random k sat two moments suffice to cross a sharp threshold many np complete constraint satisfaction problems appear to undergo a phase transition from solubility to insolubility when the constraint density passes through a critical threshold in all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment expectation of the number of solutions tends to zero we show that in the case of certain symmetric constraints considering the second moment of the number of solutions yields nearly matching lower bounds for the location of the threshold specifically we prove that the threshold for both random hypergraph 2 colorability property b and random not all equal k sat is 2 k 1 ln 2 o 1 as a corollary we establish that the threshold for random k sat is of order theta 2 k resolving a long standing open problem

on the relative merits of path dissolution and the method of analytic tableaux path dissolution is an inferencing mechanism that generalizes the method of analytic tableaux we present several results demonstrating that tableau deductions can be substantially speeded up with applications of dissolution technology we also consider the class of formulas on which the method of analytic tableaux was first shown to be intractable and prove that with the application of the ordinary distributive law standard tableau methods admit linear time proofs for this class

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

clause trees a tool for understanding and implementing resolution in automated reasoning abstract a new methodology data structure the clause tree is developed for automated reasoning based on resolution in first order logic a clause tree t on a set s of clauses is a 4 tuple n e l m where n a set of nodes divided into clause nodes and atom nodes e is a set of edges each of which joins a clause node to an atom node l is a labeling of n e which assigns to each clause node a clause of s to each atom node an instance of an atom of some clause of s and to each edge either or the edge joining a clause node to an atom node is labeled by the sign of the corresponding literal in the clause a resolution is represented by unifying two atom nodes of different clause trees which represent complementary literals the merge of two identical literals is represented by placing the path joining the two corresponding atom nodes into the set m of chosen merge paths the tail of the merge path becomes a closed leaf while the head remains an open leaf which can be resolved on the clause cl t that t represents is the set of literals corresponding to the labels of the open leaves modified by the signs of the incident edges the fundamental purpose of a clause tree t is to show that cl t can be derived from s using resolution loveland s model elimination me the selected literal procedure sl and shostak s graph construction procedure gc are explained in a unified manner using clause trees the condition required for choosing a merge path whose head is not a leaf is given this allows a clause tree to be built in one way the build ordering but justified as a proof in another the proof ordering the ordered clause set restriction and the foothold score restriction are explained using the operation on clause trees of merge path reversal a new procedure called alpoc which combines ideas from me gc and spencer s ordered clause set restriction oc to form a new procedure tighter than any of the top down procedures above is developed and shown to be sound and complete another operation on clause trees called surgery is defined and used to define a minimal clause tree any non minimal clause tree can be reduced to a minimal clause tree using surgery thereby showing that non minimal clause trees are redundant a sound procedure minalpoc that produces only minimal clause trees is given mergeless clause trees are shown to be equivalent to each of input resolution unit resolution and relative horn sets thereby giving short proofs of some known results many other new proof procedures using clause trees are discussed briefly leaving many open questions

optimizing propositional calculus formulas with regard to questions of deducibility abstract we consider propositional calculus formulas and are interested in the complexity of deciding for a clause we investigate the problem whether efficiency of an algorithm deciding can be improved by learning from queries having been answered by the algorithm before so we are looking for an optimized formula opt which is obtained from by adding suitable consequences we restrict ourselves to the following aspects of this optimization problem first we query for k 1 clauses allowing the addition of at most k clauses we prove that this problem is conp complete second we analyze various resolution strategies we establish a lower bound for the number of clauses which must be added to obtain an optimized formula finally we investigate optimization aspects of sld resolution

the complexity of gentzen systems for propositional logic abstract examples are given of valid sequents of classical propositional logic involving only the biconditional which require exponentially long proofs in a cut free gentzen system the sequents have polynomial size proofs if the cut rule is allowed

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

a kind of logical compilation for knowledge bases the forward chaining algorithm is perhaps the best known algorithm in expert systems however it is not complete because it cannot compute the two valued consequence literals of a propositional knowledge base i e set of rules with negations if the user wants to compute them he must use a particular algorithm which often takes much time we propose a compilation system for knowledge bases which we call logical compilation which allows us to compute the two valued consequence literals of a knowledge base i e set of rules using a forward chaining on the compiled base with any extensional knowledge base i e set of basic facts added we also use this compilation in a wide propositional calculus and solve the or problem in rule conclusion we present several methods with their benefits to make this compilation and we give properties on knowledge bases to avoid this compilation finally we give a theorem which defines incremental compilation

partition based logical reasoning for first order and propositional theories in this paper we show how tree decomposition can be applied to reasoning with first order and propositional logic theories our motivation is two fold first we are concerned with how to reason effectively with multiple knowledge bases that have overlap in content second we are concerned with improving the efficiency of reasoning over a set of logical axioms by partitioning the set with respect to some detectable structure and reasoning over individual partitions either locally or in a distributed fashion to this end we provide algorithms for partitioning and reasoning with related logical axioms in propositional and first order logic many of the reasoning algorithms we present are based on the idea of passing messages between partitions we present algorithms for both forward data driven and backward query driven message passing different partitions may have different associated reasoning procedures we characterize a class of reasoning procedures that ensures completeness and soundness of our message passing algorithms we further provide a specialized algorithm for propositional satisfiability checking with partitions craig s interpolation theorem serves as a key to proving soundness and completeness of all of these algorithms an analysis of these algorithms emphasizes parameters of the partitionings that influence the efficiency of computation we provide a greedy algorithm that automatically decomposes a set of logical axioms into partitions following this analysis

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

tractability of cut free gentzen type propositional calculus with permutation inference ii in arai 1996 we introduced a new inference rule called permutation to propositional calculus and showed that cut free gentzen system lk gcnf with permutation 1 satisfies the feasible subformula property and 2 proves pigeonhole principle and k equipartition polynomially in this paper we survey more properties of our system first we prove that cut free lk permutation has polynomial size proofs for nonunique endnode principle bondy s theorem second we remark the fact that permutation inference has an advantage over renaming inference in automated theorem proving since gcnf renaming does not always satisfy the feasible subformula property finally we discuss on the relative efficiency of our system vs frege systems and show that frege polynomially simulates gcnf renaming if and only if frege polynomially simulates extended frege

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

fully measurable grand lebesgue spaces abstract using variable exponents we build a new class of rearrangement invariant banach function spaces independent of the variable lebesgue spaces whose function norm is f ess sup x 0 1 p x x f where p x denotes the norm of the lebesgue space of exponent p x assumed measurable and possibly infinite and is measurable too such class contains some known banach spaces of functions among which are the classical and the grand lebesgue spaces and the exp spaces 0 we analyze the function norm and we prove a boundedness result for the hardy littlewood maximal operator via a hardy type inequality

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

cutting plane proofs in polynomial space following chvatal cutting planes may be viewed as a proof system for establishing that a given system of linear inequalities has no integral solution we show that such proofs may be carried out in polynomial workspace

tight rank lower bounds for the sherali adams proof system we consider a proof more accurately refutation system based on the sherali adams sa operator associated with integer linear programming if f is a cnf contradiction that admits a resolution refutation of width k and size s then we prove that the sa rank of f is k and the sa size of f is k 1 s 1 we establish that the sa rank of both the pigeonhole principle php n 1 n and the least number principle lnp n is n 2 since the sa refutation system rank simulates the refutation system of lovasz schrijver without semidefinite cuts ls we obtain as a corollary linear rank lower bounds for both of these principles in ls

the complexity of the hajos calculus for planar graphs the planar hajos calculus is the hajos calculus with the restriction that all the graphs that appear in the construction including a final graph must be planar we prove that the planar hajos calculus is polynomially bounded iff the hajos calculus is polynomially bounded

circuit complexity proof complexity and polynomial identity testing the ideal proof system we introduce a new and natural algebraic proof system whose complexity measure is essentially the algebraic circuit size of nullstellensatz certificates this enables us to exhibit close connections between effective nullstellensatze proof complexity and algebraic circuit complexity in particular we show that any super polynomial lower bound on any boolean tautology in our proof system implies that the permanent does not have polynomial size algebraic circuits vnp vp we also show that super polynomial lower bounds on the number of lines in polynomial calculus proofs imply the permanent versus determinant conjecture note that there was no proof system prior to ours for which lower bounds on an arbitrary tautology implied any complexity class lower bound our proof system helps clarify the relationships between previous algebraic proof systems in doing so we highlight the importance of polynomial identity testing pit in proof complexity in particular we use pit to illuminate ac0 p frege lower bounds which have been open for nearly 30 years with no satisfactory explanation as to their apparent difficulty finally we explain the obstacles that must be overcome in any attempt to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity using the algebraic structure of our proof system we propose a novel route to such lower bounds although such lower bounds remain elusive this proposal should be contrasted with the difficulty of extending ac0 p circuit lower bounds to ac0 p frege lower bounds

cocolog a conditional observer and controller logic for finite machines the problem of observation and control for partially observed input state output machines is formulated in terms of a tree of first order logical theories a set of first order languages for the description of the controlled evolution and state estimation of any given machine cal m is specified further extralogical conditional control rules are formulated so that closed loop control actions occur when extralogically specified past observation dependent conditions are fulfilled in particular conditional control rules may include commands that steer the system state from a current partially observed state estimate to a target state if such a sequence of controls can be proven to exist starting from a general theory of cal m at the initial instant observations on the input output behaviour of the system at each later instant are accepted by the system as new axioms these are then used together with the previously generated theory to generate the current theory the acronym cocolog is used to denote the family of first order conditional observer and controller logics for any given input state output system a semantics is supplied for each cocolog system in terms of interpretations of controlled transitions on a tree indexed by the possible sequences of input output observations extralogical rules including the conditional control rules relating members of the family of theories of a cocolog system are presented in the form of a set of metalevel rules following the complete definition of a cocolog system the consistency and completeness of the first order theories in a cocolog system are established decidability is obtained using a unique model property and examples of the operation of a cocolog logic control system are given

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

unrestricted resolution versus n resolution abstract an n resolution proof is a resolution proof in which the resolution rule is restricted one clause to which it is applied must only consist of negative literals n resolution is complete schoning 1987 p 109 stickel 1985 p 86 we construct an infinite family of propositional formulas and show these formulas have unrestricted resolution proofs whose length is polynomially bounded in their size all n resolution proofs of these formulas are of superpolynomial length

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

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

bayesian modeling of dynamic scenes for object detection accurate detection of moving objects is an important precursor to stable tracking or recognition in this paper we present an object detection scheme that has three innovations over existing approaches first the model of the intensities of image pixels as independent random variables is challenged and it is asserted that useful correlation exists in intensities of spatially proximal pixels this correlation is exploited to sustain high levels of detection accuracy in the presence of dynamic backgrounds by using a nonparametric density estimation method over a joint domain range representation of image pixels multimodal spatial uncertainties and complex dependencies between the domain location and range color are directly modeled we propose a model of the background as a single probability density second temporal persistence is proposed as a detection criterion unlike previous approaches to object detection which detect objects by building adaptive models of the background the foregrounds modeled to augment the detection of objects without explicit tracking since objects detected in the preceding frame contain substantial evidence for detection in the current frame finally the background and foreground models are used competitively in a map mrf decision framework stressing spatial context as a condition of detecting interesting objects and the posterior function is maximized efficiently by finding the minimum cut of a capacitated graph experimental validation of the proposed method is performed and presented on a diverse set of dynamic scenes

resolution and the integrality of satisfiability problems a satisfiability problem can be regarded as a nondisjoint union of set covering problems we show that if the resolution method of theorem proving is applied to the satisfiability problem its constraint set defines an integral polytope if and only if the constraint sets of the set covering problems do in this sense resolution reduces the integrality question for the satisfiability problem to that for the set covering problem

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

meta resolution an algorithmic formalisation we study the tseitin formulae tseitin 1970 on which resolution is known to be inefficient we first give a new proof of the satisfiability condition and precise the number of solutions if any and otherwise the minimal contradictory subformula then defining a new data structure we introduce the meta argument of that proof in the davis putnam and loveland procedure loveland 1978 and obtain a new algorithm for sat especially efficient on these formulae

universal multimode background subtraction in this paper we present a complete change detection system named multimode background subtraction the universal nature of system allows it to robustly handle multitude of challenges associated with video change detection such as illumination changes dynamic background camera jitter and moving camera the system comprises multiple innovative mechanisms in background modeling model update pixel classification and the use of multiple color spaces the system first creates multiple background models of the scene followed by an initial foreground background probability estimation for each pixel next the image pixels are merged together to form mega pixels which are used to spatially denoise the initial probability estimates to generate binary masks for both rgb and ycbcr color spaces the masks generated after processing these input images are then combined to separate foreground pixels from the background comprehensive evaluation of the proposed approach on publicly available test sequences from the cdnet and the esi data sets shows superiority in the performance of our system over other state of the art algorithms

tseitin s formulas revisited abstract g being a graph we define its cyclomatic cohesion g then using tseitin s method 1970 we construct a contradictory formula c g and prove our main theorem every resolution of c g contains at least 2 g distinct clauses a similar result was obtained by urquhart 1987 with a different method valid only for a specific family of graphs

generating hard satisfiability problems abstract we report results from large scale experiments in satisfiability testing as has been observed by others testing the satisfiability of random formulas often appears surprisingly easy here we show that by using the right distribution of instances and appropriate parameter values it is possible to generate random formulas that are hard that is for which satisfiability testing is quite difficult our results provide a benchmark for the evaluation of satisfiability testing procedures

the efficiency of resolution and davis putnam procedures we consider several problems related to the use of resolution based methods for determining whether a given boolean formula in conjunctive normal form is satisfiable first building on the work of clegg edmonds and impagliazzo in proceedings of the twenty eighth annual acm symposium on theory of computing philadelphia pa 1996 acm new york 1996 pp 174 183 we give an algorithm for unsatisfiability that when given an unsatisfiable formula of f finds a resolution proof of f the runtime of our algorithm is subexponential in the size of the shortest resolution proof of f next we investigate a class of backtrack search algorithms for producing resolution refutations of unsatisfiability commonly known as davis putnam procedures and provide the first asymptotically tight average case complexity analysis for their behavior on random formulas in particular for a simple algorithm in this class called ordered dll we prove that the running time of the algorithm on a randomly generated k cnf formula with n variables and m clauses is 2 theta n n m 1 k 2 with probability 1 o 1 finally we give new lower bounds on mbox res f the size of the smallest resolution refutation of f for a class of formulas representing the pigeonhole principle and for randomly generated formulas for random formulas chvatal and szemeredi j acm 35 1988 pp 759 768 had shown that random 3 cnf formulas with a linear number of clauses require exponential size resolution proofs and fu on the complexity of proof systems ph d thesis university of toronto toronto on canada 1995 extended their results to k cnf formulas these proofs apply only when the number of clauses is omega n log n we show that a lower bound of the form 2 n gamma holds with high probability even when the number of clauses is n k 2 4 epsilon

streams and strings in formal proofs streams are acyclic directed subgraphs of the logical flow graph of a proof representing bundles of paths with the same origin and the same end the notion of stream is used to describe the evolution of proofs during cut elimination in purely algebraic terms the algebraic and combinatorial properties of flow graphs emerging from our analysis serve to elucidate logical phenomena however the full logical significance of the combinatorics e g the absence of certain patterns within flow graphs remains unclear

complexity issues related to propagation completeness knowledge compilation is a process of adding more information to a knowledge base in order to make it easier to deduce facts from the compiled base than from the original one one type of knowledge compilation occurs when the knowledge in question is represented by a boolean formula in conjunctive normal form cnf the goal of knowledge compilation in this case is to add clauses to the input cnf until a logically equivalent propagation complete cnf is obtained a cnf is called propagation complete if after any partial substitution of truth values all logically entailed literals can be inferred from the resulting cnf formula by unit propagation the key to this type of knowledge compilation is the ability to generate so called empowering clauses a clause is empowering for a cnf if it is an implicate and for some partial substitution of truth values it enlarges the set of entailed literals inferable by unit propagation in this paper we study several complexity issues related to empowering implicates propagation completeness and its relation to resolution proofs we show several results a given a cnf and a clause it is co np complete to decide whether the clause is an empowering implicate of the cnf b given a cnf it is np complete to decide whether there exists an empowering implicate for it and thus it is co np complete to decide whether a cnf is propagation complete and c there exist cnfs to which an exponential number of clauses must be added to make them propagation complete

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

foreground segmentation with tree structured sparse rpca background subtraction is a fundamental video analysis technique that consists of creation of a background model that allows distinguishing foreground pixels we present a new method in which the image sequence is assumed to be made up of the sum of a low rank background matrix and a dynamic tree structured sparse matrix the decomposition task is then solved using our approximated robust principal component analysis arpca method which is an extension to the rpca that can handle camera motion and noise our model dynamically estimates the support of the foreground regions via a superpixel generation step so that spatial coherence can be imposed on these regions unlike conventional smoothness constraints such as mrf our method is able to obtain crisp and meaningful foreground regions and in general handles large dynamic background motion better to reduce the dimensionality and the curse of scale that is persistent in the rpca based methods we model the background via column subset selection problem that reduces the order of complexity and hence decreases computation time comprehensive evaluation on four benchmark datasets demonstrate the effectiveness of our method in outperforming state of the art alternatives

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

new properties of small lebesgue spaces and their applications we prove some new properties of the small lebesgue spaces introduced by fiorenza in f combining these properties with the poincare sobolev inequalities for the relative rearrangement see r1 we derive some new and precises estimates either for small lebesgue sobolev spaces or for quasilinear equations with data in the small lebesgue spaces

decomposition of tautologies into regular formulas and strong completeness of connection graph resolution this paper addresses and answers a fundamental question about resolution informally what is gained with respect to the search for a proof by performing a single resolution step it is first shown that any unsatisfiable formula may be decomposed into regular formulas provable in linear time by resolution a relevant resolution step strictly reduces at least one of the formulas in the decomposition while an irrelevant one does not contribute to the proof in any way the relevance of this insight into the nature of resolution and of the unsatisfiability problem for the development of proof strategies and for complexity considerations are briefly discussed the decomposition also provides a technique for establishing completeness proofs for refinements of resolution as a first application connection graph resolution is shown to be strongly complete this settles a problem that remained open for two decades despite many proff attempts the result is relevant for theorem proving because without strong completeness a connection graph resolution prover might run into an infinite loop even on the ground level

on the modeling of search in theorem proving towards a theory of strategy analysis abstract we present a model for representing search in theorem proving this model captures the notion of contraction which has been central in some of the recent developments in theorem proving we outline an approach to measuring the complexity of search which can be applied to analyze and evaluate the behaviour of theorem proving strategies using our framework we compare contraction based strategies of different contraction power and show how they affect the evolution of the respective search spaces during the derivation

space bounds for resolution we introduce a new way to measure the space needed in resolution refutations of cnf formulas in propositional logic with the former definition 1994 b h kleine and t lettman aussangenlogik deduktion und algorithmen teubner stuttgart the space required for the resolution of any unsatisfiable formula in cnf is linear in the number of clauses the new definition allows a much finer analysis of the space in the refutation ranging from constant to linear space moreover the new definition allows us to relate the space needed in a resolution proof of a formula to other well studied complexity measures it coincides with the complexity of a pebble game in the resolution graphs of a formula and as we show has relationships to the size of the refutation we also give upper and lower bounds on the space needed for the resolution of unsatisfiable formulas we show that tseitin formulas associated to a certain kind of expander graphs of n nodes need resolution space n c for some constant c measured on the number of clauses this result is the best possible we also show that the formulas expressing the general pigeonhole principle with n holes and more than n pigeons need space n 1 independent of the number of pigeons since a matching space upper bound of n 1 for these formulas exists the obtained bound is exact we also point to a possible connection between resolution space and resolution width another measure for the complexity of resolution refutations 2001 elsevier science

exponential lower bounds on the lengths of some classes of branch and cut proofs we examine the complexity of branch and cut proofs in the context of 0 1 integer programs we establish an exponential lower bound on the length of branch and cut proofs that use 0 1 branching and lift and project cuts called simple disjunctive cuts by some authors gomory chvatal cuts and cuts arising from the n0 matrix cut operator of lovasz and schrijver a consequence of the lower bound result in this paper is that branch and cut methods of the type described above have exponential running time in the worst case

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

on the chvatal rank of the pigeonhole principle we demonstrate that the cutting plane cp rank also known as the chvatal rank of the pigeonhole principle is q logn our proof uses a novel technique which allows us to demonstrate rank lower bounds for fractional points with fewer restrictions than previous methods we also demonstrate that the pigeonhole principle has a polynomially sized cp proof

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

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

density based multifeature background subtraction with support vector machine background modeling and subtraction is a natural technique for object detection in videos captured by a static camera and also a critical preprocessing step in various high level computer vision applications however there have not been many studies concerning useful features and binary segmentation algorithms for this problem we propose a pixelwise background modeling and subtraction technique using multiple features where generative and discriminative techniques are combined for classification in our algorithm color gradient and haar like features are integrated to handle spatio temporal variations for each pixel a pixelwise generative background model is obtained for each feature efficiently and effectively by kernel density approximation kda background subtraction is performed in a discriminative manner using a support vector machine svm over background likelihood vectors for a set of features the proposed algorithm is robust to shadow illumination changes spatial variations of background we compare the performance of the algorithm with other density based methods using several different feature combinations and modeling techniques both quantitatively and qualitatively

a proper hierarchy of propositional sequent calculi abstract it is known that lk a system of propositional sequent calculus without a cut rule written in tree form does not p simulate lk with a full cut rule written in tree form 1 6 it is also known that lk with a full cut rule in tree form p simulates lk in dag form in this paper we show that lk in tree form with a cut rule where the complexity of cut formulas is bounded by k 1 has an exponential speed up over the one bounded by k we also show that lk in tree form with a bounded complexity cut rule does not polynomially simulate cut free lk in dag form

tight representation of logical constraints as cardinality rules am are true then at least l of the propositions b1 bn are true the main result of the paper is that the procedure in fact provides a convex hull description

grand lebesgue spaces with respect to measurable functions abstract let 1 p given r n a measurable set of finite lebesgue measure the norm of the grand lebesgue spaces l p is given by f l p sup 0 e p 1 e 1 p e 1 f p e d x 1 p e in this paper we consider the norm f l p obtained replacing e 1 p e by a generic nonnegative measurable function e we find necessary and sufficient conditions on in order to get a functional equivalent to a banach function norm and we determine the interesting class b p of functions with the property that every generalized function norm is equivalent to a function norm built with b p we then define the l p spaces prove some embedding results and conclude with the proof of the generalized hardy inequality

on finding short resolution refutations and small unsatisfiable subsets we consider the parameterized problems of whether a given set of clauses can be refuted within k resolution steps and whether a given set of clauses contains an unsatisfiable subset of size at most k we show that both problems are complete for the class w 1 the first level of the w hierarchy of fixed parameter intractable problems our results remain true if restricted to 3 sat instances and or to various restricted versions of resolution including tree like resolution input resolution and read once resolution applying a metatheorem of frick and grohe we show that restricted to classes of sets of clauses of locally bounded treewidth the considered problems are fixed parameter tractable for example the problems are fixed parameter tractable for planar cnf formulas

asymptotic cyclic expansion and bridge groups of formal proofs abstract formal proofs even simple ones may hide an unexpected intricate combinatorics we define a new combinatorial invariant the bridge group of a proof which encodes the cyclic structure of proofs in the sequent calculus we compute the bridge groups of two infinite families of proofs and identify them with the baumslag solitar and gersten groups we observe that the distortion of cyclic subgroups in these groups equals the asymptotic growth of the procedure of elimination of lemmas from the proofs