Substructural logic

About: Substructural logic is a research topic. Over the lifetime, 1756 publications have been published within this topic receiving 48474 citations.

Linear logic

Abstract: Linear logic was introduced by Girard in 1987 [11] . Since then many results have supported Girard' s statement, \"Linear logic is a resource conscious logic,\" and related slogans . Increasingly, computer scientists have come to recognize linear logic as an expressive and powerful logic with connection s to a variety of topics in computer science . This column presents a.n intuitive overview of linear logic, some recent theoretical results, an d summarizes several applications of linear logic to computer science . Other introductions to linear logic may be found in [12, 361 .

Residuated Lattices: An Algebraic Glimpse at Substructural Logics

Abstract: The book is meant to serve two purposes. The first and more obvious one is to present state of the art results in algebraic research into residuated structures related to substructural logics. The second, less obvious but equally important, is to provide a reasonably gentle introduction to algebraic logic. At the beginning, the second objective is predominant. Thus, in the first few chapters the reader will find a primer of universal algebra for logicians, a crash course in nonclassical logics for algebraists, an introduction to residuated structures, an outline of Gentzen-style calculi as well as some titbits of proof theory - the celebrated Hauptsatz, or cut elimination theorem, among them. These lead naturally to a discussion of interconnections between logic and algebra, where we try to demonstrate how they form two sides of the same coin.We envisage that the initial chapters could be used as a textbook for a graduate course, perhaps entitled Algebra and Substructural Logics. As the book progresses the first objective gains predominance over the second. Although the precise point of equilibrium would be difficult to specify, it is safe to say that we enter the technical part with the discussion of various completions of residuated structures. These include Dedekind-McNeille completions and canonical extensions. Completions are used later in investigating several finiteness properties such as the finite model property, generation of varieties by their finite members, and finite embeddability. The algebraic analysis of cut elimination that follows, also takes recourse to completions.Decidability of logics, equational and quasi-equational theories comes next, where we show how proof theoretical methods like cut elimination are preferable for small logics/theories, but semantic tools like Rabin's theorem work better for big ones. Then we turn to Glivenko's theorem, which says that a formula is an intuitionistic tautology if and only if its double negation is a classical one. We generalise it to the substructural setting, identifying for each substructural logic its Glivenko equivalence class with smallest and largest element. This is also where we begin investigating lattices of logics and varieties, rather than particular examples.We continue in this vein by presenting a number of results concerning minimal varieties/maximal logics. A typical theorem there says that for some given well-known variety its subvariety lattice has precisely such-and-such number of minimal members (where values for such-and-such include, but are not limited to, continuum, countably many and two). In the last two chapters we focus on the lattice of varieties corresponding to logics without contraction. In one we prove a negative result: that there are no nontrivial splittings in that variety. In the other, we prove a positive one: that semisimple varieties coincide with discriminator ones. Within the second, more technical part of the book another transition process may be traced. Namely, we begin with logically inclined technicalities and end with algebraically inclined ones. Here, perhaps, algebraic rendering of Glivenko theorems marks the equilibrium point, at least in the sense that finiteness properties, decidability and Glivenko theorems are of clear interest to logicians, whereas semisimplicity and discriminator varieties are universal algebra par exellence. It is for the reader to judge whether we succeeded in weaving these threads into a seamless fabric. This book: Considers both the algebraic and logical perspective within a common framework; Is written by experts in the area; Is easily accessible to graduate students and researchers from other fields; Includes results summarized in tables and diagrams to provide an overview of the area; Is useful as a textbook for a course in algebraic logic, with exercises and suggested research directions; And provides a concise introduction to the subject and leads directly to research topics. The ideas from algebra and logic are developed hand-in-hand and the connections are shown in every level.

Efficient Induction of Logic Programs

Abstract: Recently there has been increasing interest in systems which induce rst order logic programs from examples. However, many diiculties need to be overcome. Well-known algorithms fail to discover correct logical descriptions for large classes of interesting predicates , due either to the intractability of search or overly strong limitations applied to the hypothesis space. In contrast, search is avoided within Plotkin's framework of relative least general generalisation (rlgg). It is replaced by the process of constructing a unique clause which covers a set of examples relative to given background knowledge. However, such a clause can in the worst case contain innnitely many literals, or at best grow exponentially with the number of examples involved. In this paper we introduce the concept of h-easy rlgg clauses and show that they have nite length. We also prove that the length of a certain class of \determinate" rlgg is bounded by a polynomial function of certain features of the background knowledge. This function is independent of the number of examples used to construct them. An existing implementation called GOLEM is shown to be capable of inducing many interesting logic programs which have not been demonstrated to be learnable using other algorithms.

Logic Programming with Focusing Proofs in Linear Logic

Abstract: The deep symmetry of linear logic [18] makes it suitable for providing abstract models of computation, free from implementation details which are, by nature, oriented and nonsymmetrical. I propose here one such model, in the area of logic programming, where the basic computational principle is Computation = Proof search Proofs considered here are those of the Gentzen style sequent calculus for linear logic. However, proofs in this system may be redundant, in that two proofs can be syntactically different although identical up to some irrelevant reordering or simplification of the applications of the inference rules. This leads to an untractable proof search where the search procedure is forced to make costly choices which turn out to be irrelevant. To overcome this problem, a subclass of proofs, called the 'focusing' proofs, which is both complete (any derivable formula in linear logic has a focusing proof) and tractable (many irrelevant choices in the search are eliminated when aimed at focusing proofs) is identified. The main constraint underlying the specification of focusing proofs has been to preserve the symmetry of linear logic, which is its most salient feature. In particular, dual connectives have dual properties with respect to focusing proofs. Then, a programming language, called LinLog, consisting of a fragment of linear logic, in which focusing proofs have a more compact form, is presented. Linlog deals with formulae which have a syntax similar to that of the definite clauses and goals of Horn logic, but the crucial difference here is that it allows clauses with multiple atoms in the head, connected by the 'par' (multiplicative disjunction). It is then shown that the syntactic restriction induced by LinLog is not performed at the cost of any expressive power: a mapping from full linear logic to LinLog, preserving focusing proofs, and analogous to the normalization to clausal form for classical logic, is presented.