nonperturbative tricritical exponents of trails i exact enumeration on a triangular lattice as the fugacity of intersections in trails intersecting but non overlapping lattice walks is increased the configurations change from swollen ones with the scaling exponents of self avoiding walks to compact ones separating the two regimes is a potentially new tricritical point with no perturbative renormalisation fixed point associated with it supporting evidence for the existence of a tricritical point its likely location and exponents are computed for the first time from exact enumeration of all trails up to length of fifteen lattice constants on the triangular lattice the divergence of the specific heat indicates the location of the tricritical point generalised ratio and pade methods are used to extract the scaling exponents for the number of configurations and their end to end distance

exact enumeration study of free energies of interacting polygons and walks in two dimensions we present analyses of substantially extended series for both interacting self avoiding walks isaw and polygons isap on the square lattice we argue that these provide good evidence that the free energies of both linear and ring polymers are equal above the temperature thus extending the application of a theorem of tesi et al to two dimensions below the temperature the conditions of this theorem break down in contradistinction to three dimensions but an analysis of the ratio of the partition functions for isap and isaw indicates that the free energies are in fact equal at all temperatures within at least any perceived difference can be interpreted as the difference in the size of corrections to scaling in both problems this may be used to explain the vastly different values of the crossover exponent previously estimated for isap to that predicted theoretically and numerically confirmed for isaw an analysis of newly extended neighbour avoiding self avoiding walk series is also given

the role of polymorphism in the characterisation of complexity by soft types soft type assignment systems sta sta and sta b characterise by means of reduction of terms computation in complexity classes ptime np and pspace respectively all these systems are inspired by linear logic and include polymorphism similar to the one of system f we show that the presence of polymorphism gives the undecidability of typechecking and type inference we also show that reductions in decidable monomorphic versions of these systems also capture the same complexity classes in a way sufficient for the traditional complexity theory the translations we propose show in addition that the monomorphic systems to serve as a programming language require some metalanguage support since the program which operates on data has form and type which depend on the size of the input

an arithmetic for polynomial time computation we define a restriction lha of heyting arithmetic ha with the property that all extracted programs are feasible the restrictions consist in linearity and ramification requirements

linear logic and elementary time a subsystem of linear logic elementary linear logic is defined and shown to represent exactly elementary recursive functions its choicest part consists in reducing the deductive power of the exponential also known as the bang which in linear logic is in charge of controlling duplication in the cut elimination process

phase semantics for light linear logic light linear logic girard inform comput 14 1998 175 204 is a refinement of the propositions as types paradigm to polynomial time computation a semantic setting for the underlying logical system is introduced here in terms of fibred phase spaces strong completeness is established with a purely semantic proof of cut elimination as a consequence a number of mathematical examples of fibred phase spaces are presented that illustrate subtleties of light linear logic

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

bounded linear logic a modular approach to polynomial time computability typing is a way of describing the interactive behavior of algorithms usual typing systems are mainly concerned with input output specifications e g given terms f a b and a a the computation of f a by normalization yields a result of type b however one can dream of more refined typings that would not only ensure ethereal termination but would for instance yield feasible resource bounds it seems that time complexity does not lend itself naturally to modular manipulation we seek something more primitive

specific heat and crossover exponents at the tricritical collapse of trails from the exact enumeration on a triangular lattice of trail configurations at the tricritical regime as the fugacity for intersections is increased the authors find phi t 0 68 8 for the crossover exponent and hence alpha t 0 5 2 for the specific heat exponent it indicates the collapse transition of trails to differ from that of polymer chains at the theta point in two dimensions

critical exponents for square lattice trails with a fixed number of vertices of degree 4 we prove several previously conjectured results about the number of n edge trails and n edge embeddings of eulerian graphs each with a fixed number k of degree 4 vertices in the lattice 2 in particular under the assumption that the relevant critical exponents exist we prove that the difference between the critical exponent for closed trails eulerian graph embeddings and that for self avoiding circuits polygons is exactly k the number of degree 4 vertices similarly we prove that the difference between the critical exponent for either open trails or open eulerian graph embeddings and that for self avoiding walks is also k these results are proved by establishing upper and lower bounds for the number of n edge embeddings of closed open eulerian graphs with k vertices of degree 4 in terms of the number of n edge self avoiding polygons walks the lower bounds are proved using a kesten pattern theorem argument and the upper bounds are established by developing based on a detailed case analysis a method for removing vertices of degree 4 from an embedding by altering at most a constant independent of n number of vertices and edges of the embedding the work presented here extends and improves the arguments first given in the work of zhao and lookman 1993 j phys a math gen 26 1067 76

linear types and non size increasing polynomial time computation we propose a linear type system with recursion operators for inductive datatypes which ensures that all definable functions are polynomial time computable the system improves upon previous such systems in that recursive definitions can be arbitrarily nested in particular no predicativity or modality restrictions are made

light types for polynomial time computation in lambda calculus we present a polymorphic type system for lambda calculus ensuring that well typed programs can be executed in polynomial time dual light affine logic dlal dlal has a simple type language with a linear and an intuitionistic type arrow and one modality it corresponds to a fragment of light affine logic lal we show that contrarily to lal dlal ensures good properties on lambda terms and not only on proof nets subject reduction is satisfied and a well typed term admits a polynomial bound on the length of any of its beta reduction sequences we also give a translation of lal into dlal and deduce from it that all polynomial time functions can be represented in dlal

lattice trails i exact results considers the problem of lattice trails introduced by malakis 1975 1976 the author proves the existence of a finite connective constant and establishes a result for the growth of n step trails tn analogous to the best known result for self avoiding walks tn lambda n expt o n for the honeycomb d 2 and lave s lattice d 3 the author establishes a counting theorem from which one can deduce the exact value of the connective constant lambda for the honeycomb lattice lambda 2 2 square root 2 further it follows that the trail problem is in the same universality class as the self avoiding walk problem for those lattices an exact amplitude relation between trails and self avoiding walks and between dumb bells and trails and self avoiding walks is established the non existence of a counting theorem for arbitrary lattices is established an inequality for the triangular lattice connective constant is proved a high density expansion for lambda for the d dimensional hybercubic lattice is also obtained the author argues that contrary to recent suggestions the model is in the same universality class as the self avoiding walk model

adsorption of uniform lattice animals with specified topology the author studies the effects of topology on the free energy of uniform lattice animals interacting with a surface topology is specified by fixing an abstract graph tau and the lattice animals considered are embeddings of tau in the square and simple cubic lattice he proves that such embeddings in the simple cubic lattice and interacting with a plane have the same free energy as self avoiding walks independent of the choice of tau and independent of whether or not the embeddings are restricted to be uniform for embeddings in the square lattice and interacting with a line he proves that the free energy may differ from that for walks depending on whether tau has a cut edge further restricting the embeddings to be uniform forces the free energy to be different from that for walks for all tau except the tau corresponding to walks and we obtain bounds on the free energy which depend on the number of branches cycles and vertices of degree 3 and 4 in the graph

paths based criteria and application to linear logic subsystems characterizing polynomial time abstract several variants of linear logic have been proposed to characterize complexity classes in the proofs as programs correspondence light linear logic lll ensures a polynomial bound on reduction time and characterizes in this way polynomial time p t i m e in this paper we study the complexity of linear logic proof nets and propose three semantic criteria based on context semantics stratification dependence control and nesting stratification alone entails an elementary time bound the three criteria entail together a polynomial time bound these criteria can be used to prove the complexity soundness of several existing variants of linear logic we define a decidable syntactic subsystem of linear logic s d n l l we prove that the proof nets of sdnll satisfy the three criteria which implies that s d n l l is sound for p t i m e several previous subsystems of linear logic characterizing polynomial time lll m l 4 maximal system of ms are embedded in s d n l l proving its p t i m e completeness

new pattern theorems for square lattice self avoiding walks and self avoiding polygons a general pattern theorem for weighted self avoiding polygons saps and self avoiding walks saws in is obtained the pattern theorem for saps fits into the general framework of the pattern theorem for lattice clusters introduced by madras 1999 ann comb 3 357 84 note that unlike other pattern theorems proved for saps this pattern theorem does not rely on first establishing a relationship between saps and saws these results are applied to obtain pattern theorems for self interacting saps and self interacting saws

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

the size of lp improving measures abstract a measure on the locally compact abelian group g is lp improving if for some 1 p p g l r g l space properties of such are discussed if is lp improving and non negative and f is a bounded borel function then f is lp improving the non negativity of and boundedness of f are essential as is shown in several ways in particular there exists on each infinite compact abelian group an lp improving such that is not lp improving the distribution function of every lp improving measure on the circle group is shown to satisfy a lipschitz condition but that is not sufficient for a measure to be lp improving except in restricted cases the gelfand transform on of an lp improving measure is shown to be small in a quantitative sense from that the strong continuity of lp improving measures follows the spectrum of an lp improving measure as an operator on lp is shown to be precisely the closure of related results and open problems are included

linear logic by levels and bounded time complexity we give a new characterization of elementary and deterministic polynomial time computation in linear logic through the proofs as programs correspondence girard s seminal results concerning elementary and light linear logic achieve this characterization by enforcing a stratification principle on proofs using the notion of depth in proof nets here we propose a more general form of stratification based on inducing levels in proof nets by means of indices which allows us to extend girard s systems while keeping the same complexity properties in particular it turns out that girard s systems can be recovered by forcing depth and level to coincide a consequence of the higher flexibility of levels with respect to depth is the absence of boxes for handling the paragraph modality we use this fact to propose a variant of our polytime system in which the paragraph modality is only allowed on atoms and which may thus serve as a basis for developing lambda calculus type assignment systems with more efficient typing algorithms than existing ones

lattice trails ii numerical results for pt i see ibid vol 18 p 567 1985 a numerical study of the properties of lattice trails on the honeycomb square triangular and simple cubic lattices is made critical points are estimated for all lattices and upper and lower bounds established extensive series have been obtained and series analysis of both trail generating functions and mean square end to end distance series are not inconsistent with the conclusion that the problem is in the same universality class as the self avoiding walk problem a pseudo star triangle transformation is defined and the analyticity properties of that function coupled with previous exact results clearly supports that conclusion for the triangular lattice as well as providing excellent unbiased critical point estimates the author also shows that the connective constant for d or 2 dimensional hypercubic trails is strictly greater than the corresponding quantity for saws

contacts in self avoiding walks and polygons we prove several results concerning the numbers of n edge self avoiding polygons and walks in the lattice zd which had previously been conjectured on the basis of numerical results if the number of n edge self avoiding polygons walks with k contacts is pn k cn k then we prove that 0 limn n 1 log pn k limn n 1 log cn k exists for all fixed k and is independent of k for polygons in z2 we prove that there exist two positive functions b1 and b2 independent of n but depending on k such that b1nkpn 0 pn k b2nkpn 0 for fixed k and n large also provided the limit exists we prove that 0 limn kn n 1 in addition we consider the number of polygons with a density of contacts i e k n and show that the corresponding connective constant exists and is a concave function of for d 2 we prove that lim 0 0 and the right derivative of at 0 is infinite

quasi interpretations a way to control resources this paper presents in a reasoned way our works on resource analysis by quasi interpretations the controlled resources are typically the runtime the runspace or the size of a result in a program execution quasi interpretations allow the analysis of system complexity a quasi interpretation is a numerical assignment which provides an upper bound on computed functions and which is compatible with the program operational semantics the quasi interpretation method offers several advantages i it provides hints in order to optimize an execution ii it gives resource certificates and iii finding quasi interpretations is decidable for a broad class which is relevant for feasible computations by combining the quasi interpretation method with termination tools here term orderings we obtained several characterizations of complexity classes starting from ptime and pspace

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

an abstract approach to stratification in linear logic we study the notion of stratification as used in subsystems of linear logic with low complexity bounds on the cut elimination procedure the so called light subsystems from an abstract point of view introducing a logical system in which stratification is handled by a separate modality this modality which is a generalization of the paragraph modality of girard s light linear logic arises from a general categorical construction applicable to all models of linear logic we thus learn that stratification may be formulated independently of exponential modalities when it is forced to be connected to exponential modalities it yields interesting complexity properties in particular from our analysis stem three alternative reformulations of baillot and mazza s linear logic by levels one geometric one interactive and one semantic

on the expressivity of elementary linear logic elementary linear logic is a simple variant of linear logic due to girard and which characterizes in the proofs as programs approach the class of elementary functions that is to say functions computable in time bounded by a tower of exponentials of fixed height other systems like light and soft linear logics have then been defined to characterize in a similar way the more interesting complexity class of polynomial time functions but at the price of either a more complicated syntax or of more sophisticated encodings these logical systems can serve as the basis of type systems for calculus ensuring polynomial time complexity bounds on well typed terms this paper aims at reviving interest in elementary linear logic by showing that despite its simplicity it can capture smaller complexity classes than that of elementary functions for that we carry a detailed analysis of its normalization procedure and study the complexity of functions represented by a given type we then show that by considering a slight variant of this system with type fixpoints and free weakening elementary affine logic with fixpoints we can characterize the complexity of functions of type w k 2 b where w and b are respectively types for binary words and booleans the key point is a sharper study of the normalization bounds we characterize in this way the class p of polynomial time predicates and more generally the hierarchy of classes k exp for k 0 where k exp is the union of dtime 2 k n i for i 1

non perturbative tricritical exponents of trails ii exact enumerations on square and simple cubic lattices for pt i see ibid vol 21 p 773 1988 the authors present exact enumerations of trails self intersecting but non overlapping lattice walks tabulated according to their length l number of intersection i and end to end distance for the square lattice up to l 21 i 7 and for the simple cubic lattice up to l 15 i 5 they introduce a fugacity for intersection to explore the transition to a collapsed phase through a novel tricritical point which is unaccessible by the renormalisation group approach the existence of tricritical points in both lattices is manifested by the divergence of the specific heat the values of the tricritical couplings and exponents are extracted by a dlogpade analysis they find for the three dimensional lattice nu t approximately 0 48 and gamma t approximately 0 43 nu t approximately 0 52 and gamma t approximately 1 25 for the square lattice

phase semantics and decidability of elementary affine logic light elementary and soft linear logics are formal systems derived from linear logic enjoying remarkable normalization properties in this paper we prove decidability of elementary affine logic eal the result is obtained by semantical means first defining a class of phase models for eal and then proving soundness and strong completeness following okada s technique phase models for light affine logic and soft linear logic are also defined and shown complete

collapse transition of self avoiding trails on the square lattice abstract the collapse transition of an isolated polymer has been modelled by many different approaches including lattice models based on self avoiding walks and self avoiding trails in two dimensions previous simulations of kinetic growth trails which map to a particular temperature of interacting self avoiding trails showed markedly different behaviour for what was argued to be the collapse transition than that which has been verified for models based of self avoiding walks on the other hand it has been argued that kinetic growth trails represent a special simulation that does not give the correct picture of the standard equilibrium model in this work we simulate the standard equilibrium interacting self avoiding trail model on the square lattice up to lengths over 2 000 000 steps and show that the results of the kinetic growth simulations are in fact entirely in accord with standard simulations of the temperature dependent model in this way we verify that the collapse transition of interacting self avoiding walks and trails are indeed in different universality classes in two dimensions

spectra of bernoulli convolutions as multipliers in lp on the circle it is shown that the closure of the set of fourier coefficients of the bernoulli convolution parameterized by a pisot number is countable combined with results of r salem and p sarnak this proves that for every fixed 1 the spectrum of the convolution operator f mapsto mu sb theta ast f in lp s1 where s1 is the circle group is countable and is the same for all p in 1 infty namely overline widehat mu sb theta n n in mathbb z our result answers the question raised by sarnak in 8 we also consider the sets overline widehat mu sb theta rn n in mathbb z for r 0 which correspond to a linear change of variable for the measure we show that such a set is still countable for all r in mathbb q theta but uncountable a nonempty interval for lebesgue a e r 0

the statistics of collapsing square lattice trails with a fixed number of vertices of degree 4 a trail on the square lattice with a fixed number k of vertices of degree 4 is called a k trail we model polymer collapse using k trails by incorporating an interaction energy which is proportional to the number of nearest neighbour contact edges of the trail it is known that the number of square lattice n edge closed open k trails can be bounded above and below to o nk by the number of n step self avoiding circuits walks this along with pattern theorems for self interacting self avoiding circuits and walks are used herein to establish upper and lower bounds to o nk for the collapsing free energy of k trails in terms of self avoiding circuits or walks as appropriate we also use pattern theorems to obtain bounds on the limiting nearest neighbour contact density for collapsing k trails finally we investigate k trails with a fixed density of nearest neighbour contacts and show that their limiting entropy per monomer is independent of k

statistics of collapsed lattice animals rigorous results and monte carlo simulations in this paper a two dimensional model of collapsed lattice animals with nearest neighbour contact interactions is introduced and the limiting free energy for the model is proved to exist we also establish that the partition function for collapsed lattice animals is related to the reliability polynomial for the m m grid graph and to the generating function for spanning forests on the m m grid graph from these relationships and using results about directed animals we are able to obtain rigorous upper and lower bounds on the free energy for collapsed lattice animals in addition we present monte carlo evidence that the heat capacity associated with this model remains bounded in the thermodynamic limit in fact we do not find any evidence for a second order phase transition for collapsed lattice animals

godel s system t revisited the linear lambda calculus where variables are restricted to occur in terms exactly once has a very weak expressive power in particular all functions terminate in linear time in this paper we consider a simple extension with natural numbers and a restricted iterator only closed linear functions can be iterated we show properties of this linear version of godel s tusing a closed reduction strategy and study the class of functions that can be represented surprisingly this linear calculus offers a huge increase in expressive power over previous linear versions of t which are closed at construction rather than closed at reduction we show that a linear twith closed reduction is as powerful as t

introduction to clarithmetic ii the earlier paper introduction to clarithmetic i constructed an axiomatic system of arithmetic based on computability logic and proved its soundness and extensional completeness with respect to polynomial time computability the present paper elaborates three additional sound and complete systems in the same style and sense one for polynomial space computability one for elementary recursive time and or space computability and one for primitive recursive time and or space computability

critical exponents for simple non uniform polymer networks the authors study a number of non uniform specified topologies and show rigorously that for certain topologies with cut edges the critical exponent gamma t is in agreement with a conjecture given by gaunt et al 1984 and that the exponent nu t nu the exponent for self avoiding walks the authors also find that the scaling relations gamma t gamma t1 and gamma t gamma t11 are the same as for self avoiding walks previously conjectured only for uniform networks by assigning an interaction energy to a nearest neighbour contact they prove that the collapse transition for these topologies is the same as that for self avoiding walks

realizability models for bll like languages we give a realizability model of girard scedrov scott s bounded linear logic bll this gives a new proof that all numerical functions representable in that system are polytime our analysis naturally justifies the design of the bll syntax and suggests further extensions

quantum implicit computational complexity we introduce a quantum lambda calculus inspired by lafont s soft linear logic and capturing the polynomial quantum complexity classes eqp bqp and zqp the calculus is based on the classical control and quantum data paradigm this is the first example of a formal system capturing quantum complexity classes in the spirit of implicit computational complexity it is machine free and no explicit bound e g polynomials appears in its syntax

soft linear logic and polynomial time we present a subsystem of second order linear logic with restricted rules for exponentials so that proofs correspond to polynomial time algorithms and vice versa

free energy of self interacting uniform stars we consider uniform star polymers on a variety of lattices with an additional contact interaction between pairs of vertices which are unit distance apart and are not joined by an edge of the star we present some rigorous results and other evidence which indicate that these systems have the same limiting free energy as self interacting self avoiding walks we discuss the extension of these results to other homeomorphism types and to systems with an additional surface interaction term