Extensional higher order logic programming book

We develop an extensional semantics for higher order logic programs with negation, generalizing the technique that was introduced in 2, 3 for positive higher order programs. The intricacies of 3valued extensional semantics for higherorder logic programs article pdf available in theory and practice of logic programming. In this way we provide an alternative extensional semantics for higherorder logic programs with negation to the one proposed in. Higherorder logic programming holp languages are particularly useful for various kinds of metaprogramming and theorem proving tasks because of the logical support for variable binding via abstraction.

It is usually instantiated with, or borrowed from, models of computation such as lambda calculus which make heavy use of higher order functions. Constructive negation in extensional higher order logic programming angelos charalambidis, panos rondogiannis. Logic programming is based upon an extension of first order logic. In this chapter, we develop the idea of higher order logic programming by utilizing a higher order logic as the basis for computing. In this semantics program predicates denote sets of. We propose a purely extensional semantics for higher order logic programming. It doesnt matter whether we are talking about the set of the renates or the set of the cordates, because they are the same set, and we know theyre the same set, because we define a set extensionallyi. In this speci cation, the variable n ranges over the numerals 0. Under this semantics, every program has a unique minimum herbrand. The most obvious is third, fourth, and so on order logic. He touches on a surprising range of concepts that arise in language design. Have you ever read an amazing book and did not solve the mystery itself. We develop an extensional semantics for higherorder logic programs with negation, generalizing the technique that was introduced in 2, 3 for positive higher order programs.

In this chapter, we develop the idea of higherorder logic programming by utilizing a higherorder logic as the basis for computing. We can extend higherorder logic programming with constructive negation. Wadge abstract we describe a fragment of higherorder horn logic which can be used as a higherorder extension of prolog. The intricacies of threevalued extensional semantics for higher. They have been used for a wide range of applications including theorem proving, program. Troelstra, in studies in logic and the foundations of mathematics, 1998. Higher order logic programming languages such as prolog and twelf improve the notion of first order logic programming in two ways. Pdf the intricacies of 3valued extensional semantics. Gopalan nadathur and dale miller higherorder logic programming, technical report cs199438, department of computer science, duke university,1994. Wadge abstract we describe a fragment of higher order horn logic which can be used as a higher order extension of prolog. Terms from the higher order language are defined via abstraction. Pfenning, automating higher order logic in automated theorem proving. As an immediate useful consequence of our developments, we define for the. It turns out that the extension of higherorder logic programs with constructive negation o ers a much greater versatility to extensional higherorder logic programming.

Second, they have largely imported the notion of higher order programming as it is understood within functional programming and have not examined a notion that is intrinsic to logic programming. Other articles where extensional logic is discussed. Charalambidis a, handjopoulos k, rondogiannis p and wadge w 20 extensional higher order logic programming, acm transactions on computational logic tocl, 14. Higherorder logic is based on a manysorted language with a collection of sorts or types. Pdf the intricacies of 3valued extensional semantics for. Moreover, every program has a unique minimum herbrand model which is the greatest lower bound of all herbrand models. What are the best books for improving programming logic. From frege to g odel, a source book in mathematical. Numerous applications of the close relationship between traditional logic and the algebraic language of category theory are given. Higherorder logic and intuitionistic type theory overlaps in many ways.

Secondorder and higherorder logic stanford encyclopedia of. Minimum model semantics for extensional higherorder logic programming with negation. In this paper the logic of broad necessity is explored. Completeness is proved relative to an analog of henkin models, familiar for higher type classical logics. Higherorder logic programming in twelf extends traditional rstorder logic programming in three ways. Intensional logic jaroslav peregrin 1 freges bedeutung the german mathematician and one of the founding fathers of modern logic, gottlob frege 18481925, was the first to clearly realize that semantics has little to do with psychology, and that it could be usefully explicated in mathematical terms see dummett, 1973. In other words, we provide a purely extensional theoretical framework for higher order logic programming which generalizes the familiar theory of classical first order logic programming. Part of the lecture notes in computer science book series lncs, volume.

In this semantics program predicates denote sets of ordered tuples, and two predicates are equal iff they are equal as sets. We develop an extensional semantics for higherorder logic programs with negation, generalizing the technique that was introduced in 2, 3 for positive higherorder programs. We also propose an sldresolution proof procedure which is proven sound and complete with respect to the minimum model semantics. What is the difference between intensional and extensional. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higher order logic, that there is a broadest necessity, settling one of the central questions of this investigation.

It turns out that the extension of higher order logic programs with constructive negation o ers a much greater versatility to extensional higherorder logic programming. Theory and practice of logic programming 14, 45, 725737. Higherorder programming is a style of computer programming that uses software components, like functions, modules or objects, as values. Under this semantics, every program has a unique minimum herbrand model which is the greatest lower bound of all. In the second part of the book, an automated proof procedure for extensional type theory. What is the difference between intensional and extensional logic. Annotation this book constitutes the refereed proceedings of the 5th international conference on elearning and games, edutainment 2010, held in changchun, china, in august 2010. It turns out that the extension of higher order logic programs with constructive negation o ers a much greater versatility to extensional higher order logic programming. Iclp 2017 report association for logic programming. Higher order programming is a style of computer programming that uses software components, like functions, modules or objects, as values. Some extensional term models for combinatory logics and lambdacalculi. Introduction to higherorder categorical logic cambridge. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higherorder logic, that there is a broadest necessity, settling one of the central questions of this investigation. Jul 15, 2017 the intricacies of 3valued extensional semantics for higherorder logic programs article pdf available in theory and practice of logic programming july 2017 with 24 reads how we measure reads.

Under this semantics, every program has a unique minimum herbrand model which is the greatest lower bound of all herbrand models of the program and the least fixedpoint of the immediate consequence operator of the program. Pdf we propose a purely extensional semantics for higherorder logic programming. Leibniz basic notion of the truth of a judgment was that the concepts making up the predicate were included in the concept of the subject. Higher order logic an overview sciencedirect topics. Tichy created a system of intensional logic very similar to that of montague, beginning, in english, in tichy 1971, with a detailed presentation in tichy 1988. Dynamic assumptions are permitted and can be used during the execution. This book aims to show that a programming language based on a simply typed version of higher order logic provides an elegant, declarative means for providing such a treatment. Objectoriented features require state updation and can be obtained only by destroying referential transparency. Firstorder semantics and expressiveness vernon asuncion, yan zhang, heng zhang. Download programming with higherorder logic pdf ebook.

In programming languages, firstorder functions similarly refer to functions that operate on individual data elements e. Reduce reduce is an interactive system for general algebraic computations of interest to mathematicians, sc. The wellfounded semantics is the principle of inductive definition. Section 12 contains concluding remarks and a brief discussion of future work. There are many ways to further extend secondorder logic. We propose a purely extensional semantics for higherorder logic programming. Sep 26, 2016 there is no such books on programming logic. Higherorder logic programming languages such as prolog and twelf improve the notion of first order logic programming in two ways.

Practical foundations for programming languages by robert. In this way we provide an alternative extensional semantics for higher order logic programs with negation to the one proposed in. A logic is extensional if whenever two predicates or two. Higherorder logic refers to all these logics that are more powerful than firstorder logic. Mitchell foundations for programming languages, foundations of computing, mit press,1996.

Part ii demonstrates that another formulation of higher order logic, intuitionistic type theories, is closely related to topos theory. As first order logic is well understood and can be mathematically provable it can be used for representing all computational problems. Mechanizing programming logics in higher order logic. Constructive negation in extensional higherorder logic programming angelos charalambidis, panos rondogiannis. Consequently, higherorder logic has become an important topic of research.

Higher order functions characterise most functional programming. Principles of knowledge representation and reasoning. Extensional higherorder logic programming springerlink. Charalambidis a, handjopoulos k, rondogiannis p and wadge w 20 extensional higherorder logic programming, acm transactions on computational logic tocl, 14. Citeseerx document details isaac councill, lee giles, pradeep teregowda. To appear in handbook of logic in ai and logic programming, d. Higher order logic and intuitionistic type theory overlaps in many ways. Higherorder logic 243 for their own sake, and countable models of set theory are at the base of the independence proofs. This book aims to show that a programming language based on a simply typed version of higherorder logic provides an elegant, declarative means for providing such a treatment. Citeseerx a extensional higherorder logic programming. Three broad topics are covered in pursuit of this goal. Higherorder logic programming as constraint logic programming. Unfortunately his work did not become widely known. It allows the programmer to axiomatize predicates of predicates and operations on predicates.

In part i, they show that typed lambdacalculi, a formulation of higher order logic, and cartesian closed categories, are essentially the same. Under this semantics, every program has a unique minimum herbrand model. Book title logics in artificial intelligence book subtitle 12th european conference, jelia 2010, helsinki, finland, september. Harpers book provides a comprehensive treatment of the foundations of computation. Higher order logic in relation to computing and programming. Implementing default logic via answer set programming. Second, they have largely imported the notion of higherorder programming as it is understood within functional programming and have not examined a notion that is intrinsic to logic programming. It is usually instantiated with, or borrowed from, models of computation such as lambda calculus which make heavy use of higherorder functions. But you can follow any of the programming books and there you will get better logic. Gopalan nadathur and dale miller higher order logic programming, technical report cs199438, department of computer science, duke university,1994. The distinction between intensional and extensional entities is parallel to the distinction between sense and reference. Pdf extensional higherorder logic programming researchgate. If youre looking for a free download links of programming with higherorder logic pdf, epub, docx and torrent then this site is not for you.

Part ii demonstrates that another formulation of higherorder logic, intuitionistic type theories, is closely related to topos theory. If youre looking for a free download links of programming with higher order logic pdf, epub, docx and torrent then this site is not for you. Welcome to the home page for the book programming with higherorder logic by dale miller and gopalan nadathur. This book was published by cambridge university press in june 2012. First order semantics and expressiveness vernon asuncion, yan zhang, heng zhang.

Moreover, there exists a relative simple sound proof procedure for the new language. Terms from the higherorder language are defined via abstraction. Review for theory and practice of logic programming. We can extend higher order logic programming with constructive negation. Applications of lambda prolog in higher order programming and meta programming. Applications of lambda prolog in higherorder programming and metaprogramming. Extensional semantics for higherorder logic programs with. Proof procedures of extensional higherorder logic programming. For example, higherorder hereditary harrop formulas hohh can be used to support aspects of modular programming, abstract data types, and higherorder programming. Intensional logic is an approach to predicate logic that extends firstorder logic, which has quantifiers that range over the individuals of a universe, by additional quantifiers that range over terms that may have such individuals as their value.

Automated reasoning in higherorder logic presents both a theoretical analysis of fragments of higherorder logic as well as a complete automated search procedure for an extensional form of higherorder logic. Some extensional term models for combinatory logics and lambdacalculi makes the reader have stronger analytical thinking skills. Semantics of negation in extensional higherorder logic. The general principle, already recognized by tarski 1933 1956, is that in higher order logic one can formalize the semanticsdefine truthof lower order logic. First order logics, including set theory are all extensional in this sense. Intensional logic stanford encyclopedia of philosophy. We have designed and built a logic programming system which implements hohh in much the same way prolog implements firstorder horn clauses. Intensional logic is an approach to predicate logic that extends first order logic, which has quantifiers that range over the individuals of a universe, by additional quantifiers that range over terms that may have such individuals as their value. First order logic is made up of syntax and semantics. Logic programming is based upon an extension of firstorder logic. Higher order logic programming holp languages are particularly useful for various kinds of meta programming and theorem proving tasks because of the logical support for variable binding via abstraction. Sections 7, 8 and 9 will be covered in lectures 6, 7 and 8. Modularity can be built into a pure functional language objectedorientedness. First, a prooftheoretic framework that supports a general view of logic programming is identified.