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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 71 (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
A completionbased method for mixed universal and rigid Eunification
 PROC. 12TH CONFERENCE ON AUTOMATED DEDUCTION CADE, NANCY/FRANCE, LNAI 814
, 1994
"... We present a completionbased method for handling a new version of Eunification, called “mixed” Eunification, that is a combination of the classical “universal” Eunification and “rigid” Eunification. Rigid Eunification is an important method for handling equality in Gentzentype firstorder cal ..."
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Cited by 28 (8 self)
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We present a completionbased method for handling a new version of Eunification, called “mixed” Eunification, that is a combination of the classical “universal” Eunification and “rigid” Eunification. Rigid Eunification is an important method for handling equality in Gentzentype firstorder calculi, such as freevariable semantic tableaux or matings. The performance of provers using Eunification can be increased considerably, if mixed Eunification is used instead of the purely rigid version. We state soundness and completeness results, and describe experiments with an implementation of our method.
Complete and decidable type inference for GADTs
 In Proceedings of the 14th ACM SIGPLAN international conference on Functional programming, ICFP ’09
, 2009
"... GADTs have proven to be an invaluable language extension, for ensuring data invariants and program correctness among others. Unfortunately, they pose a tough problem for type inference: we lose the principaltype property, which is necessary for modular type inference. We present a novel and simplif ..."
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Cited by 18 (4 self)
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GADTs have proven to be an invaluable language extension, for ensuring data invariants and program correctness among others. Unfortunately, they pose a tough problem for type inference: we lose the principaltype property, which is necessary for modular type inference. We present a novel and simplified type inference approach for local type assumptions from GADT pattern matches. Our approach is complete and decidable, while more liberal than previous such approaches.
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 comb ..."
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Cited by 16 (0 self)
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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...
Special Cases and Substitutes for Rigid EUnification
, 1995
"... The simultaneous rigid Eunification problem arises naturally in theorem proving with equality. This problem has recently been shown to be undecidable. This raises the question whether simultaneous rigid Eunification can usefully be applied to equality theorem proving. We give some evidence in th ..."
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Cited by 16 (0 self)
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The simultaneous rigid Eunification problem arises naturally in theorem proving with equality. This problem has recently been shown to be undecidable. This raises the question whether simultaneous rigid Eunification can usefully be applied to equality theorem proving. We give some evidence in the affirmative, by presenting a number of common special cases in which a decidable version of this problem suffices for theorem proving with equality. We also present some general decidable methods of a rigid nature that can be used for equality theorem proving and discuss their complexity. Finally, we give a new proof of undecidability of simultaneous rigid Eunification which is based on Post's Correspondence Problem, and has the interesting feature that all the positive equations used are ground equations (that is, contain no variables). Contents 1 Introduction 2 2 Paths and Spanning Sets 2 3 Critical Pairs and Rigid EUnification 4 3.1 NPCompleteness of Rigid EUnification : : :...
Proof Lengths for Equational Completion
 Information and Computation
, 1995
"... We first show that ground termrewriting systems can be completed in a polynomial number of rewriting steps, if the appropriate data structure for terms is used. We then apply this result to study the lengths of critical pair proofs in nonground systems, and obtain bounds on the lengths of critical ..."
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Cited by 15 (1 self)
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We first show that ground termrewriting systems can be completed in a polynomial number of rewriting steps, if the appropriate data structure for terms is used. We then apply this result to study the lengths of critical pair proofs in nonground systems, and obtain bounds on the lengths of critical pair proofs in the nonground case. We show how these bounds depend on the types of inference steps that are allowed in the proofs. 1 Introduction We are interested in developing theoretical techniques for evaluating the efficiency of automated inference methods. This includes bounding proof sizes, as well as bounding the size of the total search space generated. Such investigations can provide insights into the comparative strengths of various inference systems, insights that might otherwise be missed. This can also aid in the development of new methods and new inference rules, as we will show. We first consider equational deduction for systems of ground equations. We note that in general...
The Undecidability of Simultaneous Rigid EUnification
 Theoretical Computer Science
, 1995
"... Simultaneous rigid Eunification was introduced in 1987 by Gallier, Raatz and Snyder. It is used in the area of automated reasoning with equality in extension procedures, like the tableau method or the connection method. Many articles in this area assumed the existence of an algorithm for the simult ..."
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Cited by 13 (7 self)
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Simultaneous rigid Eunification was introduced in 1987 by Gallier, Raatz and Snyder. It is used in the area of automated reasoning with equality in extension procedures, like the tableau method or the connection method. Many articles in this area assumed the existence of an algorithm for the simultaneous rigid Eunification problem. There were several faulty proofs of the decidability of this problem. In this paper we prove that simultaneous rigid Eunification is undecidable. As a consequence, we obtain the undecidability of the 9 fragment of intuitionistic logic with equality. 1 Introduction Simultaneous rigid Eunification plays a crucial role in automatic proof methods for first order logic with equality based on sequent calculi, such as semantic tableaux [13], the connection method [7] (also known as the mating method [1]), model elimination [25] and a dozen other procedures. All these methods are based on the Herbrand theorem and express the idea that the proofsearch can ...
On Computational Complexity of Basic Decision Problems of Finite Tree Automata
, 1997
"... This report focuses on the following basic decision problems of finite tree automata: nonemptiness and intersection nonemptiness. There is a comprehensive proof of EXPTIMEcompleteness of the intersection nonemptiness problem, and it is shown that the nonemptiness problem is Pcomplete. A notion of ..."
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Cited by 10 (3 self)
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This report focuses on the following basic decision problems of finite tree automata: nonemptiness and intersection nonemptiness. There is a comprehensive proof of EXPTIMEcompleteness of the intersection nonemptiness problem, and it is shown that the nonemptiness problem is Pcomplete. A notion of succinctness is considered with respect to which the intersection nonemptiness problem is in fact a succinct version of the nonemptiness problem. The report includes a short survey of closely related problems which shows that there is a rule of thumb: if a decision problem for (deterministic) finite automata is complete for a certain space complexity then the same decision problem for (deterministic) finite tree automata is complete for the corresponding alternating space complexity, but alternating space is precisely deterministic time, only one exponential higher. 1 Introduction Finite tree automata [14, 51] is a natural generalization of classical finite automata to automata that acce...
A Rulebased Algorithm for Rigid Eunification
, 1994
"... We present a new rulebased method for computing complete sets of solved forms for rigid Eunifiers, improving on Gallier et al.'s method on several points: sharing of subterms is improved; substitution application and rewriting are done implicitly; the search for rigid Eunifiers is guided by the ..."
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Cited by 9 (1 self)
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We present a new rulebased method for computing complete sets of solved forms for rigid Eunifiers, improving on Gallier et al.'s method on several points: sharing of subterms is improved; substitution application and rewriting are done implicitly; the search for rigid Eunifiers is guided by the structure of the terms, and needs slightly less guessing. Our method makes extensive use of the congruence closure algorithm, and builds on it a nondeterministic procedure with six rules. We prove its soundness, its completeness  with a sharper notion of completeness than Gallier , its termination, and get as a consequence a more elementary proof of the NPcompleteness of rigid Eunification.
A practical integration of firstorder reasoning and decision procedures
 IN PROC. OF THE 14 TH INTL. CONFERENCE ON AUTOMATED DEDUCTION, VOLUME 1249 OF LNCS
, 1997
"... We present a procedure for proving the validity of firstorder formulas in the presence of decision procedures for an interpreted subset of the language. The procedure is designed to be practical: formulas can have large complex boolean structure, and include structure sharing in the form of letexp ..."
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Cited by 8 (3 self)
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We present a procedure for proving the validity of firstorder formulas in the presence of decision procedures for an interpreted subset of the language. The procedure is designed to be practical: formulas can have large complex boolean structure, and include structure sharing in the form of letexpressions. The decision procedures are only required to decide the unsatisfiability of sets of literals. However, Trefuting substitutions are used whenever they can be computed; we show how this can be done for a theory of partial orders and equality. The procedure has been implemented as part of STeP, a tool for the formal verification of reactive systems. Although the procedure is incomplete, it eliminates the need for user interaction in the proof of many verification conditions.