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

space complexity in propositional calculus we study space complexity in the framework of propositional proofs we consider a natural model analogous to turing machines with a read only input tape and such popular propositional proof systems as resolution polynomial calculus and frege systems we propose two different space measures corresponding to the maximal number of bits and clauses monomials that need to be kept in the memory simultaneously we prove a number of lower and upper bounds in these models as well as some structural results concerning the clause space for resolution and frege systems

representation of the category of tangles by kontsevich s iterated integral applying kontsevich s iterated integral for tangles we get an isotopy invariant of tangles we give a method to compute the integral of a tangle combi natorially from modified integrals of some simple tangles we localize the integral by moving the end points of the tangle to an extreme configuration and modify the integral so that it is convergent by using a similar technique we generalize kontsevich s invariant to a framed tangle

some consequences of cryptographical conjectures for s 1 2 and ef we show that there is a pair of disjoint np sets whose disjointness is provable in s 2 1 and which cannot be separated by a set in p poly if the cryptosystem rsa is secure further we show that factoring and the discrete logarithm are implicitly definable in any extension of s 2 1 admitting an np definition of primes about which it can prove that no number satisfying the definition is composite

total space in resolution we show quadratic lower bounds on the total space used in resolution refutations of random k cnfs over n variables and of the graph pigeonhole principle and the bit pigeonhole principle for n holes this answers the open problem of whether there are families of k cnf formulas of polynomial size that require quadratic total space in resolution the results follow from a more general theorem showing that for formulas satisfying certain conditions in every resolution refutation there is a memory configuration containing many clauses of large width

compressed drinfeld associators drinfeld associator is a key tool in computing the kontsevich integral of knots a drinfeld associator is a series in two non commuting variables satisfying highly complicated algebraic equations hexagon and pentagon the logarithm of a drinfeld associator lives in the lie algebra l generated by the symbols a b c modulo a b b c c a the main result is a description of compressed associators that obey the compressed pentagon and hexagon in the quotient l l l l l the key ingredient is an explicit form of campbell baker hausdorff formula in the case when all commutators commute

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 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

new high order sufficient conditions for configuration tracking in this paper we propose new conditions guaranteeing that the trajectories of a mechanical control system can track any curve on the configuration manifold we focus on systems that can be represented as forced affine connection control systems and we generalize the sufficient conditions for tracking known in the literature the new results are proved by a combination of averaging procedures by highly oscillating controls with the notion of kinematic reduction

on universal vassiliev invariants using properties of ordered exponentials and the definition of the drinfeld associator as a monodromy operator for the knizhnik zamolodchikov equations we prove that the analytic and the combinatorial definitions of the universal vassiliev invariants of links are equivalent

complexity of control affine motion planning in this paper we study the complexity of the motion planning problem for control affine systems such complexities are already defined and rather well understood in the particular case of nonholonomic or sub riemannian systems our aim is to generalize these notions and results to systems with a drift accordingly we present various definitions of complexity as functions of the curve that is approximated and of the precision of the approximation due to the lack of time rescaling invariance of these systems we consider geometric and parametrized curves separately then we give some asymptotic estimates for these quantities as a byproduct we are able to treat the long time local controllability problem giving quantitative estimates on the cost of stabilizing the system near a nonequilibrium point of the drift

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

on denominators of the kontsevich integral and the universal perturbative invariant of 3 manifolds the integrality of the kontsevich integral and perturbative invariants is discussed it is shown that the denominator of the degree n part of the kontsevich integral of any knot or link is a divisor of 2 3 n 4 n 1 we prove this by establishing the existence of a drinfeld s associator in the space of chord diagrams with special denominators we also show that the denominator of the degree n part of the universal perturbative invariant of homology 3 spheres is not divisible by any prime greater than 2n 1

quasi quantum groups knots three manifolds and topological field theory we show how to construct starting from a quasi hopf algebra or quasiquantum group invariants of knots and links in some cases these invariants give rise to invariants of the three manifolds obtained by surgery along these links this happens for a finite dimensional quasi quantum group whose definition involves a finite groupg and a 3 cocycle which was first studied by dijkgraaf pasquier and roche we treat this example in more detail and argue that in this case the invariants agree with the partition function of the topological field theory of dijkgraaf and witten depending on the same datag

on the controllability of lagrangian systems by active constraints abstract we consider a mechanical system which is controlled by means of moving constraints namely we assume that some of the coordinates can be directly assigned as functions of time by implementing frictionless constraints this leads to a system of ode s whose right hand side depends quadratically on the time derivative of the control in this paper we introduce a simplified dynamics described by a differential inclusion we prove that every trajectory of the differential inclusion can be uniformly approximated by a trajectory of the original system on a sufficiently large time interval starting at rest under a somewhat stronger assumption we show this second trajectory reaches exactly the same terminal point

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

continuous dependence with respect to the input of trajectories of control affine systems we study the continuous dependence on the input of trajectories of control affine systems belonging to the class c0 m of all systems sigma of the form sigma dot x f 0 x sum i 1 m u i t f i x where f0 fm are continuous vector fields on some open subset of mathbb r n and the control functions belong to l 1 0 t mathbb r m we give a simple necessary and sufficient condition for a control sequence u j j 1 infty to cal t 0 converge to a control u infty i e to be such that for every system sigma in c0 the trajectories generated by the uj converge as j rightarrow infty to the trajectories generated by u infty we also characterize cal t k convergence the concept of control convergence that arises when we use instead of c0 m the class ck m of systems sigma where the fi are of class ck for k ge 1 in the scalar input case and we explain how the analogous characterization for the multi input case fails to be true unless one restricts oneself to the class c k comm m of systems for which the vector fields f0 fm commute as a preliminary we define a topology of trajectory convergence or t convergence on the set of all time varying vector fields omega times i ni x t mapsto f x t in mathbb r n where omega is an open subset of mathbb r n and i is an interval and we study some of its properties this enables us to make the definition of cal t k convergence precise for sequences and more generally for nets by saying that a net u alpha alpha in a in l 1 0 t mathbb r m cal t k converges to a limit u infty if for every system sigma in c0 m the time varying vector fields x t mapsto f 0 x sum i 1 m u i alpha t f i x cal t k converge to x t mapsto f 0 x sum i 1 m u i infty t f i x

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

high order sufficient conditions for configuration tracking of affine connection control systems in this paper we study under which conditions the trajectories of a mechanical control system can track any curve on the configuration manifold we focus on systems that can be represented as forced affine connection control systems and we generalize the sufficient conditions for tracking known in the literature the sufficient conditions are expressed in terms of convex cones of vector fields defined through particular brackets of the control vector fields of the system the tracking control laws obtained by our constructions depend on several parameters by imposing suitable asymptotic conditions on such parameters we construct algorithmically one parameter tracking control laws the theory is supported by examples of control systems associated with elliptic hovercrafts and ellipsoidal submarines

on interpolation and automatization for frege systems the interpolation method has been one of the main tools for proving lower bounds for propositional proof systems loosely speaking if one can prove that a particular proof system has the feasible interpolation property then a generic reduction can usually be applied to prove lower bounds for the proof system sometimes assuming a usually modest complexity theoretic assumption in this paper we show that this method cannot be used to obtain lower bounds for frege systems or even for tc0 frege systems more specifically we show that unless factoring of blum integers is feasible neither frege nor tc0 frege has the feasible interpolation property in order to carry out our argument we show how to carry out proofs of many elementary axioms theorems of arithmetic in polynomial sized tc0 frege as a corollary we obtain that tc0 frege as well as any proof system that polynomially simulates it is not automatizable under the assumption that factoring of blum integers is hard we also show under the same hardness assumption that the k provability problem for frege systems is hard

the yang baxter equation and invariants of links on considere le fait que l on peut construire des invariants d isotopie p et f pour des liens en utilisant certaines solutions des equations de yang baxter

continuous dependence of nonmonotonic discontinuous differential equations continuous dependence of solutions for a class of nonmonotonic discontinuous differential equations is studied first a local existence theorem due to z wu is extended to a larger class then a result concerning continuous dependence for this larger class is proven this employs a type of convergence similar to gihman s convergence criterion witch is defined to be for all a b and y formule the significance of gihman s convergence criterion is that for certain classes of differential equations it has been found to be necessary and sufficient for continuous dependence finally examples are presented to motivate and clarify this continuous dependence result

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 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

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

knot polynomials and vassiliev s invariants a fundamental relationship is established between jones knot invariants and vassiliev s knot invariants since vassiliev s knot invariants have a firm grounding in classical topology one obtains as a result a first step in understanding the jones polynomial by topological methods

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