J. Goguen and J. Meseguer. Completeness of many-sorted equational logic. Houston Journal of Mathematics, 11(3):307--334, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....FINITE PRODUCT CATEGORIES AND EQUATIONAL LOGIC ATISH BAGCHI AND CHARLES WELLS Abstract. It is shown that the proof theory for sketches and forms provided in [Bagchi and Wells, 1997] is strong enough to produce all the theorems of the entailment system for multisorted equational logic provided in [Goguen and Meseguer, 1982]. 1. Introduction In [Wells, 1990] the second author introduced the notion of form, a graphbased method of specification of mathematical structures that generalizes Ehresmann s sketches. In [Bagchi and Wells, 1997] the authors produced a structure for forms which provides a uniform proof theory ....

....DMS 9022140. and the theorem that the given assertion is true (or deductible or constructible) as an actual factorization # # # # # # # # # # of the diagram (1) in which the arrow verif is constructible in SynCat[FinProd, F ] using the rules of Appendix B of [Bagchi and Wells, 1997] In [Goguen and Meseguer, 1982], Goguen and Meseguer produced a sound and complete entailment system for multisorted equational logic. In this paper, we verify that the theorems of that logic for a particular signature and equations all occur as actual factorizations in SynCat[FinProd, F ] where F is a FinProd form induced ....

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Joseph A. Goguen and Jose Meseguer. Completeness of many-sorted equational logic. Technical Report CSL-135, SRI International Computer Science Laboratory, 333 Ravenswood Ave., Menlo Park, CA 94025, USA, 1982.

....operation, and syntactic identity of terms We do not address the problem of empty sorts here and will present calculus which works under the assumption that sorts are not empty. We will usually give signatures with at least one constant for every sort but other ways of restricting the signatures [Goguen 1981, Huet 1980] or ensuring nonemptiness [Goguen 1981, Goguen 1987] can be used instead. It seems also that the most flexible approach which generalizes calculus by introducing explicit variables [Goguen 1981, Goguen 1982, Ehrig 1985] can be adapted to our framework. An equivalent (and the ....

....not address the problem of empty sorts here and will present calculus which works under the assumption that sorts are not empty. We will usually give signatures with at least one constant for every sort but other ways of restricting the signatures [Goguen 1981, Huet 1980] or ensuring nonemptiness [Goguen 1981, Goguen 1987] can be used instead. It seems also that the most flexible approach which generalizes calculus by introducing explicit variables [Goguen 1981, Goguen 1982, Ehrig 1985] can be adapted to our framework. An equivalent (and the original) formulation of the syntax and the following ....

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Goguen, J.A., Meseguer, J., "Completeness of Many-Sorted Equational Logic", SIGPLAN Notices , vol. 16, no. 7, 1981.

....has the form (X, t = t A A t k = tk t o to) with (X, t t) a Sig equation. The equational language LEq( ig) is the set of all conditional equa tions over ig. Declarations are needed to be able to define inference rules that are sound and complete even if empty sorts are allowed [18, 9]. For brevity, we gather the declarations together in one place in the examples. There is a relation Eq that defines equational deduction and a relation Eq that defines truth of a conditional equation in a structure. We don t go in details about these relations here, definitions can be found ....

J.A. Goguen and J. Meseguer. Completeness of many-sorted equational logic. SIGPLANNotices, 17(1):9-17, 1982.

....types can be automatically calculated at the level of that speci cation. Section nally discusses how abstract interpretations can be veri ed in the framework proposed in this paper. 2 Membership Equational Logic and Maude Membership equational logic [11,1] extends many sorted equational logic [8] with membership assertions t : s stating that a term t belongs to a sort s. It subsumes a wide variety of speci cation formalisms, including order sorted [7,9] and partial equational logics. Despite its generality, it still enjoys the good properties of equational logics: it is simple, eciently ....

Joseph Goguen and Jose Meseguer. Completeness of many-sorted equational logic. Houston Journal of Mathematics, 11(3):307-334, 1985.

....Technology Membership equational logic and the language Maude are introduced in this section, providing the technical background for the formalization in the rest of the paper. 3.1. Membership Equational Logic Membership equational logic [12, 1] is an extension of many sorted equational logic [7] with membership assertions t : s that state that a term t belongs to a sort s. It subsumes a wide variety of specification formalisms, including order sorted [6, 8] and partial equational logics. Despite its generality, it still enjoys the good properties of equational logics: it is simple, ....

Joseph Goguen and Jose Meseguer. Completeness of many-sorted equational logic. Houston Journal of Mathematics, 11(3):307--334, 1985.

....such as those in the OBJ family [13, 5, 8, 9] Its expressivity is probably best re ected by the fact that any computable data type can be characterized by means of a nite equational speci cation [3] Its models are just algebras, which are very simple and intuitive structures. We suggest [11, 22] for an introduction to many sorted equational logics and its completeness. There is a plethora of variants and generalizations of equational logics, ranging from unsorted [4] to partial [23] order sorted [12, 28] and hidden [10, 24] equational logics. Categorical generalizations allowed ....

Joseph Goguen and Jose Meseguer. Completeness of many-sorted equational logic. Houston Journal of Mathematics, 11(3):307-334,

.... Sigma 1 b = c (4) cannot be proved in EQ logic by using rules similar to presented in Appendix A, because the sentence b = c cannot be derived from the sentence 8 x:s :b = c (the nonemptiness of the carrier of sort s, ensured by the hidden constant a, cannot be expressed using equations, cf. GM 85] Let us try to prove judgment (4) by using rules from Appendix A and the (ae EQ FOEQ join) rule schema, where ae EQ FOEQ is the institution representation defined in Example 4. Let us notice, that the institution representation ae EQ FOEQ and the specification SP satisfy assumptions of ....

J. A. Goguen, J.Meseguer. Completeness of many--sorted equational logic. Houston Journal of Mathematics, volume 11(3), pages 307--334, 1985.

....syntax and axioms, and give only a brief impression of declarative semantics. Operational semantics is not discussed at all. Details are given elsewhere [75, 60] 5. 1 Syntax of order sorted logic We start with an exposition of order sorted logic, following the expositions by Goguen and Meseguer [19, 21, 23]. An order sorted signature Sigma = S; F; P) consists of the following sets: ffl A partially ordered set (S; of sort names. We use w as a metavariable over S (strings of sort names) and extend pointwise to strings in S of equal length. Strings over S are written as s 1 Theta Delta ....

....all function, attribute and predicate symbols to behave like a congruence with respect to equality. The explicit quantification in front of the equations is needed to avoid problems with empty sorts; completeness of the equational fragment of the logic in turn requires the (Abs) inference rule [19]. 6 Using DDL as a logic for objects, class migration and roleplaying In this section we use DDL as a logic for reasoning about objects. To do this we impose a few restrictions on signatures, and thus on the generated language. The resulting language is called Dynamic Object Language (DOL) The ....

J.A. Goguen and J. Meseguer. Completeness of many-sorted equational logic. SIGPLAN Notices, 17(1):9-- 17, 1982.

....In Section 2, the notion of interpretation system presentation as a speci cation of the intended valuations within a suitable meta logic of equational nature is introduced. The interpretation structures appear as models (algebras) of the speci cation. We adopt CEQ (conditional equational logic (Goguen and Meseguer, 1985; Meseguer, 1998) as the meta logic. Section 3 de nes the notions of unconstrained and constrained bring of interpretation system presentations. The main example, bring the paraconsistent system C 1 and the modal system KD, is discussed in Section 4 at the semantic level. Section 5 contains a ....

....Logics: Completeness Preservation 13 EXAMPLE 6. The logic systems L C1 and LKD are complete. 6. Preservation results The main goal is to establish the preservation of completeness by bring. To this end, it is convenient to take advantage of the completeness of CEQ as proved for instance in (Goguen and Meseguer, 1985; Meseguer, 1998) by encoding the deduction system of the meta logic CEQ in the object Hilbert calculus. In order to deal with local reasoning at the meta level, we shall take advantage of the following two schema variable substitutions: 1 such that 1 ( i ) i 1 for every i 1; ....

Goguen, J. and J. Meseguer: 1985, `Completeness of many-sorted equational logic'. Houston Journal of Mathematics 11(3), 307-334.

....to be empty, though is usually dropped in the notation. A sequent with exactly one formula in the consequent (m=1) is called a Horn formula, and a Horn formula with empty antecedent (n=0) is a simple formula (or a simple sequent) 1 This restriction is motivated by the fact (pointed out in [7]) that admitting empty carriers requires additional mechanisms (explicit quantification) in order to obtain sound logic. We conjecture that similar solution can be applied in our case. Singular and Plural Nondeterministic Parameters 4 All variables occurring in a sequent are implicitly ....

Goguen, J.A., Meseguer, J., "Completeness of Many-Sorted Equational Logic", SIGPLAN Notices, vol. 16, no. 7, 1981.

....and syntactic identity of terms 4 We do not address the problem of empty sorts here and will present calculus which works under the assumption that sorts are not empty. We will usually give signatures with at least one constant for every sort but other ways of restricting the signatures [Goguen 1981, Huet 1980] or ensuring nonemptiness [Goguen 1981, Goguen 1987] can be used instead. It seems also that the most flexible approach which generalizes calculus by introducing explicit variables [Goguen 1981, Goguen 1982, Ehrig 1985] can be adapted to our framework. 5 An equivalent (and the ....

....not address the problem of empty sorts here and will present calculus which works under the assumption that sorts are not empty. We will usually give signatures with at least one constant for every sort but other ways of restricting the signatures [Goguen 1981, Huet 1980] or ensuring nonemptiness [Goguen 1981, Goguen 1987] can be used instead. It seems also that the most flexible approach which generalizes calculus by introducing explicit variables [Goguen 1981, Goguen 1982, Ehrig 1985] can be adapted to our framework. 5 An equivalent (and the original) formulation of the syntax and the following ....

[Article contains additional citation context not shown here]

Goguen, J.A., Meseguer, J., "Completeness of Many-Sorted Equational Logic", SIGPLAN Notices , vol. 16, no. 7, 1981.

....of these logical systems is possible, e.g. order sorted Horn clause logic with equality which is the logic underlying Eqlog. 4.3. Category based Equational Deduction Equations are traditionally pairs of terms constructed from the symbols of a signature plus some variables. Goguen and Meseguer (Goguen and Meseguer, 1985) first made quantifiers part of the R azvan Diaconescu 14 concept of equation, for MSA. Although terms are syntactic constructs, from a model theoretic perspective they are just elements of the free term model over the set of quantified variables. Any valuation of the variables into a model ....

Goguen, J. and Meseguer, J. (1985). Completeness of many-sorted equational logic. Houston Journal of Mathematics, 11(3):307--334.

....of these logical systems is possible, e.g. order sorted Horn clause logic with equality which is the logic underlying Eqlog. 3.3 Equations, Queries and Satisfaction Equations are traditionally pairs of terms constructed from the symbols of a signature plus some variables. Goguen and Meseguer [18] first made quantifiers part of the concept of equation, for MSA. Although terms are syntactic constructs, from a model theoretic perspective they are just elements of the free term model over the set of quantified variables. Any valuation of the variables into a model extends uniquely to a model ....

Joseph Goguen and Jos'e Meseguer. Completeness of many-sorted equational logic. Houston Journal of Mathematics, 11(3):307--334, 1985.

....hypothesis that U has a leftadjoint F , U is finitary if it preserves filtered colimits (see [9, 8] or [17] for details) 3.2 Equations, Satisfaction and Completeness Equations are traditionally pairs of terms constructed from the symbols of a signature plus some variables. Goguen and Meseguer [18] first made quantifiers part of the concept of equation, for MSA. Although terms are syntactic constructs, from a model theoretic perspective they are just elements of the free term model over the set of quantified variables. Any valuation of the variables into a model extends uniquely to a model ....

Joseph Goguen and Jos'e Meseguer. Completeness of many-sorted equational logic. Houston Journal of Mathematics, 11(3):307--334, 1985.

....MAUDE [16] The near approaches [56, 21] are interested with the study of several kinds of subsort relations and overloading mechanisms and its in uence in the semantics of the equational logic. Such works o er a complete overview of such class of type systems, extending the many sorted framework [19] to a order sorted framework. The main di erence with respect to the above mentioned parametric case is that such type systems are dynamic, i.e. sorts have to be considered at the operational level as a type checking procedure at compiletime similar to the many sorted or parametric case may not ....

J. A. Goguen and J. Meseguer. Completeness of many-sorted equational logic. ACM SIGPLAN Notices, 17(1):9-17, 1982.

.... leads to the completeness theorems of [Fri75, Hen50, Sta85a] The drawback, however, is that in many computer science applications it is not appropriate to assume every type is nonempty (inhabited) This point is discussed in [MMMS87] Related discussions of multi sorted equational logic appear in [GM82, GM86]. When we reject the nonemptiness assumption and allow types to be empty, we are led to non equational principles, as in [MMMS87] which formalize reasoning by cases as above. The extended axiom system of [MMMS87] is semantically complete, but it has a very di erent avor from the traditional ....

J. Goguen and J. Meseguer. Completeness of many-sorted equational logic. SIGPLAN Notices, 17:9-17, 1982.

....modeled as a sequence of transformations. Although we do not treat many sorted logic in this paper, we note in passing that a proper treatment of many sorted equational logic similarly requires (for different reasons) an explicit indication of a set of relevant variables; see, for example, (Goguen and Meseguer 1981, 1985). Definition 2.1. A pair hA; Bi is a two element multiset of terms. A system is a finite set S of pairs together with a finite set Vars(S ) of variables, including at least the variables occurring among the terms of S. We will usually not need to explicitly indicate the set Vars(S) and may abuse ....

Goguen, J. A., and Meseguer, J. (1985), Completeness of many-sorted equational logic, Houston Journal of Mathematics, 307--334.

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Joseph Goguen and Jose Meseguer. Completeness of many-sorted equational logic. Houston Journal of Mathematics, 11(3):307-334, 1985.

....to it) is injective. We will say that a presentation ( Sigma; Gamma) is faithful if it satisfies this injectivity condition. Although they are pathological, unfaithful presentations do exist, and for them the extension with retracts is not conservative, as shown by the following example from [25]: Example 3.4 Let Sigma have sorts a; b; u with a; b u, have an operation f : a b, have no constants of sort a, have constants 0; 1 of sort b, plus ; binary infix and : unary prefix of sort b. Let Gamma have the equations : f(x) f(x) y y = y; y y = y; y ( y) 1; y) y = 1; ....

.... Gamma) For arbitrary Gamma, it is necessary and sufficient 11 Parameterized modules will be the main subject of the forthcoming Part III of this paper. 25 that Sigma has no quasi empty models, which are algebras A such that A s = for some s but A s 0 6= for some other sort s 0 [25]. For arbitrary Sigma, it is sufficient that Gamma is a set of confluent rewrite rules [56] The following model theoretic proof of the conservative extension result for faithful presentations uses naturality of the family X of morphisms, which in particular gives commutativity of the ....

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Joseph Goguen and Jos'e Meseguer. Completeness of many-sorted equational logic. Houston Journal of Mathematics, 11(3):307--334,

....are valid, including substituting one behavioral equation into another, and of course symmetry and transitivity; visible equations can also be used in such proofs. 2 Here equational deduction refers to the form with explicit universal quanti ers that was introduced by Goguen and Meseguer [49], because otherwise diculties can arise with models having empty hidden carriers. The result is easy to prove, and can be very useful. For example, if we want to prove (8 NM : Nat) 8 S : State) getx putx(N; putx(M; S) N for the theory X, we can give the following to OBJ, where the red command ....

....Church Rosser and local as rewrite rules, then the theory is consistent. 2 A proof may be found in [45] Many examples in this paper can be shown consistent using this result. A sucient condition for the Church Rosser prop 11 Those unfamiliar with the Goguen Meseguer explicit quanti er approach [49] may wish to note that what actually gets proved is the equation (8S : State) 1 = 0, which is true of all models, including the one with empty hidden carrier, but which only implies the equation (8; 1 = 0 if the hidden carrier is non empty. 17 erty is that the equations are nonoverlapping 12 ....

Joseph Goguen and Jose Meseguer. Completeness of many-sorted equational logic. Houston Journal of Mathematics, 11(3):307-334, 1985.

....and applicability, at little cost to its advantages. EL was untyped at birth [3] but later extended to many sorts in various ways, of which [1] was perhaps first and [5] notationally simplest; extensions to overloaded function symbols and conditional equations were also important; see [10] for technical and historical details. Section 2 quickly reviews many sorted EL, and Section 3 covers the next important extension, order sorted EL [11] including an inductive proof for a typical partial function. A final section discusses a further extension to hidden EL. 2 Many Subsorted ....

Joseph Goguen and Jos'e Meseguer. Completeness of many-sorted equational logic. Houston Journal of Mathematics, 11(3):307--334, 1985.

No context found.

J. Goguen and J. Meseguer. Completeness of many-sorted equational logic. Houston Journal of Mathematics, 11(3):307--334, 1985.

No context found.

J. Goguen and J. Meseguer. Completeness of many-sorted equational logic. Houston Journal of Mathematics, 11(3):307-334, 1985.

No context found.

J. A. Goguen, J.Meseguer. Completeness of many--sorted equational logic. Houston Journal of Mathematics , volume. 11(3), pages 307--334, 1985.

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Goguen, J.A., and Meseguer, J. Completeness of many-sorted equational logic. Houston Journal of Mathematics 11, 3 (1985), 307--334.

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