exact thresholds for dpll on random xor sat and np complete extensions of xor sat this paper discusses a model of constraint satisfaction problems known as uniquely extendible constraint satisfaction problems this model includes and generalizes xor sat and the model includes an np complete problem that appears to share many of the threshold characteristics of random sat in this paper we find an exact threshold in the behavior of two versions of dpll on random instances of this problem one version uses the unit clause heuristic and the other uses the generalized unit clause heuristic specifically for dpll with the unit clause heuristic we prove that there is a clause density c smaller than the satisfiability threshold such that for random instances with density smaller than this threshold dpll with unit clause will find a satisfying assignment in linear time with uniformly positive probability however for random instances with density larger than this threshold dpll with unit clause will require exponential time with uniformly positive probability to find a satisfying assignment we then find the equivalent threshold density for dpll with the generalized unit clause heuristic we also prove the analog of the 2 p sat conjecture for this class of problems

a framework for space complexity in algebraic proof systems algebraic proof systems such as polynomial calculus pc and polynomial calculus with resolution pcr refute contradictions using polynomials space complexity for such systems measures the number of distinct monomials to be kept in memory while verifying a proof we introduce a new combinatorial framework for proving space lower bounds in algebraic proof systems as an immediate application we obtain the space lower bounds previously provided for pc pcr alekhnovich et al 2002 filmus et al 2012 more importantly using our approach in its full potential we prove n space lower bounds in pc pcr for random k cnfs k 4 in n variables thus solving an open problem posed in alekhnovich et al 2002 and filmus et al 2012 our method also applies to the graph pigeonhole principle which is a variant of the pigeonhole principle defined over a constant left degree expander graph

on the power of clause learning sat solvers as resolution engines in this work we improve on existing results on the relationship between proof systems obtained from conflict driven clause learning sat solvers and general resolution previous contributions such as those by beame et al 2004 hertel et al 2008 and buss et al 2008 demonstrated that variations on conflict driven clause learning sat solvers corresponded to proof systems as powerful as general resolution however the models used in these studies required either an extra degree of non determinism or a preprocessing step that is not utilized by state of the art sat solvers in practice in this paper we prove that conflict driven clause learning sat solvers yield proof systems that indeed p simulate general resolution without the need for any additional techniques moreover we show that our result can be generalized to certain other practical variations of the solvers which are based on different learning schemes and restart policies

random constraint satisfaction easy generation of hard satisfiable instances in this paper we show that the models of random csp instances proposed by xu and li k xu w li exact phase transitions in random constraint satisfaction problems journal of artificial intelligence research 12 2000 93 103 k xu w li many hard examples in exact phase transitions with application to generating hard satisfiable instances technical report corr report cs cc 0302001 revised version in theoretical computer science 355 2006 291 302 are of theoretical and practical interest indeed these models called rb and rd present several nice features first it is quite easy to generate random instances of any arity since no particular structure has to be integrated or property enforced in such instances then the existence of an asymptotic phase transition can be guaranteed while applying a limited restriction on domain size and on constraint tightness in that case a threshold point can be precisely located and all instances have the guarantee to be hard at the threshold i e to have an exponential tree resolution complexity next a formal analysis shows that it is possible to generate forced satisfiable instances whose hardness is similar to unforced satisfiable ones this analysis is supported by some representative results taken from an intensive experimentation that we have carried out using complete and incomplete search methods

model theoretic inseparability and modularity of description logic ontologies the aim of this paper is to introduce and study model theoretic notions of modularity in description logic and related reasoning problems our approach is based on a generalisation of logical equivalence that is called model theoretic inseparability two tboxes are inseparable w r t a vocabulary s if they cannot be distinguished by the s reducts of their models and thus can equivalently be replaced by one another in any application where only vocabulary items from s are relevant we study in depth the complexity of deciding inseparability for the description logics el and alc and their extensions with inverse roles we then discuss notions of modules of a tbox based on model theoretic inseparability and develop algorithms for extracting minimal modules from acyclic tboxes finally we provide an experimental evaluation of our module extraction algorithm based on the large scale medical tbox snomed ct

a general model and thresholds for random constraint satisfaction problems in this paper we study the relation among the parameters in their most general setting that define a large class of random csp models d k csp where d is the domain size and k is the length of the constraint scopes the model d k csp unifies several related models such as the model rb and the model k csp we prove that the model d k csp exhibits exact phase transitions if klnd increases no slower than the logarithm of the number of variables a series of experimental studies with interesting observations are carried out to illustrate the solubility phase transition and the hardness of instances around phase transitions

dynamical scaling of surface growth in simple lattice models we present extensive simulations of the atomistic edwards wilkinson ew and restricted edwards wilkinson rew models in 2 1 dimensions dynamic finite size scaling analyses of the interfacial width and structure factor provide the estimates for the dynamic exponent z 1 65 0 05 for the ew model and z 2 0 0 1 for the rew model the stochastic contribution to the interface velocity u due to the deposition and diffusion of particles is characterized for both the models using a blocking procedure for the ew model the time displaced temporal correlations in u show nonexponential decay while the temporal correlations decay exponentially for the rew model dynamical scaling of the temporal correlation function for the ew model yields a value of z which is consistent with the estimate obtained from finite size scaling of the interfacial width and structure factor

finding a tree structure in a resolution proof is np complete the resolution tree problem consists of deciding whether a given sequence like resolution refutation admits a tree structure this paper shows the np completeness of both the resolution tree problem and a natural generalization of the resolution tree problem that does not involve resolution

langevin equations and surface growth departamento de fi sica universidad nacional de san luis chacabuco 917 5700 san luis argentina received 9 january 1997 the three approaches usually used to study surface growth are i master equation ii stochastic langevinequation and iii microscopic models all of them give the same scaling exponents recently vvedenskyet al phys rev e 48 852 1993 derived a stochastic langevin equation from a master equation of the birthand death type for the epitaxial growth demonstrating the equivalence of both approaches in this paper astochastic langevin equation is derived from a discrete model the results are the same as those obtained byvvedensky et al demonstrating that the three approaches are equivalent as a nontrivial example our proce dure is used to derive the kardar parisi zhang kpz equation from the ballistic deposition process thismodel with vacancies and overhangs is very dif cult to handle due to the algebraic complications that arisewhen the master equation approach is used s1063 651x 97 11905 5 pacs number s 05 40 1ji introduction

space proof complexity for random 3 cnfs abstract we investigate the space complexity of refuting 3 cnfs in resolution and algebraic systems we prove that every polynomial calculus with resolution refutation of a random 3 cnf in n variables requires with high probability n distinct monomials to be kept simultaneously in memory the same construction also proves that every resolution refutation of requires with high probability n clauses each of width n to be kept at the same time in memory this gives a n 2 lower bound for the total space needed in resolution to refute these results are best possible up to a constant factor and answer questions about space complexity of 3 cnfs the main technical innovation is a variant of hall s lemma we show that in bipartite graphs with bipartition l r and left degree at most 3 l can be covered by certain families of disjoint paths called vw matchings provided that l expands in r by a factor of 2 for 1 5

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

many hard examples for resolution for every choice of positive integers c and k such that k 3 and c 2 k 0 7 there is a positive number e such that with probability tending to 1 as n tends to a randomly chosen family of cn clauses of size k over n variables is unsatisfiable but every resolution proof of its unsatisfiability must generate at least 1 e n clauses

on sufficient conditions for unsatisfiability of random formulas a descriptive complexity approach to random 3 sat is initiated we show that unsatisfiability of any significant fraction of random 3 cnf formulas cannot be certified by any property that is expressible in datalog combined with the known relationship between the complexity of constraint satisfaction problems and expressibility in datalog our result implies that any constraint propagation algorithm working with small constraints will fail to certify unsatisfiability almost always our result is a consequence of designing a winning strategy for one of the players in the existential pebble game the winning strategy makes use of certain extension axioms that we introduce and hold almost surely on a random 3 cnf formula the second contribution of our work is the connection between finite model theory and propositional proof complexity to make this connection explicit we establish a tight relationship between the number of pebbles needed to win the game and the width of the resolution refutations as a consequence to our result and the known size width relationship in resolution we obtain new proofs of the exponential lower bounds for resolution refutations of random 3 cnf formulas and the pigeonhole principle

backjumping for quantified boolean logic satisfiability the implementation of effective reasoning tools for deciding the satisfiability of quantified boolean formulas qbfs is an important research issue in artificial intelligence many decision procedures have been proposed in the last few years most of them based on the davis logemann loveland procedure dll for propositional satisfiability sat in this paper we show how it is possible to extend the conflict directed backjumping schema for sat to the satisfiability of qbfs when applicable conflict directed backjumping allows search to skip over existentially quantified literals while backtracking we introduce solution directed backjumping which allows the same behavior for universally quantified literals we show how it is possible to incorporate both conflict directed and solution directed backjumping in a dll based decision procedure for satisfiability of qbfs we also implement and test the procedure the experimental analysis shows that because of backjumping significant speed ups can be obtained summing up we present the first algorithm that applies conflict and solution directed backjumping to qbf and demonstrate the performance of this algorithm via an empirical study

henkin quantifiers and boolean formulae a certification perspective of dqbf henkin quantifiers when applied on boolean formulae yielding the so called dependency quantified boolean formulae dqbfs offer succinct descriptive power specifying variable dependencies despite their natural applications to games with incomplete information logic synthesis with constrained input dependencies etc dqbfs remain a relatively unexplored subject however this paper investigates their basic properties including formula negation and complement formula expansion prenex and non prenex form conversions and resolution in particular the proposed dqbf formulation is established from a synthesis perspective concerned with skolem function models and herbrand function countermodels also a generalized resolution rule is shown to be sound but incomplete for dqbf evaluation

universality classification of restricted solid on solid type surface growth models we consider the restricted solid on solid rsos type surface growth models and classify them into dynamic universality classes according to their symmetry and conservation law four groups of rsos type microscopic models asymmetric a asymmetric conserved ac symmetric s and symmetric conserved sc groups are introduced and the corresponding stochastic differential equations sdes are derived analyzing these sdes using dynamic renormalization group theory we confirm the previous results that a rsos ac rsos and s rsos groups belong to the kardar parisi zhang class the villain lai das sarma class and the edwards wilkinson class respectively we also find that sc rsos group belongs to a new universality class featuring the conserved cubic nonlinearity

expansion based qbf solving versus q resolution this article introduces and studies a proof system exp res that enables us to refute quantified boolean formulas qbfs the system exp res operates in two stages it expands all universal variables through conjunctions and refutes the result by propositional resolution this approach contrasts with the q resolution calculus which enables refuting qbfs by rules similar to propositional resolution in practice q resolution enables producing proofs from conflict driven dpll based qbf solvers the system exp res can on the other hand certify certain expansion based solvers so a natural question is to ask which of the systems q resolution and exp res is more powerful the article gives several partial responses to this question on the positive side we show that exp res can p simulate tree q resolution on the negative side we show that exp res does not p simulate unrestricted q resolution in the favor of exp res we show that exp res is more powerful than a certain fragment of q resolution which is important for dpll based qbf solving

lower bounds for the weak pigeonhole principle and random formulas beyond resolution we work with an extension of resolution called res 2 that allows clauses with conjunctions of two literals in this system there are rules to introduce and eliminate such conjunctions we prove that the weak pigeonhole principle phpncn and random unsatisfiable cnf formulas require exponential size proofs in this system this is the strongest system beyond resolution for which such lower bounds are known as a consequence to the result about the weak pigeonhole principle res log is exponentially more powerful than res 2 also we prove that resolution cannot polynomially simulate res 2 and that res 2 does not have feasible monotone interpolation solving an open problem posed by krajicek

the complexity of quantified constraint satisfaction collapsibility sink algebras and the three element case the constraint satisfaction probem csp is a well acknowledged framework in which many combinatorial search problems can be naturally formulated the csp may be viewed as the problem of deciding the truth of a logical sentence consisting of a conjunction of constraints in front of which all variables are existentially quantified the quantified constraint satisfaction problem qcsp is the generalization of the csp where universal quantification is permitted in addition to existential quantification the general intractability of these problems has motivated research studying the complexity of these problems under a restricted constraint language which is a set of relations that can be used to express constraints this paper introduces collapsibility a technique for deriving positive complexity results on the qcsp in particular this technique allows one to show that for a particular constraint language the qcsp reduces to the csp we show that collapsibility applies to three known tractable cases of the qcsp that were originally studied using disparate proof techniques in different decades quantified 2 sat aspvall plass and tarjan in 1979 quantified horn sat karpinski kleine buning and schmitt in 1987 and quantified affine sat creignou khanna and sudan in 2001 this reconciles and reveals common structure among these cases which are describable by constraint languages over a two element domain in addition to unifying these known tractable cases we study constraint languages over domains of larger size

time space trade offs in resolution superpolynomial lower bounds for superlinear space we give the first time space trade off lower bounds for resolution proofs that apply to superlinear space in particular we show that there are formulas of size n that have resolution refutations of size and space t n n theta log n and like all formulas have another resolution refutation of space n but for which no resolution refutation can simultaneously have space s n t n o 1 and size t n o 1 in other words any substantial reduction in space results in a super polynomial increase in total size we also show somewhat stronger time space trade off lower bounds for regular resolution which are also the first to apply to superlinear space for any function t that is at most weakly exponential t n 2 o n 1 4 we give a tautology that has regular resolution proofs of size and space t n but no such proofs with space s n t n 1 omega 1 and size t n o 1 thus any polynomial reduction in space has a superpolynomial cost in size these tautologies

the treewidth of proofs abstract so called ordered variants of the classical notions of pathwidth and treewidth are introduced and proposed as proof theoretically meaningful complexity measures for the directed acyclic graphs underlying proofs ordered pathwidth is roughly the same as proof space and the ordered treewidth of a proof is meant to serve as a measure of how far it is from being treelike length space lower bounds for k dnf refutations are generalized to arbitrary infinity axioms and strengthened in that the space measure is relaxed to ordered treewidth

existentially restricted quantified constraint satisfaction the quantified constraint satisfaction problem qcsp is a framework for modelling pspace computational problems the general intractability of the qcsp has motivated the pursuit of restricted cases that avoid its maximal complexity in this paper we introduce and study a new model for investigating qcsp complexity in which the types of constraints given by the existentially quantified variables is restricted our primary technical contribution is the development and application of a general technology for proving positive results on parameterizations of the model of inclusion in the complexity class conp

many hard examples in exact phase transitions this paper analyzes the resolution complexity of two random constraint satisfaction problem csp models i e model rb rd for which we can establish the existence of phase transitions and identify the threshold points exactly by encoding csps into cnf formulas it is proved that almost all instances of model rb rd have no tree like resolution proofs of less than exponential size thus we not only introduce new families of csps and cnf formulas hard to solve which can be useful in the experimental evaluation of csp and sat algorithms but also propose models with both many hard instances and exact phase transitions finally conclusions are presented as well as a detailed comparison of model rb rd with the hamiltonian cycle problem and random 3 sat which respectively exhibit three different kinds of phase transition behavior in np complete problems

understanding cutting planes for qbfs abstract we study the cutting planes system cp red for quantified boolean formulas qbf obtained by augmenting propositional cutting planes with a universal reduction rule and analyse the proof theoretic strength of this new calculus while in the propositional case cutting planes is of intermediate strength between resolution and frege our findings here show that the situation in qbf is slightly more complex while cp red is again weaker than qbf frege and stronger than the cdcl based qbf resolution systems q res and qu res it turns out to be incomparable to even the weakest expansion based qbf resolution system exp res a similar picture holds for a semantic version semcp red technically our results establish the effectiveness of two lower bound techniques for cp red via strategy extraction and via monotone feasible interpolation

the qbf gallery over the last few years much progress has been made in the theory and practice of solving quantified boolean formulas qbf novel solvers have been presented that either successfully enhance established techniques or implement novel solving paradigms powerful preprocessors have been realized that tune the encoding of a formula to make it easier to solve frameworks for certification and solution extraction emerged that allow for a detailed interpretation of a qbf solver s results and new types of qbf encodings were presented for various application problems to capture these developments the qbf gallery was established in 2013 the qbf gallery aims at providing a forum to assess qbf tools and to collect new expressive benchmarks that allow for documenting the status quo and that indicate promising research directions these benchmarks became the basis for the experiments conducted in the context of the qbf gallery 2013 and follow up evaluations in this paper we report on the setup of the qbf gallery to this end we conducted numerous experiments which allowed us not only to assess the quality of the tools but also the quality of the benchmarks

finding all solution sets of piecewise linear interval equations using an integer programming solver abstract this paper presents an efficient and easily implementable method for finding all solution sets of piecewise linear interval equations using integer programming in this method the problem of finding all solution sets is formulated as a mixed integer programming problem and it is solved by a high performance integer programming solver such as cplex it is shown that all solution sets are obtained by solving a single mixed integer programming problem and related linear programming problems numerical examples are given to confirm the effectiveness of the proposed method

the complexity of constraint satisfaction games and qcsp we study the complexity of two person constraint satisfaction games an instance of such a game is given by a collection of constraints on overlapping sets of variables and the two players alternately make moves assigning values from a finite domain to the variables in a specified order the first player tries to satisfy all constraints while the other tries to break at least one constraint the goal is to decide whether the first player has a winning strategy we show that such games can be conveniently represented by a logical form of quantified constraint satisfaction where an instance is given by a first order sentence in which quantifiers alternate and the quantifier free part is a conjunction of positive atomic formulas the goal is to decide whether the sentence is true while the problem of deciding such a game is pspace complete in general by restricting the set of allowed constraint predicates one can obtain infinite classes of constraint satisfaction games of lower complexity we use the quantified constraint satisfaction framework to study how the complexity of deciding such a game depends on the parameter set of allowed predicates with every predicate one can associate certain predicate preserving operations called polymorphisms we show that the complexity of our games is determined by the surjective polymorphisms of the constraint predicates we illustrate how this result can be used by identifying the complexity of a wide variety of constraint satisfaction games

resolution for max sat max sat is the problem of finding an assignment minimizing the number of unsatisfied clauses in a cnf formula we propose a resolution like calculus for max sat and prove its soundness and completeness we also prove the completeness of some refinements of this calculus from the completeness proof we derive an exact algorithm for max sat and a time upper bound we also define a weighted max sat resolution like rule and show how to adapt the soundness and completeness proofs of the max sat rule to the weighted max sat rule finally we give several particular max sat problems that require an exponential number of steps of our max sat rule to obtain the minimal number of unsatisfied clauses of the combinatorial principle these results are based on the corresponding resolution lower bounds for those particular problems

renormalization of stochastic lattice models basic formulation we describe a general method for the multiscale analysis of stochastic lattice models beginning with a lattice langevin formulation of site fluctuations we derive stochastic partial differential equations by regularizing the transition rules of the model subsequent coarse graining is accomplished by calculating renormalization group rg trajectories from initial conditions determined by the regularized atomistic models the rg trajectories correspond to hierarchies of continuum equations describing lattice models over expanding length and time scales these continuum equations retain a quantitative connection over different scales as well as to the underlying atomistic dynamics this provides a systematic method for the derivation of continuum equations from the transition rules of lattice models for any length and time scales as an illustration we consider the one dimensional 1d wolf villain wv model europhys lett 13 389 1990 the rg analysis of this model which we develop in detail is generic and can be applied to a wide range of conservative lattice models the rg trajectory of the 1d wv model shows a complex crossover sequence of linear and nonlinear stochastic differential equations which is in excellent agreement with kinetic monte carlo simulations of this model we conclude by discussing possible applications of the multiscale method described here to other nonequilibrium systems

finding all solutions of nonlinear equations using the dual simplex method recently an efficient algorithm has been proposed for finding all solutions of systems of nonlinear equations using linear programming this algorithm is based on a simple test termed the lp test for nonexistence of a solution to a system of nonlinear equations using the dual simplex method in this letter an improved version of the lp test algorithm is proposed by numerical examples it is shown that the proposed algorithm could find all solutions of a system of 300 nonlinear equations in practical computation time

resolution proofs of generalized pigeonhole principles we extend results of haken to give an exponential lower bound on the size of resolution proofs for propositional formulas encoding a generalized pigeonhole principle these propositional formulas express the fact that there is no one one mapping from c n objects to n objects when c 1 as a corollary resolution proof systems do not p simulate constant formula depth frege proof systems

resolution lower bounds for the weak pigeonhole principle we prove that any resolution proof for the weak pigeonhole principle with n holes and any number of pigeons is of length 2ne for some global constant e 0 one corollary is that a certain propositional formulation of the statement np p poly does not have short resolution proofs

data reductions fixed parameter tractability and random weighted d cnf satisfiability data reduction is a key technique in the study of fixed parameter algorithms in the ai literature pruning techniques based on simple and efficient to implement reduction rules also play a crucial role in the success of many industrial strength solvers understanding the effectiveness and the applicability of data reduction as a technique for designing heuristics for intractable problems has been one of the main motivations in studying the phase transition of randomly generated instances of np complete problems in this paper we take the initiative to study the power of data reductions in the context of random instances of a generic intractable parameterized problem the weighted d cnf satisfiability problem we propose a non trivial random model for the problem and study the probabilistic behavior of the random instances from the model we design an algorithm based on data reduction and other algorithmic techniques and prove that the algorithm solves the random instances with high probability and in fixed parameter polynomial time o d knm where n is the number of variables m is the number of clauses and k is the fixed parameter we establish the exact threshold of the phase transition of the solution probability and show that in some region of the problem space unsatisfiable random instances of the problem have parametric resolution proof of fixed parameter polynomial size also discussed is a more general random model and the generalization of the results to the model

scaling of ballistic deposition from a langevin equation an exact lattice langevin equation is derived for the ballistic deposition model of surface growth the continuum limit of this equation is dominated by the kardar parisi zhang kpz equation at all length and time scales for a one dimensional substrate the solution of the exact lattice langevin equation yields the kpz scaling exponents without any extrapolation for a two dimensional substrate the scaling exponents are different from those found from computer simulations this discrepancy is discussed in relation to analytic approaches to the kpz equation in higher dimensions

soundness of q resolution with dependency schemes q resolution and q term resolution are proof systems for quantified boolean formulas qbfs we introduce generalizations of these proof systems named q d resolution and q d term resolution q d resolution and q d term resolution are parameterized by a dependency scheme d and use more powerful reduction and reduction rules respectively we show soundness of these systems for particular dependency schemes we prove 1 soundness of q d resolution parameterized by the reflexive resolution path dependency scheme and 2 soundness of q d term resolution parameterized by the resolution path dependency scheme these results entail soundness of the proof systems used for certificate generation in the state of the art solver depqbf

oslo rice pile model is a quenched edwards wilkinson equation the oslo rice pile model is a sandpile like paradigmatic model of self organized criticality soc in this paper it is shown that the oslo model is in fact exactly a discrete realization of the much studied quenched edwards wilkinson equation qew nattermann et al j phys ii france 2 1483 1992 this is possible by choosing the correct dynamical variable and identifying its equation of motion it establishes for the first time an exact link between soc models and the field of interface growth with quenched disorder this connection is obviously very encouraging as it suggests that established theoretical techniques can be brought to bear with full strength on some of the hitherto elusive problems of soc

langevin equations for fluctuating surfaces exact langevin equations are derived for the height fluctuations of surfaces driven by the deposition of material from a molecular beam we consider two types of model deposition models where growth proceeds by the deposition and instantaneous local relaxation of particles with no subsequent movement and models with concurrent random deposition and surface diffusion starting from a chapman kolmogorov equation the deposition relaxation and hopping rules of these models are first expressed as transition rates within a master equation for the joint height probability density function the kramers moyal van kampen expansion of the master equation in terms of an appropriate largeness parameter yields according to a limit theorem due to kurtz stoch proc appl 6 223 1978 a fokker planck equation that embodies the statistical properties of the original lattice model the statistical equivalence of this fokker planck equation solved in terms of its associated langevin equation and solutions of the chapman kolmogorov equation as determined by kinetic monte carlo kmc simulations of the lattice transition rules is demonstrated explicitly by comparing the surface roughness and the lateral height correlations obtained from the two formulations for the edwards wilkinson proc roy soc london ser a 381 17 1982 and wolfvillain europhys lett 13 389 1990 deposition models and for a model with random deposition and surface diffusion in each case as the largeness parameter is increased the langevin equation converges to the surface roughness and lateral height correlations produced by kmc simulations for all times including the crossover between different scaling regimes we conclude by examining some of the wider implications of these results including applications to heteroepitaxial systems and the passage to the continuum limit

lp narrowing a new strategy for finding all solutions of nonlinear equations an efficient algorithm is proposed for finding all solutions of systems of n nonlinear equations this algorithm is based on interval analysis and a new strategy called lp narrowing in the lp narrowing strategy boxes n dimensional rectangles in the solution domain containing no solution are excluded and boxes containing solutions are narrowed so that no solution is lost by using linear programming techniques since the lp narrowing is very powerful all solutions can be found very efficiently by numerical examples it is shown that the proposed algorithm could find all solutions of systems of 5000 50 000 nonlinear equations in practical computation time

cluster mean field approximations with the coherent anomaly method analysis for the driven pair contact process with diffusion the cluster mean field approximations are performed up to 13 cluster sizes to study the critical behavior of the driven pair contact process with diffusion dpcpd and its precedent the pcpd in one dimension critical points are estimated by extrapolating our data to the infinite cluster size limit which are in good accordance with recent simulation results within the cluster mean field approximation scheme the pcpd and the dpcpd share the same mean field critical behavior the application of the coherent anomaly method however shows that the two models develop different coherent anomalies which lead to different true critical scaling the values of the critical exponents for the particle density the pair density the correlation length and the relaxation time are fairly well estimated for the dpcpd these results support and complement our recent simulation results for the dpcpd

on the automatizability of resolution and related propositional proof systems a propositional proof system is automatizable if there is an algorithm that given a tautology produces a proof in time polynomial in the size of its smallest proof this notion can be weakened if we allow the algorithm to produce a proof in a stronger system within the same time bound this new notion is called weak automatizability among other characterizations we prove that a system is weakly automatizable exactly when a weak form of the satisfiability problem is solvable in polynomial time after studying the robustness of the definition we prove the equivalence between i resolution is weakly automatizable ii res k is weakly automatizable and iii res k has feasible interpolation when k 1 in order to prove this result we show that res 2 has polynomial size proofs of the reflection principle of resolution which is a version of consistency we also show that res k for every k 1 proves its consistency in polynomial size while resolution does not in fact we show that resolution proofs of its own consistency require almost exponential size this gives a better lower bound for the monotone interpolation of res 2 and a separation from resolution as a byproduct our techniques also give us a way to obtain a large class of examples that have small resolution refutations but require relatively large width this answers a question of alekhnovich and razborov related to whether resolution is automatizable in quasipolynomial time

a new kind of tradeoffs in propositional proof complexity we exhibit an unusually strong tradeoff in propositional proof complexity that significantly deviates from the established pattern of almost all results of this kind namely restrictions on one resource width in our case imply an increase in another resource tree like size that is exponential not only with respect to the complexity of the original problem but also to the whole class of all problems of the same bit size more specifically we show that for any parameter k e k n there are unsatisfiable k cnfs that possess refutations of width o k but such that any tree like refutation of width n1 e k must necessarily have doubly exponential size exp n k this means that there exist contradictions that allow narrow refutations but in order to keep the size of such a refutation even within a single exponent it must necessarily use a high degree of parallelism our construction and proof methods combine in a non trivial way two previously known techniques the hardness escalation method based on substitution formulas and expansion this combination results in a hardness compression approach that strives to preserve hardness of a contradiction while significantly decreasing the number of its variables

resolution for quantified boolean formulas a complete and sound resolution operation directly applicable to the quantified boolean formulas is presented if we restrict the resolution to unit resolution then the completeness and soundness for extended quantified horn formulas is shown we prove that the truth of a quantified horn formula can be decided in o rn time where n is the length of the formula and r is the number of universal variables whereas in contrast the evaluation problem for extended quantified horn formulas is conp complete for formulas with prefix further we show that the resolution is exponential for extended quantified horn formulas

characterising tree like frege proofs for qbf abstract we examine the tree like versions of qbf frege and extended frege systems while in the propositional setting tree like and dag like frege are equivalent we show that this is not the case for qbf frege where tree like systems are exponentially weaker this applies to the version of qbf frege where the universal reduction rule substitutes universal variables by 0 1 constants to show lower bounds for tree like qbf frege we devise a general technique that provides lower bounds for all tree like qbf systems of the form p red where p is a propositional system the lower bound is based on the semantic measure of strategy size corresponding to the size of countermodels for false qbfs we also obtain a full characterisation of hardness for tree like qbf frege lower bounds for this system either arise from a lower bound to propositional frege from a circuit lower bound or from a lower bound to strategy size

frege systems for quantified boolean logic we define and investigate frege systems for quantified boolean formulas qbf for these new proof systems we develop a lower bound technique that directly lifts circuit lower bounds for a circuit class c to the qbf frege system operating with lines from c such a direct transfer from circuit to proof complexity lower bounds has often been postulated for propositional systems but had not been formally established in such generality for any proof systems prior to this work this leads to strong lower bounds for restricted versions of qbf frege in particular an exponential lower bound for qbf frege systems operating with ac0 p circuits in contrast any non trivial lower bound for propositional ac0 p frege constitutes a major open problem improving these lower bounds to unrestricted qbf frege tightly corresponds to the major problems in circuit complexity and propositional proof complexity in particular proving a lower bound for qbf frege systems operating with arbitrary p poly circuits is equivalent to either showing a lower bound for p poly or for propositional extended frege which operates with p poly circuits we also compare our new qbf frege systems to standard sequent calculi for qbf and establish a correspondence to intuitionistic bounded arithmetic

a switching lemma for small restrictions and lower bounds for k dnf resolution we prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to formulas in disjunctive normal form dnfs with small terms we use this to prove lower bounds for the res k propositional proof system an extension of resolution which works with k dnfs instead of clauses we also obtain an exponential separation between depth d circuits of bottom fan in k and depth d circuits of bottom fan in k 1 our results for res k are as follows the 2n to n weak pigeonhole principle requires exponential size to refute in res k for k leq sqrt log n log log n for each constant k there exists a constant w k so that random w cnfs require exponential size to refute in res k for each constant k there are sets of clauses which have polynomial size res k 1 refutations but which require exponential size res k refutations

mean payoff games and propositional proofs we associate a cnf formula to every instance of the mean payoff game problem in such a way that if the value of the game is non negative the formula is satisfiable and if the value of the game is negative the formula has a polynomial size refutation in s 2 frege i e dnf resolution this reduces mean payoff games to the weak automatizability of s 2 frege and to the interpolation problem for s 2 2 frege since the interpolation problem for s 1 frege i e resolution is solvable in polynomial time our result is close to optimal up to the computational complexity of solving mean payoff games the proof of the main result requires building low depth formulas that compute the bits of the sum of a constant number of integers in binary notation and low complexity proofs of the required arithmetic properties

solving qbf with counterexample guided refinement abstract this article puts forward the application of counterexample guided abstraction refinement cegar in solving the well known pspace complete problem of quantified boolean formulas qbf the article studies the application of cegar in two scenarios in the first scenario cegar is used to expand quantifiers of the formula and subsequently a satisfiability sat solver is applied first it is shown how to do that for two levels of quantification and then it is generalized for arbitrary number of levels by recursion it is also shown that these ideas can be generalized to non prenex and non cnf qbf solvers in the second scenario cegar is employed as an additional learning technique in an existing dpll based qbf solver experimental evaluation of the implemented prototypes shows that the cegar driven solver outperforms existing solvers on a number of benchmark families and that the dpll solver benefits from the additional type of learning

another look at degree lower bounds for polynomial calculus abstract polynomial complexity is an algebraic proof system inspired by grobner bases which was introduced by clegg edmonds and impagliazzo alekhnovich and razborov devised a method for proving degree lower bounds for polynomial calculus we present an alternative account of their method which also encompasses a generalization due to galesi and lauria

coarse graining a restricted solid on solid model a procedure suggested by vvedensky for obtaining continuum equations as the coarse grained limit of discrete models is applied to the restricted solid on solid model with both adsorption and desorption using an expansion of the master equation discrete langevin equations are derived these agree quantitatively with direct simulation of the model from these a continuum differential equation is derived and the model is found to exhibit either edwards wilkinson or kardar parisi zhang exponents as expected from symmetry arguments the coefficients of the resulting continuum equation remain well defined in the coarse grained limit

the intractability of resolution abstract we prove that for infinitely many disjunctive normal form propositional calculus tautologies the length of the shortest resolution proof of cannot be bounded by any polynomial of the length of the tautologies we use were introduced by cook and reckhow 1979 and encode the pigeonhole principle extended resolution can furnish polynomial length proofs of these formulas

automating resolution is np hard we show that the problem of finding a resolution refutation that is at most polynomially longer than a shortest one is np hard in the parlance of proof complexity resolution is not automatable unless p np indeed we show that it is np hard to distinguish between formulas that have resolution refutations of polynomial length and those that do not have subexponential length refutations this also implies that resolution is not automatable in subexponential time or quasi polynomial time unless np is included in subexp or qp respectively

on the width of semialgebraic proofs and algorithms in this paper we study width of semialgebraic proof systems and various cut based procedures in integer programming we focus on two important systems gomory chvatal cutting planes and lovasz schrijver lift and project procedures we develop general methods for proving width lower bounds and apply them to random k cnfs and several popular combinatorial principles like the perfect matching principle and tseitin tautologies we also show how to apply our methods to various combinatorial optimization problems we establish a supercritical trade off between width and rank that is we give an example in which small width proofs are possible but require exponentially many rounds to perform them

hard examples for resolution exponential lower bounds are proved for the length of resolution refutations of sets of disjunctions constructed from expander graphs using the method of tseitin since these sets of clauses encode biconditionals they have short polynomial length refutations in a standard axiomatic formulation of propositional calculus

on the relative complexity of resolution refinements and cutting planes proof systems an exponential lower bound for the size of tree like cutting planes refutations of a certain family of conjunctive normal form cnf formulas with polynomial size resolution refutations is proved this implies an exponential separation between the tree like versions and the dag like versions of resolution and cutting planes in both cases only superpolynomial separations were known a urquhart bull symbolic logic 1 1995 pp 425 467 j johannsen inform process lett 67 1998 pp 37 41 p clote and a setzer in proof complexity and feasible arithmetics amer math soc providence ri 1998 pp 93 117 in order to prove these separations the lower bounds on the depth of monotone circuits of raz and mckenzie in combinatorica 19 1999 pp 403 435 are extended to monotone real circuits an exponential separation is also proved between tree like resolution and several refinements of resolution negative resolution and regular resolution actually this last separation also provides a separation between tree like resolution and ordered resolution and thus the corresponding superpolynomial separation of a urquhart bull symbolic logic 1 1995 pp 425 467 is extended finally an exponential separation between ordered resolution and unrestricted resolution also negative resolution is proved only a superpolynomial separation between ordered and unrestricted resolution was previously known a goerdt ann math artificial intelligence 6 1992 pp 169 184

finding all solutions of separable systems of piecewise linear equations using integer programming abstract finding all solutions of nonlinear or piecewise linear equations is an important problem which is widely encountered in science and engineering various algorithms have been proposed for this problem however the implementation of these algorithms are generally difficult for non experts or beginners in this paper an efficient method is proposed for finding all solutions of separable systems of piecewise linear equations using integer programming in this method we formulate the problem of finding all solutions by a mixed integer programming problem and solve it by a high performance integer programming software such as glpk scip or cplex it is shown that the proposed method can be easily implemented without making complicated programs it is also confirmed by numerical examples that the proposed method can find all solutions of medium scale systems of piecewise linear equations in practical computation time

on the complexity of resolution with bounded conjunctions we analyze size and space complexity of res k a family of propositional proof systems introduced by krajicek in fund math 170 1 3 2001 123 which extend resolution by allowing disjunctions of conjunctions of up to k 1 literals we show that the treelike res k proof systems form a strict hierarchy with respect to proof size and also with respect to space moreover resolution while simulating treelike res k is almost exponentially separated from treelike res k to study space complexity for general res k we introduce the concept of dynamical satisfiability which allows us to prove in a unified way all known space lower bounds for resolution and to extend them to res k

short proofs are narrow resolution made simple the width of a resolution proof is defined to be the maximal number of literals in any clause of the proof in this paper we relate proof width to proof length size in both general resolution and its tree like variant the following consequences of these relations reveal width as a crucial resource of resolution proofs in one direction the relations allow us to give simple unified proofs for almost all known exponential lower bounds on size of resolution proofs as well as several interesting new ones they all follow from width lower bounds and we show how these follow from natural expansion property of clauses of the input tautology in the other direction the width size relations naturally suggest a simple dynamic programming procedure for automated theorem proving one which simply searches for small width proofs this relation guarantees that the runnuing time and thus the size of the produced proof is at most quasi polynomial in the smallest tree like proof this algorithm is never much worse than any of the recursive automated provers such as dll used in practice in contrast we present a family of tautologies on which it is exponentially faster

the surface statistics of a granular aggregate the problem of the surface fluctuations in a settled granular material is posed a simple model is given which describes the process by which a particle settles and comes to rest on the existing surface of the packing and from this a set of langevin equations for the fourier modes of the surface are derived these equations imply that the fourier amplitudes behave like the velocities of a set of independent brownian particles we show that this results in logarithmically divergent surface fluctuations if the flux of particles onto the surface is random the divergence being removed by a more accurate description of the settling material for example by having the granules fall through a sieve

fokker planck equation for lattice deposition models an asymptotically exact fokker planck equation for the height fluctuations of lattice deposition models is derived from a van kampen expansion of the master equation using an edwards wilkinson type model as an example the solution of the equivalent langevin equation reproduces the surface roughness and lateral height correlations obtained with kinetic monte carlo kmc simulations our discrete equations of motion thereby provide an exact analytic and computational alternative to kmc simulations of these models