W. M. Farmer. Theory interpretation in simple type theory. In J. Heering et al., editor, Higher-Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96-123. SpringerVerlag, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in EHDM, in which case it must be mapped to an equivalence relation. In PVS, mappings are provided as a syntactic component of names, and are essentially an extension of theory parameters. Equality is not treated specially, but is handled by mapping a given type to a quotient type. IMPS [FGT90,Far94] also supports theory interpretations. It is similar to EHDM in that it has a special def translation form that takes a source theory, target theory, sort association list, and constant association list, and generates a theory translation. Obligations may be generated that ensure that every axiom ....

W. M. Farmer. Theory interpretation in simple type theory. In J. Heering et al., editor, Higher-Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96--123. Springer-Verlag, 1994.

....in EHDM, in which case it must be mapped to an equivalence relation. In PVS, mappings are provided as a syntactic component of names, and are essentially an extension of theory parameters. Equality is not treated specially, but is handled by mapping a given type to a quotient type. IMPS [FGT90,Far94] also supports theory interpretations. It is similar to EHDM in that it has a special def translation form that takes a source theory, target theory, sort association list, and constant association list, and generates a theory translation. Obligations may be generated that ensure that every axiom ....

W. M. Farmer. Theory interpretation in simple type theory. In J. Heering et al., editor, Higher-Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96--123. Springer-Verlag, 1994.

.... Partial Functions The solution chosen in PVS to avoid partial functions explicitly and model them as total functions on corresponding subtypes seems reasonable in particular if we look at the effort which the introduction of partiality causes in the semantical foundation of IMPS [Far90, Far93, Far94] LSL functions are also total. 4.11 Reasoning in the Large In all three systems a union of theories is performable. In IMPS we can build theory ensembles or we can define a theory with component theories which contents are then unified. PVS offers only the second way of doing a unification but ....

W. M. Farmer. Theory Interpretation in Simple Type Theory. In J. Heering et al., editor, Higher Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96--123. Springer-Verlag, 1994.

....within the respective theory. Of great importance is then the definition of a theory s interface, determining what is visible from other theories. Relations (i.e. dependences) do not only hold between complete theories (this has been examined in literature, cf. e.g. theory interpretations [3]) but between single components of theories. There are two hierarchies to be distinguished: 4.1 Logical Dependence Logical dependences characterize, which axioms (or previously proven theorems) have been used in the proof of a theorem. This information can simply be gained simultaneously to the ....

William M. Farmer, `Theory interpretation in simple type theory', in HOA'93; An International Workshop on Higher Order Algebra, Logic and Term Rewriting. CWI, Amsterdam, the Netherlands, (September 1993).

....can be used to build up the simply typed calculus. Then, we will use sections 4 to 6 to build a logical system MV, that we claim is well suited for formalizing mathematical practice. 2 Logic Morphisms and MBase Languages We will use a variant of the the theory interpretation approach proposed in [Far93] for relativization mappings, that can be used to transport meanings and proofs between logical formalisms. The fundamental logical notions for this are logical systems and logic morphisms. For the purposes of this paper, we will call a pair S = L; C) Logical System. if L is a logical language ....

....: S Gamma S 0 be a logic morphism and A an S theorem, then F 0 (A) is an S 0 theorem. This already suggests the logical structure of a mathematical knowledge base: Orthogonal to the usual theory hierarchy (induced by theory interpretation morphisms; we will not go into in this paper, see [Far93]) there is a hierarchy of logical system induced by logic morphisms.Mathematical knowledge can be specified in any of the logical systems; it can be queried and retrieved in any logical system that is downward accessible from this one. The set of logical systems in the hierarchy is not ....

William M. Farmer. Theory interpretation in simple type theory. In HOA'93, an International Workshop on Higher-order Algebra, Logic and Term Rewriting, volume 816 of LNCS, Amsterdam, The Netherlands, 1993. Springer Verlag.

....is outlined. This translation converts types and type operators to sets and functions on sets, respectively; and axioms of the abstract theory are translated to assumptions of theorems in set theory. This provides a facility offering some of the power of both the theory interpretations of IMPS [10] and abstract theories in HOL [34, 15] It is briefly illustrated using the group theory example. The experimental system obtained by adding the set theory embodied in the type V to HOL will be called HOL ST. 2 A ZF like set theory in HOL Some of the details in this section are taken from ....

....this theory is useless because there is no way that it can be applied. One might, for example, want to show that certain operations on some particular type satisfy the axioms and hence conclude the two lemmas for those operations, but without an IMPS style theory interpretation mechanism [10] (which HOL lacks) this is impossible. However, it is intuitively clear that this theory justifies the following purely definitional theory in ST. The Theory AbsGroupST Parents ST Definitions AbsGroupAxioms AbsGroupAxioms G ffl = ffl 2 G G G) 2 G G) 2 G) Assoc G ....

W. M. Farmer. Theory interpretation in simple type theory. In J. Heering, K. Meinke, B. Moller, and T. Nipkow, editors, Higher-Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96-- 123. Springer-Verlag, September 1993. First International Workshop, HOA '93, Amsterdam, The Netherlands.

No context found.

W. M. Farmer. Theory interpretation in simple type theory. In J. Heering et al., editor, Higher-Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96-123. SpringerVerlag, 1994.

....theorems of one theory to be used in another theory. Theory interpretation can also be used to show that one theory is consistent relative to another. We used mutual interpretability to show that the WM and ST theories are equivalent. The details of theory interpretation in IMPS are discussed in [F1994] and [F2000] Details of the interpretations between the ST and WM theories are discussed in chapter 7, but for the moment, it suffices to say the procedure involves translating the language of one theory to another and then proving the translations of the axioms of one theory are theorems of the ....

Farmer, W. M., "Theory interpretation in simple type theory," In: J. Heering et al., eds., Higher-Order Algebra, Logic, and Term Rewriting, Lecture Notes in Computer Science, volume 816, pages 96-123, 1994.

....E 1 and E 2 are expressions of L of the same sort, then (E 1 E 2 ) is an expression of L of sort (called an equation) that asserts the equality of E 1 and E 2 . In a standard logic, E1 E2) means that E1 and E2 denote the same value, while in a logic that admits unde nedness like lutins [17, 18, 19], the logic of imps, E1 E2) means that either E1 and E2 denote the same value or E1 and E2 are both unde ned. 5. If E 1 ; E 2 ; E 3 are expressions of L with (E 1 ) and (E 2 ) E 3 ) then if(E 1 ; E 2 ; E 3 ) is an expression of sort (called a conditional) that denotes E 2 if ....

....For each expression E of L 2 L and position p in E, E; p) is a local context in E at p with respect to (L) Many traditional logics (including propositional logic, rst order logic, and simple type theory) can be formulated as admissible logics. Logics that admit unde nedness, such as lutins [17, 18, 19], the logic of imps, and other related logics (see [23, 24] can also be formulated as admissible logics. For examples later in the paper, let K stt = L; be an admissible logic formulation of Church s simple type theory [11] 3. Theories In this section we introduce the central notion ....

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Farmer, W. M.: 1994, `Theory Interpretation in Simple Type Theory'. In: J. H. et al. (ed.): Higher-Order Algebra, Logic, and Term Rewriting, Vol. 816 of Lecture Notes in Computer Science. pp. 96-123.

....(1) D is a set fD : 2 T g of nonempty domains such that D = ft; fg. t 6= f. 2) V is a partial function 2 such that, if E 2 E and V (E) is de ned, then V (E) 2 D (E) 1 In a standard logic, E1 E2) means that E1 and E2 denote the same value, while in a partial logic like lutins [16, 17, 18], the logic of imps, E1 E2) means that either E1 and E2 denote the same value or E1 and E2 are both unde ned. 2 The domain of de nition of a function f is the set D f of values at which f is de ned, and the domain of application of f is the set D f of values to which f may be applied. A ....

....then (E 1 ) and (E 2 ) are also of the same type. An interpretation of T 1 in T 2 is a translation from T 1 to T 2 such that, for all formulas A of L 1 , if T 1 j= A and (A) is de ned, then T 2 j= A) In other words, an interpretation is a translation that maps theorems to theorems (see [15, 18, 47]) Translations and interpretations are a powerful mechanism for connecting biform theories with similar structure. They serve as conduits for passing information (in the form of formulas) from one theory to another. Translations transport problems (i.e. conjectures) while interpretations ....

W. M. Farmer. Theory interpretation in simple type theory. In J. Heering et al., editor, Higher-Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96-123. SpringerVerlag, 1994.

....The scheme is derived from an approach to recursion developed by Y. Moschovakis [12] Moschovakis presents the approach in his paper [12] using an informal second order logic that admits undefined terms and partial functions. Our scheme is presented within a formal higher order logic called LUTINS [2, 3, 4, 8] that admits undefined terms and partial functions and that contains a definite description operator. The scheme has been implemented and tested in the IMPS Interactive Mathematical Proof System [7, 8] which has LUTINS as its logic. IMPS is equipped with an automatic mechanism for syntactically ....

W. M. Farmer. Theory interpretation in simple type theory. In J. Heering et al., editor, Higher-Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96--123. Springer-Verlag, 1994.

....of the traditional approach) from free logics 3 in which there is some mechanism for reasoning about nonexistent entities such as the present king of France. In a similar way, other formalisms can be modified to support the traditional approach. For example, the logic of imps, lutins [1, 2, 3], is a variant of simple type theory which supports the traditional approach. See [4, 5] for an example of how set theory can be modified to support the traditional approach. 4 Sorts Sorts are syntactic objects similar to types that are used to organize terms in a logic that supports the ....

W. M. Farmer. Theory interpretation in simple type theory. In J. Heering et al., editor, Higher-Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96--123. Springer-Verlag, 1994.

....the underlying logic as a parameter. Our choice is Church s simple theory of types [3] denoted in this paper by C. The underlying logics of many theorem proving systems are based on C. For example, the underlying logic of imps (and its intertheory infrastructure) is a version of C called lutins [5, 6, 8]. Unlike C, lutins admits unde ned terms, partial functions, and subtypes. By virtue of its support for partial functions and subtypes, many theory interpretations can be expressed more directly in lutins than in C [8] We will give now a brief presentation of C. The missing details can be lled ....

....imps (and its intertheory infrastructure) is a version of C called lutins [5, 6, 8] Unlike C, lutins admits unde ned terms, partial functions, and subtypes. By virtue of its support for partial functions and subtypes, many theory interpretations can be expressed more directly in lutins than in C [8]. We will give now a brief presentation of C. The missing details can be lled in by consulting Church s original paper [3] or one of the logic textbooks, such as [1] which contains a full presentation of C. We will also de ne a number of logical notions in the context of C including the notions ....

W. M. Farmer. Theory interpretation in simple type theory. In J. Heering et al., editor, Higher-Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96-123. Springer-Verlag, 1994. 16

....as a network of axiomatic theories. The mathematical information in the library would be dynamically represented, and so requested information could be generated on the y. It would include both algorithmic and axiomatic mathematics. The theories would be linked via theory interpretations [7, 23] which would serve as conduits through which information from one theory could be transported to another theory [2] This would enable the library to o er multiple views of the same mathematics. 3. Creation. An iml would facilitate the creation of mathematical ideas and objects, including ....

W. M. Farmer. Theory interpretation in simple type theory. In J. Heering et al., editor, Higher-Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96-123. Springer-Verlag, 1994.

....operator, provides 2 a sort system for classifying terms by value, and includes lambda notation with term constructors for function application and function abstraction. In short, it includes both the set theoretic machinery of nbg set theory and the function theoretic machinery of lutins [1 3], the logic of the imps Interactive Mathematical Proof System [6, 7] For the purposes of this paper, the reader does not need an intimate understanding of stmm. A language of stmm contains two kinds of expressions: terms which may be unde ned and formulas which denote true or false and are ....

W. M. Farmer. Theory interpretation in simple type theory. In J. Heering et al., editor, Higher-Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96-123. Springer-Verlag, 1994.

....of the traditional approach) from free logics 3 in which there is some mechanism for reasoning about nonexistent entities such as the present king of France. In a similar way, other formalisms can be modi ed to support the traditional approach. For example, the logic of imps, lutins [1, 2, 3], is a variant of simple type theory which supports the traditional approach. See [4, 5] for an example of how set theory can be modi ed to support the traditional approach. 4 Sorts Sorts are syntactic objects similar to types that are used to organize terms in a logic that supports the ....

W. M. Farmer. Theory interpretation in simple type theory. In J. Heering et al., editor, Higher-Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96-123. Springer-Verlag, 1994.

....[13] for references) which implies that zf is consistent iff nbg is consistent. However, nbg is finitely axiomatizable, while zf is not (see [11] or [13] for a proof) stmm admits undefined terms and has the same kind 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 ....

W. M. Farmer. Theory interpretation in simple type theory. In J. Heering et al., editor, Higher-Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96--123. Springer-Verlag, 1994.

....25 07 2000; 12:57; p.2 A Set Theory with Support for Partial Functions 3 for managing the application of functions. This machinery is e ective for reasoning about total functions, but it usually can only be used to reason about partial functions in indirect and arti cial ways. lutins 1 [6, 7, 8], a version of simple type theory with partial functions, unde ned terms, and subtypes called sorts, is exceptional in this respect. As the logic of the imps Interactive Mathematical Proof System [11, 12] it has proved to be highly e ective for specifying and reasoning about partial functions. ....

W. M. Farmer. Theory interpretation in simple type theory. In J. Heering et al., editor, Higher-Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96-123. Springer-Verlag, 1994.

....the user to add new mce types to the set of mce types supported by an MMS. It employs the notions of theory interpretation and theory instantiation. An interpretation of T in T 0 is a translation from the expressions of T to the expressions of T 0 which preserves the validity of sentences (see [6, 8, 18]) Given T T 0 , an instance of T 0 via an interpretation of T in a theory U is the result of using to instantiate T in T 0 with U (see [2, 13] The following are the basic ingredients of the method. A user de ned mce type is represented as a model conservative extension (T ; T ....

....is illustrated using many sorted rstorder logic, but it works for a variety of underlying logics, especially highly expressive logics such as simple type theory. The method is partially implemented in the imps Interactive Mathematical Proof System [9, 11] which is based on the logic lutins [7, 8, 15] a version of simple type theory with partial functions and subtypes. The bulk of the paper lays the logical foundation for the method. Section 2 presents a version of many sorted rst order logic called MS. Section 3 discusses theory extension in MS. And Section 4 and Section 5 de nes theory ....

[Article contains additional citation context not shown here]

W. M. Farmer. Theory interpretation in simple type theory. In J. Heering et al., editor, Higher-Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96-123. SpringerVerlag, 1994.

....nite description operator (so DG4 is satis ed) 3. A system for classifying terms by value (so DG5 is satis ed) 4. Support for (partial and total) functions (so DG6 is satis ed) stmm.tex; 25 07 2000; 13:08; p. 5 6 Farmer 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 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 approach would be to de ne an interpretation of a theory T 1 in a theory T 2 to be a function which maps the individual constants of T 1 to appropriate terms of T 2 and the atomic sorts of T 1 to appropriate sorts, unary predicates, and terms of T 2 . Moreover, an interpretation ....

W. M. Farmer. Theory interpretation in simple type theory. In J. Heering et al., editor, Higher-Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96-123. Springer-Verlag, 1994.

No context found.

William M. Farmer. Theory interpretation in simple type theory. In HOA'93, an International Workshop on Higher-order Algebra, Logic and Term Rewriting, volume 816 of LNCS, Amsterdam, The Netherlands, 1993. Springer Verlag.

No context found.

William M. Farmer. Theory interpretation in simple type theory. In HOA'93, an International Workshop on Higher-order Algebra, Logic and Term Rewriting, volume 816 of LNCS, Amsterdam, The Netherlands, 1993. Springer Verlag.

No context found.

W. M. Farmer. Theory interpretation in simple type theory. In J. Heering et al., editor, Higher-Order Algebra, Logic, and Term Rewriting, volume 816 of Lecture Notes in Computer Science, pages 96-123. SpringerVerlag,

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