bck combinators and linear l terms have types abstract a term l is linear when each abstract x m in it has at most one free x in m and each free variable of l occurs free only once bck combinators correspond exactly to linear terms without free variables the main theorem says that all terms of each kind are stratified i e are typable this result is not deep it merely confirms an intuition that non stratification is caused partly by multiple occurrences of variables for bck combinators it was stated as theorem 1 by bunder and meyer in 1 but the proof in 1 seems to contain a gap it uses induction on a definition of level of a combinator but omits to show that the level concept is defined for all bck combinators the proof below proceeds by translating to linear terms for which the induction becomes simply one on length in fact the proof is essentially just an amalgamation of elements from an unpublished proof of m coppo and m dezani ciancaglini and elements from 1 the type schemes of linear terms form a very primitive linear logic and linear logic is of current interest see girard 3 so we shall set out the basic facts about linear terms and bck combinators in some detail these basic facts are not new some appearin komori 6 for example and some are implicit in curry and feys 2

type theories normal forms and d lambda models abstract a non standard inverse limit model d is constructed which has a non hilbert post complete theory moreover in d a simple semantic characterization of normalizable terms is given these results are proved using the properties of a generalized type assignment system which yields a filter model barendregt coppo and dezani ciancaglini 1983 j symbolic logic 48 931 940 coppo dezani ciancaglini honsell and longo 1983 pp 241 262 logic colloquium 82 north holland amsterdam isomorphic to d the type assignment system is also proved complete with respect to an interpretation of types in the term model of equality based only on normalization properties as an application a class of maximal monoids of normalizable terms is characterized

lambda terms definable as combinators abstract it is well known that for each term there is a corresponding combinatory term formed using the combinators k and s instead of the operator similarly for every combinatory term there is a term for weaker sets of combinators such as b c and k or b b i and w we show how such a correspondence or translation can be formulated and we determine in the case of several such sets of combinators the sets of terms that can be translated using them as combinators can represent hilbert style proofs of theorems of implicational logic and terms natural deduction style proofs this work allows us to formulate natural deduction systems equivalent to the hilbert style systems for various implicational logics and can form a basis for proof generating programs for these logics

principal type scheme and unification for intersection type discipline abstract the intersection type discipline for the calculus itd is an extension of the classical functionality theory of curry in the itd a term satisfying a given property has a principal type scheme in an extended meaning i e there is a type scheme deducible for it from which all and only the type schemes deducible for it are reachable by means of suitable operations the problem of finding the principal type scheme for a term if it exists is semidecidable in the paper a procedure is shown building the principal type scheme of a term through the construction of the most general unifier for intersection type schemes

duality for vector optimization of set valued functions in this note a general cone separation theorem between two subsets of image space is presented with the aid of this optimality conditions and duality for vector optimization of set valued functions in locally convex spaces are discussed

intersection types for combinatory logic abstract two different translations of the usual formulation of intersection types for calculus into combinatory logic are proposed in the first one the rule is unchanged while in the second one the rule is replaced by three new rules and four axiom schemes which seem to be simpler than rule itself

cone directed higher order generalized adjacent derivative and higher order mond weir duality in this paper cone directed higher order generalized adjacent derivative for a set valued map is defined by virtue of the cone directed higher order generalized adjacent derivative a higher order mond weir type dual problem for a constrained set valued optimiza tion problem is introduced and its weak duality strong duality and converse duality properties are established the results obtained improve the corresponding results in the literatures

new existence results for equilibrium problems we consider equilibrium problems in the framework of the formulation proposed by blum and oettli which includes variational inequalities nash equilibria in noncooperative games and vector optimization problems for instance as particular cases we establish new sufficient and or necessary conditions for existence of solutions of such problems our results are based upon the relation between equilibrium problems and certain auxiliary convex feasibility problems together with extensions to equilibrium problems of gap functions for variational inequalities then we apply our results to some particular instances of equilibrium problems obtaining results which include among others a new lemma of the alternative for convex optimization problems

characterization theorems for a filter lambda model the filter model is a model of the calculus based on a system of type assignment which extends the basic functionality theory of curry invented in order to give a completeness proof for scott s semantics of curry s type assignment in this paper the local structure of this model i e the syntactical characterization of the equality and inclusion relation is given

principal type schemes for an extended type theory we consider the extension of curry s basic functionality theory presented by barendregt et al to appear and we define for any term x a principal type scheme p t s we prove that all and the only type schemes deducible for x can be obtained from the p t s of x by suitable operations

non differentiable second order symmetric duality in mathematical programming with f convexity abstract a pair of wolfe type non differentiable second order symmetric primal and dual problems in mathematical programming is formulated the weak and strong duality theorems are then established under second order f convexity assumptions symmetric minimax mixed integer primal and dual problems are also investigated

wolfe type duality involving b η invex functions for a minmax programming problem abstract a sufficient optimality theorem is proved for a certain minmax programming problem under the assumptions of proper b invexity conditions on the functions involved in the objective and in the constraints next a dual is presented for such a problem and duality theorems relating the primal and the dual are proved the dual for a minimax generalized fractional programming problem is presented as special case of the main problem considered in the paper

higher order optimality conditions for henig efficient solutions in set valued optimization in this paper higher order generalized contingent epiderivative and higher order generalized adjacent epiderivative of set valued maps are introduced necessary and sufficient conditions for henig efficient solutions to a constrained set valued optimization problem are given by employing the higher order generalized epiderivatives

strongly convergent algorithms by using new adaptive regularization parameter for equilibrium problems abstract two new algorithms are proposed in this paper for solving an equilibrium problem whose associated bifunction is monotone and satisfies a lipschitz type condition in a hilbert space in the first algorithm it is assumed that the value of the lipschitz constant of the bifunction is known while in the second one the prior knowledge of this constant is not explicitly requested the proposed algorithms are constructed around the proximal like mapping and the regularized method and use some new variable stepsize rules strong convergence theorems are established under some mild conditions imposed on bifunction and control parameters finally several numerical results are provided to illustrate the behavior of the new algorithms and to compare them to well known algorithms

normalization results for typeable rewrite systems in this paper we introduce curryfied term rewriting systems and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts and three operations on types substitution expansion and lifting are used to define type assignment and are proved to be sound with this result the system is proved closed for reduction using a more liberal approach to recursion we define a general scheme for recursive definitions and prove that for all systems that satisfy this scheme every term typeable without using the type constant is strongly normalizable we also show that under certain restrictions all typeable terms have a weak head normal form and that terms whose type does not contain are normalizable

the tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions abstract we extend the tikhonov regularization method widely used in optimization and monotone variational inequality studies to equilibrium problems it is shown that the convergence results obtained from the monotone variational inequality remain valid for the monotone equilibrium problem for pseudomonotone equilibrium problems the tikhonov regularized subproblems have a unique solution only in the limit but any tikhonov trajectory tends to the solution of the original problem which is the unique solution of the strongly monotone equilibrium problem defined on the basis of the regularization bifunction

higher order mond weir duality for set valued optimization in this paper we introduce a higher order mond weir dual for a set valued optimization problem by virtue of higher order contingent derivatives and discuss their weak duality strong duality and converse duality properties

principal types of bck lambda terms abstract bck terms are the terms in which each variable occurs at most once the principal type of a term is the most general type of the term in this paper we prove that if two bck terms in normal form have the same principal type then they are identical this solves the following problem y komori 1987 in more general form if two closed normal form bck terms are assigned to the same minimal bck formula are they identical a minimal bck formula is the most general formula among bck provable formulas with respect to substitutions for type variables to analyze type assignment the notion of connection is introduced a connection is a series of occurrences of a type in a type assignment figure connected occurrences of a type have the same meaning the occurrences of the type in distinct connection classes can be rewritten separately as a result we have more general type assignment by formulae as types correspondence the result implies the uniqueness of the normal proof figure for principal bck formulas the result is valid for bci logic or implicational fragment of linear logic as well

on certain conditions for the existence of solutions of equilibrium problems the main purpose of this paper is the study of sufficient and or necessary conditions for existence of solutions of equilibrium problems we discuss some of the assumptions of the problem under which the introduced conditions are sufficient and or necessary and also analyze the effect of these assumptions on the connection between the solution sets of the equilibrium problem and of a related convex feasibility problem

normalization approximation and semantics for combinator systems this paper studies normalization of typeable terms and the relation between approximation semantics and filter models for combinator systems it presents notions of approximants for terms intersection type assignment and reduction on type derivations the last will be proved to be strongly normalizable with this result it is proved that every typeable term has an approximant with the same type and a characterization of the normalization behaviour of terms using their assignable types is given then the two semantics are defined and compared and it is shown that the approximants semantics is fully abstract but the filter semantics is not

on principal types of combinators in this paper we study in some cases the relationship between the combinatory completeness of a set of typable combinators with simple types for a system of calculus and the axiomatic completeness under substitution and modus ponens of the respective set of principal types for the corresponding logical system we show that combinatory completeness is a necessary but not sufficient condition for axiomatic completeness in the k and in the i calculus while the two problems become equivalent for the bck as well as for the bci calculus furthermore we present an algorithm which whenever b is a principal pair for some normal bck term m reconstructs m up to conversion and which fails if there is no normal bck term for which b is a principal pair from the correctness proof of the algorithm we also obtain another proof for the fact that each bck term in normal form is completely determined by its principal pairs

most general first order theorems are not recursively enumerable abstract an e substitution is a uniform replacement of n ary predicates qn by n ary complex predicates that is by formulas containing all the name form variables of qn in propositional logic e substitutions coincide standard substitutions for propositional variables formula a is more e general than b if a is an e substitution instance of b but not vice versa every theorem of a given logic l is an e instance of a most e general l theorem gene l denotes the class of l s most e general theorems is gene l axiomatizable we prove that for any recursively enumerable r e logic l extending intuitionistic proportional or first order logic gene l is r e only if l is decidable the theorem still holds if l includes modal operators one of its consequences is that most general first order theorems are not axiomatizable

projection algorithms for solving nonmonotone equilibrium problems in hilbert space we propose two projection algorithms for solving an equilibrium problem where the bifunction is not required to be satisfied any monotone property under assumptions on the continuity convexity of the bifunction and the nonemptyness of the solution set of the minty equilibrium problem we show that the sequences generated by the proposed algorithms converge weakly and strongly to a solution of the primal equilibrium problem respectively