adsorption of pulled directed paths in a slanted boundary in this paper the generating function free energy and phase diagram of a class of models of directed paths related to the models of paths adsorbing in the boundary y nx n 2 for and pulled by a vertical force in its endpoint are determined these models are related to the special classes of models of random walks in the quarter plane and which interact with the x axis generating functions are determined by solving a functional recurrence using the kernel method and are algebraic the generating function is determined for general values of n and the solutions for n 1 and n 2 are examined explicitly together with the derivation of the phase diagram and locations of critical curves three phases including a free or bulk phase an adsorbed phase and a ballistic or stretched phase are identified and the nature of the transitions between these phases are examined order parameters are defined and asymptotic expressions are determined for the critical adsorption surface as a function of the pulling force some of these results are also briefly generalized to arbitrary values of n

adsorbing and collapsing directed animals a model of a self interacting directed animal which also interacts with a solid wall is studied as a model of a directed branched polymer which can undergo both a collapse and an adsorption transition the directed animal is confined to a 45 wedge and it interacts with one of the walls of this wedge the existence of a thermodynamic limit in this model shown and the presence of an adsorption transition is demonstrated by using constructive techniques by comparing this model to a process of directed percolation we show that there is also a collapse or transition in this model we examine directed percolation in a wedge to show that there is a collapse phase present for arbitrary large values of the adsorption activity the generating function of adsorbing directed animals in a half space is found next from which we find the tricritical exponents associated with the adsorption transition a full solution for a collapsing directed animal seems intractible so instead we examine the collapse transition of a model of column convex directed animals with a contact activity next

universal randomized switching in this paper we consider a competitive approach to sequential decision problems suitable for a variety of signal processing applications where at each of a succession of times a selection must be made from among a fixed set of strategies or outcomes for each such decision and outcome pair loss is incurred and it is the time accumulation of these losses that is sought to be minimized rather than using a statistical performance measure our goal in this pursuit is to sequentially accumulate loss that is no larger than that of the best loss that could be obtained through a partitioning of the sequence of observations into an arbitrary fixed number of segments and independently selecting a different strategy for each segment for this purpose we introduce a randomized sequential algorithm built upon that of kozat and singer that asymptotically achieves the performance of a noncausal algorithm that would be able to choose the number of segments and the best algorithm for each segment based on observing the whole observation process a priori in addition to improving upon the bounds of kozat and singer as well as gyorgy the results we provide hold for more general loss functions than the square error loss studied therein

a zero delay sequential scheme for lossy coding of individual sequences we consider adaptive sequential lossy coding of bounded individual sequences when the performance is measured by the sequentially accumulated mean squared distortion the encoder and the decoder are connected via a noiseless channel of capacity r and both are assumed to have zero delay no probabilistic assumptions are made on how the sequence to be encoded is generated for any bounded sequence of length n the distortion redundancy is defined as the normalized cumulative distortion of the sequential scheme minus the normalized cumulative distortion of the best scalar quantizer of rate r which is matched to this particular sequence we demonstrate the existence of a zero delay sequential scheme which uses common randomization in the encoder and the decoder such that the normalized maximum distortion redundancy converges to zero at a rate n sup 1 5 log n as the length of the encoded sequence n increases without bound

knotting in stretched polygons the knotting in a lattice polygon model of ring polymers is examined when a stretching force is applied to the polygon by examining the incidence of cut planes in the polygon we prove a pattern theorem in the stretching regime for large applied forces this theorem can be used to examine the incidence of entanglements such as knotting and writhing in particular we prove that for arbitrarily large positive but finite values of the stretching force the probability that a stretched polygon is knotted approaches 1 as the length of the polygon increases in the case of writhing we prove that for stretched polygons of length n and for every function the probability that the absolute value of the mean writhe is less than f n approaches 0 as n for sufficiently large values of the applied stretching force

domain permutation reduction for constraint satisfaction problems this paper is concerned with the constraint satisfaction problem csp it is well known that the general csp is np hard however there has been significant success in discovering subproblems which are tractable polynomial time solvable one of the most effective ways to obtain a tractable subproblem has been to force all of the constraint relations to lie in some tractable language in this paper we define a new way of identifying tractable subproblems of the csp let p be an arbitrary csp instance and c be any tractable language suppose there exists for each variable of p a permutation of the domain such the resultant permuted constraint relations of p all lie in c the domain permuted instance is then an instance of a tractable class and can be solved by the polynomial time algorithm for c solutions to p can be obtained by inverting the domain permutations the question for a given class of instances and language c whether such a set of domain permutations can be found efficiently is the key to this method s tractability one of the important contributions made in this paper is the notion of a lifted constraint instance which is a powerful tool to study this question we consider the open problem of discovering domain permutations which make instances max closed we show that for bounded arity instances over a boolean domain this problem is tractable while for domain size three it is intractable even for binary instances we give a simple proof verifying the tractability of discovering domain permutations which make instances row convex we refute a published result by giving a simple proof of the intractability of discovering domain permutations which make instances even with domain size four connected row convex we demonstrate that triangulated and stable marriage instances are reducible via domain permutations to max closed instances this provides a simple explanation for arc consistency deciding these instances we verify with a simple direct proof the tractability of identification of renamable horn instances and the intractability of finding the largest renamable horn subset of clauses of an instance of sat we describe natural tractable classes which properly extend the maximal relational classes arising from tractable constraint languages we believe that domain permutation reductions have a significant chance of being useful in practical applications

a short note on some tractable cases of the satisfiability problem it is shown that the tractable class of cnf formulas solvable by linear autarkies properly contains the class of q horn formulas and that it is incomparable with slur

adsorption and collapse of self avoiding walks in three dimensions a monte carlo study we consider self avoiding walks on the simple cubic lattice with a unit degree vertex at the origin and confined to the half space in addition the walks interact with the plane z 0 and have a vertex vertex interaction within the walk so that the walk can adsorb at the surface z 0 and also undergo collapse we investigate the interaction between collapse and adsorption using monte carlo methods and explore the form of the phase diagram

exact evaluation of the collapse phase boundary for two dimensional directed polymers the phase boundary of the collapsed phase for a directed polymer on a square lattice in the presence of an attractive wall and monomer monomer interactions is calculated exactly using transfer matrix techniques the position of the multicritical point is also identified

statistics of lattice animals the authors investigate the large n behaviour of the number of lattice animals with n vertices having alpha cycles per vertex they prove concavity and continuity properties for the corresponding growth constant and in particular show that lattice trees are exponentially scarce in the set of lattice animals they also consider the corresponding generating function and prove a number of theorems which bound it and set other limits on its possible behaviour

composite models of polygons a composite polygon is composed of a lattice polygon in the square lattice which contains in its interior an internal structure which may also be a lattice polygon or a lattice tree or a lattice animal or a lattice disc or a collection of these the properties of composite polygons are considered in this manuscript in particular i shall consider the growth constants and generating functions of these models as well as the statistical mechanics of interacting models of composite polygons it is shown that there is an adsorption transition of the internal structure on the containing polygon and a transition which corresponds to the inflation of the containing polygon by the internal structure

study of the θ point by enumeration of self avoiding walks on the triangular lattice the author reports series expansion analyses of self avoiding walks with nearest neighbour bond interactions the estimates 2 nu t 1 07 or 0 05 and phi 0 64 or 0 05 for the correlation and crossover exponents at the theta point were obtained by examining the number of walks and the end to end distance data up to 16 steps on the triangular lattice

on the minimum description length principle for sources with piecewise constant parameters universal lossless coding in the presence of finitely many abrupt changes in the statistics of the source at unknown points is investigated the minimum description length mdl principle is derived for this setting in particular it is shown that for any uniquely decipherable code for almost every combination of statistical parameter vectors governing each segment and for almost every vector of transition instants the minimum achievable redundancy is composed from 0 5 log n n bits for each unknown segmental parameter and log n n bits for each transition where n is the length of the input string this redundancy is shown to be attainable by a strongly sequential universal encoder i e an encoder that does not utilize the knowledge of a prescribed value of n

surface critical exponents for models of polymer collapse and adsorption the universality of the theta and theta points the surface critical exponents are estimated both in ordinary and in special regimes from exact series expansions of up to 28 terms for a model of the theta point they are found to be in agreement with those derived for the theta point confirming the conjecture that the theta and theta points lie in the same universality class some essential features of the phase diagram for a self interacting self avoiding polymer in solution are also calculated to a higher degree of accuracy than previously obtained

a class of logic problems solvable by linear programming in propositional logic several problems such as satisfiability max sat and logical inference can be formulated as integer programs in this paper we consider sets of clauses for which the corresponding integer programs can be solved as linear programs we prove that balanced sets of clauses have this property

the linking number and the writhe of uniform random walks and polygons in confined spaces random walks and polygons are used to model polymers in this paper we consider the extension of the writhe self linking number and linking number to open chains we then study the average writhe self linking and linking number of random walks and polygons over the space of configurations as a function of their length we show that the mean squared linking number the mean squared writhe and the mean squared self linking number of oriented uniform random walks or polygons of length n in a convex confined space are of the form o n2 moreover for a fixed simple closed curve in a convex confined space we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of n edges is of the form our numerical studies confirm those results they also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons of n edges each is of the form o n equilateral random walks and polygons are used to model polymers in conditions we use numerical simulations to investigate how the self linking and linking number of equilateral random walks scale with their length

universal switching linear least squares prediction in this paper we consider sequential regression of individual sequences under the square error loss we focus on the class of switching linear predictors that can segment a given individual sequence into an arbitrary number of blocks within each of which a fixed linear regressor is applied using a competitive algorithm framework we construct sequential algorithms that are competitive with the best linear regression algorithms for any segmenting of the data as well as the best partitioning of the data into any fixed number of segments where both the segmenting of the data and the linear predictors within each segment can be tuned to the underlying individual sequence the algorithms do not require knowledge of the data length or the number of piecewise linear segments used by the members of the competing class yet can achieve the performance of the best member that can choose both the partitioning of the sequence as well as the best regressor within each segment we use a transition diagram f m j willems 1996 to compete with an exponential number of algorithms in the class using complexity that is linear in the data length the regret with respect to the best member is o ln n per transition for not knowing the best transition times and o ln n for not knowing the best regressor within each segment where n is the data length we construct lower bounds on the performance of any sequential algorithm demonstrating a form of min max optimality under certain settings we also consider the case where the members are restricted to choose the best algorithm in each segment from a finite collection of candidate algorithms performance on synthetic and real data are given along with a matlab implementation of the universal switching linear predictor

multiple markov chain monte carlo study of adsorbing self avoiding walks in two and in three dimensions a self avoiding walk adsorbing on a line in the square lattice and on a plane in the cubic lattice is studied numerically as a model of an adsorbing polymer in dilute solution the walk is simulated by a multiple markov chain monte carlo implementation of the pivot algorithm for self avoiding walks vertices in the walk that are visits in the adsorbing line or plane are weighted by e t he critical value of where the walk adsorbs on the adsorbing line or adsorbing plane is determined by considering energy ratios and approximations to the free energy we determine that the critical values of are c 0 565 0 010 in the square lattice 0 288 0 020 in the cubic lattice in addition the value of the crossover exponent is determined 0 501 0 015 in the square lattice 0 5005 0 0036 in the cubic lattice metric quantities including the mean square radius of gyration are also considered as well as rescaling of the specific heat and free energy as the critical point is approached

computing minimal models by partial instantiation abstract unlike sets of definite horn clauses logic programs with disjunctions of atoms in clause heads are often interpreted in terms of minimal models it is also well known that the minimal models of logic programs are closely related to the so called stable models of logic programs with nonmonotonic negation in clause bodies as well as to circumscription methods to compute minimal models of logic programs are becoming increasingly important as an intermediate step in the computation of structures associated with nonmonotonic logic programs however to date all these techniques have been restricted to the case of propositional logic programs which means that an ordinary disjunctive logic program must be grounded out prior to computation grounding out in this manner leads to a combinatorial explosion in the number of clauses and hence is unacceptable in this paper we show how given any method m which correctly computes the set of minimal models of a propositional logic program we can develop a strategy to compute truth in a minimal model of a disjunctive logic program p the novel feature of our method is that it works on an instantiate by need basis and thus avoids unnecessary grounding

self avoiding walks interacting with a surface discusses two models of polymer adsorption in one model a self avoiding walk on a d dimensional lattice interacts with a d 1 dimensional hyperplane in the other model this walk must also lie in or on one side of this hyperplane both models exhibit non analytic behaviour corresponding to a phase transition but these phase transitions do not occur at the same point the authors give numerical estimates of the locations of these transitions

interacting self avoiding walks and polygons in three dimensions self interacting walks and polygons on the simple cubic lattice undergo a collapse transition at the point we consider self avoiding walks and polygons with an additional interaction between pairs of vertices which are unit distance apart but not joined by an edge of the walk or polygon we prove that these walks and polygons have the same limiting free energy if the interactions between nearest neighbour vertices are repulsive the attractive interaction regime is investigated using monte carlo methods and we find evidence that the limiting free energies are also equal here in particular this means that these models have the same point in the asymptotic limit the dimensions and shapes of walks and polygons are also examined as a function of the interaction strength

on the topology of a polymer ring first we survey the literature on knots and links in theoretical physics next we report a numerical study in which equilibrium configurations of ring polymers in an infinite space or confined to the interior of a sphere are generated by using a new algorithm the a priori probability for the occurrence of a knot is determined numerically the results are compatible with scaling laws of striking simplicity we also study the mutual entanglement of links comparing the gauss invariant with the alexander polynomial

collapse of two dimensional linear polymers we study the collapse transition of a two dimensional very long polymer the model we consider is a lattice model where the chain is represented by a self avoiding walk with nearest neighbor attraction by using the transfer matrix technique we calculate exactly the thermal and geometrical properties of the polymer on strips of finite width we then use finite size scaling to determine the values of the tricritical point exponentsv1 0 55 0 01vu 1 15 0 15 1 v1 1 80 0 05 we compare these results to the other values already proposed in the literature

extended horn sets in propositional logic the class of horn clause sets in propositional logic is extended to a larger class for which the satisfiability problem can still be solved by unit resolution in linear time it is shown that to every arborescence there corresponds a family of extended horn sets where ordinary horn sets correspond to stars with a root at the center these results derive from a theorem of chandresekaran that characterizes when an integer solution of a system of inequalities can be found by rounding a real solution in a certain way a linear time procedure is provided for identifying hidden extended horn sets extended horn but for complementation of variables that correspond to a specified arborescence finally a way to interpret extended horn sets in applications is suggested

entanglement complexity of self avoiding walks self avoiding walks on three dimensional lattices are flexible linear objects which can be self entangled the authors discuss several ways to measure entanglement complexity for n step walks and prove that these complexity measures tend to infinity with n for small n they use monte carlo methods to estimate and compare the n dependence of two of these complexity measures

collapsing and adsorbing polygons the adsorption transition in the phase diagram of a self interacting lattice polygon is examined the polygon has a nearest neighbour contact fugacity and the interaction between the polygon and an impenetrable wall is modelled by a visit fugacity which is conjugate to the number of vertices of the polygon incident with the wall the partition function of this model is z c y z d p v cp c v c y v z c where p c v c is the number of polygons with c nearest neighbour contacts v visits to the wall and n edges and counted up to translations parallel to the wall the limiting free energy of this model is f c y z d limn 1 1 logz c y z and it is known to be a non analytic function of y for each z 1 for all z2 0 1

tracking the best quantizer an algorithm is presented for online prediction that allows to track the best expert efficiently even when the number of experts is exponentially large provided that the set of experts has a certain additive structure as an example we work out the case where each expert is represented by a path in a directed graph and the loss of each expert is the sum of the weights over the edges in the path these results are then used to construct universal limited delay schemes for lossy coding of individual sequences in particular we consider the problem of tracking the best scalar quantizer that is adaptively matched to the source sequence with piecewise different behavior a randomized algorithm is presented which can perform on any source sequence asymptotically as well as the best scalar quantization algorithm that is matched to the sequence and is allowed to change the employed quantizer for a given number of times the complexity of the algorithm is quadratic in the sequence length but at the price of some deterioration in performance the complexity can be made linear analogous results are obtained for sequential multiresolution and multiple description scalar quantization of individual sequences

the weighted majority algorithm we study the construction of prediction algorithms in a situation in which a learner faces a sequence of trials with a prediction to be made in each and the goal of the learner is to make few mistakes we are interested in the case where the learner has reason to believe that one of some pool of known algorithms will perform well but the learner does not know which one a simple and effective method based on weighted voting is introduced for constructing a compound algorithm in such a circumstance we call this method the weighted majority algorithm we show that this algorithm is robust in the presence of errors in the data we discuss various versions of the weighted majority algorithm and prove mistake bounds for them that are closely related to the mistake bounds of the best algorithms of the pool for example given a sequence of trials if there is an algorithm in the pool a that makes at most m mistakes then the weighted majority algorithm will make at most c log a m mistakes on that sequence where c is fixed constant

tracking the best linear predictor in most on line learning research the total on line loss of the algorithm is compared to the total loss of the best off line predictor u from a comparison class of predictors we call such bounds static bounds the interesting feature of these bounds is that they hold for an arbitrary sequence of examples recently some work has been done where the predictor ut at each trial t is allowed to change with time and the total on line loss of the algorithm is compared to the sum of the losses of ut at each trial plus the total cost for shifting to successive predictors this is to model situations in which the examples change over time and different predictors from the comparison class are best for different segments of the sequence of examples we call such bounds shifting bounds they hold for arbitrary sequences of examples and arbitrary sequences of predictors naturally shifting bounds are much harder to prove the only known bounds are for the case when the comparison class consists of a sequences of experts or boolean disjunctions in this paper we develop the methodology for lifting known static bounds to the shifting case in particular we obtain bounds when the comparison class consists of linear neurons linear combinations of experts our essential technique is to project the hypothesis of the static algorithm at the end of each trial into a suitably chosen convex region this keeps the hypothesis of the algorithm well behaved and the static bounds can be converted to shifting bounds

knots in self avoiding walks discusses the existence of knots in random self avoiding walks on a lattice using kesten s 1963 pattern theorem it is shown that almost all sufficiently long self avoiding walks on the three dimensional simple cubic lattice contain a knot

universal linear prediction by model order weighting a common problem that arises in adaptive filtering autoregressive modeling or linear prediction is the selection of an appropriate order for the underlying linear parametric model we address this problem for linear prediction but instead of fixing a specific model order we develop a sequential prediction algorithm whose sequentially accumulated average squared prediction error for any bounded individual sequence is as good as the performance attainable by the best sequential linear predictor of order less than some m this predictor is found by transforming linear prediction into a problem analogous to the sequential probability assignment problem from universal coding theory the resulting universal predictor uses essentially a performance weighted average of all predictors for model orders less than m efficient lattice filters are used to generate the predictions of all the models recursively resulting in a complexity of the universal algorithm that is no larger than that of the largest model order examples of prediction performance are provided for autoregressive and speech data as well as an example of adaptive data equalization

entanglement complexity of lattice ribbons we consider a discrete ribbon model for double stranded polymers where the ribbon is constrained to lie in a three dimensional lattice the ribbon can be open or closed and closed ribbons can be orientable or nonorientable we prove some results about the asymptotic behavior of the numbers of ribbons withn plaquettes and a theorem about the frequency of occurrence of certain patterns in these ribbons we use this to derive results about the frequency of knots in closed ribbons the linking of the boundary curves of orientable closed ribbons and the twist and writhe of ribbons we show that the centerline and boundary of a closed ribbon are both almost surely knotted in the infinite n limit for an orientable ribbon the expectation of the absolute value of the linking number of the two boundary curves increases at least as fast as n and similar results hold for the twist and writhe

building tractable disjunctive constraints many combinatorial search problems can be expressed as constraint satisfaction problems this class of problems is known to be np hard in general but a number of restricted constraint classes have been identified which ensure tractability this paper presents the first general results on combining tractable constraint classes to obtain larger more general tractable classes we give examples to show that many known examples of tractable constraint classes from a wide variety of different contexts can be constructed from simpler tractable classes using a general method we also construct several new tractable classes that have not previously been identified

on some tractable classes in deduction and abduction abstract we address the identification of propositional theories for which entailment is tractable so that every query about logical consequences of the theory can be answered in polynomial time we map tractable satisfiability classes to tractable entailment classes including hierarchies of tractable problems and show that some initially promising conditions for tractability of entailment proposed by esghi 1993 and del val 1994 surprisingly only identify a subset of renamable horn we then consider a potential application of tractable entailment through a reduction due to esghi 1993 of certain abduction problems to a sequence of entailment problems besides clarifying the range of applicability of esghi s results from the semantic point of view we show that the reduction can almost trivially fail to be in any of the basic tractable classes discussed in the first part of the paper we leave open the question of how to more broadly identify tractable entailment classes as our examples suggest that there is room for progress in this area

solving peptide sequencing as satisfiability this paper presents an approach for determining the amino acid sequence of a peptide through the solution of propositional satisfiability problems data obtained from the mass spectrometry analysis of a peptide are used to build a propositional logic formula whose models represent coherent interpretations of that set of data and can be used to generate all possible correct results of the analysis itself some computational results on real world peptide analysis problems are reported which show the effectiveness of our approach

the writhe of a self avoiding walk the writhe of a self avoiding walk in a three dimensional space is the average over all projections onto a plane of the sum of the signed crossings we compute this number using a monte carlo simulation our results suggest that the average of the absolute value of the writhe of self avoiding walks increases as nalpha where n is the length of the walks and alpha approximately 0 5 the mean crossing number of walks is also computed and found to have a power law dependence on the length of the walks in addition we consider the effects of solvent quality on the writhe and mean crossing number of walks

adsorption and collapse of self avoiding walks and polygons in three dimensions we consider self avoiding walks and polygons on the simple cubic lattice confined to the half space and interacting with the plane z 0 in addition there is a short range vertex vertex interaction in the walk or polygon which can lead to a collapse transition we explore the interaction between collapse and adsorption in these systems and discuss the form of the phase diagram key results include a proof of the existence of an adsorption transition for polygons for every value of the vertex vertex interaction a corresponding proof for walks when the vertex vertex interaction term is repulsive and a proof that if polygons exhibit a collapse transition then the phase boundary between the expanded and desorbed phase and the collapsed and desorbed phase must be a straight line

self avoiding walks and polymer adsorption edge and vertex weighting self avoiding walks interacting with a surface are a useful model of polymer adsorption and one can choose to weight the walk according to the number of vertices or the number of edges in the surface we consider a combined model of self avoiding walks on the d dimensional hypercubic lattice with weights associated with the numbers of vertices and edges in the surface and investigate its phase diagram we also derive inequalities relating the critical points in the edge and vertex weighting schemes

random linking of lattice polygons we study the linking probability of polygons on the simple cubic lattice in particular we consider two polygons each having n edges confined to a cube of side l and ask for the linking probability as a function of n and l we also consider other situations in which the polygons are restricted to be not too far apart but not necessarily confined to a cube we prove several rigorous results and use monte carlo methods to address some questions which we are unable to answer rigorously an interesting feature is that the linking probability is a function of l nv where v is the exponent characterizing the radius of gyration of a polygon

coding for a binary independent piecewise identically distributed source two weighting procedures are presented for compaction of output sequences generated by binary independent sources whose unknown parameter may occasionally change the resulting codes need no knowledge of the sequence length t i e they are strongly sequential and also the number of parameter changes is unrestricted the additional transition redundancy of the first method was shown to achieve the merhav lower bound i e log t bits per transition for the second method we could prove that additional transition redundancy is not more than 3 2 log t bits per transition which is more than the merhav bound however the storage and computational complexity of this method are also more interesting than those of the first method simulations show that the difference in redundancy performance between the two methods is negligible

the statistical mechanics of random copolymers random copolymers are polymers with two or more types of monomer where the monomer sequence is determined by some random process once determined the sequence is fixed so random copolymers are an example of a system with quenched randomness we review the statistical mechanics of random copolymers focusing on self avoiding walk models where there are two types of monomers a and b which are randomly distributed along the polymer chain theoretical approximate and numerical results are reviewed for models of the random copolymer adsorption localization and collapse phase transitions we concentrate on what is known about the existence of phase transitions the morita approximation and results about self averaging we also discuss in less detail the replica trick and numerical methods including monte carlo methods exact enumeration and transfer matrix methods important open problems are identified throughout and highlighted in the conclusions

the satisfiability problem for a class consisting of horn sentences and some non horn sentences in porportional logic in this paper the satisfiability problem for a class of proportional sentences is considered here a sentence is a set of clauses a clause is a set of literals first it is proposed that a class s 0 of propositional sentences which properly includes the class of propositional horn sentences a sentence c 1 c n is in s 0 if there are sets p 1 p n of positive literals such that 1 p 1 p 2 p n 2 p i c i for 1 i n and 3 c i p i is a horn clause for 1 i n then it is proposed that a new inference rule based on the resolution principle by which un satisfiability for s 0 in polynomial time can be decided

an almost linear time algorithm for graph realization given a 0 1 matrix m the graph realization problem for m is to find a tree t such that the columns of m are incidence vectors of paths in t or to show that no such t exists an algorithm is presented for this problem the time complexity of which is very nearly linear in the number of ones in m