Dale miller we argue that a logic programming language with a higher order intuitionistic logic as its foundation can be used both to naturally specify and implement theorem provers. Finally, a methodology for programming with specifications is exposed by showing how several computations over formal objects such as logical formulas, functional programs, and. Formal systems that describe computations over syntactic structures occur frequently in computer science. Higherorder programming is a style of computer programming that uses software components, like functions, modules or objects, as values. Logic for reasoning with higherorder abstract syntax conference paper pdf available in proceedings symposium on logic in computer science january 1997. In a higher order setting, we can achieve the same e ciency gain, but remain declarative, and indeed retain reversibility. Logic programming in a fragment of intuitionistic linear logic. Higher order logic programming in twelf extends traditional rst order logic programming in three ways. Deduction logic programming is a particular way to approach. The logic is for a language with an imperative semantics and. In this paper we consider the problem of extending prolog to include predicate and function variables and typed. In the standard or full semantics, quantifiers over highertype objects range over all possible objects of that. Thus, higherorder logic in this sense is basically a model theoretic study sha85. The teyjus system is an efficient implementation of the higher order logic programming language lambda prolog.
Mechanizing programming logics in higher order logic. He suggested that they were not necessary by modelling higher order concepts in logic programs. For example, higherorder hereditary harrop formulas hohh can be used to support aspects of modular programming, abstract data types, and higherorder programming. We give a set of typing rules which both provide a formal definition of type correctness and define the sourcetosource transformation we have used to. Dale miller we argue that a logic programming language with a higherorder intuitionistic logic as its foundation can be used both to naturally specify and implement theorem provers. Higher order logic programming holp languages 17 typically use a typed calculus as their domain of computation. In other words, i claim, that if two people started using secondorder logic for formalizing mathematical proofs, person f with the full secondorder logic and person hwith the henkin secondorder logic, we would not be able to see any di. These languages are particularly useful for various kinds of meta. He suggested that they were not necessary by modelling higherorder concepts in logic programs. We do this so that we can sketch the language of higherorder logic, a simple yet amazingly expressive language.
Amy felty, a logic programming approach to implementing higherorder term rewriting, conference proceedings on extensions of logic programming, p. Programming with higherorder logic focuses on how logic programming can exploit higherorder intuitionistic logic. We also discuss administrative details of the course, although these are not included here, but can be found on the course web page. Modularity can be built into a pure functional language objectedorientedness.
Logic programming frank pfenning august 29, 2006 in this. Such expansion trees can be defined with or without the use of skolem functions. The aim of this course is to introduce you to the theory and practice of logic programming. Specifying and implementing theorem provers in a higher order logic programming language amy p. This paper proposes an extension of hoare logic 17 for callbyvalue higherorder functionswith mllike new reference generation 1, 2, and demonstrates its use through nontrivial reasoning examples. The replication of common subproofs leads to redundancy in the. In particular automation of higherorder logic, the topic of this article, has often been considered as too challenging, at least until recently.
The sixth annual ieee symposium on logic, volume 6, part 1991 the sixth annual. Welcome to the home page for the book programming with higherorder logic by dale miller and gopalan nadathur. The authors emphasize using higher order logic programming to declaratively specify a range of applications. Higher order logic programming holp as the programming paradigm best suited for the three tasks above. Prolog is developed by applying this view to higherorder logic. In this speci cation, the variable n ranges over the numerals 0. In first order logic, a consistent theory which have infinite models will have models in every cardinality. Citeseerx document details isaac councill, lee giles, pradeep teregowda. They have been used for a wide range of applications including theorem proving, program. Types and programming languages, 2002, 623 pages, benjamin c. Programming with higherorder logic pdf download free.
Expansion trees are defined as generalizations of herbrand instances for formulas in a nonextensional form of higher order logic based on churchs simple theory of types. There are two possible semantics for higher order logic. A higherorder logic programming language with constraints. First, we point out that calln is not the way to go, despite its recent popularity as. It is usually instantiated with, or borrowed from, models of computation such as lambda calculus which make heavy use of higherorder functions. Implementing hol in an higher order logic programming language. As pointed out by necula and lee, typical safety proof in the context of certi ed code commonly have repeated subproofs that should be hoisted out and proved only once.
Theorem proving in higher order logics 12th international conference, tphols99, nice, france. Higher order logic programming extending the logic. Higherorder logic programming holp languages 17 typically use a typed calculus as their domain of computation. Specifying and implementing theorem provers in a higherorder logic programming language amy p. To a proof theorist, all logics correspond to formal systems that are recursively presented and a higherorder logic is no di.
We present an implementation of a program logic of objects, extending that al of abadi and leino. Program logics for certified compilers separation logic is the twentyfirstcentury variant of hoare logic that permits verification of pointermanipulating programs. Our aim is to improve the expressiveness of traditional logic programming with the benefits of both fields. For this purpose, we use a higherorder logic to describe a generalization to firstorder horn clauses.
In particular automation of higher order logic, the topic of this article, has often been considered as too challenging, at least until recently. Logic programming provides a natural framework for encoding and animating such systems. In this paper we make use of a higherorder logic programming language, aprolog 14, to implement programs transformations because this language employs higherorder unification in a direct fashion. The term higherorder logic is often a source of confusion and so it is relevant to explain the sense in which it is used here. Higherorder logic is the union of first, second, third, n thorder logic.
The main strength of lambda prolog, a language developed by dale miller and gopalan nadathur, is the support it offers for what is known as lambda tree syntax, a new and effective way to view the structure of complex syntactic objects. Counterexample generation for higherorder logic using. Higherorder logic programming as constraint logic programming. Programming with higherorder logic, by dale miller and. Gordon, why higherorder logic is a good formalism for specifying and verifying hardware, in g. We do this so that we can sketch the language of higherorder logic, a simple yet amazingly expressive language for mathematical statements. Specifying and implementing theorem provers in a higher. Section 11 explains how vdmstyle speci cations, weakest preconditions and dynamic logic can be represented in higher order logic.
Objectoriented features require state updation and can be obtained only by destroying referential. The combined functional and logic programming features of curry have been shown useful in diverse applications. Implementing a program logic of objects in a higherorder. Acm transactions in computational logic, 74, october 2006. Programming with higherorder logic, by dale miller and gopalan nadathur, cambridge university press, 2012, hardcover, isbn10. In the case of prolog 18 it is the simplytyped calculus, while in the case of elf 22 it is a dependently typed calculus. Finally, a methodology for programming with specifications is exposed by showing how several computations over formal objects such as logical formulas, functional programs, and terms and calculus expressions can be encoded in prolog. Download programming with higherorder logic pdf ebook. In contrast, in computing, there is hardly any relationship between, for example, turing machines as a model of computation and relational algebra as a model of database queries. In mathematics and logic, a higherorder logic is a form of predicate logic that is distinguished from firstorder logic by additional quantifiers and, sometimes, stronger semantics.
Scott 1986 cambridge lcf paulson 1987 linear logic girard et al. Higher order logic programming extending the logic extending. It is used for preliminary assessment of efficiency. Implementing a program logic of objects in a higher order logic theorem prover martin hofmann and francis tang laboratory for foundations of computer science division of informatics university of edinburgh scotland, uk abstract. When logic programming is based on the proof theory of intuitionistic logic, it is natural to allow implications in goals and in the bodies of clauses. Second, an actual language called prolog is developed by applying this view to higher order logic. All code fragments from this book below is a version of the book in which most of the code and sectioning information chapters, sections, subsection is kept. Preliminary version presented at the ninth international. A higherorder language for reasoning about objects, inheritance, and scheme article pdf available in acm sigmod record 182 july 1997 with 356 reads how we measure reads. Semester 1, 201516 announcements programming exam a pdf version of the sicstus manual will be available during the exam also a short description of some builtin predicates. Additional mechanized proof generating tools in hol are also described.
To appear in handbook of logic in ai and logic programming, d. We describe the design and implementation of a higherorder tabled logic programming interpreter where some redundant and in. The result is denoted higher order hereditary harrop formulas with constraints. Below is a version of the book in which most of the code and sectioning information. Functional logic programming april 2010 communications. Pierce, 0262162091, 9780262162098, mit press, 2002.
How is this paper different from some of the others. Expansion trees are defined as generalizations of herbrand instances for formulas in a nonextensional form of higherorder logic based on churchs simple theory of types. Any program written in a logic programming language is a set of sentences in logical form, expressing facts and rules about some problem domain. Pdf logic for reasoning with higherorder abstract syntax. In a higherorder setting, we can achieve the same e ciency gain, but remain declarative, and indeed retain reversibility.
Implementing a program logic of objects in a higherorder logic theorem prover martin hofmann and francis tang laboratory for foundations of computer science division of informatics university of edinburgh scotland, uk abstract. Sections 7, 8 and 9 will be covered in lectures 6, 7 and 8. Leino2 1 computer science department, university of california at santa cruz, ca, usa 2 microsoft research, redmond, wa, usa abstract. What is the actual difference between 1st order and higher. Second, an actual language called prolog is developed by applying this view to higherorder logic.
The authors emphasize using higherorder logic programming to declaratively specify a range of applications. The situation described above is bad in two respects. However, these systems often embody variable binding, a notion that must be treated carefully at a. Philosophers of mathematics usually divide logic into. The teyjus system is an efficient implementation of the higherorder logic programming language lambda prolog. This book covers practical and theoretical aspects of separation logic at a level accessible to beginning graduate students interested in software verification. We develop a logic for reasoning about objectoriented programs. The main strength of lambda prolog, a language developed by dale miller and gopalan nadathur, is the support it offers for what is known as lambda tree syntax, a new and effective way to view the structure of complex syntactic objects such as types, formulas, proofs and programs. Higherorder 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. Mar 21, 2001 we present a framework for the combination of constraint logic programming ticlp and higher order hereditary harrop formulas tihohh. Higherorder logics with their standard semantics are more expressive, but their modeltheoretic properties are less wellbehaved than those of firstorder logic the term higherorder logic, abbreviated as hol. Higherorder logics with their standard semantics are more expressive, but their modeltheoretic properties are less wellbehaved than those of firstorder logic the term higherorder logic, abbreviated as.
Logic programming is a programming paradigm which is largely based on formal logic. Higher order functions characterise most functional programming. Welcome to the home page for the book programming with higher order logic by dale miller and gopalan nadathur. This is yet another paper which tells logic programmers what functional programmers have known and practiced for a long time. Semester 1, 201516 announcements programming exam a pdf version of the sicstus manual will be available during the exam. One of the most recent developments in higherorder logic programming is teyjus, which is an improved version of the prolog language. An essential difference between first order and higher order logic is lowenheimskolem. Higher order logic in relation to computing and programming. One of the most recent developments in higher order logic programming is teyjus, which is an improved version of the prolog language. Specifying and implementing theorem provers in a higherorder. Introduction reference generation in higherorder programming.
Acm transactions on programming languages and systems, 265, september 2004. Reduce reduce is an interactive system for general algebraic computations of interest to mathematicians, sc. The improved computation model handles higherorder unification as well as dynamic terms. 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. This book was published by cambridge university press in june 2012. These trees store substitution terms and either critical. Download pdf programming with higherorder logic free pdf. Section 12 contains concluding remarks and a brief discussion of future work. The improved computation model handles higher order unification as well as dynamic terms. Implementing hol in an higher order logic programming.