A. Quaife. Automated Development of Fundamental Mathematical Theories. PhD thesis, University of California at Berkeley, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....extensive experimentation with theories of number and sets specified in L , and we are eager to compare the results of our experiments with the work of others. Otter is attractive in this respect, because it has been the system underlying experiments of the kind we have in mind, as reported in [16, 17]. Moreover, the fact that Otter encompasses full first order logic paves the way to combined reasoning tactics that, e.g. perform resolution of 1l ffi P ffi 1l=1l against P= 8 Conclusions The language L may look distasteful to reading, but it ought to be clear that techniques for moving ....

A. Quaife. Automated development of fundamental mathematical theories. Kluwer Academic Publishers, 1992.

....it is understood in the sense that paramodulation permits a program to treat equality as understood . Often, the set theoretic aspects of a question present a serious obstacle with regard to efficient reasoning by a program. Eventually, a breakthrough will occur: the groundbreaking work of A. Quaife [1992] has recently been extended by J. Belinfante [1999a, 1999b] and the Mizar group (see Section 10) is also active in the study of set theory. Strikingly different from the preceding problem is that concerning the automated generation of models and counterexamples. Two programs, MACE [McCune, 2001] ....

A. Quaife. Automated development of fundamental mathematical theories. Kluwer Academic Publishers, Dordrecht, 1992.

....Finally I manipulated this representation (e.g. to give a Godel numbering system) Let us now proceed to see if this kind of activity occurs when machines prove such things as Godel I. 4 A Computerized Proof of Godel I A mechanized proof of Godel I has recently been engineered by Art Quaife [13]. This proof was was carried out by OTTER, a purely syntactic resolution based theorem prover particularly well suited to reasoning in first order extensional logic. The trick that allows OTTER to prove a deep meta mathematical theorem like Godel I is Quaife s encoding of this theorem in the ....

Quaife, A. (1992) Automated Development of Fundamental Mathematical Theories (Dordrecht, The Netherlands).

....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 method described in this dissertation. The method used by IPR is essentially the implementation of the axiom schema ....

.... McAllester s interactive Ontic prover checks proofs in set theory using a combination of fast 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 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 ....

Art Quaife. Automated Development of Fundamental Mathematical Theories, volume 2 of Automated Reasoning Series. Kluwer Academic Publishers, 1992.

....( Q 1 R ) P = P P 1 1 = P ( P 2 Q ) 1 = Q 1 2 P 1 ( P [Q ) R = QR [ P R P 1 ( R nPQ ) Q = 1l P = P 1 2 f4; g and 2 2 f ; g Fig. 1. Operators and axioms for map algebra. Whatever formalism is used to describe classes or sets, very soon one will be confronted (see, e.g. [15]) with some design choices regarding the characterization of ordered pairs and associated projections. In a theory like ZermeloSkolem Fraenkel [9] many notions of ordered pairs compete with one another. The best known of them, due to Kuratowski, 2 is: ha; bi 1 = Def f a g; f b; a g . ....

A. Quaife. Automated development of fundamental mathematical theories. Kluwer Academic Publishers, 1992.

....conjunction implicational and disjunction implicational paradoxes. The paradox free 869 Journal of Computing and Information, Vol. 2, No. 1, 1996, pp. 853 873. relevant logics Tc, Ec, and Rc are hopeful candidates for the fundamental logic underlying ATF. Indeed, we have taken NBG set theory [3, 20] as the starting point of our experiments on ATF by entailment calculus and got some success, i.e. there is no paradoxical empirical theorems in the theorems of NBG set theory found automatically by EnCal, which is an automated forward deduction system for general purpose entailment calculus, ....

A. Quaife, "Automated Development of Fundamental Mathematical Theorems," Kluwer Academic, 1992.

....found this proof in the presence of a knowledge base of over 100 sequents, each taken from earlier sections of Kelley s text. In the second and following examples, the proof itself is rather complex. Example 1 The challenge is part of the 101st labeled theorem from John Kelley s General Topology [3]. This is Theorem 19 on page 147. If a product is locally compact, then each coordinate space is locally compact. 5 This is formalized as (8X) 8A) locally compact( Y A X) oe (8a)locally compact(X a ) This is true in the following theory: 8X) 8A) 8a)continuous from to( a ; Q A X;X a ....

....and the input for these examples are available from the author. References [1] Richard L. Bishop and Samuel I. Goldberg. Tensor Analysis on Manifolds. Dover, 1980. 2] Frank M. Brown. Towards the automation of set theory and its logic. Artificial Intelligence, 10(3) 281 316, 1978. [3] John Kelley. General Topology. The University Series in Higher Mathematics. D. Van Nostrand, 1955. 4] James R. Munkres. Topology: A First Course. Prentice Hall, 1975. 5] F. Oppacher and E. Suen. HARP: A tableau based theorem prover. J. Automated Reasoning, 4:69 100, 1988. 8 Presentation of ....

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Art Quaife, Automated Development of Fundamental Mathematical Theories, Ph.D. thesis, University of California at Berkeley, Kluwer Academic Publishers, Dordrecht, the Netherlands, 1992.

.... symbol S, any dist(S) the length of the shortest path from a clause containing the symbol S to a clause in the set of support; res dist(S) the length of the shortest path, consisting of type 1 edges only, from a clause containing the symbol S to a clause in the set of support; 2 See also [18]. para dist(S) the length of the shortest path, consisting of type 2 edges only, from a clause containing the symbol S to a clause in the set of support. Finally, we can de ne a weighting function [14] in terms of these relevance measures. weight(S) w 1 any dist(S) w 2 res dist(S) w 3 ....

Quaife, A., Automated Development of Fundamental Mathematical Theories, Dordrecht: Kluwer Academic Publishers, 1992.

....the set theory has been regarded as the ultimate proving ground for automated theorem proving programs. This is also true in ATF. We take set theory as the starting point of our experiments on ATF with EnCal and are finding new and interesting theorems in NBG set theory (Boyer et al. 1986, and Quaife 1992) by EnCal. The underlying logic we adopted is Tcqe which is an extension of Tc such that it has quantifier and equality and relative axiom schemata. Using EnCal, we have found the following: 1) There are 15 1st degree theorems, 46 2nd degree theorems, and 71 3rd degree theorems which are ....

Quaife A. 1992. Automated Development of Fundamental Mathematical Theorems. Kluwer Academic.

.... A A hd hd 1 C B hd Y B tl tl 1 A C tl Z Y C: After this change, a translation is found by the algorithm, namely ( hd tl tl 1 ) hd hd 1 hd ) tl ) 1 : 2 The following three examples, which also refer to the betweenness relation in elementary geometry (cf [Qua92], pp.95 96) suggest ways in which the translation technique could be enhanced when conjugated projections hd, tl as in the preceding example are available. Example 6.3 The lower dimension axiom for plane geometry states that ( 9X;Y;Z ) X [Y jZ] Y [ZjX] Z [XjY ] To prepare for ....

A. Quaife. Automated development of fundamental mathematical theories. Kluwer Academic Publishers, 1992.

....paradoxes from the classical mathematical logic and or its various extensions is not practical. Since almost all mathematics can be formulated in the language of set theory, the set theory has been regarded as the ultimate proving ground for automated theorem proving programs (Boyer et al. 1986, Quaife, 1992). This should be also true in automated theorem finding. We take set theory as the starting point of our experiments on automated theorem finding with EnCal and are finding new and interesting theorems in NBG set theory by EnCal (Cheng, 1995) The underlying logic we adopted is Tcqe which is an ....

....reasoned out by EnCal automatically is paradoxical. We are continuing the experiment using the 3rd degree fragment of Tcqe. We are also doing a comparison of the theorems reasoned out by us using EnCal automatically and the theorems proved by Quaife using OTTER automatically or semiautomatically (Quaife, 1992). Other experiments we are doing using EnCal include automated theorem finding in Peano arithmetic and number theory, new reasoning rule generation in Lenat s AM system, and development of an autonomous information system to support theorem finding in mathematics. 5 CONCLUDING REMARKS We have ....

Quaife, A. (1992) Automated Development of Fundamental Mathematical Theorems, Kluwer Academic.

....some logic systems which are free of not only implicational paradoxes but also conjunction implicational and disjunction implicational paradoxes. The paradox free relevant logics Tc, Ec, and Rc are hopeful candidates for the fundamental logic underlying ATF. Indeed, we have taken NBG set theory [3, 20] as the starting point of our experiments on ATF by entailment calculus and got some success, i.e. there is no paradoxical empirical theorems in the theorems of NBG set theory found automatically by EnCal, which is an automated forward deduction system for general purpose entailment calculus, ....

A. Quaife, "Automated Development of Fundamental Mathematical Theorems," Kluwer Academic, 1992.

....While Nuprl uses a constructive logic based on Martin Lof s theory of types, Imps [5] and TPS [1] are based on Church s simple theory of types. In systems like Muscadet [15] and FORREnMat [9] most of the mathematical knowledge is encoded into rules. In contrast Ontic [12] as well as Quaife s work [17] are based on set theory as the basic formalism. The system with the biggest knowledge base of ver 3 M. Kerber ified mathematical knowledge may be the Mizar system [18] It is based on some set theory too. An advanced approach for integrating visual information into mechanised reasoning system ....

Art Quaife. Automated Development of Fundamental Mathematical Theories, volume 2 of Automated Reasoning Series. Kluwer Academic Publishers, 1992.

....experimentation with theories of numbers and sets specified in L Theta , and we are eager to compare the results of our experiments with the work of others. Otter is attractive in this respect, because it has been the system underlying experiments of the kind we have in mind, as reported in [19, 20]. Moreover, the fact that Otter encompasses full first order logic paves the way to combined reasoning tactics that, e.g. perform resolution of 1l ffi P ffi 1l=1l against P= a. P ffi =P right unit for ffi b. P 4 1l ) 4 1l=P double complementation law c. P 4 P= periodicity of 4 d. P ....

A. Quaife. Automated development of fundamental mathematical theories. Kluwer Academic Publishers, 1992.

....deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids. In the TPTP the Geometry domain deals mainly with plane geometry, based on Tarski s axiom system for Euclidean geometry. Indices : DDC 516; MSC 51. References : General [Tar51, Tar59] ATP [Qua92b] GRA Graph Theory. A graph consists of a finite non empty set of points together with a prescribed set of pairs of points. Indices : DDC 510.09; MSC 05CXX, 68R10. References : General [Har69, BB70] ATP . GRP Group Theory. A group is a set G and a binary operation :GxG G which is ....

....MSC 90XX. References : General , ATP [PM94, PBMON94] MSC Miscellaneous. A collection of problems which do not fit well into any other domain. NUM Number Theory. Number theory is the study of integers and their properties. Indices : DDC 512.7; MSC 11YXX. References : General [HW92] ATP [Qua92b] PLA Planning. Planning is the process of determining the sequence of actions to be performed by an agent, to reach a desired state. The initial state and the desired state are provided. Indices : DDC 006.3; MSC 68T99. References : General [AKPT91] ATP [Pla81, Pla82] PRV Program ....

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A. Quaife. Automated Development of Fundamental Mathematical Theories. Kluwer Academic Publishers, 1992.

....to the availability of complete and understandable axiomatizations of various theories in mathematics. These theories are expressed formally in mathematics texts, and can often be easily translated to the logical form required for use in ATP. A classic example is Art Quaife s translation [Qua92a, Qua92b] of parts of Mendelson s text on mathematical logic [Men87] 2.2 Problem Versions and Standard Axiomatizations. There are often many ways to formulate a problem for presentation to an ATP system. Thus, in the TPTP, there are often alternative presentations of a problem. The alternative ....

....is a set with a system of operations defined on it. The Algebra domain is one of the smallest of the TPTP s mathematical domains. The first of the two abstract problems is a well known one in ATP, which is to prove that the composition of homomorphisms is a homomorphism [BLM 86, Qua92a, Qua92b] The domain contains one standard axiomatization, based on Neumann BernaysG odel Set Theory axioms [BLM 86] see Section 3.13) The problems in this domain are all non Horn. Indices : DDC 512; MSC 06XX, 20XX. References : General [Bou89, BM65, BB70] ATP . 3.2 Analysis Analysis is ....

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A. Quaife. Automated Development of Fundamental Mathematical Theories. Kluwer Academic Publishers, 1992.

....to the outermost level. This gives greater scope to the quantifiers, and may therefore increase the arity of the Skolem functions created. Quaife Art Quaife developed an effective clausifier as part of his investigation into the use of ATP in set theory and formal mathematical theories [24, 25]. The clausifier does the following steps: ffl Remove implications and equivalences, and Move negations in. Note that these two operations are done together. ffl Mini scope ffl Skolemize ffl Distribute disjunctions ffl Convert to CNF This algorithm tries to limit the number of symbols ....

A Quaife. Automated Development of Fundamental Mathematical Theories. Kluwer Academic Publishers, 1992.

....: DDC 512.32; MSC 12XX. References : General [Ada82] ATP [Dra93] GEO Geometry. Geometry is a branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids. Indices : DDC 516; MSC 51. References : General [Tar51, Tar59] ATP [Qua92b] GRA Graph Theory. A graph consists of a finite non empty set of vertices together with a prescribed set of edges, each edge connecting a pair of vertices. Indices : DDC 510.09; MSC 05CXX, 68R10. References : General [Har69, BB70] ATP . GRP Group Theory. A group is a set G and a binary ....

....MSC 90XX. References : General , ATP [PM94, PBMON94] MSC Miscellaneous. A collection of problems which do not fit well into any other domain. NUM Number Theory. Number theory is the study of integers and their properties. Indices : DDC 512.7; MSC 11YXX. References : General [HW92] ATP [Qua92b] PLA Planning. Planning is the process of determining the sequence of actions to be performed by an agent, to reach a specified desired state from a specified initial state. Indices : DDC 006.3; MSC 68T99. References : General [AKPT91] ATP [Pla81, Pla82] PRV Program Verification. Program ....

[Article contains additional citation context not shown here]

A. Quaife. Automated Development of Fundamental Mathematical Theories. Kluwer Academic Publishers, 1992.

....that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids. In the TPTP the Geometry domain deals mainly with plane geometry, based on Tarski s axiom system for Euclidean geometry. Indices : DDC 516; MSC 51. References : General [112, 113] ATP [86]. GRA Graph Theory. A graph consists of a finite non empty set of vertices together with a prescribed set of edges, each edge connecting a pair of vertices. Indices : DDC 510.09; MSC 05CXX, 68R10. References : General [34, 10] ATP . GRP Group Theory. A group is a set G and a binary operation ....

....References : General , ATP [41] MSC Miscellaneous. THE TPTP PROBLEM LIBRARY 15 A collection of problems which do not fit well into any other domain. NUM Number Theory. Number theory is the study of integers and their properties. Indices : DDC 512.7; MSC 11YXX. References : General [35] ATP [86]. PLA Planning. Planning is the process of determining the sequence of actions to be performed by an agent, to reach a desired state. The initial state and the desired state are provided. Indices : DDC 006.3; MSC 68T99. References : General [2] ATP [79, 80] PRV Program Verification. Program ....

[Article contains additional citation context not shown here]

\Phi A. Quaife. Automated Development of Fundamental Mathematical Theories. Kluwer Academic Publishers, 1992.

....experimentation with theories of number and sets specified in L Theta , and we are eager to compare the results of our experiments with the work of others. Otter is attractive in this respect, because it has been the system underlying experiments of the kind we have in mind, as reported in [16, 17]. Moreover, the fact that Otter encompasses full first order logic paves the way to combined reasoning tactics that, e.g. perform resolution of 1l ffi P ffi 1l=1l against P= 8 Conclusions The language L Theta may look distasteful to reading, but it ought to be clear that techniques for ....

A. Quaife. Automated development of fundamental mathematical theories. Kluwer Academic Publishers, 1992.

....to be forced to do so. It is vital that automated reasoning programs be able to easily handle set theoretic statements. Robert Boyer et al. 2] 3] showed how Kurt Godel s finite axiomatization for set theory [5] can be employed to prove theorems in set theory within first order logic. Art Quaife [4] greatly simplified this formalism. The author has reproduced Quaife s elegant exposition with a view toward extending this theory. The purpose of this brief note is to present a complete list of corrections of misprints and other minor errors found in Chapter 2 and Appendix 2 of Art Quaife s ....

Art Quaife, Automated Development of Fundamental Mathematical Theories, Ph.D. thesis, University of California at Berkeley, Kluwer Academic Publishers, Dordrecht, the Netherlands, (1992).

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A. Quaife. Automated Development of Fundamental Mathematical Theories. PhD thesis, University of California at Berkeley, 1990.

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Quaife, A. (1992). Automated development of fundamental mathematical theories. Kluwer Academic Publishers.

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A. Quaife. Automated Development of Fundamental Mathematical Theories. Kluwer Academic, 1992.

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Quaife, A., Automated Development of Fundamental Mathematical Theories, Kluwer Academic Publishers, (1992).

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