Combinatory logic an overview sciencedirect topics. Functioning as an entire provide for current developments of combinatory logic, this book is the one one amongst its selection to cowl outcomes of the ultimate four many years. All this links combinatory logic to mathematical logic, philosophical logic and computational. The formalism has since been extended with types and used in functional programming haskell, clean and proof assistants coq, isabelle, hol, used in designing and verifying it products and mathematical proofs.
This is a quick summary of the third, fourth, and fifth sections of the stanford encyclopedia of philosophy article on combinatory logic by katalin bimbo, with occasional comments of my own. What is the combinatory logic equivalent of intuitionistic. History of lambdacalculus and combinatory logic department of. Illative combinatory logic encyclopedia of mathematics. I generated those combinatory types using the bracket abstraction translation from the relevant type theory terms. The download link provided above is randomly linked to our ebook promotions or third. Pure, applied and typed discrete mathematics and its applications pdf, epub, docx and torrent then this site is not for you.
In combinatory logic one chooses as basic the concepts of a oneplace function and the operation of applying a function to an argument application. Typeassignment systems are often formulated as natural deduction systems. Data types as lattices siam journal on computing vol. Download fulltext pdf equivalences between illative combinatory logics and pure type systems article pdf available january 1997 with 17 reads. Bcl has applications in the theory of programsize complexity kolmogorov complexity. I think many thoughts from john hughes why functional programming matters can be applied to programming in combinatory logic. The results of this paper extend some of the intimate relations that are known to obtain between combinatory logic and certain substructural logics to establish a general characterization theorem that applies to a very broad family of such logics. We establish that the typability testing and detailed type restoration can be done in polynomial time and that the derivability relation for rcl is decidable and pspacecomplete. Download it once and read it on your kindle device, pc, phones or tablets.
Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. Cl is an elegant and powerful logical theory that is connected to many areas of logic, and has found applications in other disciplines, especially, in computer science and mathematics cl was originally invented as a continuation of the reduction of the set of logical constants to a singleton set in classical firstorder logic fol. Combinatory logic, lambda calculus, and type theories. We study artemovs reflective combinatory logic rcl. It was introduced by moses schonfinkel and haskell curry, and has more. In this book, the authors focus on three classes of typing for lambda terms. Combinatory logic is one of the most versatile areas within logic that is tied to parts of philosophical, mathematical, and computational logic. Types the calculus is good tool for exploring type systems, invaluable both in the foundations of mathematics and for practical programming languages. The lambdacalculus, combinatory logic, and type systems.
I can write the types of the combinators, fully abstracted over their parameters, as follows. This places combinatory logic next to recursion theory and computer science. Pure, applied and typed by katalin bimbo free downlaod publisher. Jan 22, 2016 combinatory logic combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. Combinatory logic stanford encyclopedia of philosophy. Combinatory logic is a powerful and concise programming language. Systems of illative combinatory logic complete for firstorder propositional and predicate calculus barendregt, henk, bunder, martin, and dekkers, wil, journal of symbolic logic, 1993. And almost all concepts used in the haskell world catamorphisms etc.
Turning from pure to applied cl and although typetheory made important. Since it has been understood that the curryhoward isomorphism relating proofs and programs can be extended to classical logic, various systems have been introduced. Introduction to mathematical logic and type theory. Guide for authors annals of pure and applied logic issn. Under the propositions as types correspondence, combinatory logic corresponds to the presentation of logic using a hilbert system instead of natural deduction or sequent calculus. For combinatory logic there are introductions in hindley and seldin, 1986. In pure combinatory logic there is a set of terms built by application from variables and two constants and, and there are two conversion rules. A merge doesnt make sense because the presnt article on combinatory logic is already too long.
The constants are again the combinators \\textsfs\ and \\textsfk\. Here the concept of a function is regarded as primitive, instead of that of a set, and is generalized in such a way that a function can be applied to objects at the same level with it. In the presence of the rule of extensionality, the theory is equivalent with untyped lambdacalculus cf. Typing in reflective combinatory logic sciencedirect. Pure, applied and typed discrete mathematics and its applications kindle edition by bimbo, katalin. One of the aims of combinatory logic is to study the most basic properties of functions and other concepts, with as few restrictions as possible. The calculus a pure calculus of functional abstraction and function application, with applications throughout logic and computer science. Each combinator is like a function or lambda abstraction, but without any free variables. Pure, applied and typed discrete mathematics and its applications pdf,, download ebookee alternative successful tips for a best ebook reading. Combinatory logic and the semantics of substructural logics. To show how i did it, and make this post not entirely pointless, let me offer my equipment. Combinatory logic and lambda calculus were first introduced in curry 9 and. Curry and his coworkers continued the work on illative combinatory logic 107, 108, i.
In practice, however, a sophisticated applied theoretician and an applied programmer must have a thorough grasp of what the system of objects does in response to an evaluation request. These processes are taken for granted in most formulations of logic, but they are complex, and since a fundamental part of the resulting theory is recursively undecidable the analysis. It was introduced by moses schonfinkel and haskell curry, and has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages. Combinatory logic and lambdacalculus, in their typefree version. It seems to be written in a very clear style, explaining the rationale behind the study of combo logic and its historical roots. Consequently, cl is susceptible to godeltype incompleteness theorems. Research on combinatory logic has been continued essentially by currys students, hindley and seldin 1986.
If youre looking for a free download links of combinatory logic. Pure, applied and typed discrete mathematics and its. Systems trc and trcu of illative combinatory logic are introduced and shown to be equivalent in consistency strength and expressive power to quines set theory new. Binary combinatory logic bcl is a formulation of combinatory logic using only the symbols 0 and 1. The calculus of combinators was then studied as an independent subject, and a wealth of results was obtained. Systems of combinatory logic related to quines new foundations. We provide the explicit definition of types for rcl and prove that every wellformed term has a unique type. Combinatory logic is a model by which logical statements can be described as a combination of a small number of primitive elements called combinators. In the intended ap plications, combinatory type environments model repositories of. Programs may be applied to input data and of course result in data, which may again be. Combinatory logic shows that bound variables can be eliminated without loss of expressiveness.
Combinatory logic combinatory logic is a branch of mathematical logic that analyzes certain processes, such as substitution, which are associated with variables. Published july 27, 2011 by chapman and hallcrc 357 pages 10 bw illustrations request inspection copy. Some philosophical aspects of combinatory logic sciencedirect. Functioning as a comprehensive source for current developments of combinatory logic, this book is the only one of its kind to cover results of the last four decades. Pure type systems with more liberal rules bunder, martin and dekkers, wil, journal of symbolic logic, 2001. However, our decision procedure for this logic, which we presented in earlier papers, relies on two sequent calculi and it does not. Keywords and phrases illative combinatory logic, term rewriting, firstorder logic. There are versions of combinatory logic corresponding to linear logic and affine logic, generally called bci logic and bck logic after the combinators they use. Discrete dynamics of combinatory logic and morphoca systems. Use features like bookmarks, note taking and highlighting while reading combinatory logic. On the other hand, typed combinatory logic straightforwardly connects with intuitionistic logic and other nonclassical logics, for example, relevance logics and linear logic.
Combinatory logic is a branch of mathematical logic which is con cerned with the. Pure, applied and typed discrete mathematics and its applications pdf. That is, even if two expressions applied to the same expression are. Contents preface ix 1 elementsofcombinatorylogic 1 1. The first axiom and the first two rules constitute equational logic. Download guide for authors in pdf aims and scope scope of the journal the journal annals of pure and applied logic publishes high quality papers in all areas of mathematical logic as well as applications of logic in mathematics, in theoretical computer science and in other related disciplines. The article on combinatory logic should probably be broken down into a number of separate articles.