Probability logic and optimization SAT: The PSAT and CPA models

Both probabilistic satisfiability (PSAT) and the check of coherence of probability assessment (CPA) can be considered as probabilistic counterparts of the classical propositional satisfiability problem (SAT). Actually, CPA turns out to be a particular case of PSAT; in this paper, we compare the computational complexity of these two problems for some classes of instances. First, we point out the relations between these probabilistic problems and two well known optimization counterparts of SAT, namely Max SAT and Min SAT. We then prove that Max SAT with unrestricted weights is NP-hard for the class of graph formulas, where Min SAT can be solved in polynomial time. In light of the aforementioned relations, we conclude that PSAT is NP-complete for ideal formulas, where CPA can be solved in linear time.     

General default logic

In this paper, we propose a new nonmonotonic logic called general default logic. On the one hand, it generalizes Reiter’s default logic by adding to it rule-like operators used in logic programming. On the other hand, it extends logic programming by allowing arbitrary propositional formulas. We show that with this new logic, one can formalize naturally rule constraints, generalized closed world assumptions, and conditional defaults. We show that under a notion of strong equivalence, sentences of this new logic can be converted to a normal form. We also investigate the computational complexity of various reasoning tasks in the logic, and relate it to some other nonmonotonic formalisms such as Lin and Shoham’s logic of GK and Moore’s autoepistemic logic.     

The Definition and Numerical Method of Final Value Problem and Arbitrary Value Problem

Many Engineering Problems could be mathematically described by FinalValue Problem, which is the inverse problem of InitialValue Problem. Accordingly, the paper studies the final value problem in the field of ODE problems and analyses the differences and relations between initial and final value problems. The more general new concept of the endpoints-value problem which could describe both initial and final problems is proposed. Further, we extend the concept into inner-interval value problem and arbitrary value problem and point out that both endpoints-value problem and inner-interval value problem are special forms of arbitrary value problem. Particularly, the existence and uniqueness of the solutions of final value problem and inner-interval value problem of first order ordinary differential equation are proved for discrete problems. The numerical calculation formulas of the problems are derived, and for each algorithm, we propose the convergence and stability conditions of them. Furthermore, multivariate and high-order final value problems are further studied, and the condition of fixed delay is also discussed in this paper. At last, the effectiveness of the considered methods is validated by numerical experiment.     

Space-Efficient Counting in Graphs on Surfaces

We consider the problem of counting the number of spanning trees in planar graphs. We prove tight bounds on the complexity of the problem, both in general and especially in the modular setting. We exhibit the problem to be complete for Logspace when the modulus is 2^k, for constant k. On the other hand, we show that for any other modulus and in the non-modular case, our problem is as hard in the planar case as for the case of arbitrary graphs. The techniques used are algebraic topological that may be useful in many other problems involving planar or higher genus graphs – such as higher genus graph recognition in Logspace. In the spirit of counting problems modulo 2^k, we also exhibit a highly parallel ?L\oplus {\bf L} algorithm for finding the value of a permanent modulo 2^k. Previously, the best known result in this direction was Valiant’s result that this problem lies in P. We also show that we can count the number of perfect matchings modulo 2^k in an arbitrary graph in P. This extends Valiant’s result for the permanent, since the Permanent may be modeled as counting the number of perfect matchings in bipartite graphs.     

The Parameterized Approximability of TSP with Deadlines

Modern algorithm theory includes numerous techniques to attack hard problems, such as approximation algorithms on the one hand and parameterized complexity on the other hand. However, it is still uncommon to use these two techniques simultaneously, which is unfortunate, as there are natural problems that cannot be solved using either technique alone, but rather well if we combine them. The problem to be studied here is not only natural, but also practical: Consider TSP, generalized as follows. We impose deadlines on some of the vertices, effectively constraining them to be visited prior to a given point of time. The resulting problem DlTSP (a special case of the well-known TSP with time windows) inherits its hardness from classical TSP, which is both well known from practice and renowned to be one of the hardest problems with respect to approximability: Within polynomial time, not even a polynomial approximation ratio (let alone a constant one) can be achieved (unless P = NP). We will show that DlTSP is even harder than classical TSP in the following sense. Classical TSP, despite its hardness, admits good approximation algorithms if restricted to metric (or near-metric) inputs. Not so DlTSP (and hence, neither TSP with time windows): We will prove that even for metric inputs, no constant approximation ratio can ever be achieved (unless P = NP). This is where parameterization becomes crucial: By combining methods from the field of approximation algorithms with ideas from the theory of parameterized complexity, we apply the concept of parameterized approximation. Thereby, we obtain a 2.5-approximation algorithm for DlTSP with a running time of k! · poly(|G|), where k denotes the number of deadlines. Furthermore, we prove that there is no fpt-algorithm with an approximation guarantee of 2-ε for any ε > 0, unless P = NP. Finally, we show that, unlike TSP, DlTSP becomes much harder when relaxing the triangle inequality. More precisely, for an arbitrary small violation of the triangle inequality, DlTSP does not admit an fpt-algorithm with approximation guarantee ((1-ε)/2)|V| for any ε > 0, unless P = NP.     

The “log rank” conjecture for modular communication complexity

The “log rank” conjecture involves the question of how precisely the deterministic communication complexity of a problem can be described in terms of algebraic invariants of the communication matrix of this problem. We answer this question in the context of modular communication complexity. We show that the modular communication complexity can be exactly characterised in terms of the logarithm of a certain rigidity function of the communication matrix. Thus, we are able to exactly determine the modular communication complexity of several problems, such as, e.g., set disjointness, comparability, and undirected graph connectivity. From the bounds obtained for the modular communication complexity we deduce exponential lower bounds on the size of depth-two circuits having arbitrary symmetric gates at the bottom level and a MOD _m -gate at the top. Received: April 14, 2000.     

Bases for Boolean co-clones

The complexity of various problems in connection with Boolean constraints, like, for example, quantified Boolean constraint satisfaction, have been studied recently. Depending on what types of constraints may be used, the complexity of such problems varies. A very interesting observation of the recent past has been that the thus derived classification of constraints can be explained with the help of universal algebra. More precisely, the difficulty of such a constraint problem often depends on the co-clone the constraints are from. A co-clone is a set of Boolean relations that is closed under very natural closure operations. Nearly all these co-clones can be generated by said operators out of a finite set of relations, a so-called base. Knowing a, preferably simple, base for each co-clone can therefore be of great value when studying the complexity of Boolean constraint problems, since this knowledge reduces the infinitely many cases of equivalent problems to a single one—the constraint satisfaction problem for this base. In this paper we give a finite and simple base for every Boolean co-clone, where this is possible. We give evidence that the presented bases are as easy as possible.     

Probability logic and optimization SAT: The PSAT and CPA models

Both probabilistic satisfiability (PSAT) and the check of coherence of probability assessment (CPA) can be considered as probabilistic counterparts of the classical propositional satisfiability problem (SAT). Actually, CPA turns out to be a particular case of PSAT; in this paper, we compare the computational complexity of these two problems for some classes of instances. First, we point out the relations between these probabilistic problems and two well known optimization counterparts of SAT, namely Max SAT and Min SAT. We then prove that Max SAT with unrestricted weights is NP-hard for the class of graph formulas, where Min SAT can be solved in polynomial time. In light of the aforementioned relations, we conclude that PSAT is NP-complete for ideal formulas, where CPA can be solved in linear time.     

Logical considerations on default semantics

We consider a reinterpretation of the rules of default logic. We make Reiter’s default rules into a constructive method of building models, not theories. To allow reasoning in first‐order systems, we equip standard first‐order logic with a (new) Kleene 3‐valued partial model semantics. Then, using our methodology, we add defaults to this semantic system. The result is that our logic is an ordinary monotonic one, but its semantics is now nonmonotonic. Reiter’s extensions now appear in the semantics, not in the syntax. As an application, we show that this semantics gives a partial solution to the conceptual problems with open defaults pointed out by Lifschitz [V. Lifschitz, On open defaults, in: Proceedings of the Symposium on Computational Logics (1990)], and Baader and Hollunder [F. Baader and B. Hollunder, Embedding defaults into terminological knowledge representation formalisms, in: Proceedings of Third Annual Conference on Knowledge Representation (Morgan‐Kaufmann, 1992)]. The solution is not complete, chiefly because in making the defaults model‐theoretic, we can only add conjunctive information to our models. This is in contrast to default theories, where extensions can contain disjunctive formulas, and therefore disjunctive information. Our proposal to treat the problem of open defaults uses a semantic notion of nonmonotonic entailment for our logic, related to the idea of “only knowing”. Our notion is “only having information” given by a formula. We discuss the differences between this and “minimal‐knowledge” ideas. Finally, we consider the Kraus–Lehmann–Magidor [S. Kraus, D. Lehmann and M. Magidor, Nonmonotonic reasoning, preferential models, and cumulative logics, Artificial Intelligence 44 (1990) 167–207] axioms for preferential consequence relations. We find that our consequence relation satisfies the most basic of the laws, and the Or law, but it does not satisfy the law of Cut, nor the law of Cautious Monotony. We give intuitive examples using our system, on the other hand, which on the surface seem to violate these two laws. We make some comparisons, using our examples, to probabilistic interpretations for which these laws are true, and we compare our models to the cumulative models of Kraus, Lehmann, and Magidor. We also show sufficient conditions for the laws to hold. These involve limiting the use of disjunction in our formulas in one way or another. We show how to make use of the theory of complete partially ordered sets, or domain theory. We can augment any Scott domain with a default set. We state a version of Reiter’s extension operator on arbitrary domains as well. This version makes clear the basic order‐theoretic nature of Reiter’s definitions. A three‐variable function is involved. Finding extensions corresponds to taking fixed points twice, with respect to two of these variables. In the special case of precondition‐free defaults, a general relation on Scott domains induced from the set of defaults is shown to characterize extensions. We show how a general notion of domain theory, the logic induced from the Scott topology on a domain, guides us to a correct notion of “affirmable sentence” in a specific case such as our first‐order systems. We also prove our consequence laws in such a way that they hold not only in first‐order systems, but in any logic derived from the Scott topology on an arbitrary domain. This revised version was published online in June 2006 with corrections to the Cover Date.     

Consensus and collision detectors in radio networks

We consider the fault-tolerant consensus problem in radio networks with crash-prone nodes. Specifically, we develop lower bounds and matching upper bounds for this problem in single-hop radios networks, where all nodes are located within broadcast range of each other. In a novel break from existing work, we introduce a collision-prone communication model in which each node may lose an arbitrary subset of the messages sent by its neighbors during each round. This model is motivated by behavior observed in empirical studies of these networks. To cope with this communication unreliability we augment nodes with receiver-side collision detectors and present a new classification of these detectors in terms of accuracy and completeness. This classification is motivated by practical realities and allows us to determine, roughly speaking, how much collision detection capability is enough to solve the consensus problem efficiently in this setting. We consider nine different combinations of completeness and accuracy properties in total, determining for each whether consensus is solvable, and, if it is, a lower bound on the number of rounds required. Furthermore, we distinguish anonymous and non-anonymous protocols—where “anonymous” implies that devices do not have unique identifiers—determining what effect (if any) this extra information has on the complexity of the problem. In all relevant cases, we provide matching upper bounds. This work is supported by MURI–AFOSR SA2796PO 1-0000243658, USAF–AFRL #FA9550-04-1-0121, NSF Grant CCR-0121277, NSF-Texas Engineering Experiment Station Grant XS64961-CS, and DARPA F33615-01-C-1896.     

Dichotomies in the Complexity of Solving Systems of Equations over Finite Semigroups

We consider the problem of testing whether a given system of equations over a fixed finite semigroup S has a solution. For the case where S is a monoid, we prove that the problem is computable in polynomial time when S is commutative and is the union of its subgroups but is NP-complete otherwise. When S is a monoid or a regular semigroup, we obtain similar dichotomies for the restricted version of the problem where no variable occurs on the right-hand side of each equation. We stress connections between these problems and constraint satisfaction problems. In particular, for any finite domain D and any finite set of relations Γ over D, we construct a finite semigroup S_Γ such that CSP(Γ) is polynomial-time equivalent to the satifiability problem for systems of equations over S_Γ.     

On a lower bound on the computational complexity of a parallel implementation of the branch-and-bound method

We study parallel complexity of the branch-and-bound method for optimization problems. We consider a standard implementation scheme for the branch-and-bound method on a parallel system, in which first only one processor is working, and then the resulting subtasks are given out to other processors. For this scheme, we give a lower bound on the parallel complexity independent of the problem. We study the complexity of this scheme for the Boolean knapsack problem. For a classical algorithmically hard example, we obtain parallel complexity bounds and show that these bounds coincide in order with each other and with the common lower bound on parallel complexity. Thus, we show that the common lower bound is achieved, in the order, for some optimization problems.     

Implementing a Test for Tractability

The question of determining which sets of constraints give rise to NP-complete problems, and which give rise to tractable problems, is an important open problem in the theory of constraint satisfaction. It has been shown in previous papers that certain sufficient conditions for tractability and NP-completeness can be identified using algebraic properties of relations, and that these conditions can be tested by solving a particular form of constraint satisfaction problem (the so-called indicator problem).This paper describes a program which can solve the relevant indicator problems for arbitrary sets of constraints over small domains, and for some sets of constraints over larger domains. The main innovation in the program is its ability to deal with the many symmetries present in the problem; it also has the ability to preserve symmetries in cases where this speeds up the solution.Using this program, we have systematically investigated the complexity of all individual binary relations over a domain of size four or less, and of all individual ternary relations over a domain of size three or less. This automated analysis includes the derivation of more than 450 000 new NP-completeness results, and precisely identifies the small set of individual relations which cannot be classified as either tractable or NP-complete using the algebraic conditions presented in previous papers.     

Complexity theoretic hardness results for query learning

We investigate the complexity of learning for the well-studied model in which the learning algorithm may ask membership and equivalence queries. While complexity theoretic techniques have previously been used to prove hardness results in various learning models, these techniques typically are not strong enough to use when a learning algorithm may make membership queries. We develop a general technique for proving hardness results for learning with membership and equivalence queries (and for more general query models). We apply the technique to show that, assuming , no polynomial-time membership and (proper) equivalence query algorithms exist for exactly learning read-thrice DNF formulas, unions of halfspaces over the Boolean domain, or some other related classes. Our hardness results are representation dependent, and do not preclude the existence of representation independent algorithms.?The general technique introduces the representation problem for a class F of representations (e.g., formulas), which is naturally associated with the learning problem for F. This problem is related to the structural question of how to characterize functions representable by formulas in F, and is a generalization of standard complexity problems such as Satisfiability. While in general the representation problem is in , we present a theorem demonstrating that for "reasonable" classes F, the existence of a polynomial-time membership and equivalence query algorithm for exactly learning F implies that the representation problem for F is in fact in co-NP. The theorem is applied to prove hardness results such as the ones mentioned above, by showing that the representation problem for specific classes of formulas is NP-hard. Received: December 6, 1994     

Dempster–Shafer clustering using Potts spin mean field theory

 In this article we investigate a problem within Dempster–Shafer theory where 2 ^q −1 pieces of evidence are clustered into q clusters by minimizing a metaconflict function, or equivalently, by minimizing the sum of weight of conflict over all clusters. Previously one of us developed a method based on a Hopfield and Tank model. However, for very large problems we need a method with lower computational complexity. We demonstrate that the weight of conflict of evidence can, as an approximation, be linearized and mapped to an antiferromagnetic Potts spin model. This facilitates efficient numerical solution, even for large problem sizes. Optimal or nearly optimal solutions are found for Dempster–Shafer clustering benchmark tests with a time complexity of approximately O(N ²log² N). Furthermore, an isomorphism between the antiferromagnetic Potts spin model and a graph optimization problem is shown. The graph model has dynamic variables living on the links, which have a priori probabilities that are directly related to the pairwise conflict between pieces of evidence. Hence, the relations between three different models are shown.     

Inapproximability Results for Guarding Polygons and Terrains

Past research on art gallery problems has concentrated almost exclusively on bounds on the numbers of guards needed in the worst case in various settings. On the complexity side, fewer results are available. For instance, it has long been known that placing a smallest number of guards for a given input polygon is NP -hard. In this paper we initiate the study of the approximability of several types of art gallery problems. Motivated by a practical problem, we study the approximation properties of the three art gallery problems VERTEX GUARD, EDGE GUARD, and POINT GUARD. We prove that if the input polygon has no holes, there is a constant δ >0 such that no polynomial time algorithm can achieve an approximation ratio of 1+δ , for each of these guard problems. We obtain these results by proposing gap-preserving reductions from 5-OCCURRENCE-MAX-3-SAT. We also prove that if the input polygons are allowed to contain holes, then these problems cannot be approximated by a polynomial time algorithm with ratio ((1-ɛ)/12) \ln n for any ɛ > 0 , unless NP \subseteq TIME(n ^{O (log log n)} ) , where n is the number of vertices of the input polygon. We obtain these results by proposing gap-preserving reductions from the SET COVER problem. We show that this inapproximability for the POINT GUARD problem for polygons with holes carries over to the problem of covering a 2.5-dimensional terrain with a minimum number of guards that are placed at a certain fixed height above the terrain. The same holds if the guards may be placed on the terrain itself. Received September 22, 1999; revised December 5, 1999.     

The Complexity of Polynomial-Time Approximation

In 1996 Khanna and Motwani proposed three logic-based optimization problems constrained by planar structure, and offered the hypothesis that these putatively fundamental problems might provide insight into characterizing the class of optimization problems that admit a polynomial-time approximation scheme (PTAS). The main contribution of this paper is to explore this program from the point of view of parameterized complexity. Problems of optimization are naturally parameterized by the parameter k = 1/ε. An optimization problem admits a PTAS if and only if there is an algorithm Φ, that, given an input of size n, and a parameter value k = 1/ε, produces a solution that is within a multiplicative factor of (1+ ε) of optimal, in a running time that is polynomial for every fixed value of ε. This definition admits the possibility that the degree of the polynomial that bounds the running time of Φ may increase, even quite dramatically, as a function of k = 1/ε. In fact, amongst the PTASs that are currently known, for an error of 20%, polynomial-time bounds no better than O(n²⁰⁰⁰) are quite common. Viewing k = 1/ε as a problem parameter, in the sense of parameterized complexity, leads naturally to the question of whether an efficient PTAS (EPTAS) might be possible for a given optimization problem. An EPTAS is simply a fixed-parameter tractable algorithm with respect to this parameter. We offer a number of results concerning the problems Planar TMIN, Planar TMAX and Planar MPSAT defined by Khanna and Motwani: (1) We show that each of these problems of approximation, naturally parameterized by k = 1/ε, is hard for W[1], and thus it is highly unlikely that they admit EPTASs. (One could also interpret this as indicating that PTASs for these problems are unlikely to be a useful way of coping with intractability in the sense of Garey and Johnson.) (2) We show that there are EPTASs for some subproblems described by syntactic restrictions, and establish some limits on how far these positive results can be extended.     

Nonmonotonic reasoning: from complexity to algorithms

The purpose of this paper is to outline various results regarding the computational complexity and the algorithms of nonmonotonic entailment in different coherence‐based approaches. Starting from a (non necessarily consistent) belief base E and a pre‐order on E, we first present different mechanisms for selecting preferred consistent subsets. Then we present different entailment principles in order to manage these multiple subsets. The crossing point of each generation mechanism m and each entailment principle p defines an entailment relation which we study from the computational complexity point of view. The results are not very encouraging since the complexity of all these nonmonotonic entailment relations is, in most restricted languages, larger than the complexity of monotonic entailment. So, we decided to extend Binary Decision Diagrams technics, which are well suited to the task of solving NP‐hard logic‐based problems. Both theoretical and experimental results are described along this line in the last sections. This revised version was published online in June 2006 with corrections to the Cover Date.     

Termination Proofs for String Rewriting Systems via Inverse Match-Bounds

Annotating a letter by a number, one can record information about its history during a rewrite derivation. In each rewrite step, numbers in the reduct are updated depending on the redex numbering. A string rewriting system is called match-bounded if there is a global upper bound to these numbers. Match-boundedness is known to be a strong sufficient criterion for both termination and preservation of regular languages. We show that the string rewriting systems whose inverse (left and right hand sides exchanged) is match-bounded, also have exceptional properties, but slightly different ones. Inverse match-bounded systems need not terminate; they effectively preserve context-free languages; their sets of normalizable strings and their sets of immortal strings are effectively regular. These languages can be used to decide the normalization, the uniform normalization, the termination and the uniform termination problem for inverse match-bounded systems. We also prove that the termination problem is decidable in linear time, and that a certain strong reachability problem is decidable, thereby solving two open problems of McNaughton’s. Like match-bounds, inverse match-bounds entail linear derivational complexity on the set of terminating strings.