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1. Peter B. Andrews
Research is in mathematical logic, especially Higherorder logic (type theory) and automated theorem proving. It is directed toward enabling computers to
Peter B. Andrews
Professor of Mathematics
Department of Mathematical Sciences

Carnegie Mellon University

Pittsburgh, Pa. 15213
412-268-2554 (office)
412-268-2545 (Math. Dept.)
412-268-6380 (Fax to Math. Dept.)
Ph.D., Princeton University, 1964. Advisor: Alonzo Church
Herbrand Award
Research is in mathematical logic, especially higher-order logic (type theory) and automated theorem proving. It is directed toward enabling computers to construct and check proofs of theorems of mathematics and other disciplines formalized in type theory or first-order logic and to assist humans engaged in these tasks. Potential applications of automated theorem proving include hardware and software verification, partial automation of various mathematical activities, promoting development of formal theories in a wide variety of disciplines, deductive information systems for these disciplines, expert systems which can reason, and certain aspects of artificial intelligence. The research is based on an approach to automated theorem proving involving expansion proofs and matings. Expansion proofs and matings for a theorem represent the essential combinatorial structure of various proofs of the theorem. They can be found by heuristic search, and natural deduction proofs can be constructed from them without further search. A computer implementation of these ideas called TPS (Theorem Proving System) handles theorems of type theory as well as theorems of first-order logic. The system can be used in automatic or interactive mode, and is available to interested parties. It runs in Common Lisp on Unix (including Linux) platforms. A subsystem of TPS called

2. Logic Higher Order
Higherorder logic and intuitistic type theory overlaps in many ways. The textbook (Nordstrom, Petersson, Smith, 1990) is a good source for intuitistic
A short article for the Encyclopedia of Artificial Intelligence : Second Edition by Dale Miller, February 1991
While first-order logic has syntactic categories for individuals, functions, and predicates, only quantification over individuals is permitted. Many concepts when translated into logic are, however, naturally expressed using quantifiers over functions and predicates. Leibniz's principle of equality, for example, states that two objects are to be taken as equal if they share the same properties; that is, a b P P a P b )]. Of course, first-order logic is very strong and it is possible to encode such a statement into it. For example, let app be a first-order predicate symbol of arity two that is used to stand for the application of a predicate to an individual. Semantically, app P x ) would mean P satisfies x or that the extension of the predicate P contains x P app P a app P b )] (appropriate axioms for describing app app . Higher-order logics arise from not doing this kind of encoding: instead, more immediate and natural representation of higher-order quantification are considered. Indeed naturalness of higher-order quantification is part of the reason why higher-order logics were initially considered by Frege and Russell as a foundation for mathematics.

3. JSTOR Categorical Logic And Type Theory
The following chapters cover the basic semantics of simple type theory, equational logic, and firstorder and Higher-order predicate logic.<225:CLATT>2.0.CO;2-P

4. Chad E Brown: Materials
Chad E Brown Automated Reasoning in Higherorder logic Set Comprehension and Extensionality in Church s type theory College Publications.
Published and unpublished material written by Chad E Brown
See also the web page for slides
My Book on Higher Order Automated Reasoning
Chad E Brown Automated Reasoning in Higher-Order Logic: Set Comprehension and Extensionality in Church's Type Theory College Publications. Studies in Logic: Logic and Cognitive Systems, volume 10. 2007
DeTSeT (Dependently Typed Set Theory) Related Material
One page memo describing the reason for changing the foundation of Omega to DeTSeT. ps pdf Dependently Typed Set Theory. (A SEKI Report specifying DeTSeT a ZFC-style set theory encoded in a dependent type theory with proof irrelevance) ps pdf
Scunak Related Papers
Paper presented at IJCAR 2006: Combining Type Theory and Untyped Set Theory ps pdf bib Paper presented at LFMTP Workshop 2006: Encoding Functional Relations in Scunak pdf bib Paper presented at MKM 2006: Verifying and Refuting Textbook Proofs using Scunak ps pdf bib MKM 2007 Paper with Feryal Fulya Horozal: Formal Representation of Mathematics in a Dependently Typed Set Theory Scunak Users Manual (in progress)
Automated Theorem Proving
For my thesis work, I studied set comprehension in higher-order theorem proving.

5. Type Theory And Higher Order Logic? - Object Mix
This is maybe OT but I wonder if anyone can interpret this sentence from Wikipedia s article Secondorder logic In mathematical logic,

6. Type Theory, Set Theory And Domain Theory
The logic of Nuprl is a constructive type theory, but its types include . This theory can be generalized to Intuitionistic Higherorder logic, say IHOL.
Type Theory, Set Theory and Domain Theory
The logic of Nuprl is a constructive type theory , but its types include those of partial objects, so it is also a domain theory . Its primitives are rich enough to build a substantial set theory on top of the type theory, one equivalent in expressive power to ZFC. This section first looks back to the origins of type theory and then traces its development through domain theory and to Nuprl. The first chapter of Aristotle's classical book on logic, the Organon (Logic), is Categories ; it is Aristotle's type theory. He clearly introduces the notion that some assertions make sense only when their subjects are of the right type. In translation he says, The differentia of genera which are different and not subordinate one to the other are themselves different in kind. For example, animal and knowledge: footed, winged, aquatic, two-footed, are differentia of animal, but none of these is a differentia of knowledge. In Nuprl terms, Aristotle is saying that if the underling types (genera) are different, say A is not equal to B , then no subtype of A B A is a subtype of B (subordinate genera of B ), then we can predicate

7. Higher-order Semantics And Extensionality
Peter B. andrews Resolution in type theory, Journal of Symbolic logic, vol. . Gopalan Nadathur and Dale Miller Higherorder logic programming,
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    Higher-order semantics and extensionality
    Source: J. Symbolic Logic Volume 69, Issue 4 (2004), 1027-1088.
    In this paper we re-examine the semantics of classical higher-order logic with the purpose of clarifying the role of extensionality. To reach this goal, we distinguish nine classes of higher-order models with respect to various combinations of Boolean extensionality and three forms of functional extensionality. Furthermore, we develop a methodology of abstract consistency methods (by providing the necessary model existence theorems) needed to analyze completeness of (machine-oriented) higher-order calculi with respect to these model classes. Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text Links and Identifiers Permanent link to this document:

8. OCC - Open Calculus Of Constructions
To close the gap between these two different paradigms of equational logic and Higherorder type theory we are currently investigating the open calculus of
OCC - An Open Calculus of Constructions
Mark-Oliver Stehr Keywords: Calculus of Constructions, Higher Order Logic, Typed Lambda Calculus, Equational Logic, Rewriting Logic Rewriting logic together with it's membership equational sublogic favors the use of abstract specifications. It has a flexible computation system based on conditional rewriting modulo equations, and via its initial semsntics it supports a very liberal notion of inductive definitions. Pure type systems, on the other hand, in particular the calculus of constructions, provide higher-order (dependent) types, but they are based on a fixed notion of computation, namely beta-reduction. This unsatisfying situation has been addressed by addition of inductive definitions (Ch. Pauline-Mohring 1993, Z. Luo 1994) and algebraic extensions in the style of abstract data type systems (Blanqui, Jouannaud, Okada 1999). Also, the idea of overcoming these limitations using some combination of membership equational logic with the calculus of constructions has been suggested as a long-term goal by Jouannaud 1998. To close the gap between these two different paradigms of equational logic and higher-order type theory we are currently investigating the open calculus of constructions (OCC) an equational variant of the calculus of constructions with an open computational system and a flexible universe hierarchy. Using

9. Linear Type Theories, Semantics And Action Calculi
We show that the logic and typetheory DILL arise as a Higher-order instance of our general framework. We then define the Higher-order extension of any
Linear Type Theories, Semantics and Action Calculi
Andrew Graham Barber Abstract: In this thesis, we study linear type-theories and their semantics. We present a general framework for such type-theories, and prove certain decidability properties of its equality. We also present intuitionistic linear logic and Milner's action calculi as instances of the framework, and use our results to show decidability of their respective equality judgements. Firstly, we motivate our development by giving a split-context logic and type-theory, called dual intuitionistic linear logic DILL ), which is equivalent at the level of term equality to the familiar type-theory derived from intuitionistic linear logic ( ILL ). We give a semantics for the type-theory DILL , and prove soundness and completeness for it; we can then deduce these results for the type-theory ILL by virtue of our translation. Secondly, we generalise DILL to obtain a general logic, type-theory and semantics based on an arbitrary set of operators , or general natural deduction rules. We again prove soundness and completeness results, augmented in this case by an initiality result. We introduce Milner's

10. Type Theory And Term Rewriting, Sept 1996
I shall outline the future I forsee for the type theory and term rewriting group at . Inductive types in Higherorder logic and type theory Christine
International School on Type Theory and Term Rewriting
Glasgow University Sept 12-15 1996
ULTRA group
Useful Logics, Types, Rewriting,
and Applications UKII
UK Institute of Informatics,
The Engineering and Physical Sciences Research Council (EPSRC)
In the past twenty years, a very rich body of results has emerged in type theory and term rewriting systems . This is unsurprising since these topics are at the heart of the theory and implementation of programming languages. Type theory enables increasing expressiveness in programming languages and in theorem proving, whilst retaining strong theoretical and applied properties. Understanding termination and reduction strategies in term rewriting systems provides insights into increasingly efficient implementations with safe theoretical foundations. Furthermore, the two notions of type theory and term rewriting are very general and cover various topics including termination, calculi of substitutions, subtyping, reduction, unification, interaction and process calculi, the logic of programs and theorem proving. Unfortunately, collaboration between type theory and term rewriting systems is not as strong as it should be despite their common goals and extreme importance. The

11. TPHOLs 2001 List Of Presentations
Nested General Recursion and Partiality in type theory Ana Bove and Mechanizing in Higherorder logic Proofs of Correctness and Completeness for a Set
The 14th International Conference on Theorem Proving in Higher Order Logics
3-6 September 2001, Edinburgh, Scotland Home Page Conference History Call for Papers Guide for Authors ... TPHOLs 2002
TPHOLs 2001 List of Presentations
Invited Speakers
Bart Jacobs , University of Nijmegen
JavaCard Program Verification
Steven D. Johnson , Indiana University
View from the Fringe of the Fringe
(Joint with CHARME 2001
Natarajan Shankar , SRI International
Using Decision Procedures with a Higher-Order Logic
Accepted Category A (Full Research) Papers
Computer Algebra Meets Automated Theorem Proving: Integrating Maple and PVS
Andrew Adams, Martin Dunstan, Hanne Gottliebsen, Tom Kelsey, Ursula Martin, and Sam Owre
An Irrational Construction of R from Z
Rob D. Arthan
HELM and the Semantic Math-Web
Andrea Asperti, Luca Padovani, Claudio Sacerdoti Coen, and Irene Schena
Calculational Reasoning Revisited (An Isabelle/Isar Experience)
Gertrud Bauer and Markus Wenzel
Mechanical Proofs about a Non-repudiation Protocol
Giampaolo Bella and Lawrence C. Paulson

12. Phil 514: Math Logic II
MATHEMATICAL logic II Spring 2006 Kevin C. Klement NBG), Quine’s “New Foundations” and related systems, Higherorder logic and type theory, and others,
Spring 2006
Kevin C. Klement
Course description:
Prerequisite: Phil 310 (Intermediate Logic) or equivalent, or consent of instructor
Course Handouts and other materials:
All content below is available in Adobe Acrobat (.PDF) format. In order to view them, you will need either Acrobat or Acrobat Reader installed. To download Adobe Acrobat Reader, click here

13. The Computer Journal -- Sign In Page
The underlying type theory of Coq is called the Calculus of Inductive Constructions, essentially a typed lambda calculus extended with a Higherorder logic.
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Y VES B ERTOT AND P IERRE C ASTERAN Interactive Theorem Proving and Program Development...
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14. Handbook Of Automated Reasoning - Elsevier
Contents Part V. Higherorder logic and logical frameworks. Chapter 15. Classical type theory (Peter B. andrews). 1. Introduction to type theory.
Home Site map Elsevier websites Alerts ... Handbook of Automated Reasoning Book information Product description Author information and services Ordering information Bibliographic information Conditions of sale Volume information Volume II Book-related information Submit your book proposal Other books in same subject area About Elsevier Select your view HANDBOOK OF AUTOMATED REASONING
Volume II: Handbook of Automated Reasoning, Volume II
Edited By
Alan Robinson
, 96 Highland Avenue, Greenfield, Massachusetts, USA
Andrei Voronkov , University of Manchester, Computer Science Department, Oxford Road, Manchester, M13 9LP, UK.
Part V. Higher-order logic and logical frameworks.
Chapter 15. Classical Type Theory (Peter B. Andrews).
1. Introduction to type theory.
2. Metatheoretical foundations.
3. Proof search. 4. Conclusion. Bibliography. Index. Chapter 16. Higher-Order Unification and Matching (Gilles Dowek). 1. Type Theory and Other Set Theories. 2. Simply Typed λ-calculus. 3. Undecidability. 4. Huet's Algorithm. 5. Scopes Management.

15. RAIRO - Theoretical Informatics And Applications (RAIRO: ITA)
E. Giménez, Structural recursive definitions in type theory, in Automata, B. Pientka, Termination and reduction checking for Higherorder logic programs
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16. Mechanized Reasoning Systems
NUPRL is a proof system for an intuitionistic type theory based on Martin TPS is a theorem proving system for first and Higher-order logic with both
Database of Existing Mechanized Reasoning Systems
This page represents the current state of an ongoing effort to collect information about existing automated reasoning systems. One objective is to provide concise useful information for people who have need for such a system and don't want to `roll their own'. Another objective is to provide a single place where information about existing systems can be accessed, thus providing an overview of the state of the art. This page is split into two parts: available systems and others . Entries in the available systems part are restricted to reasoning systems and tools that are implemented and available to outside users. Systems falling outside that category are listed in the others part. Click here or here for a general mechanized reasoning page. We attempt to provide for each entry: a one sentence description; a link to a brief overview; contact information; a link to a home page for the entry; and possibly other relevant information or links. In addition to individual there is a list of automated reasoning related mailing lists, and a list of related pages. Please send email to Michael Kohlhase

17. Citations Of The Clausal Theory Of Types
Chapter 3 Higherorder logic. Peter B. andrews. Classical type theory, in Alan Robinson and andrei Voronkov, Eds., Handbook of Automated Reasoning,
Some Citations of
The Clausal Theory of Types
D.A. Wolfram
Cambridge Tracts in Theoretical Computer Science
Volume 21. viii+ 124 pages. ISBN 521 39538 0.
Cambridge University Press

BiBTeX Entry
lausal heory ypes ", PUBLISHER = " Cambridge University Press", YEAR = "1993", VOLUME = "21", SERIES = " Cambridge
Chapter 2 Simply Typed lambda-Calculus
  • Manuel M. T. Chakravarty Yike Guo , Martin Köhler and Hendrik Lock.
    Goffin : Higher-order functions meet concurrent constraints, Science of Computer Programming Robert W. Hasker.
    The Replay of Program Derivations , PhD Dissertation, University of Illinois at Urbana-Champaign, 1995, 225 pp. Jan Malolepszy Malgorzata Moczurad , and Marek Zaionc
    Schwichtenberg -style lambda definability is undecidable Typed Lambda Calculus and Applications , Lecture Notes in Computer Science , (Springer, Berlin, 1997) 267-283. Malgorzata Moczurad Jerzy Tyszkiewicz and Marek Zaionc
    Statistical properties of simple types, School of Computer Science and Engineering, The University of New South Wales
  • 18. Higher-order Logic - Wikipedia, The Free Encyclopedia
    Another way in which Higherorder logic differs from first-order logic is in the constructions allowed in the underlying type theory.
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    Higher-order logic
    From Wikipedia, the free encyclopedia
    Jump to: navigation search In mathematics higher-order logic is distinguished from first-order logic in a number of ways. One of these is the type of variables appearing in quantifications ; in first-order logic, roughly speaking, it is forbidden to quantify over predicates . See second-order logic for systems in which this is permitted. Another way in which higher-order logic differs from first-order logic is in the constructions allowed in the underlying type theory . A higher-order predicate is a predicate that takes one or more other predicates as arguments. In general, a higher-order predicate of order n takes one or more ( n − 1)th-order predicates as arguments, where n higher-order functions. Higher-order logics are more expressive, but their properties, in particular with respect to model theory , make them less well-behaved for many applications. By a result of G¶del , classical higher-order logic does not admit a ( recursively axiomatized ) sound and complete proof calculus ; however, such a proof calculus does exist which is sound and complete with respect to

    19. Categorical Logic And Type Theory - Elsevier
    Dependent predicate logic, categorically. Polymorphic dependent type theory. Strong and very strong sum and equality. Full higher order dependent type
    Home Site map Elsevier websites Alerts ... Categorical Logic and Type Theory Book information Product description Author information and services Ordering information Bibliographic and ordering information Conditions of sale Book-related information Submit your book proposal Other books in same subject area About Elsevier Select your view CATEGORICAL LOGIC AND TYPE THEORY
    To order this title, and for more information, click here
    B. Jacobs
    , Computing Science Institute, University of Nijmegen, The Netherlands
    Included in series
    Studies in Logic and the Foundations of Mathematics, 141

    This book is an attempt to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category. Its intended audience consists of logicians, type theorists, category theorists and (theoretical) computer scientists.
    Preface. Contents. Preliminaries. Prospectus.

    Logic, type theory, and fibred category theory. The logic and type theory of sets. Introduction to fibred category theory. Fibrations. Some concrete examples: sets, ω-sets and PERs. Some general examples. Cloven and split fibrations. Change-of-base and composition for fibrations. Fibrations of signatures. Categories of fibrations. Fibrewise structure and fibred adjunctions. Fibred products and coproducts. Indexed categories.

    20. Church's Type Theory (Stanford Encyclopedia Of Philosophy)
    Tarski (1923) noted that in the context of Higherorder logic, one can define .. of elementary type theory is analogous to first-order logic in certain
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    Church's Type Theory
    First published Fri 25 Aug, 2006 Church's type theory is a formal logical language which includes first-order logic , but is more expressive in a practical sense. It is used, with some modifications and enhancements, in most modern applications of type theory. It is particularly well suited to the formalization of mathematics and other disciplines and to specifying and verifying hardware and software. A great wealth of technical knowledge can be expressed very naturally in it. With possible enhancements, Church's type theory constitutes an excellent formal language for representing the knowledge in automated information systems, sophisticated automated reasoning systems, systems for verifying the correctness of mathematical proofs, and certain projects involving logic and artificial intelligence . Some examples are given in Section 1.2.2 below. Type theories are also called higher-order logics, since they allow quantification not only over individual variables (as in first-order logic), but also over function, predicate, and even higher order variables. Type theories characteristically assign types to entities, distinguishing, for example, between numbers, set of numbers, functions from numbers to sets of numbers, and sets of such functions. As illustrated in Section 1.2.2 below, these distinctions allow one to discuss the conceptually rich world of sets and functions without encountering the paradoxes of naive set theory.

    21. Paperback Announcement: Categorical Logic And Type Theory
    Prospectus Introduction to fibred category theory Simple type theory Equational logic First order predicate logic Higher order predicate logic The effective
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    paperback announcement: Categorical Logic and Type Theory Bart Jacobs, Dep. Comp. Sci., Univ. Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands. Email: PS. Please pass this information on to your institute's librarian. Jacobsflyer.doc

    22. Workshop On Lambda-Calculus, Type Theory, And Natural Language, 2005
    This page describes the second workshop on Lambda Calculus, type theory ``Hyperintensional Semantics in A Higherorder logic with Definable Subtypes
    Please go to:

    23. Classical Type Theory
    58 Gerard Pierre Huet, Constrained resolution a complete method for Higherorder logic., 1972. 59 HUET G. P. 1973a, A Mechanization of type theory,

    24. HOG
    Pollard, Carl (2005) Hyperintensional semantics in a higher order logic with grammar with a type theory based on Lambek and Scott s higher order
    What is HOG?
    Key features:
    • There is a propositional logic of types , which denote sets of linguistic (phonological, syntactic, or semantic) entities. For example, the type NP denotes the syntactic category (or form class) of noun phrases.
    • HOG maintains Curry's distinction between tectogrammatical structure (abstract syntax) and phenogramatical structure (concrete syntax).
    • Abstract syntactic entities are identified with structuralist (Bloomfield-Hockett) free forms (words and phrases). For example, the NP your cat is an NP, distinct from its phonology or its semantics.
    • Concrete syntax is identified with phonology, broadly construed to include word order.
    • The modelling of Fregean senses is broadly similar to Montague's, but with intensions replaced by finer-grained hyperintensions
    • There is a (Curry-Howard) proof term calculus , whose terms denote linguistic (phonological, syntactic, or semantic) entities.
    • The term calculus is embedded in a classical higher-order logic (HOL).
    • The syntax-phonology and syntax-semantics interfaces are expressed as axiomatic theories in the HOL.

    25. The Choice Of A Foundational System
    Simple type theory (higher order or \omegaorder logic) is a direct descendant of Russell s type theory, as simplified by chwistek, ramsey-fm and
    The choice of a foundational system
    As already indicated, the intellectual trend has in the past moved away from the idea of actually using formal systems towards using them at one remove. This is responsible for the almost exclusive concentration on first order logic, despite its obvious defects. In general we should not be awed by existing foundational systems (as Kant was by Aristotelean logic) but should be prepared to invent our own new ideas. Even quite trivial syntactic changes, like making quantifiers bind more weakly than other connectives (most logic books adopt the opposite convention) can help with the usability of the system! don't have the property x x) it was simply too big to be a set. Indeed, Cantor had already distinguished between 'consistent' and 'inconsistent' multiplicities, more or less on the basis of size (whether they were equinumerous with the class of all ordinals). This is called the 'limitation of size' doctrine. Russell, on the other hand, felt that the problem was with the circular nature of the definition (considering whether a set is a member of itself), and proposed separating mathematical objects into layers, so that it only makes sense to ask whether an object at layer n is a member of an object of layer n + 1, making the Russell set meaningless on syntactic grounds. In fact the distinction between set theory and type theory is not clear-cut. Fraenkel added the Axiom of Foundation to Zermelo's set theory, and this gave rise to an intuitive picture of the set-theoretic universe as built up layer by layer in a wellfounded way. (A picture which is nowadays

    26. Edinburgh Research Archive : Item 1842/1203
    In this thesis we study them in the context of dependent type theory. them are the firstorder Nominal logic, the Higher-order logic FM-HOL, the theory
    Search DSpace Advanced Search Home Browse Communities Titles Authors By Date Sign on to: Receive email updates My DSpace authorized users Edit Profile Help About DSpace Edinburgh Research Archive ... Foundations of Computer Science PhD thesis collection Please use this identifier to cite or link to this item:
    Title: Names and Binding in Type Theory Authors: Sch¶pp, Ulrich Supervisors: Stark, Ian Issue Date: May-2006 Publisher: University of Edinburgh. College of Science and Engineering. School of Informatics. Abstract: URI: Type: Thesis or Dissertation; Doctoral; Doctor of Philosophy (PHD(R)) Appears in Collections: Foundations of Computer Science PhD thesis collection Files in This Item: File Description Size Format th.pdf Adobe PDF View/Open DSpace Software MIT and Hewlett-Packard Feedback

    27. Combining HOL With Isabelle
    Some time may be spent investigating alternative logics for formal reasoning, such as set theory. The type system of Higherorder logic catches many errors,
    Combining HOL with Isabelle
    Lawrence C. Paulson and Michael J. C. Gordon
    Computer Laboratory
    , University of Cambridge Funded by the EPSRC (1 Sep 1992 - 31 Aug 1995).
    HOL is a theorem prover for higher-order logic. Since 1984 it has been used for a series of demanding proofs, mainly for hardware verification. Isabelle is a generic theorem prover; it can handle higher-order logic, set theory, and many other logics. It is of much more recent design and is yet to be tested on really large examples. HOL and Isabelle are based on many of the same principles. However, Isabelle also provides support for unification and embedded logics, and could provide a better environment for large proofs than HOL. We propose to investigate whether this really is the case. We shall evaluate Isabelle on major hardware/system verifications. Task-specific tools are often better than generic ones. However, generic tools based on new technology may easily have the edge: automatically generated LR(1) parsers are usually better than hand-coded parsers. Researchers at Chalmers Univeristy adopted Isabelle to do proofs in Constructive Type Theory [ ] when one might have expected them to prefer Nuprl, which supports this logic specifically.

    28. Introduction To Higher-Order Categorical Logic - Cambridge
    Part II demonstrates that another formulation of Higherorder logic, (intuitionistic) type theories, is closely related to topos theory.

    29. Logic Matters: Negative Type Theory
    But while the idea of a negative type theory is a formally natural one – and Are we entitled to draw from this the conclusion that higher order or
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    Logic Matters
    Logical reflections and prejudices: enthusiasms and sceptical thoughts
    Sunday, August 05, 2007
    Negative Type Theory
    Stephen Simpson repeatedly talks about certain subsystems of second-order arithmetic as 'natural' or as 'arising naturally' ( SOAS , e.g. pp. 33, 43, etc.). In a similar context, John Burgess contrasts 'artificial examples' with 'theories that it is natural to consider' ( Fixing Frege , p. 54). But what idea of naturalness is at work here? There's more than one sort of naturalness that can be in play when we talk about mathematical theories. Which is a trite point, but which bears some discussion. In fact, there's a number of things to be said. But here's a nice case which is perhaps relatively unfamiliar but which very vividly illustrates one basic, preliminary, distinction we need to draw.
    As background, recall the structure of a simple theory of types. The entities in its universe are divided into levels. At level 0, there are individuals. At level 1, there are sets of individuals. Then at level 2, sets of level 1 sets; at level 3, sets of level 2 sets; and so on up an unending but non-cumulative hierarchy. The two key principles structuring this hierarchy are an extensionality principle for sets,and a comprehension principle to the effect that given any condition satisfied by zero or more level entities, there exists a level

    30. FLoC 2006 - IJCAR
    A stable proposal for extending TPTP3 to include Higherorder logic is presented. logic - in our case a sequent calculus for classical type theory
    IJCAR 2006 3rd International Joint Conference on Automated Reasoning
    Seattle, August 17 - 20, 2006 FLoC Home About FLoC MEETINGS CAV ICLP IJCAR LICS ... Workshops (by conf.) PROGRAM Room Assignments FLoC at a glance Social Events Invited Talks ... Workshop Proceedings FACILITIES Conference Hotel Event Space Internet Access SEATTLE Travel to/in Seattle Dining Guide Sightseeing in Seattle ORGANIZATION Steering Committee Program Committee Organizing Committee Sponsors MISCELLANEOUS Related Events Site Design OUT-OF-DATE Registration Visa Information Student Travel Support
    IJCAR on Friday, August 18th
    Chair: Mark Stickel
    Location: Metropolitan B
    Jürgen Zimmer (Universität des Saarlandes)
    Serge Autexier (DFKI)
    The MathServe Framework for Semantic Web Reasoning Services
    Predrag Janicic

    Pedro Quaresma
    (Department of Mathematics, School of Science and Technology, University of Coimbra)
    System Description: GCLCprover + GeoThms
    We present a system consisting of dynamic geometry tool and an automated theorem prover. We show how a tight integration of these modules can be achieved, and how, together with a database of geometry theorems, they provide a framework for exploring geometry problems on different levels. Joe Hendrix (University of Illinois at Urbana-Champaign)
    Jose Meseguer (University of Illinois at Urbana-Champaign)
    Hitoshi Ohsaki
    (National Institute of Advanced Industrial Science and Technology) A Sufficient Completeness Checker for Linear Order-Sorted Specifications Modulo Axioms

    31. Herman Geuvers - Research Page
    The Calculus of Constructions and Higher Order logic, in The CurryHoward isomorphism, ed. The connection between type theory and logic, notably via the
    Research of Herman Geuvers
    There is a complete list of my publications . My topics of research are
    • Type theory : type systems, typed (and untyped) lambda calculus, its connection with logic and (interactive) theorem proving. Formalizing Mathematics : interactive theorem proving, systems for representing mathematics, doing and studying (large) formalizations Computer Mathematics : integrating the mathematical activities of defining, proving and computing into a system that also supports the presentation of the mathematical content
    Some publications with an overview nature on these topics are:
    • Logics and Type systems , PhD. Thesis, University of Nijmegen, September 1993. With H. Barendregt, Proof Assistants using Dependent Type Systems chapter of the Handbook of Automated Reasoning, eds. A. Robinson and A. Voronkov, Elsevier 2001. With R.P. Nederpelt, Untyped lambda-calculus, Typed lambda-calculus, sections 32 and 33 in Logic: Mathematics, Language, Computer Science and Philosophy, Volume II, H.C.M. de Swart et al., Peter Lang, Frankfurt am Main, 1994, pp 132199. Edited together with R.P. Nederpelt and R.C. de Vrijer

    32. Summer School And Workshop On Proof Theory, Computation And Complexity
    Semantics and Cutelimination for Church s (Intuitionistic) theory of Types, with Applications to Higher-order logic Programming
    Summer School and Workshop on
    Proof Theory, Computation and Complexity
    June 23-July 4, 2003 Call for Participation
    (Dresden) and For attending courses, we ask for a fee of 100 EUR (to be paid in cash at the school). Registration is requested before May 25, 2003; please send an email to PTEvent@Janeway.Inf.TU-Dresden.DE , making sure you include a very brief bio (5-10 lines) stating your experience, interests, home page, etc. We select applicants in case of excessive demand. A limited number of grants covering all expenses is available. Applications for grants must include an estimate for travel costs and they should be sent together with the registration. We provide assistance in finding an accommodation in Dresden. Week 1, June 23-27: courses on
    • Denotational Semantics of Lambda Calculi
      Achim Jung
      (Birmingham, UK)
    • Proof Theory with Deep Inference
      Alessio Guglielmi
      (Dresden, Germany)
    • Semantics and Cut-elimination for Church's (Intuitionistic) Theory of Types, with Applications to Higher-order Logic Programming
      Jim Lipton
      (Wesleyan, USA)

    33. An Intensional Type Theory: Motivation And Cut-Elimination | Lambda The Ultimate
    An Intensional type theory Motivation and CutElimination, Paul C. Gilmore. By the theory TT is meant the higher order predicate logic with the following
    @import "misc/drupal.css"; @import "themes/chameleon/ltu/style.css";
    Lambda the Ultimate
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    An Intensional Type Theory: Motivation and Cut-Elimination
    An Intensional Type Theory: Motivation and Cut-Elimination , Paul C. Gilmore. By the theory TT is meant the higher order predicate logic with the following recursively defined types: (1) 1 is the type of individuals and [] is the type of the truth values;
    (2) [τ1, ..., τn] is the type of the predicates with arguments ofthe types τ1, ..., τn. The theory ITT described in this paper is an intensional version of TT. The types of ITT are the same as the types of TT, but the membership of the type 1 of individuals in ITT is an extension of the membership in TT. The extension consists of allowing any higher order term, in which only variables of type 1 have a free occurrence, to be a term of type 1. This feature of ITT is motivated by a nominalist interpretation of higher order predication. In ITT both well-founded and non-well-founded recursive predicates can be defined as abstraction terms from which all the properties of the predicates can be derived without the use of non-logical axioms. The elementary syntax, semantics, and proof theory for ITT are defined. A semantic consistency proof for ITT is provided and the completeness proof of Takahashi and Prawitz for a version of TT without cut is adapted for ITT; a consequence is the redundancy of cut.

    34. Publications
    Coercive subtyping in type theory. Proc. of CSL 96, the 1996 Annual Conference of conservativity of calculus of constructions over Higherorder logic.
    Shortcuts to pre-1992
    • P. Callaghan, Z. Luo, J. McKinna, R. Pollack (Eds.): Types for Proofs and Programs (Selected papers from the Annual Meeting (2000) of the TYPES Working Group .) Published by Springer-Verlag as LNCS 2277, and accessible through Springer's LINK system
    • Y. Luo and Z. Luo: Coherence and Transitivity in Coercive Subtyping . In: Proc. 8th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR'01), Havana, Cuba, R. Nieuwenhuis and A. Voronkov (eds), Springer LNAI 2250, 2001. [ ps file available.]
    • Precise but flexible programming with coercions . Submitted to Special issue of Journal of Functional Programming: Dependent type theory meets programming practice, December 2001. [ pdf file available.]
    • P. Callaghan, Z. Luo, and J. Pang. Object languages in a type-theoretic meta-framework . In: Proc. Workshop on Proof Transformations, Proof Presentations and Complexity of Proofs (PTP'01), June 2001. [ ps file available.]

    35. IngentaConnect TPS: A Theorem-Proving System For Classical Type Theory
    Keywords Higherorder logic; type theory; mating; connection; expansion proof; Document type Regular paper. Affiliations 1 Mathematics Department,
    var tcdacmd="dt";

    36. AARNEWS - June 2003
    The last three chapters of the book provide an introduction to type theory (Higherorder logic). The author shows how various mathematical concepts can be
    From the AAR President

    Announcement of the 2003 Herbrand Award

    Call for Nominations: CADE Trustees Election

    Call for Submissions
    ... Web site. The board of trustees currently includes (in alphabetical order): Franz Baader, CADE 2003 program chair, Germany
    David Basin, IJCAR 2004 program co-chair, Switzerland
    Maria Paola Bonacina, Secretary, Italy
    Gilles Dowek, elected 9/2001, France
    Ulrich Furbach, President, elected 8/1997 and 10/2000, Germany
    Harald Ganzinger, former program chair, elected 10/1999 and 10/2002, Germany
    John R. Harrison, elected 9/2001, U.S.A. Michael Kohlhase, elected 10/2000, U.S.A. David McAllester, former program chair, elected 10/2000, U.S.A. Neil V. Murray, Treasurer, U.S.A. Frank Pfenning, Vice-President, elected 10/1998 and 9/2001, U.S.A. Andrei Voronkov, former program chair, elected 10/2002, U.K. Toby Walsh, elected 10/2002, Ireland The terms of Ulrich Furbach, Michael Kohlhase and David McAllester expire after CADE 2003, because CADE Trustees are elected for three years. The term of office of Franz Baader also expires with CADE 2003, but his position is taken by David Basin as IJCAR 2004 program co-chair. Thus, there are three trustees to elect (the IJCAR 2004 program co-chairs are David Basin and Michael Rusinowitch; the CADE trustees offered them to serve on the board, and they agreed that David will). Among the outgoing trustees, Michael Kohlhase and David McAllester are eligible to be nominated for a second term, and Franz Baader is eligible to be nominated for a first term as elected trustee; Ulrich Furbach is not eligible because one cannot be elected for three consecutive terms.

    37. Publications By Z. Luo
    Weyl s predicative classical mathematics as a logicenriched type theory. adequacy conservativity of calculus of constructions over Higher-order logic.
    Type theory and logical framework
    • R. Adams and Z. Luo. Weyl's predicative classical mathematics as a logic-enriched type theory. In Types for Proofs and Programs, Proc. of Inter. Conf. of TYPES'06. LNCS 4502. 2007. [ pdf-file available.]
      Z. Luo. A type-theoretic framework for formal reasoning with different logical foundations. Proc of the 11th Annual Asian Computing Science Conference. LNCS 4435. 2007. [ pdf-file available.]
      Z. Luo. PAL+: a lambda-free logical framework. Journal of Functional Programming, 13(2), pp. 317-338. 2003. [ ps file of the final version available.]
      Z. Luo. PAL+: a lambda-free logical framework. LFM'2000 , Inter Workshop on Logical Frameworks and Meta-languages, Santa Barbara, California, 2000. [ ps file available.]
      Z. Luo. Logical truths in constructive type theory (abstract). Logic Colloquium 96. 1996. [ ps file available.]
      H. Goguen and Z. Luo. Inductive data types: well-ordering types revisited. In G. Huet and G. Plotkin (eds.), Logical Environments, Cambridge University Press, 1993. Also as ECS-LFCS-92-209, Dept of Computer Science, Edinburgh Univ. [ ps file available.]

    38. FOM: Questions On Higher-order Logic
    FOM Questions on Higherorder logic 3) Is the set of validities for 3rd-order-logic or for type theory stronger under Turing reducibility than the
    FOM: Questions on higher-order logic
    Robert M. Solovay solovay at
    Sun Sep 3 22:32:23 EDT 2000 Dear Joe, I am able to give answers to four of the five questions you raise. On Thu, 31 Aug 2000 JoeShipman at Here are some precisely posed questions that might help make the current discussion more clearly focused. In the following, "standard semantics" is assumed, and I am working in first-order ZFC, using the definition of "second-order validity in standard semantics" given in the first chapter of Manzano's book. 1) For which ordinals alpha is the the truth set for V(alpha) Turing reducible to the set of second-order validities? 2) Is the set of second-order validities reducible to the truth set of V(alpha) for any alpha? This seems unlikely at first, because GCH, a statement about arbitrarily high ranks of sets, is equivalent to the validity

    39. LICS Newsletter 14
    Proof theory of type systems, logic and type systems, typed lambda calculi as models of (higher order) computation, semantics of type systems,
    Newsletter 14, March 15, 1994

    40. Team-Parsifal:
    A proof theory for generic judgments, in ACM Trans. on Computational logic, October 2005 . The type system of a Higherorder logic programming language,
    Team Parsifal Members Overall Objectives Scientific Foundations Application Domains Software New Results Other Grants and Activities Dissemination Bibliography Inria ...
    Team: Parsifal
    Major publications by the team in recent years
    J. Despeyroux , P. Leleu
    Recursion over Objects of Functional Type , in: Special issue of MSCS on "Modalities in Type Theory" , August 2001, vol. 11, n o J. Despeyroux , F. Pfenning , C. Schürmann
    Primitive Recursion for Higher-Order Abstract Syntax , in: Theoretical Computer Science (TCS) , September 2001, vol. 266, n

    41. Powell's Books - Cambridge Tracts In Theoretical Computer Science #42: Basic Sim
    type theory is one of the most important tools in the design of higherlevel Computational Learning theory An Introduction Higher Order logic and

    42. Publications
    Implementing a Program logic of Objects in a Higherorder logic Theorem Prover . It is shown that extensional Martin-Löf type theory is a conservative

    • Proof-theoretic approach to description logic
      Presented at Symp. Logic in Computer Science (LICS 2005) .pdf
    • Static prediction of heap space usage for first-order functional programs
      Presented at Symposium on Principles of Programming Languages (POPL),
      In Proceedings of the 30th ACM SIGPLAN-SIGACT 2003, 185-197
    • The strength of non size-increasing computation .ps .pdf
      Presented at POPL'02. Here are the slides from 3 lectures at the 2002 Oberwolfach meeting on Mathematical Logic. .ps .pdf
      ACM SIGPLAN Notices 37(1), 260 - 269
    • Realizability models for BLL-like languages .ps .pdf
      with Phil Scott
      To appear in Theoretical Computer Science. Accepted for the LICS affiliated workshop Implicit Computational Complexity (ICC), Santa Barbara, 28-29 June 2000.
    • A new "feasible arithmetic" .ps .pdf with Steve Bellantoni Appeared in Journal of Symbolic Logic 67(1), 104-116, March 2002
      A classical quantified modal logic is used to define a ``feasible'' arithmetic whose provably total functions are exactly the polynomial-time computable functions. The crucial restrictions are (1) that induction is limited to modality-free formulas and (2) that an induction hypothesis may be used at most once (in the sense of linear logic). The logic is defined without any reference to bounding terms, and admits induction over formulas having arbitrarily many alternations of unbounded quantifiers.

    43. Computer Science Logic - Mathematical Logic And Formal Languages Journals, Books
    Computer Science logic Foundations of Computing. Higher-order logic, nonmonotonic reasoning, as well as logics and type systems for biology.

    44. J Roger Hindley : Research
    These two systems were invented in the 1920s by mathematicians for use in Higherorder logic, and came to be applied in programming theory from the 1970s
    Swansea University Physical Sciences Mathematics Department J Roger Hindley
    J Roger Hindley : Research
    Fields of Interest
    Mathematical logic; particularly lambda-calculus, combinatory logic and type-theories, with a current bias towards historical aspects. Lambda-calculus and combinatory logic are formal systems, to some extent rivals, used in the construction and study of programming languages which are higher-order (i.e. in which programs may change other programs). These two systems were invented in the 1920s by mathematicians for use in higher-order logic, and came to be applied in programming theory from the 1970s onward, when that theory expanded to cover higher-order computations. In a type-theory, types are labels which may be attached to certain programs to show what other programs they can change. A type-system is a particular set of rules for attaching types; the rules themselves are usually reasonably simple, but such questions as what programs are typable, what set of types a program may receive, and whether a typable computation can continue indefinitely, are not always easy to answer and have occupied many researchers.
    Main Publications
    J R Hindley, J P Seldin

    45. Type Theory
    Simply typed lambda calculus; Church s higher order logic; Isabelle; Lambda calculus with A permodel of dependent type theory (Pierre Hyvernat)
    Type Theory
    During March and April 2003 there will be three courses on type theory in the department:
    Type Theory: Impredicative Part
    A one-week postgraduate course given by Alexandre Miquel. First meeting on Monday 3 March, 13.15 - 15.00 in MD8. The content of the course will be roughly the following
  • The erasing function; second-order quantification as an intersection
  • The strong normalisation theorem
  • Normalisation of HA2 and consistency
  • System Fw and Church's theory of simple types
  • Inconsistent extensions (system U, *:*) : the failure of parametricity
    Dependent Type Theory
  • Friday 14 March, 13.15 - 15.00 in S4. Introduction (Thierry Coquand). Slides
    • Historical remarks
    • NuPurl , Coq, Alfa
    • What we can hope from type theory: applications, certificate, computer algebra, polytyping, formalised documents
    • Reading: Martin-Löf 1972. (See list below)
  • 46. Practical Foundations Of Mathematics
    In higher order logic, predicates (or, by comprehension, subsets) are first The type of propositions Even though set theory can be presented in a first
    Practical Foundations of Mathematics
    Paul Taylor
    Higher Order Logic
    Well-foundedness is a second order property because it is defined by quantifying the induction scheme over all predicates q x ]. In higher order logic, predicates (or, by comprehension, subsets) are first class citizens, allowing quantification over predicates on predicates. First order schemes Before ascending to the second order, let us note first that there is a tradition (with almost a strangle-hold over twentieth century logic [ ]) of reading any quantification over predicates or types as a scheme to be instantiated by each of the formulae which can be defined in the first order part. This has a profound qualitative effect. R EMARK 2.8.1 The completeness of first order model theory - the fact that the syntax and semantics exactly match (Remark ) - has strange corollaries for the cardinality of its models. If a theory has arbitrarily large finite models then it has an infinite one (the compactness theorem ), and in this case there are models of any infinite cardinality (the ). Second order logic has no such property.

    47. Jørgen Villadsen
    Supralogic Using Transfinite type theory with type Variables for A Paraconsistent Higher Order logic. Springer Lecture Notes in Computer Science
    Jørgen Villadsen
    Associate Professor
    Director of Studies - MSc in Computer Science and Engineering - Program Coordinator
    Student Ideas Attend the European Summer School in Logic, Language and Information in 2008!
    PhD MSc - Computer Science
    Informatics and Mathematical Modelling - Technical University of Denmark IMM DTU Mail jv(a) CV Danish Bio Contact Information Research Area: Logic for Computer Science and Artificial Intelligence Automated Reasoning Natural Language Processing IT Security Isabelle Teaching Competence: Computer Science and Engineering Logic Algorithms AI Programming Semantics
    5th International Workshop on Constraints and Language Processing CSLP 2008 Program Committee Co-Chair International Conference on Intelligent Systems and Agents ISA 2008 Program Committee Member MSc in Computer Science and Engineering CSE Master 2007-2008
    Recent Events
    CSLP@Context07: 4th International Workshop on Constraints and Language Processing CSLP 2007 Program Committee Co-Chair International Workshop on Hybrid Logic at European Summer School in Logic, Language and Information HyLo 2007 Organization Committee Member - 6-10 August 2007, Dublin, Ireland International and Interdisciplinary Conference on Modeling and Using Context CONTEXT 2007 Organization Committee Member - 20-24 August 2007, Roskilde, Denmark

    48. Course Information
    Dependent type theory II lambdaP Higher Order logic lambda-HOL (a type theory for Higher Order logic) Extensions to lambda-HOL the Lambda Cube
    Master Class 2006/2007 on Logic.
    Course Information
    Below, you find some preliminary descriptions of the courses. This information page is, as yet, still under construction.
    MODEL THEORY (Wim Veldman)
    In mathematics one often studies the class of structures satisfying a given set of formal axioms, for instance the class of groups, the class of fields, or the class of linear orders.
    In Model Theory one starts to study the rather general case that the axioms are formulated in a first-order or elementary language. This means that, when interpreting the formulas of such a language, one only quantifies over the domain of the structure, and not, for instance, over the power set of the domain.
    The pivotal notion of model theory is the notion of a formula being true in a mathematical structure. This notion has been given a formal definition by A. Tarski.
    Axiomatizing a structure is closely related to finding a method to decide which sentences are true in the structure. We shall discuss Tarski's quantifier elimination results. Given a formal theory, what can we say about the class of its countable models? We give a characterization, due to several mathematicians independently, of theories that have exactly one countable model.

    49. DBLP: Thierry Coquand
    30, Thierry Coquand Program Construction in Intuitionistic type theory (Abstract). Thomas Ehrhard An Equational Presentation of Higher Order logic.
    Thierry Coquand
    List of publications from the DBLP Bibliography Server FAQ Coauthor Index - Ask others: ACM DL Guide CiteSeer CSB ... EE Thierry Coquand, Arnaud Spiwack : Towards Constructive Homological Algebra in Type Theory. Calculemus/MKM 2007 EE Andreas Abel , Thierry Coquand, Peter Dybjer : Normalization by Evaluation for Martin-Lof Type Theory with Typed Equality Judgements. LICS 2007 EE Thierry Coquand, Arnaud Spiwack : A proof of strong normalisation using domain theory CoRR abs/0709.1401 EE Thierry Coquand: The Completeness of Typing for Context-Semantics. Fundam. Inform. 77 EE Andreas Abel Fundam. Inform. 77 Thierry Coquand, Henri Lombardi : Mathematics, Algorithms, Proofs, 9.-14. January 2005 EE Thierry Coquand, Arnaud Spiwack : A Proof of Strong Normalisation using Domain Theory. LICS 2006 EE Thierry Coquand: Alfa/Agda. The Seventeen Provers of the World 2006 EE Bernhard Banaschewski , Thierry Coquand, Giovanni Sambin : Preface. Ann. Pure Appl. Logic 137 EE Gilles Barthe , Thierry Coquand: Remarks on the equational theory of non-normalizing pure type systems. J. Funct. Program. 16

    50. DBLP: Frank Pfenning
    26 Gopalan Nadathur, Frank Pfenning The type System of a Higherorder logic Programming Language. types in logic Programming 1992 245-283
    Frank Pfenning
    List of publications from the DBLP Bibliography Server FAQ Coauthor Index - Ask others: ACM DL ACM Guide CiteSeer CSB ... Home Page Frank Pfenning: Automated Deduction - CADE-21, 21st International Conference on Automated Deduction, Bremen, Germany, July 17-20, 2007, Proceedings Springer 2007 EE Frank Pfenning: Subtyping and intersection types revisited. ICFP 2007 EE Uluc Saranli , Frank Pfenning: Using Constrained Intuitionistic Linear Logic for Hybrid Robotic Planning Problems. ICRA 2007 EE Frank Pfenning: On a Logical Foundation for Explicit Substitutions. RTA 2007 EE Frank Pfenning: On a Logical Foundation for Explicit Substitutions. TLCA 2007 Frank Pfenning: Term Rewriting and Applications, 17th International Conference, RTA 2006, Seattle, WA, USA, August 12-14, 2006, Proceedings Springer 2006 EE Deepak Garg , Frank Pfenning: Non-Interference in Constructive Authorization Logic. CSFW 2006 EE Deepak Garg Lujo Bauer ... Kevin D. Bowers , Frank Pfenning, Michael K. Reiter : A Linear Logic of Authorization and Knowledge. ESORICS 2006 EE Kaustuv Chaudhuri , Frank Pfenning, Greg Price : A Logical Characterization of Forward and Backward Chaining in the Inverse Method.

    51. New Foundations Home Page
    There is a theorem prover Watson whose higher order logic is an untyped Richard Kaye has worked on the theory KF which is a subtheory of both NF and
    New Foundations home page
    Note, added March 30, 2005 After systematic neglect for some years, I'm about to overhaul the page (done) and update the bibliography. Any comments from NFistes would be useful at this point... For new information about the mailing list, look in the Mailing List and Links to NF Fans section.
    This page is (permanently) under construction by Randall Holmes The subject of the home page which is developing here is the set theory "New Foundations", first introduced by W. V. O. Quine in 1937 . This is a refinement of Russell's theory of types based on the observation that the types in Russell's theory look the same, as far as one can apparently prove. To see Thomas Forster's master bibliography for the entire subject, as updated and HTML'ed by Paul West, click here . References in this page also refer to the master bibliography. We are very grateful to Thomas Forster for allowing us to use his bibliography. An all purpose reference for this field (best for NF) is

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