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TPS: A TheoremProving System for Classical Type Theory
, 1996
"... . This is description of TPS, a theoremproving system for classical type theory (Church's typed #calculus). TPS has been designed to be a general research tool for manipulating wffs of first and higherorder logic, and searching for proofs of such wffs interactively or automatically, or in a ..."
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. This is description of TPS, a theoremproving system for classical type theory (Church's typed #calculus). TPS has been designed to be a general research tool for manipulating wffs of first and higherorder logic, and searching for proofs of such wffs interactively or automatically, or in a combination of these modes. An important feature of TPS is the ability to translate between expansion proofs and natural deduction proofs. Examples of theorems that TPS can prove completely automatically are given to illustrate certain aspects of TPS's behavior and problems of theorem proving in higherorder logic. AMS Subject Classification: 0304, 68T15, 03B35, 03B15, 03B10. Key words: higherorder logic, type theory, mating, connection, expansion proof, natural deduction. 1. Introduction TPS is a theoremproving system for classical type theory ## (Church's typed #calculus [20]) which has been under development at Carnegie Mellon University for a number years. This paper gives a general...
Using Reflection to Explain and Enhance Type Theory
 Proof and Computation, volume 139 of NATO Advanced Study Institute, International Summer School held in Marktoberdorf, Germany, July 20August 1, NATO Series F
, 1994
"... The five lectures at Marktoberdorf on which these notes are based were about the architecture of problem solving environments which use theorem provers. Experience with these systems over the past two decades has shown that the prover must be extensible, yet it must be kept safe. We examine a way to ..."
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The five lectures at Marktoberdorf on which these notes are based were about the architecture of problem solving environments which use theorem provers. Experience with these systems over the past two decades has shown that the prover must be extensible, yet it must be kept safe. We examine a way to safely add new decision procedures to the Nuprl prover. It relies on a reflection mechanism and is applicable to any tacticoriented prover with sufficient reflection. The lectures explain reflection in the setting of constructive type theory, the core logic of Nuprl.
Types in logic and mathematics before 1940
 Bulletin of Symbolic Logic
, 2002
"... Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, thou ..."
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Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege’s Grundgesetze der Arithmetik for which Russell applied his famous paradox 1 and this led him to introduce the first theory of types, the Ramified Type Theory (rtt). We present rtt formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from rtt leading to the simple theory of types stt. We present stt and Church’s own simply typed λcalculus (λ→C 2) and we finish by comparing rtt, stt and λ→C. §1. Introduction. Nowadays, type theory has many applications and is used in many different disciplines. Even within logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. But, before 1903 when Russell first introduced
A Metamodel for the Unified Modeling Language
 Martinus Nijhoff Publishers Natanson, Maurice 1973 Edmund Husserl: Philosopher of Infinite Tasks, Evanston: Northwestern University Press Reid, Thomas 1971 Essays on the Intellectual Powers of Man
, 2002
"... Nowadays models, rather than code, become the key artifacts of software development. Consequently, this raises the level of requirements for modeling languages on which modeling practitioners should rely in their work. A minor inconsistency of a modeling language metamodel may cause major problems i ..."
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Nowadays models, rather than code, become the key artifacts of software development. Consequently, this raises the level of requirements for modeling languages on which modeling practitioners should rely in their work. A minor inconsistency of a modeling language metamodel may cause major problems in the language applications; thus with the model driven systems development the solidness of modeling languages metamodels becomes particularly important. In its current state the UML metamodel leaves a significant area for improvement. We present an alternative metamodel that was inspired by the RMODP standard and that solves the problems of UML. RMODP was mentioned in UML specifications as a framework that has already influenced UML.
A Formal Foundation of the RMODP Conceptual Framework
, 2001
"... This paper presents an approach for formalizing the Reference Model for Open Distributed Processing (RMODP), an ISO and ITU standard for the modeling of distributed system. The goals of this formalization are to clarify the RMODP modeling framework thus making it more accessible to modelers such ..."
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This paper presents an approach for formalizing the Reference Model for Open Distributed Processing (RMODP), an ISO and ITU standard for the modeling of distributed system. The goals of this formalization are to clarify the RMODP modeling framework thus making it more accessible to modelers such as system architects, designers and implementers and to open the way for the formal verification of RMODP models (either within an ODP viewpoint or across multiple ODP viewpoints). RMODP officially declared as one of its goals to create a formal representation of Part 2: Foundations. The result of our work is a complete and truly consistent formal representation, until now nonexistent, of clauses 5, 6, 8 and 9 of part 2 of RMODP in their interrelations.
Näıve computational type theory
 Proof and SystemReliability, Proceedings of International Summer School Marktoberdorf, July 24 to August 5, 2001, volume 62 of NATO Science Series III
, 2002
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On the Logic and Learning of Language
, 2002
"... algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.1 Homomorphisms and free generators . . . . . . . . . . . . 34 3.1.2 Quotient algebras . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.3 Reducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Algebras of la ..."
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Cited by 7 (2 self)
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algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.1 Homomorphisms and free generators . . . . . . . . . . . . 34 3.1.2 Quotient algebras . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.3 Reducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Algebras of languages . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.1 The algebra of formulae . . . . . . . . . . . . . . . . . . . 38 3.2.2 Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.3 Associated algebras . . . . . . . . . . . . . . . . . . . . . . 40 3.2.4 Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.5 LindenbaumTarski quotient algebras . . . . . . . . . . . . 42 3.3 Algebras of deductive systems . . . . . . . . . . . . . . . . . . . . 44 3.3.1 Determining a class of algebras . . . . . . . . . . . . . . . 45 3.3.2 Algebra of a sequent calculus . . . . . . . . . . . . . . . . . 46 3.3.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4 Subsuming special cases: an example . . . . . . . . . . . . . . . . 49 3.4.1 The sequent system GL . . . . . . . . . . . . . . . . . . . . 49 3.4.2 The equivalent system t(GL) . . . . . . . . . . . . . . . . . 51 3.4.3 Algebraic models for GL . . . . . . . . . . . . . . . . . . . 52 3.5 Kripke semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4 Categorial type logics 61 4.1 The typed lambda calculus . . . . . . . . . . . . . . . . . . . . . . 62 4.2 Categorial grammar . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.3 Forms of Lambek's calculus . . . . . . . . . . . . . . . . . . . . . . 69 4.3.1 Classical CG revisited . . . . . . . . . . . . . . . . . . . . . 70 4.3.2 The nonassociative productfree system . . . . . . . . . . . 70 4.3.3 Addin...
Information theory, evolutionary computation, and Dembski’s “complex specified information”
, 2003
"... Intelligent design advocate William Dembski has introduced a measure of information called “complex specified information”, or CSI. He claims that CSI is a reliable marker of design by intelligent agents. He puts forth a “Law of Conservation of Information” which states that chance and natural laws ..."
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Intelligent design advocate William Dembski has introduced a measure of information called “complex specified information”, or CSI. He claims that CSI is a reliable marker of design by intelligent agents. He puts forth a “Law of Conservation of Information” which states that chance and natural laws are incapable of generating CSI. In particular, CSI cannot be generated by evolutionary computation. Dembski asserts that CSI is present in intelligent causes and in the flagellum of Escherichia coli, and concludes that neither have natural explanations. In this paper we examine Dembski’s claims, point out significant errors in his reasoning, and conclude that there is no reason to accept his assertions.
Fundamentals of Fuzzy Quantification Plausible Models, Constructive Principles, and Efficient Implementation
, 2003
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Higher Order Modal Logic
 Handbook of Modal Logic, Studies in Logic and Practical Reasoning
, 2006
"... A logic is called higher order if it allows for quantication (and possibly abstraction) over higher order objects, such as functions of individuals, relations between individuals, functions of functions, relations between functions, etc. ..."
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A logic is called higher order if it allows for quantication (and possibly abstraction) over higher order objects, such as functions of individuals, relations between individuals, functions of functions, relations between functions, etc.