Peter B. Andrews, Dale A. Miller, Eve Longini Cohen, Frank Pfenning, "Automating HigherOrder Logic," in Automated Theorem Proving: After 25 Years, edited by W. W. Bledsoe and D. W. Loveland, Contemporary Mathematics series, vol. 29, American Mathematical Society, 1984, 169-192.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....has an element which is minimal w.r.t. inclusion (cf. 20] p. 49) Intuitively speaking, in fact, the set formed by all infinite cs in the power set ( a ) of a has no minimal elements when a is This was one of the first major theorems whose proof was automatically found by a theorem prover, cf. [1]. This achievement originally took place in the framework of typed lambda calculus. 146 infinite, because every such c remains infinite after a single element removal. Conversely, if a belongs to some b which has no minimal elements, then the intersection of b with ( a ) has no minimal elements ....

P. B. Andrews, D. Miller, E. Longini Cohen, and F. Pfenning. Automating higher-order logic. In W. W. Bledsoe and D. W. Loveland eds., Automated theorem proving: After 25 years, 169--192. American Mathematical Society, Contemporary Mathematics vol.29, 1984.

....There are several reasons for implementing higher order logic in Isabelle. Gordon and others have used higher order logic, with great success, for hardware verification [6, 14] They have developed a theorem prover called hol, based on lcf. Another implementation of higher order logic is tps [2]. How well does Isabelle perform against specialized systems like these Zermelo Fraenkel set theory [36] is intended as a foundation of mathematics, but is inconvenient for formal proof even set theorists use intuition and diagrams. Yet set theory is the basis of the Z specification language ....

....from their higher order definitions. A standard example of higher order logic is Cantor s Theorem that every set has more subsets than elements, which can be expressed as follows: #g bool ) #f bool . #j : #.f= g j (There is no onto function from # to # bool . While tps [2] can prove Cantor s Theorem automatically, Isabelle must be guided towards the proof. The proof procedures for first order logic work directly with the natural deduction rules, as sketched in Chapter 2 of my book [29] Although none of the procedures 22 is complete or fast, they can prove many ....

Peter B. Andrews, Dale A. Miller, Eve L. Cohen, and Frank Pfenning. Automating higher-order logic. In W. W. Bledsoe and D. W. Loveland, editors, Automated Theorem Proving: After 25 Years, pages 169--192. American Mathematical Society, 1984.

....its clause form, which renders a formula unintelligible. What about non classical logics Each must be treated individually. Intuitionistic logic di#ers from classical logic by the lack of the one rule A , but it requires a completely di#erent approach. TPS proves theorems in higher order logic [Andrews et al. 1984]. It uses general matings (essentially Bibel s method) and higher order unification. Lincoln Wallen [1990] has developed matrix methods for several modal logics and intuitionistic logic through a careful analysis of their semantics. A remarkable system exploits an e#cient decision procedure for a ....

Peter B. Andrews, Dale A. Miller, Eve L. Cohen, and Frank Pfenning. Automating higher-order logic. In W. W. Bledsoe and D. W. Loveland, editors, Automated Theorem Proving: After 25 Years, pages 169--192, American Mathematical Society.

....substantially and prevents some searches from diverging. TPS uses general matings rather than resolution. The mating approach unifies subformulas against each other without reducing everything to clause form. TPS can automatically prove Cantor s Theorem: every set has more subsets than elements [2]. Unification discovers the diagonalization function. The EKL proof checker uses higher order matching of rewrite rules [20] N. G. de Bruijn s AUTOMATH project has investigated several higher order # calculi, reminiscent of Martin Lof s type theory, as languages for machine checked proof [5] ....

P. B. Andrews, D. A. Miller, E. L. Cohen, F. Pfenning, Automating higher-order logic, in: W. W. Bledsoe and D. W. Loveland, editors, Automated Theorem Proving: After 25 Years, American Mathematical Society (1984), pages 169--192.

....1) a,b = c,d a=c b=d No subgoals Given the lemma, the total time to prove this theorem is about three seconds. All Isabelle timings are on a Sun SPARCstation ELC. 5. 2 Cantor s Theorem Cantor s Theorem is one of the few major results in mathematics that can be proved automatically [1]. It is easily expressed and its proof, although deep, is short. goal ZF Rule.thy ALL f: A Pow(A) EX S: Pow(A) ALL x:A. f x=S ; ALL f:A Pow(A) EX S:Pow(A) ALL x:A. f x=S 1. ALL f:A Pow(A) EX S:Pow(A) ALL x:A. f x=S We begin by routine rule applications, using the ....

Peter B. Andrews, Dale A. Miller, Eve L. Cohen, and Frank Pfenning. Automating higher-order logic. In W. W. Bledsoe and D. W. Loveland, editors, Automated Theorem Proving: After 25 Years, pages 169--192. American Mathematical Society, 1984.

....allows more complex sentences to be more easily expressed and proved. However, when the language is extended in this way, reasoning typically becomes much more complex. Several methods have been developed for handling higher order reasoning. Some have developed provers for full higher order logic [2, 4]. Others have implemented Gdel s axioms in order to 6 keep set theory completely within first order logic [102] Others have despaired of handling the problem automatically and resorted to proof checkers [96] The method used by Frank Brown [39] turns out to be rather closely related to the ....

.... and a language that is based on the theory of English grammar [84] Quaife used the Otter program of Wos and McCune to prove many theorems of set theory based on Gdel s first order axioms [102] 14 Peter Andrews developed a completely automatic theorem prover for full higher order logic [4]. The fact that his prover, TPS, uses full higher order logic allows many complex statements about sets and functions to be stated more simply than is possible in the language of first order logic with set theory. Human like techniques. Solow [114] and Polya [98] studied general heuristics used ....

[Article contains additional citation context not shown here]

Peter B. Andrews, Dale A. Miller, Eve Longini Cohen, and Frank Pfenning. Automating higherorder logic. In W. W. Bledsoe and D. W. Loveland, editors, Automatic Theorem Proving: After 25 Years, volume 29 of Contemporary Mathematics, pages 169--192. American Mathematical Society, 1984.

....M. Schmidt Schau [19] A. Cohn [6] C. Weidenbach [22] and others. On the other hand there is an increasing interest in deduction systems for higher order logic, since many problems in mathematics are inherently higherorder. Current automated deduction systems for higher order logic like TPS [2] are rather weak on the rst order fragment, which is in part due to the fact that many of the advances of rst order deduction (like sorted calculi) have not yet been transported to higher order logic. Thus the question about the behavior of This work was supported by the Deutsche ....

Peter B. Andrews, Eve Longini-Cohen, Dale Miller, and Frank Pfenning. Automating higher order logics. Contemp. Math, 29:169-192, 1984.

....the logic embedded in it. If disambiguation between these is needed I will call the former the HOL system and the latter the HOL logic . Various other projects to automate Higher Order Logic are in progress. These include the TPS theorem prover being developed at Carnegie Mellon University [Andrews et al. and the EKL proof checker at Stanford [Ketonen Weening] The idea of using Higher Order Logic for hardware speci cation and veri cation is due to Keith Hanna of the University of Kent [Hanna Daeche] The HOL logic is a version of Church s simple theory of types [Church] modi ed by: ....

P. Andrews, D. Miller, E. Longini Cohen and F. Pfenning. Automating Higher Order Logic. Contemporary Mathematics 29, 1984.

....new rules of inference. Isabelle is an attempt to support this style. 2 The meta logic M Alonzo Church developed higher order logic, also called hol or simple type theory. It is based on the typed # calculus [16] Gordon [13] built his hol theorem prover from lcf; another theorem prover is tps [2]. Andrews [1] has written a book covering higher order logic. Here is a brief sketch of a fragment called M, which will be our meta logic. 2.1 Syntax of the meta logic The types 1 consist of basic types and function types of the form # # # . Let the Greek letters #, # , and # stand for ....

P. B. Andrews, D. A. Miller, E. L. Cohen, and F. Pfenning, Automating higherorder logic, in: W. W. Bledsoe and D. W. Loveland, editors, Automated Theorem Proving: After 25 Years (American Mathematical Society, 1984), pages 169--192.

.... between syntax and semantics for a variety of higher order deduction calculi, whereas the calculus development up to now has been guided by Andrew s Unifying Principle for Type Theory [And71] This model existence theorem has set the completeness standard for higher order calculi such as [Hue73, ALCMP84] even though it is weaker than the intuitive one given by Henkin Models. The semantical notions in Section 3 come from the attempt to achieve completeness with respect to Henkin models for higher order tableaux [Koh95, Koh98] and higher order resolution [Koh94a, Ben97, BK97b] A model existence ....

Peter B. Andrews, Eve Longini-Cohen, Dale Miller, and Frank Pfenning. Automating higher order logics. Contemp. Math, 29:169-192, 1984.

....allows more complex sentences to be more easily expressed and proved. However, when the language is extended in this way, reasoning typically becomes much more complex. Several methods have been developed for handling higher order reasoning. Some have developed provers for full higher order logic [2, 4]. Others have implemented Godel s axioms in order to keep set theory completely within first order logic [102] Others have despaired of handling the problem automatically and resorted to proof checkers [96] The method used by Frank Brown [39] turns out to be rather closely related to the method ....

.... procedures and a language that is based on the theory of English grammar [84] Quaife used the Otter program of Wos and McCune to prove many theorems of set theory based on Godel s first order axioms [102] Peter Andrews developed a completely automatic theorem prover for full higher order logic [4]. The fact that his prover, TPS, uses full higher order logic allows many complex statements about sets and functions to be stated more simply than is possible in the language of first order logic with set theory. Human like techniques. Solow [114] and Polya [98] studied general heuristics used ....

[Article contains additional citation context not shown here]

Peter B. Andrews, Dale A. Miller, Eve Longini Cohen, and Frank Pfenning. Automating higher-order logic. In W. W. Bledsoe and D. W. Loveland, editors, Automatic Theorem Proving: After 25 Years, volume 29 of Contemporary Mathematics, pages 169--192. American Mathematical Society, 1984.

....C. Walther [26] M. Schmidt Schau [22] A. G. Cohn [8] and others. On the other hand there is an increasing interest in deduction systems for higherorder logic, since many problems in mathematics are inherently higher order. Current automated deduction systems for higher order logic like TPS [4] are rather weak on the first order fragment. The question about the behavior of higher order logic under the constraints of a full order sorted type structure is a natural one to ask, in particular since calculi in this system promise the development of more powerful deduction systems for real ....

Peter B. Andrews, Eve Longini-Cohen, Dale Miller, and Frank Pfenning. Automating higher order logics. Contemp. Math, 29:169--192, 1984.

....results of running the prover on some benchmarks: 22] and [17] The prover described in [22] is limited to propositional calculus. In recent years we observe a renewed interest in proof search for intuitionistic logic, originating mostly from research in intuitionistic type theories (see [1] and [2] for early research in automating type theories) Although type theories are essentially higher order, there exist useful fragments of type theories which can be directly encoded in first order intuitionistic logic, with no additional axioms or axiom schemes required, see e.g. 20] In fragments ....

Andrews, P. B., Miller, D. A, Longini Cohen, E., Pfenning, F. Automating Higher Order Logic. In Automated Theorem Proving: After 25 Years, pages 169--192, Contemporary Mathematics Series, vol. 29, American Matehmatical Society, 1984.

....M. Schmidt Schau [19] A. Cohn [6] C. Weidenbach [22] and others. On the other hand there is an increasing interest in deduction systems for higher order logic, since many problems in mathematics are inherently higherorder. Current automated deduction systems for higher order logic like TPS [2] are rather weak on the first order fragment, which is in part due to the fact that many of the advances of first order deduction (like sorted calculi) have not yet been transported to higher order logic. Thus the question about the behavior of higher order logic under the constraints of a full ....

Peter B. Andrews, Eve Longini-Cohen, Dale Miller, and Frank Pfenning. Automating higher order logics. Contemp. Math, 29:169--192, 1984.

....logic, and type theory, we think our analyses apply more generally. Introduction Currently, theorem provers are used in the verification of both hardware and software [GM93, ORS92, BM90, HRS90, FFMH92] the formalization of informal mathematical proofs [FGT90, CH85, Pau90b] the teaching of logic[AMC84] and as tools of mathematical and metamathematical research [WWM 90, CAB 86] 1 In this paper we describe important facilities that one might want to find in a theorem prover. Our perspective comes from using proof systems for the verification of computer systems. Verification ....

Peter Andrews, Dale Miller, and Eve Longini Cohen. Automating higher order logic. In Woody Bledsoe and Donald Loveland, editors, Automated Theorem Proving: After 25 Years, volume 29 of Contemporary Mathematics Series, pages 169--192. American Mathematical Society, 1984.

....by using normalization properties of the calculus. We conclude in Section 7. 2 Related Works and Intuitions Higher order unification was pioneered by Darlington [8] and was used first to extend resolution to second order [31] and higher order logic [32] It was also used in the method of matings [2], and in higher order logic programming [27] for instance. A more complete survey can be found in [14] It consists in, given two simply typed terms t and t 0 of the same type, finding complete sets of substitutions oe such that toe = fi t 0 oe (resp. toe = fij t 0 oe) where = fi is ....

P. B. Andrews, D. Miller, E. Cohen, and F. Pfenning. Automating higher-order logic. Contemporary Mathematics, 29:169--192, 1984.

....of formulas as higher type functions is used extensively in Computer Science in formal calculi such as the Calculus of Constructions [ Coquand and Huet, 1985 ] as well as in programming systems such as Edinburgh lego, L. Paulson s isabelle [ Paulson, 1989; Paulson, 1990 ] Andrews s TPS [ Andrews et al. 1984 ] and Miller s Prolog [ Nadathur and Miller, 1988; Nadathur and Miller, 1990; Miller, 1993 ] 3.7 Truth definitions revisited It is rewarding to relate finite order logic to the issue of truth definitions mentioned in Section 2.6 above. The proof of Theorem 2.6.2 can be easily adapted to ....

....metamathematical interest, and suitable for the formalization of much of mathematics, of computer science, and of cognitive science. Moreover, these formalisms satisfy important syntactic properties 46 which permit a natural adaptation to these formalisms of methods of automated theorem proving [ Andrews et al. 1984 ] The expressive power of second order logic stems from the standard interpretation of the second order quantifiers as ranging over all relations and functions over the structure in hand. It is not surprising that this range cannot be enforced by a deductive formalism, since each such formalism, ....

[Article contains additional citation context not shown here]

P. B. Andrews, D. Miller, E. Cohen, and F. Pfenning. Automating higher-order logic. Contemporary Mathematics, 29:169--192, 1984.

....2 a,b = c,d a = c b = d No subgoals Given the lemma, the total time to prove this theorem is about three seconds. 8.2 Cantor s Theorem Cantor s Theorem is one of the few major results in mathematics that can be proved automatically. Its proof, although deep, is short. The prover TPS [2] can prove it in higher order logic, where its statement is almost trivial. Some set theory systems can prove it too [4, 10] goal ZF.thy ALL f: A Pow(A) EX S: Pow(A) ALL x:A. f x=S ; Level 0 ALL f:A Pow(A) EX S:Pow(A) ALL x:A. f x = S 1. ALL f:A Pow(A) EX S:Pow(A) ALL x:A. ....

Peter B. Andrews, Dale A. Miller, Eve L. Cohen, and Frank Pfenning. Automating higher-order logic. In W. W. Bledsoe and D. W. Loveland, editors, Automated Theorem Proving: After 25 Years, pages 169--192. American Mathematical Society, 1984.

....an element which is minimal w.r.t. inclusion (cf. 20] p. 49) Intuitively speaking, in fact, the set formed by all infinite cs in the power set ( a ) of a has no minimal elements when a is 2 This was one of the first major theorems whose proof was automatically found by a theorem prover, cf. [1]. This achievement originally took place in the framework of typed lambda calculus. infinite, because every such c remains infinite after a single element removal. Conversely, if a belongs to some b which has no minimal elements, then the intersection of b with ( a ) has no minimal elements ....

P. B. Andrews, D. Miller, E. Longini Cohen, and F. Pfenning. Automating higher-order logic. In W. W. Bledsoe and D. W. Loveland eds., Automated theorem proving: After 25 years, 169--192. American Mathematical Society, Contemporary Mathematics vol.29, 1984.

.... surveys [DJ90] Klo91] Pla93] and the existence of powerful first order deduction systems ( OS89] Sti90] LSBB92] Lus92] the inherently higher order nature of many problems whose solutions one would like to deduce automatically has sparked a growing interest in higher order deduction ([ALMP84], Gor85] Pau90] Mil91] On the other hand, the fact that human reasoning naturally assumes an intrinsically structured universe, in which one typically wants to make assertions about every object in a certain class rather than about every object in the entire domain of discourse, often aids ....

P. B. Andrews, E. Longini-Cohen, D. Miller, and F. Pfenning. Automating Higher-order Logics. Contemporary Mathematics 29, pp. 169 -- 192, 1984.

....a rapidly increasing number of applications. In spite of the undecidability of the problem [7] experience has shown that Huet s semidecision algorithm [13] is quite usable in practice. The first applications were to theorem proving in higher order logic, using resolution [12] and later matings [1]. Another was to use to encode the syntax of programming languages, and in particular the scopes of variable bindings. One can then express program transformation rules without many of the previously required complicated side conditions, and use HOU to apply them [14,22,8,4] A related area of ....

Peter B. Andrews, Dale Miller, Eve Cohen, and Frank Pfenning. Automating higherorder logic. Contemporary Mathematics, 29:169--192, August 1984.

....( Sch89] and others. Despite the existence of powerful first order deduction systems (see, e.g. OS89] Lus92] the inherently higher order nature of many problems whose solutions one would like to deduce automatically has sparked an increasing interest in higher order deduction ([ALMP84], Gor85] Pau90] Mil91] Certainly any system intended for automating real mathematics must concern itself with higher order logic, as suggested by van Dalen s observation that analysis is just another word for second order arithmetic ( van91] The behavior of sorted higher order calculi, ....

P. B. Andrews, E. Longini-Cohen, D. Miller, and F. Pfenning. Automating Higher-order Logics. Contemporary Mathematics 29, pp. 169 -- 192, 1984.

No context found.

Peter B. Andrews, Dale A. Miller, Eve Longini Cohen, Frank Pfenning, "Automating HigherOrder Logic," in Automated Theorem Proving: After 25 Years, edited by W. W. Bledsoe and D. W. Loveland, Contemporary Mathematics series, vol. 29, American Mathematical Society, 1984, 169-192.

....them to other approaches to theorem proving, so a procedure for automatically transforming acceptable matings into proofs in natural deduction style was developed [4] The ideas in [4, 5] and [32] were implemented in a theorem proving system which we now call TPS1. This was described in [37] and [6]. It automatically proved certain theorems of type theory (higher order logic) as well as first order logic, and embodied a proof procedure which was in principle complete for first order logic, though not for type theory. As a start toward extending the mating method to a complete method for ....

....(higher order logic) as well as first order logic, and embodied a proof procedure which was in principle complete for first order logic, though not for type theory. As a start toward extending the mating method to a complete method for proving theorems of higher order logic, it was shown in [6] that a sentence is a theorem of elementary type theory (the system of [1] and [2] if and only if it has a tautologous development, where a development is the analogue of a Herbrand expansion of a sentence of first order logic. Once one has found a tautologous development for a theorem, one can ....

[Article contains additional citation context not shown here]

Andrews, P. B., Miller, D. A., Cohen, E. L., and Pfenning, F.: Automating higher-order logic, in W. W. Bledsoe and D. W. Loveland (eds), Automated Theorem Proving: After 25 Years, Contemporary Mathematics Series, Vol. 29, Amer. Math. Soc., 1984, pp. 169--192.

....than in first order logic as claimed in the last point, the nature of proofs in higherorder logic is far from mysterious. For example, higherorder resolution [1] and unification [8] has been developed, and based on these principles, several theorem provers for various higher order logics (see [2] and its references) have been built and tested. The experience with such systems shows that theorem proving in such a logic is difficult. It is not clear, however, that the difficulty is inherent in the language chosen to express a theorem rather than in the theorem itself. In fact, expressing a ....

Peter B. Andrews, Dale A. Miller, Eve Longini Cohen, Frank Pfenning, "Automating Higher-Order Logic" in Automated Theorem Proving: After 25 Years, AMS Contemporary Mathematics Series 29 (1984).

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