W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little theories. 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....modularization of representations can be found in the area of theorem proving. Farmer and colleagues promote the use of combinations of Little Theories , representations of a specific mathematical structure in order to reason about complex Ontology Infrastructure for the Semantic Web problems [16]. They show the advantages of this modular approach in terms of reusability and reduced modelling effort. The idea of reusing and combining chunks of knowledge rather than building knowledge bases from scratch has later been adopted by the knowledge engineering community for building real world ....

W. Farmer, J. Guttman, and F. Thayer. Little theories. In D. Kapur, editor, Proceedings of the Eleventh International Conference on Automated Deduction, volume 607 of Lecture Notes in Computer Science, pages 567--581. Springer Verlag, 1992.

....40 the additive and multiplicative monoid of a given ring, you have to de ne two distinct type classes. 6.1. 2 IMPS The Interactive Mathematical Proof System [FGT95] is based on salient characteristics of the actual mathematician practice [FGF96] This system promotes the use of little theories [FGT92] in a sense very closed to the one used by mathematicians when speaking about set theory, group theory, and reminiscent of Bourbaki s work [Bou70] A structure is a tuple of terms enjoying some properties. Application of a theory to a structure is done by giving it the tuple of values ....

William M. Farmer, Joshua D. Guttman, and F. Javier Thayer. Little theories. In Deepak Kapur, editor, 11th International Conference on Automated Deduction, volume 607 of Lecture Notes in Computer Science, pages 567-581, Saratoga Springs, NY, June 1992. Springer-Verlag.

....classes 2 2 Systems The Interactive Mathematical Proof System IMPS [FGT92a, FGT93a] developed at MITRE is mainly designed to the interactive machine supported proof of mathematical reasoning. It tries to emphasize the linking of axiomatic theories as the main method of mathematical reasoning [FGT92b] The heart of IMPS is its higher order logic LUTINS, a Logic of Undefined Terms for Inference in a Natural Style. The speciality of LUTINS compared to other logics based on simple type theory, like HOL, TPS, EHDM, PVS, and Isabelle is its explicit notion of partial functions. IMPS proofs are a ....

....j 8i(i 2 x = i 2 y) For example, it could be derived add(i, add(j, s) add(j, add(i, s) from the partitioned by clause without needing further axioms about the order of elements in sets. 3 Theories 3. 1 IMPS Theories The philosophy of IMPS may be shortly characterized by Little Theories [FGT92b] Mathematical reasoning in the sense of IMPS is at least as much a matter of relating theories or transporting theorems from one theory to another as is the explicit reasoning in one large theory, like Zermelo Fraenkel set theory. Thus, a notion of theory is emphasized in which theories and ....

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W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little Theories. In D. Kapur, editor, Automated Deduction---CADE-11, volume 607 of Lecture Notes in Computer Science, pages 567--581. Springer-Verlag, 1992.

....to provide the necessary formal background to come up with an adequate translation of various languages for structured specifications into the notion of a development graph. Structured data bases for theorem provers have been used in the IMPS system. An axiomatic method called little theories [8] is used to distribute reasoning over a network of theories linked by theory interpretations. Unlike our approach this technique is basically used to transport theorems between different theories rather than to reason about the relations between different theories. The Specware system [13] ....

W. Farmer, J. Guttman, and F. Thayer. Little theories. In Proceedings 11th International Conference on Automated Deduction, CADE-11. Springer-Verlag, LNCS 607, 1992.

....1 It follows that they are not theorems in first or higher order logic and their proofs rely on the use of axioms, theorems and definitions putatively satisfiable from a knowledge base. Theorems proved in this way are only known to be true in the models of the theorems used in the proof [3, 8]. IPR can be run in an informative, interactive mode in which the user may control decisions made in the theorem proving process. However, this report is concerned with IPR s success when it is running completely automatically. The output of IPR, when successful, is an English proof at a ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little theories. In D. Kapur, editor, Proceedings of the Eleventh International Conference on Automated Deduction, volume 607 of Lecture Notes in Computer Science, pages 567--581. Springer Verlag, New York, 1992.

....by Basin and Constable [2] However the user is obliged, with each new theory declaration, to prove new metatheorems relating it to previously constructed theories. The IMPS prover also provides support for the formalization of individual little theories and interpretations between them [4]. Theory interpretation allows us to apply, for example, abstract theorems of group theory to a more concrete mathematical structure like the real numbers, after showing that multiplication over nonzero elements forms a group. In many respects, theory interpretation is more general than the kind ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little theories. In D. Kapur, editor, Proc. CADE{11, pages 567-581. Springer, Berlin, 1992.

....that they are no longer capable of proving some theorems if further (irrelevant) axioms are added. In interactive systems, structured theories are more commonly used. Isabelle [15, 16] HOL [8] and IMPS [6, 4] for instance use a layered approach to knowledge bases (cf. little theories in IMPS [5]) Theories, however, are in a sense atomic, so that for a proof one must either explicitly enter a theory and use its methods solely, or gets the sum of all theories logically underneath. In the sequel it is shown why a selective theory expansion is preferable. 2 Learning from Mathematical ....

William M. Farmer, Joshua D. Guttman, and F. Javier Thayer, `Little theories', in Automated Deduction --- CADE11; Proceedings of the 11th International Conference on Automated Deduction, ed., D. Kapur, number 607 in Lecture Notes in Artificial Intelligence, pp. 567--581, Saratoga Springs, N.Y., (June 1992). Springer.

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W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little theories. 1992.

....J. Guttman and J. Thayer. The creators of IMPS had several goals in mind for their system. Among them is the creation of a system to support rigorous mathematical development, the development of a mathematical database and a general mathematics laboratory. IMPS is based on the little theories [FGT1992] version of the axiomatic method in which mathematical knowledge is distributed over a network of theories. The benefit of this approach and how it is used in IMPS is discussed in [FGT1992] 3.1 Why use IMPS A benefit obtained by using IMPS, is its ability to catch inconsistencies or omissions ....

....of a mathematical database and a general mathematics laboratory. IMPS is based on the little theories [FGT1992] version of the axiomatic method in which mathematical knowledge is distributed over a network of theories. The benefit of this approach and how it is used in IMPS is discussed in [FGT1992]. 3.1 Why use IMPS A benefit obtained by using IMPS, is its ability to catch inconsistencies or omissions in a language definition. We came across an example of this in chapter 1 of [S2002] The author introduces rules defining strings but confuses the labels of elements with the elements ....

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Farmer, W. M., Guttman, J. D., and Thayer, F. J., "Little theories," In D. Kapur, editor, Automated Deduction - CADE-11, Lecture Notes in Computer Science, volume 607, pages 567-581. Springer-Verlag, 1992.

....while interpretations transport solutions (i.e. theorems) A translation may not preserve meaning, but an interpretation connects an abstract theory to a more concrete theory, or an equally abstract theory, in a meaning preserving way. Interpretations enable the little theories method [27], in which mathematical knowledge and reasoning is distributed across a network of theories, to be applied to biform theories. EXAMPLE 6.1. Let M be a biform theory of monoids, and let 0; and 1; be interpretations of M in T pa (see Example 3.3) that interpret the unit and binary operator of ....

Farmer, W. M., J. D. Guttman, and F. J. Thayer: 1992, `Little Theories'. In: D. Kapur (ed.): Automated Deduction|CADE-11, Vol. 607 of Lecture Notes in Computer Science. pp. 567-581.

....conjectures) while interpretations transport results (i.e. theorems) A translation may not preserve meaning, but an interpretation connects an abstract theory to a more concrete theory, or an equally abstract theory, in a meaning preserving way. Interpretations enable the little theories method [24], in which mathematical knowledge and reasoning is distributed across a network of theories, to be applied to biform theories. In the rest of this section, let be a translation from T 1 to T 2 . Proposition 5.1 (Relative Satis ability) Let be an interpretation of T 1 in T 2 . If T 2 is satis ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little theories. In D. Kapur, editor, Automated Deduction|CADE-11, volume 607 of Lecture Notes in Computer Science, pages 567-581. Springer-Verlag, 1992. 35

....definite description operator I. LUTINS is also equipped with a system of sorts for classifying terms by value which is an extension of the system of types. LUTINS closely corresponds to mathematics practice and has proven to be an effective logic for formalizing traditional mathematics (e.g. see [6]) The application of a term denoting a partial function to a term that denotes an argument outside of the domain of the partial function is undefined. For example, 2=0 and p 3 are undefined in a standard theory of real arithmetic. The application of a term denoting a partial function to an ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little theories. In D. Kapur, editor, Automated Deduction---CADE-11, volume 607 of Lecture Notes in Computer Science, pages 567--581. SpringerVerlag, 1992.

....are not included here, but are publically available along with the imps distribution. Some of the preliminary mathematical development is also omitted in this presentation. This includes the theory of metric space topology up to the Contractive Fixed Point Theorem. For a discussion of this see [5]. This paper does assume a basic knowledge of the imps system as described in the manual [7] In particular, the reader should be forwarned that terms may be unde ned and functions may be partial, that is de ned on only part of their syntactic domain. 2 Preliminaries We begin with the de nition ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little theories. In D. Kapur, editor, Automated Deduction|CADE-11, volume 607 of Lecture Notes in Computer Science, pages 567-581. Springer-Verlag, 1992.

....and of hardware dependencies. 1 Introduction imps, an Interactive Mathematical Proof System [6] aims to provide mechanical support for traditional methods and activities of mathematics, and for traditional styles of classical mathematical proof. The bulk of imps work has focused on mathematics [7, 5]. However, the same broadly understandable techniques are also valuable for formal methods. This paper will illustrate that imps provides an attractive and exible modeling framework for formal methods, and that imps provides adequate interactive theorem proving power to expose errors in speci ....

....multi threaded implementation may conform. Moreover, the example takes a form more reminiscent of Multics s style of virtual addressing [12] than of Mach s. IMPS as a Speci cation language. imps supports mathematics, and formal methods, using the little theories version of the axiomatic method [5]. The imps user develops a collection of axiomatic theories, all within a single xed logic. Theories may be related in two main ways: a theory may extend a number of other theories, and a theory interpretation may translate one theory into another. A particular case study may introduce several ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little theories. In D. Kapur, editor, Automated Deduction|CADE-11, volume 607 of Lecture Notes in Computer Science, pages 567-581. Springer-Verlag, 1992.

....concrete theories, or indeed to other equally abstract theories. As a result, the big theory is replaced with a network of theories which can include both small compact theories and large powerful theories. The little theories approach has been used in both mathematics and computer science (see [10] for references) In [10] we argue that the little theories method o ers important advantages for mechanized mathematics. Many of these advantages have been demonstrated by the imps Interactive Mathematical Proof System [9, 11] which supports the little theories method. A mechanized mathematics ....

....to other equally abstract theories. As a result, the big theory is replaced with a network of theories which can include both small compact theories and large powerful theories. The little theories approach has been used in both mathematics and computer science (see [10] for references) In [10] we argue that the little theories method o ers important advantages for mechanized mathematics. Many of these advantages have been demonstrated by the imps Interactive Mathematical Proof System [9, 11] which supports the little theories method. A mechanized mathematics system based on the little ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little theories. In D. Kapur, editor, Automated Deduction|CADE-11, volume 607 of Lecture Notes in Computer Science, pages 567-581. Springer-Verlag, 1992.

....of axiomatic theories. The theories in the network are linked together by theory interpretations which serve as conduits to pass results from one theory to another. This way of formalizing mathematics the little theories version of axiomatic method has advantages for mechanized mathematics [19]. In particular, it fosters the reuse of theories and their constituents. Section 3 discusses the little theories approach in imps. Proofs. In contrast to the formal proofs described in logic textbooks, imps proofs are a blend of computation and high level inference. Consequently, they resemble ....

.... The restriction to predicative de nitions may or may not be an advantage; from the point of view of developing classical analysis, for instance, it is certainly an impediment [44, 50, 49, 21] 3 Little Theories Approach imps supports the little theories version of the axiomatic method [19] as well as the big theory version in which all reasoning is performed within a single powerful and highly expressive axiomatic theory, such as ZermeloFraenkel set theory. In the little theories version, a number of theories are used in the course of developing a portion of mathematics. Di erent ....

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W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little theories. In D. Kapur, editor, Automated Deduction|CADE-11, volume 607 of Lecture Notes in Computer Science, pages 567-581. Springer-Verlag, 1992.

.... Theorem macetes are created for all theorems, even those that are not conditional equalities [5] A theorem may be applied as a macete using ordinary matching or using translation matching , an inter theory form of expression matching which allows a theorem to be applied outside of its home theory [3]. Atomic macetes also include simpli cation and beta reduction. Compound macetes are speci ed using an extremely simple language for determining control of the process of applying atomic macetes. This language provides a few simple constructors for sequencing and iteration of arbitrary macetes. ....

....from an axiomatic theory T into itself; for instance, the right cancellation law in groups follows from the left cancellation law, using the symmetry (interpretation) that maps the group operation to x; y : y x. The imps mechanisms supporting this form of reasoning are discussed in [4, 3]. In this section, we will instead focus on cases which do not easily t that pattern, but in which portions of a proof are symmetrical with each other. The formula shown in Figure 6 involving the oor function 3 supplies a very simple example: In proving the right hand side from the left hand ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little theories. In D. Kapur, editor, Automated Deduction|CADE-11, volume 607 of Lecture Notes in Computer Science, pages 567-581. Springer-Verlag, 1992.

...., Lecture Notes in Computer Science, vol. 607, SpringerVerlag, 1992, pp. 701 705. 1 other equally abstract theories. Theory interpretations provide the mechanism for transporting theorems. The little theories style of the axiomatic method is employed extensively in mathematical practice; in [4], we discuss its bene ts for mechanical theorem provers, and how the approach is used in imps. Logic. Standard mathematical reasoning in many areas focuses on functions and their properties, together with operations on functions. For this reason, imps is based on a version of simple type theory. ....

....of an expression gives some immediate information about the expression s value, if it is de ned at all. Moreover, many theorems have restrictions that can be stated in terms of the subtype of a value, and the theorem prover can be programmed to handle these assertions using special algorithms [4]. This subtyping mechanism interacts well with the type theory only because functions may be partial. If 0 is a subtype of 0 , while 1 is a subtype of 1 , then 0 1 is a subtype of 0 1 . In particular, it contains just those partial functions that are never de ned for arguments ....

[Article contains additional citation context not shown here]

W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little theories. In D. Kapur, editor, Automated Deduction|CADE-11, volume 607 of Lecture Notes in Computer Science, pages 567-581. Springer-Verlag, 1992.

....that in Simple Pre Scheme programs the primitive operator always represents subtraction, and not negation. 3 Several aspects of imps makes it especially well suited as an environment in which to analyze numerical programs: 1) imps supports the little theories version of the axiomatic method [1]. Hence numerical objects can be formalized as members of abstract numerical datatypes, and one formalization can be related to an alternative formalization by means of theory interpretation. 2) imps contains a well developed theory of the real numbers called H OReal Arithmetic (which is ....

.... and p implies (p implies p) backchain with(p:prop,forall(k:zz,p and p implies (p implies p) simplify direct inference (block (script comment direct inference at (0) instantiate universal antecedent with(p:prop,forall(k:zz,p and p implies (p implies p) [ 1] m ) simplify antecedent with(r:rr,not(0 =r) simplify antecedent with(m:zz,r:rr,not(r m) simplify antecedent with(n:zz,r:rr,not(r =n) block (script comment direct inference at (1) instantiate universal antecedent with(p:prop,forall(k:zz,p and p implies (p implies p) ....

[Article contains additional citation context not shown here]

W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little theories. In D. Kapur, editor, Automated Deduction|CADE-11, volume 607 of Lecture Notes in Computer Science, pages 567-581. Springer-Verlag, 1992.

.... of machinery for reasoning with functions and for classifying terms as lutins [4, 5, 6] the logic of the imps Interactive Mathematical Proof System [10] lutins closely corresponds to mathematics practice and has proven to be an effective logic for formalizing traditional mathematics (e.g. see [9]) In particular, stmm is equipped with operators for forming definite descriptions, function applications, and function abstractions and a sort system for classifying terms by value. stmm is essentially the same as a logic called nbg described in [7] and defined in [8] Defined in stages, nbg ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little theories. In D. Kapur, editor, Automated Deduction---CADE-11, volume 607 of Lecture Notes in Computer Science, pages 567--581. Springer-Verlag, 1992.

....system or body of knowledge is described by a network of theories linked by interpretations. The little theories method is an old idea that has been used in mathematics since the late 1800s and has been advocated in recent years by a number of computer scientists for various purposes (e.g. see [1, 2, 5, 6, 7, 8]) The imps theorem proving system [3] supports the little theories method and contains a large database of traditional mathematics organized as a network of theories. Suppose that an intelligent agent is trying to solve a goal G in a some theory T within a theory network. How can an intelligent ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little theories. In D. Kapur, editor, Automated Deduction|CADE-11, volume 607 of Lecture Notes in Computer Science, pages 567-581. Springer-Verlag, 1992.

....within a single powerful and highly expressive axiomatic theory, such as Zermelo Fraenkel set theory; in the little theories version, reasoning is distributed across a network of theories linked by interpretations which serve as conduits to pass results from one theory to another. We argue in [10] that this way of organizing mathematics is very advantageous for managing complex mathematics within mechanized mathematics systems. 19 ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little theories. In D. Kapur, editor, Automated Deduction|CADE-11, volume 607 of Lecture Notes in Computer Science, pages 567-581. Springer-Verlag, 1992. 20

....are useful for organizing and supporting mathematical reasoning in automated reasoning systems such as mechanical theorem provers and computer system speci cation and veri cation environments. Interpretations are used extensively with success in the imps Interactive Mathematical Proof System [10, 11, 12]. They are also a fundamental component in the following programming and veri cation environments: ehdm [27] m eves [5] and eves [6] iota [24] and obj3 [14] Theory interpretation has primarily been studied and applied in the context of rst order predicate logic. Logic textbooks like Enderton ....

....Moreover, interpretations of this sort are fundamental to the little theories version of the axiom method in which mathematical reasoning is performed over a network of theories linked by interpretations instead of entirely within one single big theory such as Zermelo Fraenkel set theory. In [11], we describe the little theories approach and argue in favor of its implementation in mechanical theorem provers. 1 also establishes that TO is a re nement of PO. Actually, TO re nes PO in two ways: 1) the models of TO are a special kind of partial order, namely a total order, and (2) the ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little theories. In D. Kapur, editor, Automated Deduction|CADE-11, volume 607 of Lecture Notes in Computer Science, pages 567-581. Springer-Verlag, 1992.

.... This machinery has the the same avor as the special machinery of lutins [11, 12, 13] the logic of the imps Interactive Mathematical Proof System [16, 18] lutins closely corresponds to mathematics practice and has proven to be an e ective logic for formalizing traditional mathematics (e.g. see [17]) The following are the major ingredients of stmm. In contrast to zf and nbg, the underlying logic of stmm is Partial First Order Logic (pfol) a version of rst order logic that admits partial functions and unde ned terms (see [14, 15] stmm contains the usual vocabulary and axioms of nbg. It ....

....in imps. This machinery could also be used in an implementation of stmm. 6.3. Little Theories The little theories method is a version of the axiomatic method in which a complex system or body of knowledge is described as a network of axiomatic theories linked by theory interpretations. The paper [17] argues that the little theories approach is highly desirable for mechanized mathematics and describes how imps supports it. stmm could be implemented with little theories much like imps. One would have to de ne the notion of an interpretation of one stmm theory in another (see [13] A simple ....

W. M. Farmer, J. D. Guttman, and F. J. Thayer. Little theories. In D. Kapur, editor, Automated Deduction|CADE-11, volume 607 of Lecture Notes in Computer Science, pages 567-581. Springer-Verlag, 1992.

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W. Farmer, J. Guttman, and F. Thayer. Little theories. In D. Kapur, editor, Proc. of the 11th CADE, pages 567-581, Saratoga Springs, New York, USA, June 1992. Springer Verlag, Berlin, Germany, LNAI 607.

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