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1 edition of A resolution style proof procedure for higher-order logic found in the catalog.

A resolution style proof procedure for higher-order logic

Lawrence Joseph Henschen

# A resolution style proof procedure for higher-order logic

## by Lawrence Joseph Henschen

Written in English

Subjects:
• First-order logic,
• Automatic theorem proving

• Edition Notes

The Physical Object ID Numbers Statement by Lawrence Joseph Henschen Series Report no. 452, Report (University of Illinois (Urbana-Champaign campus). Dept. of Computer Science) -- no. 452. Pagination iv, 79 leaves; Number of Pages 79 Open Library OL25464105M OCLC/WorldCa 3705232

In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.. First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, also quantifies over relations. "A proof procedure for the logic of Hereditary Harrop formulas", by Gopalan Nadathur. This paper covers the basics of universes, environments, and Lambda Prolog-style proof search. Quite readable. "A new formulation of tabled resolution with delay", by Theresa Swift. This paper gives a kind of abstract treatment of the SLG formulation that is. We show that the proof of termination of normalization for the usual formulation of higher-order logic can be adapted to a proof of termination of normalization for its first-order formulation. The “hard part” of the proof, that cannot be carried out in higher-order logic itself, i.e. Cited by: The Foundation of a Generic Theorem Prover Lawrence C Paulson Computer Laboratory University of Cambridge Abstract Isabelle [28, 30] is an interactive theorem prover that supports.

Now modern proof assistants are often less concerned with foundations from the paradigm of Principia Mathematia and are more useful for theorem proving for everyday work, and so they have some support for fragments of higher order logic, SAT/SMT solving, type theories, and other more informal and less foundational approaches. Higher-order logic was chosen, since most concepts and theorems in mathematics can be expressed very naturally in type theory. It is my belief that this ease of expression will eventually lead to powerful and user-friendly tools assisting the student of mathematics and logic and, in the farther future, the mathematician in his research. Code and resources for "Handbook of Practical Logic and Automated Reasoning" The code available on this page was written by John Harrison to accompany his textbook on logic and automated theorem proving, published in March by Cambridge University Press. For more information about the book, click the picture on the right. cal proofs, the best formalization of it so far is the Henkin second-order logic. In other words, I claim, that if two people started using second-order logic for formalizing mathematical proofs, person F with the full second-order logic and person Hwith the Henkin second-order logic, we would not be able to .

Higher-order Logic: Foundations Bibliography •M. J. C. Gordon and T. F. Melham, Introduction to HOL: A theorem proving environment for higher order logic, Cambridge University Press, •Peter B. Andrews, An Introduction to Mathematical Logic and Type . Key words: Fitch style, natural deduction, proof simpliﬁcation, denotational proof languages, NDL, assumption bases, detours. 1. Introduction This paper is concerned with the problem of simplifying proofs in Fitch-style natural-deduction systems. The hallmark of such systems is the idea of Bmaking. Self-Formalisation of Higher-Order Logic 3 A version of this paper [16] was originally published in the conference proceedings of ITP , and describes a semantics for Stateless HOL [34]. By contrast, the semantics presented in this paper works directly on standard HOL and uses a theory context to handle deﬁnitions (in the style of Arthan [3]).Cited by: discussion of higher-order reasoning and automated methods of proof search in that setting would have been natural in this context; given that HOL Light is based on higher-order logic, it seems likely that the omission is simply due to time and space constraints. One of the most striking features of the book (and one of the factors contribut-File Size: 62KB.

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### A resolution style proof procedure for higher-order logic by Lawrence Joseph Henschen Download PDF EPUB FB2

Dag Prawitz. Hauptsatz for Higher Order Logic. The Journal of Symbolic Logic, Bd. 33, S. – - Dag Prawitz. Completeness and Hauptsatz for Second Order Logic. Theoria, Bd. 33, S. – - Moto-o Takahashi. A Proof of Cut-Elimination in Simple Type-Theory. Journal of the Mathematical Society of Japan, Bd.

19, S. –   A resolution style proof procedure for higher-order logic. PhD thesis, University of Illinois at Urbana-Champaign, PhD thesis, University of Illinois at Author: Wenchang Fang, Jung-Hong Kao.

In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from A resolution style proof procedure for higher-order logic book logic by additional quantifiers and, sometimes, stronger -order logics with their standard semantics are more expressive, but their model-theoretic properties are A resolution style proof procedure for higher-order logic book well-behaved than those of first-order logic.

The term "higher-order logic", abbreviated as HOL. •Polynomial-time inference procedure exists when KB is expressed as Horn clauses: where the P i and Q are non-negated atoms. –First-Order logic •Godel’s completeness theorem showed that a proof procedure exists •But none was demonstrated until Robinson’s resolution algorithm.

A A resolution style proof procedure for higher-order logic book style proof procedure for higher-order logic / (Urbana, IL.: University of Illinois, ), by Lawrence Joseph Henschen (page images at HathiTrust) Theorem proving with abstraction, part I / (Urbana: Dept.

of Computer Science, University of Illinois at Urbana-Champaign, ), by David A. Plaisted (page images at HathiTrust). A.S. Troelstra, in Studies in Logic and the Foundations of Mathematics, Formulation of HAH.

Higher-order logic is based on a many-sorted language with a collection of sorts or types; we use σ, σ′, τ, τ′, for arbitrary are variables (x σ, y σ, z σ,) for each type, and an equality symbol = σ for each σ.

Relation symbols and function symbols may take. Geometric Resolution: A Proof Procedure Based on Finite Model Search. with Higher Order Logic (HOL) theorem provers. Both SAT solvers generate resolution-style proofs for (instances of.

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.

Three broad topics are covered in pursuit of this goal. First, a proof-theoretic framework that supports a general view of logic programming is by: First-Order Logic At the end of the last lecture, I talked about doing deduction and propositional logic in the natural deduction, high-school geometry style, and then I promised you that we would look at resolution, which is a propositional-logic proof system used by computers.

We present a mechanised semantics for higher-order logic (HOL), and a proof of soundness for the inference system, including the rules for making definitions, implemented by the kernel of the HOL.

HIGHER-ORDER LOGIC for their own sake, and countable models of set theory are at the base of the inde-pendence proofs: ﬁrst-order logic’s loss thus can often be the mathematician’s or philosopher’s gain.

Extensions When some reasonable notionfalls outsidethe scope of ﬁrst-orderlogic, one ratherFile Size: KB. Proofs in Higher-Order Logic. Abstract. Expansion trees are defined as generalizations of Herbrand instances for formulas in a nonextensional form of higher-order logic based on Church's simple theory of types.

Such expansion trees can be defined with or without the use of skolem functions. These trees store substitution terms and either critical. Propositional Logic Propositional logic consists of a set of atomic propositional symbols (e.g. Socrates, Father, etc), which are often referred to by letters p, q, r etc.

(Note that these letters aren't variables as such, as propositio. This volume is a self-contained introduction to interactive proof in higher-order logic (HOL), using the proof assistant Isabelle. It is written for potential users rather than for our colleagues in the research world.

The book has three parts. { The rst part, Elementary Techniques, shows how to model functional programs in higher-order Size: 1MB. First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer -order logic uses quantified variables over non-logical objects and allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man.

If we think of a predicate as a function to truth values, then a higher-order predicate is a function on a power set or a function set. Typed higher-order logic may be called higher-order type theory.

Typed higher-order intuitionistic logic is often identified with the internal logic of a topos. Related concepts. propositional logic (0th order). Leo Bachmair, Christopher Lynch, in Handbook of Automated Reasoning, Refutational Theorem Proving.

Theorem provers are procedures that can be used to check whether a given formula F (the “goal”) is a logical consequence of a set of formulas N (the “theory”). Refutational theorem provers deal with the equivalent problem of showing that the set N∪{¬F} is inconsistent.

The procedures of the latter paper are based on a higher-order formulation of hybrid logic involving the simply typed lambda calculus. The article Hansen, Bolander, and Braüner () gives a tableau-based decision procedure for many-valued hybrid logic, that is, hybrid logic where the two-valued classical logic basis has been generalized to a.

This volume constitutes the refereed proceedings of the Higher-Order Logic User's Group Workshop, held at the University of British Columbia in August The workshop was sponsored by the Centre for Integrated Computer System Research.

It was the sixth in the series of annual international. Resolution proof procedurecombined a perfected pattern-matching algorithm with a variant of modus ponens to give the most-used proof procedure to date.

Model-elimination proof procedurewhen augmented with resolution concepts, led to the inference engine used in the programming language Prolog. Prolog has its roots in first-order logic, a formal logic, and unlike pdf other programming languages, Prolog pdf intended primarily as a declarative programming language: the program logic is expressed in terms of relations, represented as facts and rules.A computation is initiated by running a query over these relations.

The language was developed and implemented in Marseille, France, in Designed by: Alain Colmerauer, Robert Kowalski. A Hilbert-style proof system Here is a simple proof system for propositional logic. There are countless similar systems. They are often called Hilbert systems after the logician David Hilbert, although they existed before him.

13 A Hilbert-style proof system This proof system provides rules for implication only.work based on higher order logic. The central contribution of this ebook is the extension of Isabelle with a calculus of primitive proof terms, in which proofs are represented using λ-terms in the spirit of the Curry-Howard isomorphism.

Primitive proof terms allow for an independent veriﬁcation of proofs constructed.