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tps: A theorem proving system for classical type theory
 Journal of Automated Reasoning
, 1996
"... This is a description of TPS, a theorem proving system for classical type theory (Church’s typed λcalculus). 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 comb ..."
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Cited by 70 (6 self)
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This is a description of TPS, a theorem proving system for classical type theory (Church’s typed λcalculus). 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 which TPS can prove completely automatically are given to illustrate certain aspects of TPS’s behavior and problems of theorem proving in higherorder logic. 7
The TPS theorem proving system
 9th International Conference on Automated Deduction, Argonne, Illinois
, 1988
"... TPS is a theorem proving system for first and higherorder logic which runs in Common Lisp and can operate in automatic, semiautomatic, and interactive modes. As its logical language TPS uses the typed Acalculus [6], in which most theorems of mathematics can be expressed very directly. TPS can be ..."
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Cited by 27 (5 self)
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TPS is a theorem proving system for first and higherorder logic which runs in Common Lisp and can operate in automatic, semiautomatic, and interactive modes. As its logical language TPS uses the typed Acalculus [6], in which most theorems of mathematics can be expressed very directly. TPS can be used to search for an expansion proof [10, 11] of a theorem, which represents in a nonredtmdant way the basic combinatorial information required to construct a proof of
System Description: TPS: A Theorem Proving System for Type Theory
, 2000
"... Introduction This is a brief update on the Tps automated theorem proving system for classical type theory, which was described in [3]. Manuals and information about obtaining Tps can be found at http://gtps.math.cmu.edu/tps.html. In Section 2 we discuss some examples of theorems which Tps can now ..."
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Cited by 14 (2 self)
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Introduction This is a brief update on the Tps automated theorem proving system for classical type theory, which was described in [3]. Manuals and information about obtaining Tps can be found at http://gtps.math.cmu.edu/tps.html. In Section 2 we discuss some examples of theorems which Tps can now prove automatically, and in Section 3 we discuss an example which illustrates one of the many challenges of theorem proving in higherorder logic. We rst provide a brief summary of the key features of Tps . Tps uses Church's type theory [8] (typed calculus) as its logical language. Ws are displayed on the screen and in printed proofs in the notation of this system of symbolic logic. One can use Tps in automatic, semiautomatic, or interactive mode to construct proofs in natural deduction style, and a mixture of these modes of operation is most useful fo
Solving for Set Variables in HigherOrder Theorem Proving
 Proceedings of the 18th International Conference on Automated Deduction
, 2002
"... In higherorder logic, we must consider literals with exible (set variable) heads. Set variables may be instantiated with logical formulas of arbitrary complexity. An alternative to guessing the logical structures of instantiations for set variables is to solve for sets satisfying constraints. U ..."
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Cited by 6 (1 self)
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In higherorder logic, we must consider literals with exible (set variable) heads. Set variables may be instantiated with logical formulas of arbitrary complexity. An alternative to guessing the logical structures of instantiations for set variables is to solve for sets satisfying constraints. Using the KnasterTarski Fixed Point Theorem [ 15 ] , constraints whose solutions require recursive de nitions can be solved as xed points of monotone set functions. In this paper, we consider an approach to higherorder theorem proving which intertwines conventional theorem proving in the form of mating search with generating and solving set constraints.
TPS: An Interactive and Automatic Tool for Proving Theorems of Type Theory
 Higher Order Logic Theorem Proving and Its Applications: 6th International Workshop, HUG '93, volume 780 of Lecture Notes in Computer Science
, 1994
"... This is a demonstration of TPS, a theorem proving system for classical type theory (Church's typed lcalculus). TPS can be used 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 ..."
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Cited by 1 (1 self)
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This is a demonstration of TPS, a theorem proving system for classical type theory (Church's typed lcalculus). TPS can be used 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. CATEGORY: Demonstration 1. Introduction This presentation is a demonstration of TPS, a theorem proving system for classical type theory (Church's typed 3 lcalculus [14]) which has been under development at Carnegie Mellon University for a number of years. TPS is based on an approach to automated theorem proving called the mating method [2], which is essentially the same as the connection method developed independently by Bibel [13]. The mating method does not require reduction to clausal form. TPS handles two sorts of proofs, natural deduction proofs and expansion proofs. Natural deduction proofs are humanreadable formal proofs. An example of such a proof which was produced aut...
October 1990 A short article for the
"... tax of a higherorder logic is to introduce some kind of typing scheme. One approach types firstorder individuals with #, sets of individuals with ###, sets of pairs of individuals with ####, sets of sets of individuals with #####, etc. Such a typing scheme does not provide types for function ..."
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tax of a higherorder logic is to introduce some kind of typing scheme. One approach types firstorder individuals with #, sets of individuals with ###, sets of pairs of individuals with ####, sets of sets of individuals with #####, etc. Such a typing scheme does not provide types for function symbols. Since in some treatments of higherorder logic, functions can be represented by their graphs, i.e. certain kinds of sets of ordered pairs, this lack is not a serious restriction. Identifying functions up to their graphs does, of course, treat functions extensionally, something that might be 1 too strong in some applications. (A logic is extensional if whenever two predicates or two functions are equal on all their arguments, they themselves are equal.) A more general approach to typing is that used in the Simple Theory of Types (Church, 1940). Here again, the type # is used to denote the set of firstorder individuals, and the type o is used to denote the sort of booleans, false
A Systematic Approach to Canonicity in the Classical Sequent Calculus
"... The sequent calculus is often criticized for requiring proofs to be laden with large volumes of lowlevel syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, cutfree sequent proofs can separate closely related steps—such ..."
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The sequent calculus is often criticized for requiring proofs to be laden with large volumes of lowlevel syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, cutfree sequent proofs can separate closely related steps—such as instantiating a block of quantifiers—by irrelevant noise. Moreover, the sequential nature of sequent proofs forces proof steps that are syntactically noninterfering and permutable to nevertheless be written in some arbitrary order. The sequent calculus thus lacks a notion of canonicity: proofs that should be considered essentially the same may not have a common syntactic form. To fix this problem, many researchers revolt against the sequent calculus and replace it with proof structures that are more parallel or geometric. Proofnets, matings, and atomic flows are examples of such revolutionary formalisms. In this paper, we propose taking, instead, an evolutionary approach to recover canonicity within the sequent calculus, an approach we illustrate for classical firstorder logic. We use a multifocused sequent system as our means of abstracting away the details from classical sequent proofs. We then show that, among the focused sequent proofs, the maximally multifocused proofs, which make the foci as parallel as possible, are canonical. Moreover, such proofs are isomorphic to expansion tree proofs—a well known, simple, and parallel generalization of Herbrand disjunctions—for classical firstorder logic. We thus provide a systematic method of recovering the essence of any sequent proof without abandoning the sequent calculus. 1
PathFocused Duplication: rocedure for
"... The mating paradigm for automated theorem provers was proposed by Andrews to avoid some of the shortcomings in resolution. It facilitates automated deduction in higherorder and nonclassical logics. Moreover, there are procedures which translate back and forth between refutations by the mating meth ..."
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The mating paradigm for automated theorem provers was proposed by Andrews to avoid some of the shortcomings in resolution. It facilitates automated deduction in higherorder and nonclassical logics. Moreover, there are procedures which translate back and forth between refutations by the mating method and proofs in a natural deduction system. We describe a search procedure, called pathfocused duplication, for finding refutations by the mating method. This procedure, which is a complete strategy for the mating method, addresses two crucial issues (inadequately handled in current implementations) that arise in the search for refutations: when and how to expand the search space. It focuses on a particular path that seems to cause an impasse in the search and expands the search space relative to this path in a way that allows the search to immediately resolve the impasse. The search space grows and shrinks dynamically to respond to the requirements that have arisen or have been met in the search process, thus avoiding an explosion in the size of the search space. We have implemented a prototype of this procedure and have been able to easily solve many problems that an earlier program found difficult.
A short article for the Encyclopedia of Artificial Intelligence: Second Edition “Logic, Higherorder”
, 1991
"... While firstorder 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, ..."
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While firstorder 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 can be defined as ∀P [P (a) ≡ P (b)]. Of course, firstorder logic is very strong and it is possible to encode such a statement into it. For example, let app be a firstorder 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. In this case, the quantified expression could be rewritten as the firstorder expression ∀P [app(P, a) ≡ app(P, b)] (appropriate axioms for describing app are required). Such an encoding is often done in a multisorted logic setting, where one sort is for individuals and another sort is for predicates over individuals. Settheory is another firstorder language that encodes such higherorder concepts