Narrow Results

In a recent article, the class of functions from the integers to the integers computable in polynomial time has been characterized using discrete ordinary differential equations (ODE), also known as finite differences. Doing so, the authors pointed out the fundamental role of linear (discrete) ODEs and classical ODE tools such as changes of variables to capture computability and complexity measures, or as a tool for programming.

In this article, we extend the approach to a characterization of functions from the integers to the reals computable in polynomial time in the sense of computable analysis. In particular, we provide a characterization of such functions in terms of the smallest class of functions that contains some basic functions, and that is closed by composition, linear length ODEs, and a natural effective limit schema.

This research paper aims to redefine the complexity factor in $f(R,G)$ gravity and the appearance of electric charge, where $R$ is the Ricci scalar and $G$ is the Gauss–Bonnet term. In this context, we intend to analyze the splitting of the Riemann tensor after considering the anisotropic distribution of the charged fluid related to the spherically symmetric spacetime. We interpret $Y_{\mathrm{\text{TF}}}$ as the complexity factor among all the determined structural scalars that encompasses the characteristics of anisotropic pressure and the effective representation of the energy density. The correction terms associated with modified theory are considered to calculate some significant results related to the Weyl scalar, Tolman mass, and the complexity factor (CF). Moreover, the expression for CF is established by using the structure scalars determined in our paper, and the diminishing complexity restraint is utilized to determine the solutions for the various models. The celestial object having non-uniform energy density and anisotropic pressure asserts the maximum intricacy. But, if the effects of non-uniform energy density and anisotropic distribution of pressure are eradicated due to the presence of dark source terms associated with modified gravity then these fluids may not exhibit any complexity. Consequently, it is revealed that the constituents of effective and electromagnetic parts directly influence the structure scalars and CF.

We formulate a microscopic model of the stock market and study the resulting macroscopic phenomena via simulation. In a market of homogeneous investors periodic booms and crashes in stock price are obtained, When there are two types of investors in the market, differing only in their memory spans, we observe sharp irregular transitions between eras where one population dominates the market and eras where the other population dominates. When the number of investor subgroups is three the market undergoes a dramatic qualitative change — it becomes complex. We show that complexity is an intrinsic property of the stock market. This suggests an alternative to the widely accepted but empirically questionable random walk hypothesis.

We present a constraint system, OF, of feature trees that is appropriate to specify and implement type inference for first-class messages. OF extends traditional systems of feature constraints by a selection constraint x <y> z, "by first-class feature tree" y, which is in contrast to the standard selection constraint x[f]y, "by fixed feature" f. We investigate the satisfiability problem of OF and show that it can be solved in polynomial time, and even in quadratic time if the number of features is bounded. We compare OF with Treinen's system EF of feature constraints with first-class features, which has an NP-complete satisfiability problem. This comparison yields that the satisfiability problem for OF with negation is NP-hard. Further we obtain NP-completeness, for a specific subclass of OF with negation that is useful for a related type inference problem. Based on OF we give a simple account of type inference for first-class messages in the spirit of Nishimura's recent proposal, and we show that it has polynomial time complexity: We also highlight an immediate extension of this type system that appears to be desirable but makes type inference NP-complete.

Recently, researchers have proposed many models using reconfigurable optically pipelined buses. We present simulations for a number of these models and establish that they possess the same complexity, so that any of these models can simulate a step of one of the other models in constant time with a polynomial increase in size. Specifically, we determine the complexity of three optical models (the PR-Mesh, APPBS, and AROB) to be the same as the well known LR-Mesh and the CF-LR-Mesh.

We study the learnability of representation classes in Angluin's exact learning model. In particular, we consider the following three query types: equivalence queries, equivalence and membership queries, and membership queries only. We show in all three cases that polynomial query complexity implies already polynomial-time learnability, provided that the learner additionally has access to an oracle in . It follows that boolean circuits are polynomial-time learnable with equivalence queries and the help of an oracle in .^a

The aim of this paper is to compute Shapley's and Banzhaf's values of cooperative games restricted by a combinatorial structure. There have been previous models developed to study the problem of games with partial cooperation. Games restricted by a communication graph were introduced by Myerson and Owen. Another type of combinatorial structure introduced by Gilles, Owen and van den Brink is equivalent to a subclass of antimatroids. Cooperative games in which the set of players is a partially ordered set, that is, games on distributive lattices was investigated by Faigle and Kern. We introduce a new combinatorial structure called augmenting system which is a generaligation of the antimatroid structure and the system of connected subgraphs of graph. We present new algorithmic procedures for computing values of games under augmenting systems restrictions and we show that there exist problems with polynomial algorithm complexity.

Several complexity and decidability results for automatic monoids are shown: (i) there exists an automatic monoid with a P-complete word problem, (ii) there exists an automatic monoid such that the first-order theory of the corresponding Cayley-graph is not elementary decidable, and (iii) there exists an automatic monoid such that reachability in the corresponding Cayley-graph is undecidable. Moreover, it is shown that for every hyperbolic group the word problem belongs to LOGCFL, which improves a result of Cai [8].

A fragment of linear time temporal logic (LTL) is presented. It is proved that the satisfiability problem for this fragment is NP-complete. The fragment is larger than previously known NP-complete fragments. It is obtained by prohibiting the use of until operator and requiring to use only next operators indexed by a letter.

We study completeness in differential approximability classes. In differential approximation, the quality of an approximation algorithm is the measure of both how far is the solution computed from a worst one and how close is it to an optimal one. We define natural reductions preserving approximation and prove completeness results for the class of the NP optimization problems (class NPO), as well as for DAPX, the differential counterpart of APX, and for a natural subclass of DGLO, the differential counterpart of GLO. We also define class 0-APX of the NPO problems that are not differentially approximable within any ratio strictly greater than 0 unless P = NP. This class is very natural for differential approximation, although has no sense for the standard one. Finally, we prove the existence of hard problems for a subclass of DPTAS, the differential counterpart of PTAS.

This paper deals with an important problem of mobile communication. The objective is to place k base stations of equal range on the boundary of a convex polygonal region P such that each point inside P is covered by at least one base station. We name this problem as region-cover(k) problem. A simplified form of this problem is the vertex-cover(k) problem, where the objective is to communicate with only the vertices of P instead of covering the entire polygon. We first present efficient algorithms for vertex-cover(2) and region-cover(2) problems, where the base stations are to be installed on a pair of specified edges. The time complexity of these algorithms are O(n log n) and O(n²) respectively, where n is the number of vertices in the polygon P. Next, we consider the case where k ≥ 3. We first concentrate on a restricted version of the vertex-cover(k) and region-cover(k) problems, where all the k base stations are to be installed on the same edge of P. Our proposed algorithm for the restricted vertex-cover(k) problem produces optimum result in O(min(n²,nk log n)) time, whereas the algorithm for the restricted region-cover(k) problem produces an (1 + ∊)-factor approximation result in time. Finally, we propose efficient heuristic algorithm for the unrestricted version of the region-cover(k) problem for k ≥ 3. Experimental results demonstrate that our proposed algorithm runs fast and produces near optimum solutions.

In this paper we consider the computational complexity of deciding the existence of a perfect matching in certain classes of dense k-uniform hypergraphs. It has been known that the perfect matching problem for the classes of hypergraphs H with minimum ((k - 1)–wise) vertex degreeδ(H) at least c|V(H)| is NP-complete for and trivial for c ≥ ½, leaving the status of the problem with c in the interval widely open. In this paper we show, somehow surprisingly, that ½ is not the threshold for tractability of the perfect matching problem, and prove the existence of an ε > 0 such that the perfect matching problem for the class of hypergraphs H with δ(H) ≥ (½ - ε)|V(H)| is solvable in polynomial time. This seems to be the first polynomial time algorithm for the perfect matching problem on hypergraphs for which the existence problem is nontrivial. In addition, we consider parallel complexity of the problem, which could be also of independent interest.

A famous theorem of Kuratowski states that, in a topological space, at most 14 distinct sets can be produced by repeatedly applying the operations of closure and complement to a given set. We re-examine this theorem in the setting of formal languages, where by "closure" we mean either Kleene closure or positive closure. We classify languages according to the structure of the algebras they generate under iterations of complement and closure. There are precisely 9 such algebras in the case of positive closure, and 12 in the case of Kleene closure. We study how the properties of being open and closed are preserved under concatenation. We investigate analogues, in formal languages, of the separation axioms in topological spaces; one of our main results is that there is a clopen partition separating two words if and only if the words do not commute. We can decide in quadratic time if the language specified by a DFA is closed, but if the language is specified by an NFA, the problem is PSPACE-complete.

Transient algebra is a multi-valued algebra for hazard detection in gate circuits. Sequences of alternating 0's and 1's, called transients, represent signal values, and gates are modeled by extensions of boolean functions to transients. Formulas for computing the output transient of a gate from the input transients are known for NOT, AND, OR and XOR gates and their complements, but, in general, even the problem of deciding whether the length of the output transient exceeds a given bound is NP-complete. We propose a method of evaluating extensions of general boolean functions. We study a class of functions for which, instead of evaluating the extensions on a given set of transients, it is possible to get the same values by using transients derived from the given ones, but having length at most 3. We prove that all functions of three variables, as well as certain other functions, have this property, and can be efficiently evaluated.

The branch of coding theory that is based on formal languages has produced several methods for defining code properties, including word relations, dependence systems, implicational conditions, trajectories, and language inequations. Of those, the latter three can be viewed as formal methods in the sense that a certain formal expression can be used to denote a code property. Here we present a formal method which is based on transducers. Each transducer of a certain type defines/describes a desired code property. The method provides simple and uniform decision procedures for the basic questions of property satisfaction and maximality for regular languages. Our work includes statements about the hardness of deciding some of the problems involved. It turns out that maximality can be hard to decide even for "classical" code properties of finite languages. We also present an initial implementation of a LAnguage SERver capable of deciding the satisfaction problem for a given transducer code property and regular language.

Interval temporal logics (ITLs) are logics for reasoning about temporal statements expressed over intervals, i.e., periods of time. The most famous temporal logic for intervals studied so far is probably Halpern and Shoham's HS, which is the logic of the thirteen Allen's interval relations. Unfortunately, HS and most of its fragments are undecidable. This discouraged the research in this area until recently, when a number non-trivial decidable ITLs have been discovered. This paper is a contribution towards the complete classification of all different fragments of HS. We consider here different combinations of the interval relations begins (B), meets (A), later (L) and their inverses , and . We know from previous work that the combination is decidable only when finite domains are considered (and undecidable elsewhere), and that is decidable over the natural numbers. In the present paper we show that, over strongly discrete linear models (e.g. finite orders, the naturals, the integers), decidability of can be further extended to capture the language , which lies strictly in between and . The logic turns out to be maximal w.r.t decidability over the considered classes, and its satisfiability problem is EXPSPACE-complete. In this paper we also provide an optimal non-deterministic decision procedure, and we show that the language is powerful enough to polynomially encode metric constraints on the length of the current interval.

The quotient complexity, also known as state complexity, of a regular language is the number of distinct left quotients of the language. The quotient complexity of an operation is the maximal quotient complexity of the language resulting from the operation, as a function of the quotient complexities of the operands. The class of star-free languages is the smallest class containing the finite languages and closed under boolean operations and concatenation. We prove that the tight bounds on the quotient complexities of union, intersection, difference, symmetric difference, concatenation and star for star-free languages are the same as those for regular languages, with some small exceptions, whereas 2ⁿ-1 is a lower bound for reversal.

In this paper, we investigate the central problem of finding recombination events. It is commonly assumed that a present population is a descendent of a small number of specific sequences called founders. The recombination process consists in given two equal length sequences, generates a third sequence of the same length by concatenating the prefix of one sequence with the suffix of the other sequence. Due to recombination, a present sequence (called a recombinant) is thus composed of blocks from the founders. A major question related to founder sequences is the so-called Minimum Mosaic problem: using the natural parsimony criterion for the number of recombinations, find the "best" founders. In this article, we prove that the Minimum Mosaic problem given haplotype recombinants with no missing values is NP-hard when the number of founders is given as part of the input and propose some exact exponential-time algorithms for the problem, which can be considered polynomial provided some extra information. Notice that Rastas and Ukkonen proved that the Minimum Mosaic problem is NP-hard using a somewhat unrealistic mutation cost function. The aim of this paper is to provide a better complexity insight of the problem.

Our work is to study the Minimum Latency Broadcast Scheduling problem in the geometric SINR model with power control. With power control, sensor nodes have the ability to adjust transmitting power. While existing works studied the problem assuming a uniform power assignment or allowing unlimited power levels, we investigate the problem with a more realistic power assignment model where the maximum power level is bounded. To the best of our knowledge, no existing work formally proved the NP-hardness, though many researchers have been assuming that this fact holds true. In this paper, we provide a solid proof for this result.

Given an undirected, connected, simple graph G = (V,E), two vertex labelings L_V and L'_V of the vertices of G, and a label flip operation that interchanges a pair of labels on adjacent vertices, the Vertex Relabeling Problem is to transform G from L_V into L'_V using the flip operation. Agnarsson et al. showed solving the Vertex Relabeling Problem on arbitrary graphs can be done in θ(n²), where n is the number of vertices in G. In this article we study the Vertex Relabeling Problem on graphs K_m,m and introduce the concept of parity and precise labelings. We show that, when we consider the parity labeling, the problem on graphs K_m,m can be solved quickly in O(log m) time using m processors on an EREW PRAM. Additionally, we also show that the number of processors can be further reduced to in this case while the time complexity does not change. When the labeling is precise, the parallel time complexity increases by a factor of log m while the processor complexities remain m and . We also show that, when graphs are restricted to K_m,m, this problem can be solved optimally in O(m) time when the labeling is parity, and can be solved in O(m log m) time when the labeling is precise, thereby improving the result in Agnarsson et al. for this specific case. Moreover, we generalize the result in the case of precise labeling to the cases when L_V and L'_V can be any configuration. In the end we give a conclusion and a list of some interesting open problems.