The semantics of predicate logic university of waterloo. This paper clarifies how this fixpoint theory can define the stable and wellfounded semantics of logic. Approximation fixpoint theory was developed as a fixpoint theory of lattice operators that provides a uniform formalization of four main semantics of three major nonmonotonic reasoning formalisms. A very desirable datalog extension investigated by many researchers in the last 30 years consists in allowing the use of the basic sql aggregates min, max, count and sum in recursive rules. Scotland abstract sentences in firstorder predicate logic can be usefully interpreted as programs in this paper the. A fixpoint semantics and an sldresolution calculus for.
A fixpoint semantics for disjunctive logic programs. Specifically, we show that the familiar operational and least fixpoint semantics can be given to these clauses. Knowledge compilation of logic programs using approximation. Pdf fixpoint 3valued semantics for autoepistemic logic. Introduction to logic lecture 2 syntax and semantics of propositional logic.
He was a professor at city university of new york, lehman college and the graduate center. The key difference with the semantics proposed by moore is that we consider approximations of possibleworld structures by pairs of. It is concluded that operational semantics is a part of proof theory and that fixpoint semantics is a special case of modeltheoretic semantics. A pure logic programming based semantics is not adequate to account for the behavior of sieve. A constructive semantic characterization of aggregates in. This generalizes the conventional semantics, and agrees with it on successes for horn clause programs. In some contexts, we call the input data the extensional database and the program the intensional database. The variety of semantical approaches that have been invented for logic programs is quite broad, drawing on classical and manyvalued logic, lattice theory, game theory, and topology. A fixpoint semantics and an sldresolution calculus for modal. Logic programming and functional programming are often lumped together.
The study of semantics of logic programs has shown strong. We are here concerned with fixpoint semantics rather than 3valued models of clarks completions, to denote the success and failure sets by sldnf resolutions with nonsafe rules. Mathematical aspects of logic programming semantics 1st. Logic programming for knowledge representation miroslaw truszczynski. Then, a denotational semantics of the equivalent, propositional, and infinite dlp program groundp, provides a. Unfolding and fixpoint semantics of concurrent constraint. Execution of a logic program is a theorem proving process. We give a fixpoint semantics and an sldresolution calculus for mprolog in all of the basic serial modal logics kd, t, kdb, b, kd. A note on logic programming fixedpoint semantics eprints soton. In this paper, we propose a simple comprehensive solution that extends the declarative least fixpoint semantics of horn clauses, along with the optimization techniques used in the bottomup implementation. The welldefined least fixpoint semantics based on the 3valued logic approach is included in the wellfounded semantics 6. It significantly extends the tools and methods from traditional order theory to include nonconventional. We introduce a derivation operator and define the semantics as its least fixpoint. In contrast, a variable in a logic program is a variable in the m athem atical sense, i.
Fixpoint semantics for logic programming a survey melvin fittinga. This paper clarifies how this fixpoint theory can define the stable and wellfounded semantics of logic programs. In other words, an ideal of logic programming is purely declarative programming. Show full abstract modeltheoretic semantics, a fixpoint semantics, andasemantics based on a reduction to ordinary logic programming with function symbols. Since logic programming computation is proof search, to study logic programming means to study proofs.
Pdf the stable model semantics for logic programming. Together these results provide an elegant algebraic, fixpoint and logical semantics for pure logic programs. Melvin mel fitting born january 24, 1942 is a logician with special interests in philosophical logic and tableau proof systems. The semantics of logic refers to the approaches that logicians have introduced to understand and determine that part of meaning in which they are interested. The semantics is 3valued in the sense that, for some formulas, the least fixpoint does not specify whether they are believed or not. Fixpoint semantics for logic programs cs240b notes notes based on section 8. Pdf a fixpoint semantics is given for logic programming using domain theory, with. Leeuven, celestijnenlaan 200a, b3001 heverlee, belgium 2 department of computer science, university of kentucky, lexington, ky 405060046, usa dedicated to ray reiter on his 60th birthday abstract.
It has been successfully applied to unify all common semantics of logic programs, autoepistemic logic, default logic. Fixpoint 3valued semantics for autoepistemic logic marc denecker1, v. Cristian molinaro the use of logic in databases started in the late 1960s. It significantly extends the tools and methods from traditional order theory to include nonconventional methods from mathematical analysis that depend on. In this paper the operational and fixpoint semantics of predicate logic programs are defined, and the connections with the proof theory and model theory of logic are investigated. In this paper, we propose a simple comprehensive solution that extends the declarative least fixpoint semantics of horn clauses, along with the optimization techniques used in the bottomup. This monograph provides an intensive course for graduate students in computer science, as well as others interested in extensions of logic programming, on the. In particular, static causal laws of the following form are considered. For logic programming, the realists and purists are far apart. The computational model has been proved to be sound. Objectives the main objective of both editions of this textbook is to provide a uniform account of both the foundations of logic programming and simple programming techniques in the programming. We introduce new notions and modify the classical semantics, i. But you can follow any of the programming books and there you will get better logic.
The semantics of predicate logic as a programming language m. Approximation fixpoint theory and the semantics of logic and. In this paper, we present an account of classical logic programming fixedpoint semantics in terms of two standard categorical constructions in which the least. F is a set of wellformed formulas c is a class of possible interpretations models. On the semantics of negations in logic programming. In the early 1970s codd formalized databases in terms of the relational calculus and the relational algebra.
To compute the truth value of a query, a computational model which directly manipulates linguistic terms is provided. Mathematical aspects of logic programming semantics. This paper describes a simpler way for programmers to reason about the correctness of their code. Conclusion for arbitrary sentences x and y of firstorder predicate logic, proof theory determines when x y and model theory determines when x y. The semantics of predicate logic as a programming language. The result of both the unfolding and the fixpoint semantics is a set of reactive behaviors, which are trees abstractly representing all the possible computations of a program, including deadlocks and. Part of the lecture notes in computer science book series lncs, volume 6125. Pdf groundedness in logics with a fixpoint semantics. Prolog programming in logic is a representative logic language. Pdf predicate introduction for logics with a fixpoint. A constructive semantic characterization of aggregates in answer set programming volume 7 issue 3 tran cao son, enrico pontelli skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites.
Pdf approximation fixpoint theory and the semantics of. Theoretical foundations and semantics of logic programming. Nonetheless, much of the work on logic programming semantics seems to exist side by side with similar work done for imperative and functional programming, with relatively minimal contact between communities. Wellfounded and stable semantics of logic programs with. This course is a prerequisite for types part ii, denotational semantics part ii, and topics in concurrency part ii.
Approximation fixpoint theory aft is an abstract algebraical unifying framework that aims at exposing these principles by formalising them in lattice theory. Approximation fixpoint theory and the wellfounded semantics. Since logic programming involves both logic and programming, it should not be surprising that several varieties of semantics have been developed for it. We have argued that m the procedural interpretation, operational semanttcs s proof theory and fixpoint semantics is model theory. A fixpoint semantics and an sldresolution calculus for modal logic programs linh anh nguyen institute of informatics university of warsaw ul. The above result can be proved by means of the fixpoint theorems of knastertarski. Available formats pdf please select a format to send. It will show how to specify the meaning of typical programming. Smith 2 1 computer laboratory, university of cambridge, cambridge cb2 3qg, england.
In this paper, a logic program under the stable semantics is constructed to capture the meaning of theories in the action language al. May 30, 2018 bibtex does not have the right entry for preprints. An effective fixpoint semantics for linear logic programs. The book is subtitled an elementary introduction using structural operational semantics and as such is a very good introduction to many of the key topics in this course, presented in a more leisurely and detailed way than winskels book. Department of computer science university of kentucky september 10, 2007 iclp 2007, porto university of kentuckylogic programming for kr 9102007 1 55. Pdf fixpoint semantics for logic programming a survey. Predicate introduction for logics with a fixpoint semantics. Bibtex does not have the right entry for preprints. New semantic tools for logic programming springerlink.
We propose a modal logic programming language called mprolog, which is as expressive as the general modal horn fragment. Operational semantics and proof theory distinguished dissertations in computer science andrews, james h. Indeed, it will allow both prime2, 3 and prime3, 3 to be part of the meaning of prime. To our knowledge, this is the first attempt to define an effective fixpoint semantics for linear logic programs. The semantics we present, starting in the next section, is still a fixpoint semantics.
A fixpoint 3valued semantics for autoepistemic logic our semantics for autoepistemic logic is defined in terms of possibleworld structures and fixpoint conditions. An excellent survey of fixpoint semantics for logic programming is fit99. The paper presents a constructive 3valued semantics for autoepistemic logic ael. As sufficient conditions for termination, we show that the fixpoint computation is guaranteed to converge for propositional lo. A sound and complete semantics for a similaritybased logic programming language. Programs are written in the language of some logic. It is concluded that operational semantics is a part of proof theory and that fixpoint semantics is a special case of modeltheoretic. So far we have kept syntax and semantics rather informal but, in metalogic we want to prove things about logic this requires us to get really precise about syntax and semantics we are going to give syntax and semantics of propositional logic a mathematical treatment this is called formal syntax and formal semantics. The paper is a general overview of our approach to the semantics of logic programs. We then generalize the wellfounded w p operator to disjunctive logic programs, give a fixpoint semantics for disjunctive stable models and present an algorithm for computing the stable models of functionfree programs. This is a hack for producing the correct reference. Syntax, semantics, and proof introduces students to the fundamental concepts, techniques, and topics involved in deductive reasoning. Unfortunately, this has not yet been achieved with current logic programming systems. Answerset programming for the semantic web kbs tu wien.
Fixpoint semantics and optimization of recursive datalog programs with aggregates. It significantly extends the tools and methods from traditional order theory to include nonconventional methods from mathematical. The semantics of constraint logic programs sciencedirect. Agler guides students through the basics of symbolic logic by explaining the essentials of two classical systems, propositional and predicate logic. Pdf on semantics, syntactics and fixpoints of general. Pdf logic programming semantics using a compact data structure. In this paper, after recalling the modeltheoretic and fixpoint semantics for a pure subset of this language. This paper presents a fixpoint semantics for fuzzy linguistic logic programs and based on it proves the completeness of the computational model. We extend the concept of the herbrand base of a logic program to consist of all positive clauses that may be formed using the atoms in the herbrand base. One natural generalization of logic programs is to allow different unification mechanisms in the operational semantics.
Approximation fixpoint theory and the semantics of logic and answers set programs. Although he studied logic as a basis for functional programming rather than logic programming, his ideas are more fundamental and therefore equally applicable in both paradigms. Department of mathematics and computer science lehman. Mathematical aspects of logic programming semantics crc. Fixpoint semantics and optimization of recursive datalog. To appear in theory and practice of logic programming tplp, proceedings of iclp 2015 recent advances in knowledge compilation introduced techniques to compile \emphpositive logic programs into propositional logic, essentially exploiting the constructive nature of the least fixpoint computation. Mathematical logic for computer science is a mathematics textbook, just as a. A sound and complete semantics for a similaritybased. Unfolding is also used to define an immediate consequences operator, and, therefore, a fixpoint semantics in the typical logic programming style. Unfolding is also used to define an immediate consequences operator and, therefore, a fixpoint semantics in the typical logic programming style. The algorithms soundness and completeness are proved and some complexity.
The book of lloyd provides a detailed introduction to the semantics of logic programs. It significantly extends the tools and methods from tradi. Fixpoint 3valued semantics for autoepistemic logic. Approximation fixpoint theory and the semantics of logic. Information theory language has many uses, only one of which is to convey information but surely transferring information is important we can measure information in a limited, technical, and very useful, sense. An application of the fixpoint operator can be computed algorithmically. Mathematical aspects of logic programming semantics crc press book covering the authors own stateoftheart research results, mathematical aspects of logic programming semantics presents a rigorous, modern account of the mathematical methods and tools required for the semantic analysis of logic. Fixpoint semantics for logic programming a survey sciencedirect. Introduction to logic lecture 2 syntax and semantics of.
What are the best books for improving programming logic. Pdf a fixpoint semantics for disjunctive logic programs. It is intended for undergraduates, probably beginning a linguisticsrelated course, who find themselves having to deal with semantics for the first time. Fixpoint semantics for active integrity constraints. In this respect, l ogic p rogram m ing is therefore m uch closer to m athem atical intuition than im perative program m ing. Fixpoint semantics and completeness of the computational. Sldnf resolution with nonsafe rule and fixpoint semantics. A fixpoint semantics is given for logic programming using domain theory, with undefined as one truth value, allowing negation, and arbitrary data structures. Nonmonotonic logic is now seen as a close relative of logic programming, and developments in either area tend to a. Sentences in firstorder predicate logic can be usefully interpreted as programs. Semantics ix using this book this book is intended to meet the need for a genuinely introductory course book in semantics.