Goguen J.A., Meseguer J. : \Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations", Technical Report SRI-CSL89 -10, SRI, July 1989.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....n , called the interpretation of g, with the property that, for every admissible type T 1 ; T n S of g and every (a 1 ; a n ) 2 A T 1 A Tn , g (a 1 ; a n ) is de ned and contained in U S AU . We follow Reynolds [32] in contrast to Goguen and Meseguer [13, 17], in not requiring A S A T when S T . However Reynolds handles subtyping by means of an implicit coercion mapping between the domains A S and A T when S T , and this has essentially the e ect of requiring A S and A T to be disjoint when S 6= T . We take a middle ground. The domains of ....

....2 Z; y 2 Z; c 2 Zg. The interpretations of the operations are given in Figure 2. Note that the operations movex and movey change the color when their argument is of type CPt. It is convenient, in some contexts, to consider the more restricted notion of Goguen and Meseguer s order sorted algebras [13, 17]. A algebra A is called order sorted if A S A T when S T . Thus, in order sorted algebras, carrier sets satisfy the following property: S T A S = A T . The algebra PT is not order sorted, since A CPt 6 A Pt . However, it is possible to transform PT , into an order sorted algebra by ....

Joseph Goguen and Jose Meseguer, Order-Sorted Algebra I: Equational Deduction for Multiple Inheritance, Overloading, Exceptions and Partial Operations, Theor. Comp. Sci. 105(1987), 217-273

....a TRS with the order sort, we can obtain weaker condition on which each normal form is a normal form. Definitions An order sorted signature is formalized as a triple (S, #) where S is a set of sorts, # is an S # S sorted family #w,s w S # and s S and S is a partial order [DF98, GM92]. We write f : w s for f #w,s and call w its arity, s its value (or result or coarity or sort) Especially c : s is written instead of c : # s. A variable V is an S sorted family V S of countably infinite sets. A set of order sorted terms T(#, V ) or T ) is constructed as the ....

....and f : s 1 . s n s # . We write t : s for T s and call s a sort of t. A rewrite rule is a pair of terms l : s r : s # where s or s # s. Other subjects are defined straightforwardly, term rewriting, context sensitive rewriting and so on. More details are found in the literatures [DF98, GM92]. Hierarchical order For order sorted TRS, we propose a method for obtaining suitable context sensitive rewriting in which some waste search for a redex can be eliminated. #) be an order sorted signature. We define a hierarchical order # as the least quasi order on S satisfying following ....

J.A. Goguen and J. Meseuer, Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations, Theoretical Computer Science, 105(2), 217-273, 1992.

....in associative (and associative commutative) function symbols, the domain and range sorts must lie in the same connected sort component and thus, for simplicity, we can assume that the sort structure consists of just a single connected sort component S. If an order sorted signature is pre regular [7], for each n ary function symbol f there exists a order sorting function S which given a tuple of argument sorts s 1 ; s n yields the least sort of any term f(ff 1 ; ff n ) where ff i has least sort s i . Let f be is an associative (associative commutative) We insist that ....

J. Goguen and J. Meseguer. Order sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science, 105:217--273, 1992. 19

....to make changes in the types of attributes, for example an ontology may be refined by refining the types of some of its attributes, so we consider these types to be a part of ontological specifications. We formalise this part of specifications using the notion of order sorted algebraic theory ; [21, 17] give details of order sorted algebra, the following is an example of a theory for the natural numbers, in the notation of the language OBJ [23, 17] th NAT is sorts Nat NonZeroNat . subsort NonZeroNat Nat . op 0 : Nat . op s : Nat NonZeroNat . op p : NonZeroNat Nat . var N : Nat ....

....of T ; this allows us to define morphisms of data domains as follows: Definition 2 A morphism of data domains : T ; D) T ; D ) is a pair = OE; h) where OE : T T is an order sorted theory morphism and h : OED D is a T homomorphism. 2 Order sorted theory morphisms (see [21]) are pairs (f; g) where f is a monotonic map from sort names to sort names, and g maps operation symbols to operation symbols. Data morphisms go from coarse to fine structures. For example, one ontology might specify an attribute with values in a data type shade, interpreted in one domain as ....

Joseph A. Goguen and Jos'e Meseguer. Ordersorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science, 105(2):217--273, 1992.

....object stating that this object has characteristics simultaneously or in sequence so different that they would normally be attributed to different objects. However, in the computing community, polymorphic usually denotes a property of an operator or function symbol, namely that its meaning [28] or associated behaviour [51] is determined by its operands or parameters (rather than the symbol alone) Strachey, who introduced the term to the computing field [68] made a distinction between what he called ad hoc polymorphism and parametric polymorphism. According to his definition, ad hoc ....

J.A. Goguen, J. Meseguer, Order-sorted Algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations, Theoretical Computer Science 105 (2) (1992) 217473.

....disjoint. So Xs N Xs2 0 even if 81 82 The set of all variables is X [JsesXs. A ig term of sort s is either a variable of sort s s or a term of the form f(tl, t) where f is declared as f: Slx s so with so s and the sort of ti is less than or equal to si, i 1, n [19, 44]. T(Sig)s(X) is the set of ig terms containing variables from X. The set of closed ig terms of sort s is T(Sig)s. We omit ig if the signature is understood or irrelevant. Goguen and Meseguer [19] show that under certain weak condi tions, called regularity, the terms of an equational signature ....

....as f: Slx s so with so s and the sort of ti is less than or equal to si, i 1, n [19, 44] T(Sig)s(X) is the set of ig terms containing variables from X. The set of closed ig terms of sort s is T(Sig)s. We omit ig if the signature is understood or irrelevant. Goguen and Meseguer [19] show that under certain weak condi tions, called regularity, the terms of an equational signature always have a unique least sort. In the definitions, we assume that all specifications satisfy these conditions. Definition 2 (Equational language) Let Sig: S, IF) be an equa tional ....

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Joseph Goguen and Jos Meseguer. Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Technical Report SRI-CSL-89-10, SRI International, Computer Science Lab, July 1989.

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Joseph Goguen and Jose Meseguer. Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science, 105(2):217-273, 1992. Drafts exist from as early as 1985.

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Joseph Goguen and Jose Meseguer. Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science, 105(2):217--273, 1992.

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J. A. Goguen and J. Meseguer. Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions, and partial operations. Theoretical Computer Science, 105:217--273, 1992.

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Joseph Goguen and Jos'e Meseguer. Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science, 105:217--273, 1992.

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J. A. Goguen and J. Meseguer, Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions, and partial operations, Theoretical Computer Science 105 (1992) 217--273.

....(keyword: comm) operators, which can have also an identity element (keyword: id: This means that equational simpli cation takes place not just between terms, but between equivalence classes of terms modulo such equational axioms. Furthermore, the equational theories can be order sorted [20]. That is, we can declare both sorts (or types) and subsorts, that is, set theoretic sort inclusions. The operators can then be subsort overloaded. That is, an operator can have several di erent typings, related in the subsort ordering. Finally, operator syntax is user de nable. It can be pre x, ....

....variable x, say of kind k, in X a term (x) 2 T (Y ) k By de nition, T =E is initial among all ( E) models i for each ( E) model A there exists a unique homomorphism h : T =E A. Intuitively assuming con uence, termination, and a syntactic condition on called (pre )regularity [20, 2] if the axioms E are sort decreasing, then, the canonical form canE (t) of a term t contains the most precise sort information about t, in the sense that we can compute the smallest sort s possible for t by repeated application to canE (t) of a more specialized version of the above inference ....

J. Goguen and J. Meseguer. Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science, 105:217-273, 1992.

....to the encapsulation of modules, and in Section 4.1, we will see that it is an important part of the formalisation of logical system called an institution by Burstall and Goguen [6, 17, 18] 2.5 Other Topics Section 3. 1 assumes familiarity with basic order sorted algebra, as given for example in [22]. For simplicity of exposition, this paper will rst treat the many sorted case, and then treat the order sorted case more brie y afterwards. Some examples assume familiarity with some basics of term rewriting, including con uence, termination and narrowing; these are explained, for example, in ....

....c(z) top z c(z) top pop z c(z) top pop pop z These contexts respectively select the top, second and third elements on a stack, if they exist. Strictly speaking, the above are not really contexts for STACK, but rather for the enrichment of STACK by retracts, as described for example in [22], and implemented in OBJ3. 2 We are now ready to de ne hidden order sorted signatures and algebras; these extend our previous de nitions for the many sorted case by adding an ordering relation on sorts. De nition 13: Given an order sorted signature (V; with set V of visible sorts and ....

Joseph Goguen and Jose Meseguer. Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science, 105(2):217-273, 1992.

....features [77] which made the system more convenient for users. OBJ2 [35, 36] was implemented using parts of OBJ1 during 1984 85 at SRI by Kokichi Futatsugi and Jean Pierre Jouannaud, following a design in which Jos e Meseguer and Joseph Goguen also participated, based on order sorted algebra [44, 76, 62, 73, 141] rather than error algebra; also, OBJ2 provided Clear like parameterized modules, theories, and views, although not in full generality. Another influence on OBJ3 s design and implementation was the HISP system [38, 39, 30] of Kokichi Futatsugi. OBJ3 was first developed at SRI by Timothy Winkler, ....

....sorted algebra, and that essentially all results generalize without difficulty from the many sorted to the order sorted case. Although this paper omits the technical details, OSA is a rigorous mathematical theory. OSA was originally suggested by Goguen in 1978 [44] and is further developed in [76] and [73] some alternative approaches have been given by Gogolla [41, 42] Mosses [122] Poigne [133, 134] Reynolds [135] Smolka et al. 140, 141] Wadge [150] and others. A survey as of 1993 appears in [61] along with some new generalizations. Meseguer has recently proposed a new ....

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Joseph Goguen and Jos'e Meseguer. Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science, 105(2):217--273, 1992.

.... are three kinds of module, for defining data, objects, and properties, each with its appropriate interpretation for equality, which are initial algebra semantics for data, hidden semantics for objects, and both for first order theories, in each case supporting subtypes through order sorted algebra [28]. First order sentences can occur only in modules having loose semantics. As in CafeOBJ, behavioral modules support the notion of coherence, as introduced by Diaconescu [5] but in a generalized form [56] called behavioral congruence, that allows more than one hidden argument in behavioral ....

Joseph Goguen and Jos'e Meseguer. Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science, 105(2):217--273, 1992.

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Goguen J.A., Meseguer J. : \Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations", Technical Report SRI-CSL89 -10, SRI, July 1989.

No context found.

J. A. Goguen and J. Meseguer. Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theor. Comput. Sci., 105(2):217--273, 1992.

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Joseph A. Goguen and Jose Meseguer. Ordersorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science, 105(2):217--273, November 1992.

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J. Goguen and J. Meseguer. Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science, 105:217--273, 1992.

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Joseph A. Goguen and Jose Meseguer. Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science, 105(2):217--173, 1992.

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J. A. Goguen and J. Meseguer. Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science, 105:217#273, 1992.

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frm-eJoseph Goguen and Jose Meseguer. Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science, 105(2):217-273, 1992.

No context found.

J. Goguen and J. Meseguer. Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science, 105:217--273, 1992.

No context found.

J. Goguen and J. Meseguer. Order sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science, 105(2):217-273, November 1992.

No context found.

J. A. Goguen and J. Meseguer. Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science, 105:217-273, 1992.

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