In 3rd international conference on logic and argumentation clar 2020, proceedings, lecture. Under this semantics, every program has a unique minimum herbrand model. Pdf extensional higherorder logic programming researchgate. We define a novel, extensional, threevalued semantics for higherorder logic programs with negation.
It turns out that the extension of higherorder logic programs with constructive negation o ers a much greater versatility to extensional higherorder logic programming. Higher order logic in relation to computing and programming. Higherorder programming is a style of computer programming that uses software components, like functions, modules or objects, as values. It is usually instantiated with, or borrowed from, models of computation such as lambda calculus which make heavy use of higherorder functions. Higherorder logic programming languages such as prolog and twelf improve the notion of first order logic programming in two ways. In logic programming, there is a gap between the metaprogramming practice and its theory. Unfortunately his work did not become widely known. Proof procedures of extensional higherorder logic programming. The syntax of the language extends that of prolog by supporting higherorder constructs such as higherorder predicate variables, partial application and lambda terms. Fortunately, a while back i discovered a subset of extensional higher order horn logic that works as logic programming and has a minimum model semantics. Fourth, we present solutions to some slightly more challenging programming tasks. The interaction between higher order programming and some of the unique features of logic programming is illustrated and some important programming techniques are discussed.
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. A semantics for intuitionistic extensional higherorder logic supporting higherorder abstract syntax. Hilog has a higherorder syntax and allows arbitrary terms to appear in places where predicates, functions, and atomic formulas occur in predicate calculus. As first order logic is well understood and can be mathematically provable it can be used for representing all computational problems. Third, we compare higher order programming with the skeletons and techniques approach. Pdf extensional higherorder logic programming angelos. Download ladder logic programming software for free. Moreover, there exists a relative simple sound proof procedure for the new language. Mechanizing programming logics in higher order logic. First order logic is made up of syntax and semantics.
Leoiii sb19,s18,sb18 is an automated theorem prover for polymorphic higherorder logic which supports all common tptp dialects, including thf, tff and fof as well as their rank1. The new semantics is based on interpreting the types of the source language as threevalued fittingmonotonic functions at all levels of the type hierarchy. In 24th european conference on artificial intelligence ecai 2020, proceedings, 2020. The key idea behind this paradigm is that the predicates defined in a program essentially denote sets. Firstorder semantics and expressiveness va, yz, hz. Wadge, university of victoria we propose a purely extensional semantics for higher order logic programming. To see the problem with unrestricted higher order extensional horn logic, consider the following clauses.
We show that higherorder programming techniques are already in use in logic programming albeit in a limited form and that these can be generalized. In this semantics program predicates denote sets of ordered tuples. Higher order unification 30 years later proceedings of. In this chapter, we develop the idea of higherorder logic programming by utilizing a higherorder logic as the basis for computing. Higherorder hornlogic programming semantic scholar. We can extend higherorder logic programming with constructive negation. Extensional higherorder logic programming angelos charalambidis, konstantinos handjopoulos, and panagiotis rondogiannis, university of athens william w. This book was published by cambridge university press in june 2012. Section 12 contains concluding remarks and a brief discussion of future work.
Logic programming is based upon an extension of firstorder logic. In this speci cation, the variable n ranges over the numerals 0. Extensional higherorder paramodulation in leoiii alexander steen, christoph benzmuller alexander. Intensional logic stanford encyclopedia of philosophy. Logic programming is exceptionally suitable to programming language implementation. Pdf we propose a purely extensional semantics for higherorder logic programming. What is the difference between intensional and extensional. In this paper we consider the problem of extending prolog to include predicate and function variables and typed. Wadge, university of victoria we propose a purely extensional semantics for higherorder logic programming. This book builds up the ideas behind lambda prolog progressively starting with traditional prolog presented as a logic and then adding in more powerful features. And adding clauses to a program might break it, because it might change the scope of predicate variables. Fruehwirthtype inference by program transformation and partial evaluation.
For this purpose, we use a higherorder logic to describe a generalization to firstorder horn clauses. The representation of program synthesis in higher order logic. A generalization of horn clauses to a higherorder logic is described and examined as a basis for logic programming. We also formalize an mset model where the monoid m is a type of substitutions and a. The key idea is to restrict what can appear on the left hand side. Welcome to the home page for the book programming with higherorder logic by dale miller and gopalan nadathur. Intensional logic jaroslav peregrin 1 freges bedeutung the german mathematician and one of the founding fathers of modern logic, gottlob frege 18481925, was the. 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. Charalambidis a, handjopoulos k, rondogiannis p and wadge w 20 extensional higherorder logic programming, acm transactions on computational logic tocl, 14. We propose a purely extensional semantics for higherorder logic programming. An automated theorem prover for classical higherorder logic with choice. 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. The talk will present a survey of higher order unification, covering an outline of its historical development, a summary of its applications to three fields. Extensional higherorder logic programming springerlink.
One aim of this paper is to illustrate higher order programming techniques in prolog. Secondorder and higherorder logic stanford encyclopedia of. Higherorder functions, as programprocessing programs are called in functional programming, and metaprograms, as they are called in logic programming, are widespread declarative programming techniques. An intensional tool applied to french language educational software. We formalize a higherorder logic with church style typing. Higherorder logic is based on a manysorted language with a collection of sorts or types.
Logic programming definition and meaning collins english. In this paper we reexamine the semantics of classical higherorder logic with the purpose of clarifying the role of extensionality. Compositional semantics for composable continuations. Prologs dcgs are the only parsing dsl ive ever used that truly felt like a firstclass language construct, presumably because theyre just a small amount of syntax sugar over normal prolog. Implementing hol in an higher order logic programming. The distinction between intensional and extensional entities is parallel to the distinction between sense and reference. Reasoning with primitive equality is enabled by adapting paramodulationbased proof search to. Tomer libal, alexander steen, towards an executable methodology for the formalization of legal texts. William wadge university of victoria, victoria uvic. We give examples to show the interaction between higher order programming and other features of prolog such as nondeterminism, logic variables, flexible modes, meta programming, definite clause grammar notation and in some systems. Maidaknowing intensional individuals and reasoning about knowing intensional individuals. 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. Secondorder logic has a subtle role in the philosophy of mathematics.
In this semantics program predicates denote sets of ordered tuples, and. We describe a novel logic, called hilog, and show that it provides a more suitable basis for logic programming than does traditional predicate logic. An intensional logic, however we do need to know what the terms mean, because we want to ask not just about which things do, as a matter of contingent fact, happen to fall under those terms. International conference on functional programming, gothenburg, sweden icfp2014. Two features of logic programming play important role in this modelling. In this semantics program predicates denote sets of ordered tuples, and two predicates are equal iff they are equal as sets. The lectures will first focus on a reconstruction of cic from its syntactic components lambdacalculus, recursion and corecursion, weak and strong dependent patternmatching, types, sorts, conversion rule and their expressivity higherorder logic, leibniz equality, pitypes, base types, wtypes and settheoretic universes. Extensional higherorder logic programming has been proposed 10, 1, 2, 7, 5, 4 as a p romising general ization of classical logic progr amming. Dynamic assumptions are permitted and can be used during the execution. In this paper we develop principles of a formal theory on reasoning about programs and program construction within a uni. Extensional higherorder logic programming has been proposed 10 1 2 7 5 4 as a promising generalization of classical logic programming. The syntax of first order logic is a formal language that is used to express concepts.
Completeness is proved relative to an analog of henkin models, familiar for higher type classical logics. Higher order logic an overview sciencedirect topics. Under this semantics, every program has a unique minimum herbrand. To reach this goal, we distinguish nine classes of higherorder models with respect to various combinations of boolean extensionality and three forms of functional extensionality. Download ladder logic programming software for free windows. Troelstra, in studies in logic and the foundations of mathematics, 1998. Education software downloads ladder logic simulator by triangle research international, inc and many more programs are available for instant and free download. Hopes is a prototype interpreter for a higherorder prologlike language. We show that this extension possesses certain desirable computational properties. Citeseerx a extensional higherorder logic programming. Terms from the higherorder language are defined via abstraction. Higher order logic programming holp as the programming paradigm best suited for the three tasks above. Higherorder programming with effects, vancouver, bc, canada hope2015.
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