. This work was mainly carried out while T. Jurdziński was visiting the University of Kassel, supported by a grant from the Deutsche Forschungsgemeinschaft (DFG). F. Mráz and M. Plátek were partially supported by the Grant Agency of the Czech Republic under Grant-No. 201/04/2102 and by the program ‘Information Society’ under project 1ET100300517. F. Mráz was also supported by the Grant Agency of Charles University in Prague under Grant-No. 358/2006/A-INF/MFF.     

Cardinal interpolation with periodic polysplines on strips

Abstract  We obtain a multivariate extension of a classical result of Schoenberg on cardinal spline interpolation. Specifically, we prove the existence of a unique function in , polyharmonic of order p on each strip , , and periodic in its last n variables, whose restriction to the parallel hyperplanes , , coincides with a prescribed sequence of n-variate periodic data functions satisfying a growth condition in . The constructive proof is based on separation of variables and on Micchelli’s theory of univariate cardinal -splines. Keywords: cardinal -splines, polyharmonic functions, multivariable interpolation Mathematics Subject Classification (2000): 41A05, 41A15, 41A63