This article examines dynamic energy consumption caused by data during software execution on deeply embedded microprocessors, which can be significant on some devices. In worst-case energy consumption analysis, energy models are used to find the most costly execution path. Taking each instruction’s worst-case energy produces a safe but overly pessimistic upper bound. Algorithms for safe and tight bounds would be desirable. We show that finding exact worst-case energy is NP-hard, and that tight bounds cannot be approximated with guaranteed safety. We conclude that any energy model targeting tightness must either sacrifice safety or accept overapproximation proportional to data-dependent energy.
Formally, following the presentation of @cite_9 , define @math to be a set of literals, and @math a set of disjunctive form clauses: where each @math is a Boolean variable. A truth assignment defines each @math or its negation to be true. A clause is deemed to be satisfied if at least one literal in the clause is assigned true. A MAXSAT problem is a set of literals and set of clauses @math , such that the solution is the truth assignment that causes the maximal number of clauses to be satisfied.
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Abstract In this paper, we present improved exact and parameterized algorithms for the maximum satisfiability problem. In particular, we give an algorithm that computes a truth assignment for a boolean formula F satisfying the maximum number of clauses in time O(1.3247 m | F |), where m is the number of clauses in F , and | F | is the sum of the number of literals appearing in each clause in F . Moreover, given a parameter k , we give an O(1.3695 k +| F |) parameterized algorithm that decides whether a truth assignment for F satisfying at least k clauses exists. Both algorithms improve the previous best algorithms by Bansal and Raman for the problem. We analyze the pure literal rule heuristic for computing a satisfying assignment to a random 3-CNF formula with n variables. We show that the pure literal rule by itself finds satisfying assignments for almost all 3-CNF formulas with up to 1.63n clauses, but it fails for more than 1.7n clauses. As an aside we show that the value of maximum satisfiability for random 3-CNF formulas is tightly concentrated around its mean. Let I be a random 3CNF formula generated by choosing a truth assignment φ for variables x 1 , ..., x n uniformly at random and including every clause with i literals set true by φ with probability p i , independently. We show that for any 0 ≤ η 2 , η 3 ≤ 1 there is a constant d min so that for all d ≥ d min , a spectral algorithm similar to the graph coloring algorithm of [1] will find a satisfying assignment with high probability for p 1 = d n2, p 2 = η 2 d n2, and p3 = η 3 d n2. Appropriately setting η 2 and η 3 yields natural distributions on satisfiable 3CNFs, not-all-equal-sat 3CNFs, and exactly-one-sat 3CNFs. An algorithm for learning the class of Boolean formulas that are expressible as conjunctions of Horn clauses is presented. (A Horn clause is a disjunction of literals, all but at most one of which is a negated variable). The algorithm uses equivalence queries and membership queries to produce a formula that is logically equivalent to the unknown formula to be learned. The amount of time used by the algorithm is polynomial in the number of variables and the number of clauses in the unknown formula. > We present an exact algorithm that decides, for every fixed r ≥ 2 in time O(m) + 2O(k2) whether a given set of m clauses of size r admits a truth assignment that satisfies at least ((2r - 1)m + k) 2r clauses. Thus Max-r-Sat is fixed-parameter tractable when parameterized by the number of satisfied clauses above the tight lower bound (1--2-r)m. This solves an open problem of Mahajan, Raman and Sikdar (J. Comput. System Sci., 75, 2009). Our algorithm is based on a polynomial-time data reduction procedure that reduces a problem instance to an equivalent algebraically represented problem with O(k2) variables. This is done by representing the instance as an appropriate polynomial, and by applying a probabilistic argument combined with some simple tools from Harmonic analysis to show that if the polynomial cannot be reduced to one of size O(k2), then there is a truth assignment satisfying the required number of clauses. Combining another probabilistic argument with tools from graph matching theory and signed graphs, we show that if an instance of Max-2-Sat with m clauses has at least 3k variables after application of certain polynomial time reduction rules to it, then there is a truth assignment that satisfies at least (3m + k) 4 clauses. We also outline how the fixed-parameter tractability result on Max-r-Sat can be extended to a family of Boolean Constraint Satisfaction Problems. Model checking and counter-example guided abstraction refinement are examples of applications of SAT solving requiring the production of models for satisfiable formulas. Better than giving a truth value to every variable, one can provide an implicant, i.e. a partial assignment of the variables such that every full extension is a model for the formula. An implicant is prime if every assignment is necessary. Since prime implicants contain no literal irrelevant for the satisfiability of the formula, they are considered as highly refined information. We here propose a novel algorithm that uses data structures found in modern CDCL SAT solvers to efficiently compute prime implicants starting from an existing model. The original aspects are (1) the algorithm is based on watched literals and a form of propagation of required literals, adapted to CDCL solvers (2) the algorithm works not only on clauses, but also on generalized constraints (3) for clauses and, more generally for cardinality constraints, the algorithm complexity is linear in the size of the constraints found. We implemented and evaluated the algorithm with the Sat4j library. Random constraint satisfaction problems (CSPs) are known to exhibit threshold phenomena: given a uniformly random instance of a CSP with @math variables and @math clauses, there is a value of @math beyond which the CSP will be unsatisfiable with high probability. Strong refutation is the problem of certifying that no variable assignment satisfies more than a constant fraction of clauses; this is the natural algorithmic problem in the unsatisfiable regime (when @math ). Intuitively, strong refutation should become easier as the clause density @math grows, because the contradictions introduced by the random clauses become more locally apparent. For CSPs such as @math -SAT and @math -XOR, there is a long-standing gap between the clause density at which efficient strong refutation algorithms are known, @math , and the clause density at which instances become unsatisfiable with high probability, @math . In this paper, we give spectral and sum-of-squares algorithms for strongly refuting random @math -XOR instances with clause density @math in time @math or in @math rounds of the sum-of-squares hierarchy, for any @math and any integer @math . Our algorithms provide a smooth transition between the clause density at which polynomial-time algorithms are known at @math , and brute-force refutation at the satisfiability threshold when @math . We also leverage our @math -XOR results to obtain strong refutation algorithms for SAT (or any other Boolean CSP) at similar clause densities. Our algorithms match the known sum-of-squares lower bounds due to Grigoriev and Schonebeck, up to logarithmic factors. Additionally, we extend our techniques to give new results for certifying upper bounds on the injective tensor norm of random tensors. Let @math be a uniformly distributed random @math -SAT formula with @math variables and @math clauses. For clauses variables ratio @math the formula @math is satisfiable with high probability. However, no efficient algorithm is known to provably find a satisfying assignment beyond @math with a non-vanishing probability. Non-rigorous statistical mechanics work on @math -CNF led to the development of a new efficient "message passing algorithm" called [M ', Science 2002]. Experiments conducted for @math suggest that the algorithm finds satisfying assignments close to @math . However, in the present paper we prove that the basic version of Survey Propagation Guided Decimation fails to solve random @math -SAT formulas efficiently already for @math with @math almost a factor @math below @math . Let @math be a uniformly distributed random @math -SAT formula with @math variables and @math clauses. We present a polynomial time algorithm that finds a satisfying assignment of @math with high probability for constraint densities @math , where @math . Previously no efficient algorithm was known to find satisfying assignments with a nonvanishing probability beyond @math [A. Frieze and S. Suen, J. Algorithms, 20 (1996), pp. 312-355]. We present a simple probabilistic algorithm for solving k-SAT and more generally, for solving constraint satisfaction problems (CSP). The algorithm follows a simple local search paradigm (S. , 1992): randomly guess an initial assignment and then, guided by those clauses (constraints) that are not satisfied, by successively choosing a random literal from such a clause and flipping the corresponding bit, try to find a satisfying assignment. If no satisfying assignment is found after O(n) steps, start over again. Our analysis shows that for any satisfiable k-CNF-formula with n variables this process has to be repeated only t times, on the average, to find a satisfying assignment, where t is within a polynomial factor of (2(1-1 k)) sup n . This is the fastest (and also the simplest) algorithm for 3-SAT known up to date. We consider also the more general case of a CSP with n variables, each variable taking at most d values, and constraints of order l, and analyze the complexity of the corresponding (generalized) algorith m. It turns out that any CSP can be solved with complexity at most (d spl middot (1-1 l)+ spl epsiv ) sup n . Finding a satisfying assignment for a k-CNF formula (k ≥ 3), assuming such exists, is a notoriously hard problem. In this work we consider the uniform distribution over satisfiable k-CNF formulas with a linear number of clauses (clause-variable ratio greater than some constant). We rigorously analyze the structure of the space of satisfying assignments of a random formula in that distribution, showing that basically all satisfying assignments are clustered in one cluster, and agree on all but a small, though constant proportion, number of variables. This observation enables us to describe a polynomial time algorithm that finds whp a satisfying assignment for such formulas, thus asserting that most satisfiable k-CNF formulas are easy (whenever the clause-variable ratio is greater than some constant). This should be contrasted with the setting of very sparse k-CNF formulas (which are satisfiable whp), where experimental results show some regime of clause density to be difficult for many SAT heuristics. One explanation for this phenomena, backed up by partially non-rigorous analytical tools from statistical physics, is the complicated clustering of the solution space at that regime, unlike the more “regular” structure that denser formulas possess. Thus in some sense, our result rigorously supports this explanation. key words: computational and structural complexity, algorithms and data structures, message passing algorithms, SAT. Abstract We propose an extension of the SLDNF proof procedure for checking integrity constraints in deductive databases. To achieve the effect of the simplification methods investigated by Nicolas [1982], Lloyd, Sonenberg, and Topor [1986], and Decker [1986], we use clauses corresponding to the updates as top clauses for the search space. This builds in the assumption that the database satisfied its integrity constraints prior to the transaction, and that, therefore, any violation of the constraints in the updated database must involve at least one of the updates in the transaction. Different simplification methods can be simulated by different strategies for literal selection and search. The SLDNF proof procedure needs to be extended to use as top clause any arbitrary deductive rule, denial, or negated fact, to incorporate inference rules for reasoning about implicit deletions resulting from other deletions and additions, and to allow an extended resolution step for reasoning forward from negated facts. Abstract Many AI problems, when formalized, reduce to evaluating the probability that a propositional expression is true. In this paper we show that this problem is computationally intractable even in surprisingly restricted cases and even if we settle for an approximation to this probability. We consider various methods used in approximate reasoning such as computing degree of belief and Bayesian belief networks, as well as reasoning techniques such as constraint satisfaction and knowledge compilation, that use approximation to avoid computational difficulties, and reduce them to model-counting problems over a propositional domain. We prove that counting satisfying assignments of propositional languages is intractable even for Horn and monotone formulae, and even when the size of clauses and number of occurrences of the variables are extremely limited. This should be contrasted with the case of deductive reasoning, where Horn theories and theories with binary clauses are distinguished by the existence of linear time satisfiability algorithms. What is even more surprising is that, as we show, even approximating the number of satisfying assignments (i.e., “approximating” approximate reasoning), is intractable for most of these restricted theories. We also identify some restricted classes of propositional formulae for which efficient algorithms for counting satisfying assignments can be given. We consider a random instance I of k-SAT with n variables and m clauses, where k=k(n) satisfies k—log2 n→∞. Let m 0=2 k nln2 and let ∈=∈(n)>0 be such that ∈n→∞. We prove that @math We study the satisfiability of randomly generated formulas formed by M clauses of exactly K literals over N Boolean variables. For a given value of N the problem is known to be most difficult when α = M N is close to the experimental threshold αc separating the region where almost all formulas are SAT from the region where all formulas are UNSAT. Recent results from a statistical physics analysis suggest that the difficulty is related to the existence of a clustering phenomenon of the solutions when α is close to (but smaller than) αc. We introduce a new type of message passing algorithm which allows to find efficiently a satisfying assignment of the variables in this difficult region. This algorithm is iterative and composed of two main parts. The first is a message-passing procedure which generalizes the usual methods like Sum-Product or Belief Propagation: It passes messages that may be thought of as surveys over clusters of the ordinary messages. The second part uses the detailed probabilistic information obtained from the surveys in order to fix variables and simplify the problem. Eventually, the simplified problem that remains is solved by a conventional heuristic. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005 Abstract This paper deals with the problem of removing redundant literals from a given clause. We first consider condensing , a weak type of redundancy elimination. A clause is condensed if it does not subsume any proper subset of itself. It is often useful (and sometimes necessary) to replace a non-condensed clause C by a condensation, i.e., by a condensed subset of C which is subsumed by C . After studying the complexity of an existing clause condensing algorithm, we present a more efficient algorithm and provide arguments for the optimality of the new method. We prove that testing whether a given clause is condensed is co-NP-complete and show that several problems related to clause condensing belong to complexity classes that are, probably, slightly harder than NP. We also consider a stronger version of redundancy elimination: a clause C is strongly condensed iff it does not contain any proper subset C ′ such that C logically implies C ′. We show that the problem of testing whether a clause is strongly condensed is undecidable. We study the problem of satisfiability of randomly chosen clauses, each with K Boolean variables. Using the cavity method at zero temperature, we find the phase diagram for the @math case. We show the existence of an intermediate phase in the satisfiable region, where the proliferation of metastable states is at the origin of the slowdown of search algorithms. The fundamental order parameter introduced in the cavity method, which consists of surveys of local magnetic fields in the various possible states of the system, can be computed for one given sample. These surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems. One such algorithm is shown here for the @math satisfiability problem, with very good performances. Let @math be a nontrivial @math -ary predicate. Consider a random instance of the constraint satisfaction problem @math on @math variables with @math constraints, each being @math applied to @math randomly chosen literals. Provided the constraint density satisfies @math , such an instance is unsatisfiable with high probability. The problem is to efficiently find a proof of unsatisfiability. We show that whenever the predicate @math supports a @math - probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree @math (which runs in time @math ) refute a random instance of @math . In particular, the polynomial-time SOS algorithm requires @math constraints to refute random instances of CSP @math when @math supports a @math -wise uniform distribution on its satisfying assignments. Together with recent work of [LRS15], our result also implies that polynomial-size semidefinite programming relaxation for refutation requires at least @math constraints. Our results (which also extend with no change to CSPs over larger alphabets) subsume all previously known lower bounds for semialgebraic refutation of random CSPs. For every constraint predicate @math , they give a three-way hardness tradeoff between the density of constraints, the SOS degree (hence running time), and the strength of the refutation. By recent algorithmic results of [AOW15] and [RRS16], this full three-way tradeoff is , up to lower-order factors. Current literature offers a number of different approaches to what could generally be called “probabilistic logic programming”. These are usually based on Horn clauses. Here, we introduce a new formalism, Logic Programs with Annotated Disjunctions, based on disjunctive logic programs. In this formalism, each of the disjuncts in the head of a clause is annotated with a probability. Viewing such a set of probabilistic disjunctive clauses as a probabilistic disjunction of normal logic programs allows us to derive a possible world semantics, more precisely, a probability distribution on the set of all Herbrand interpretations. We demonstrate the strength of this formalism by some examples and compare it to related work. We study the satisfiability of random Boolean expressions built from many clauses with K variables per clause (K-satisfiability). Expressions with a ratio α of clauses to variables less than a threshold α c are almost always satisfiable, whereas those with a ratio above this threshold are almost always unsatisfiable. We show the existence of an intermediate phase below α c , where the proliferation of metastable states is responsible for the onset of complexity in search algorithms. We introduce a class of optimization algorithms that can deal with these metastable states; one such algorithm has been tested successfully on the largest existing benchmark of K-satisfiability.