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.

....[6] list a number of features which they argue are essential for OO database systems. Parallel to this, a number of formalizations of OO models have been proposed, such as COL [1, 2, 3] F logic [33] HILOG [13] ILOG [29] IQL [4, 5] formalizations of 02 [36, 35] OBJ and related languages [20, 21, 22, 23, 25, 38], OBLOG [15, 19, 17, 43] and CMSL [50, 46] Beeri [9] gives a survey of some issues. Even though these papers are top down, there is not yet any root concept from which they develop their formalization of object orientation. The trees of possibilities they explore have different roots and are ....

....to CMSL that have been realized or are planned for the future. The central role of oid s in this formalization of objects is also discussed. Section 5 concludes the paper. We assume the theory of order sorted equational ADT specification developed, among others, by Goguen and Meseguer [24, 25, 44]. This theory is now reaching textbook status [16] I summarize it very briefly, in order to see how it meets our shopping list. In order sorted equational specification, an ADT is a class of models of an equational specification, and an equational specification is a set of axioms in order sorted ....

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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. 20

....[2, 3] CafeOBJ has several kinds of imports, parameterised programming (also allowing integration of CafeOBJ specifications with executable code in a lower level language) views, and module expressions. Powerful type system The type system that allows subtypes based on order sorted algebra [22, 15] (abbreviatted OSA) The method of retracts, a mathematically rigorous form of runtime type checking and error handling, gives CafeOBJ a syntactic flexibility comparable to that of untyped languages, while preserving all the advantages of strong typing. The order sortedness of CafeOBJ not only ....

....between specification programming paradigms and logics as they appear in the actual version of CafeOBJ, also pointing to some basic references. ABBREVIATION LOGIC SPEC PGM PARADIGM BASIC REF. MSA many sorted algebraic specification [13] algebra OSA order sorted algebraic specification [13, 22, 15] algebra with subtypes HSA hidden sorted behavioural specification [16] algebra HOSA hidden order sorted behavioural specification [16, 1] algebra with subtypes RWL rewriting logic concurrent [27] algebraic specification OSRWL order sorted concurrent rewriting logic algebraic ....

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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.

....types (ADTs) and that a number of standard ADTs are available to the formal specification, such as NATURAL, STRING and MONEY. In this paper, we assume order sorted equational logic with initial algebra semantics as the mechanism to specify ADTs. We refer to the literature for the formal details [9, 10, 15] and here give an informal indication of what this means. ffl An ADT is a set, called a sort, with a number of operations. ffl ADTs are partially ordered to represent the subtype relation on ADTs. The subtype relation implies a subset relation on sorts. For example, in the partial ordering of ....

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.

....In [11] we extended the completeness results for consistency tests in form of saturation procedures to R logics, as illustrated in Section 6. 15 Order sorted algebras, ETL and Unified Algebras: From an algebraic point of view, R logics compare best with many and order sorted algebras [7], already implemented via rewriting for instance in OBJ 3 [15] However simple G logics provide arbitrary terms as sorts and thus achieve a greater expressivity. Polymorphic order sorted algebras can be seen as a fragment of simple logic. In order to compare R logics with ETL and ....

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

....needed. A specification of an information model in LCM consists of a collection of value type specifications and a collection of class specifications. 3.3. 1 Value type specifications The value type specification part of LCM is an order sorted equational language that is a simplified form of OBJ [17, 18]. Figure 3.3 contains some example value type specifications. The value type ZERO contains one element, 0, and is a subtype of NATURAL. Another subtype of NATURAL, not defined here, is POSINTEGER. In a meaningful specification, the specification of POSINTEGER must be included. A more elaborate ....

....then the retract axiom does not tell us that Storage tank(s) is true; we need the mode predicate axiom for this. A problem with the retract functions is that they are partial functions, which gives a problem with the initial semantics. This problem would be avoided if we use sort constraints [17] rather than axioms to define model classes. If we view the mode predicate axiom as a sort constraint, then given a particular model, it defines the extension of the STORAGE TANK sort in that model. Since we are not aware of a sound and complete axiom system or an operational semantics for sort ....

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.

....who is permitted, forbidden, or obligated to do what. Section 6 sketches how to specify objects in this formalism, and in particular how active objects are specified. Finally, in section 8 we summarize the paper. 2. Static constraints and ADT s We use the equational paradigm of specifying ADT s [8, 12, 11, 30]. A simple example is datatype spec Persons Booleans PERSON operations p0 : PERSON next : PERSON PERSON eq : PERSON x PERSON BOOL [E1] p0 eq p0 = true [E2] next(p) eq next(p) p eq p [E3] p eq next(p) false [E4] next(p) eq p = false end spec Persons Booleans is a ....

....eq p0 = true [E2] next(p) eq next(p) p eq p [E3] p eq next(p) false [E4] next(p) eq p = false end spec Persons Booleans is a specification of the set of Boolean values, called BOOL, and their operations. Sets of data elements are called sorts. They may be partially ordered as defined in [12]. For example, we may declare EMPLOYEE to be a subsort of PERSON, by saying EMPLOYEE PERSON. All operations declared for a supersort are inherited by its subsorts. The declarations of the sorts, their ordering, and the operations on the sorts form an ADT signature, and an ADT signature with a ....

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J.A. Goguen and J. Meseguer, Order-Sorted Algebra I: Equational Deduction for Multiple Inheritance, Overloading, Exceptions and Partial Operations, Programming Research Group, Oxford, and SRI International, Menlo Park (June 19, 1989).

....The semantics of equational value specifications is called initial, which means, roughly, that 1. all elements of all sorts are named by closed terms of the sorts, and 2. closed terms name equal sort elements iff they can be proved equal from the equations in the specification. See [5, 6, 10] for initial algebra semantics of ADT specifications. The specification of the natural numbers and integers in appendix B is based on a specification given by Smolka et al. 16] The number sorts form a poset illustrated in figure 2.1. In general, VSL specifications can be parametrized and can ....

.... is conceptual model spec Example Persons, AccountLifeCycle initialization [1] exists(db) true [2] exists(p0) true [3] exists(a0) true [4] age(p0) 30 [5] name(p0) Piet [6] address(p0) a0 [7] city(a0) Amsterdam [8] zip(a0) 1234 [9] street(a0) De Boelelaan [10] nr(a0) 1081 end spec Example 4.2 Implementation The finite set of equations in the initialization is a database state, and can therefore be implemented as a database state or even a file that is loaded into a database. The CMSL implementation must then verify that the set of equations ....

Joseph Goguen and Jos'e 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

.... provide support for the concept of subsorts and overloading (e.g. to allow the specification of natural numbers as a subsort of the integers, with the usual functions on natural numbers being extended to integers) but the notion of regularity of signatures, as adopted in order sorted algebras [5], was found to have some drawbacks. Finally, it was decided to put no conditions at all on the declarations of overloaded functions, but instead to require that any uses of overloaded functions in terms should be sufficiently disambiguated, ensuring that different parses of the same term ....

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

....variables we can construct data terms in the usual way. T#D ,s (X) denotes the set of data terms of sort s over #D (X) T#D (#) is the family of closed terms, also written T#D . Terms denote a certain value, so they can be evaluated under a given interpretation. We refer the interested reader to [5] for a detailed presentation of order sorted signatures, their interpretation structures (algebras) and categorical results. As an example of the latter, we can define morphisms between order sorted signatures in such a way that the signatures and morphisms define a category. For the purpose of ....

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.

....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 implementable, and 3 Fischer and Ros u admits sound and complete deduction as well as free models. In this section we informally present membership equational ....

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.

....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, efficiently implementable, and admits sound and complete deduction as well as free models. In this section we informally present membership equational logic, referring ....

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.

....seen an increasing interest in order sorted algebras, and its operational counterpart order sorted rewriting. Order sortedness has been introduced to term rewriting to provide for a more powerful type concept, allowing us to express partiality of functions, error handling and subtype inheritance [8]. In languages like OBJ3 [9] and ABEL [1] order sorted specifications are given an operational semantics based on term rewriting. One of the most important properties a rewrite system mayhaveistermi nation,which means that no infinite computation in the system can take place. In general it is ....

....paper by briefly mentioning yet another extension to our proposed method and then indicate a differentapproach to order sorted termination not involving any transformation into an unsorted system. 2 Basic Notions We shall only introduce the syntactic aspects of order sorted algebra and refer to [8] and [13] for semantics of order sorted specifications. The paper [13] compares the two main semantic variations of order sorted algebra overloading vs. nonoverloading. In this paper, where the rewrite relation is based on overloaded algebra semantics, notions and notations mainly follow[8] and ....

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J. A. Goguen and J. Meseguer. Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operators. Technical report, Programming Research Group, Oxford University Computing Laboratory, 1989.

....verifier control the structural consistency of thetotal proof. Using OBJ3 meantdefiningtheinvariantsasfunctions whose semantics are given by the initial algebra of some algebraic specification. A detailed survey of this method may be found in [EM85] andthe extension to order sorted algebras in [GM89]and [OD91] It does, however, suffice to knowthattodefineaninvariant weneeded togiverewrite rules suchthatapplyingthese rules repeatedly toany of the global states we are considering will lead toeither TRUE or FALSE. Even if OBJ3 is used bothforthedefinition of the transition system andof ....

J. A. Goguen and J. Meseguer. Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operators. Technical report, Programming Research Group, Oxford University Computing Laboratory, 1989.

....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 proving common results, such as variety and quasi variety theorems, only once [2, 21, 25] These categorical approaches abstract equational satisfaction by injectivity, which turn out to be equivalent concepts in concrete ....

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,

....of derivation so as to capture, e.g. provisional reasoning and sub proof sharing. In this paper we focus on a subclass of RThs called equational RThs [7] Equational RThs are characterized by sequents being equivalence classes (w.r.t. a set of equations) of terms of an order sorted signature [10]. In the sequel, since we will only deal with equational RThs, we will omit the adjective equational , for brevity. We rst de ne a sequent system, as a quadruple Ssys = h ; X; E; Qi. hS; Oi is an order sorted signature: S is a set of sort symbols, a partial order over S, and O an (S ....

Joseph Goguen and Jose 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 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.

....theories are envelope invariant, where T 2 Th MEL is sortable i its membership axioms consisting of the subsort inclusions and the operator declarations can prove (8X) u : k , for any subterm u in any axiom in T . Such proofs are decidable, because order sorted parsing of terms is decidable [10]; yet S(T ) may still have models in which some operations are partial. A broad class of practical interest is the class of those T 2 Th MEL that are Church Rosser and terminating in the precise sense of [2] This is a key class of executable speci cations, for which we can use tools like ....

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, 1992.

....is the most concrete, the types as algebras approach is a modest abstraction of it, and the types as theories approach supports genuine independence of representation, while still encompassing the other approaches. Another natural step extends from many sorted algebra to order sorted algebra [24], in which a partial order is given on the set of sorts, interpreted semantically as subset inclusion. This permits a nice treatment of exception handling, and of partial and overloaded operations, as illustrated later in this paper. See also the unified algebra of Mosses [50] Another extension ....

....: S S 0 and an S Theta S indexed family of functions g w;s : Sigma w;s Sigma 0 f(w) f(s) Sigma models are Sigma algebras, Sigma sentences are (conditional) Sigma equations, and Sigma satisfaction is as usual. Actually, the main examples will use order sorted equational logic [24]. 2.2 Theories This subsection develops theories over an arbitrary institution, following the semantics of Clear [3, 18] Definition 1 A theory consists of a signature Sigma and a set E of Sigma sentences, i.e. it is a pair h Sigma; Ei. We may also call h Sigma; Ei a Sigma theory. Given a ....

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Joseph Goguen and Jos'e 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

....is now obvious. We can view each of the labeled arcs of the Petri net as a rewrite rule in a rewrite theory having a binary associative, commutative operator Omega (multiset union) with identity 1 so that rewriting happens modulo ACI, that is, is multiset rewriting. Adopting an order sorted [65] version of rewriting logic, we can gather together the places ; q; a; c into a type Place and view the states of the net, usually called markings, as elements of a supertype Marking containing Place and endowed with a multiset union operator. Using the syntax of the Maude language, the rewrite ....

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.

....give recommendations for its use and some examples. equations . 4.1 Church Rosser Order Sorted Speci cations In this section we introduce the notion of Church Rosser order sorted speci cation. We assume speci cations of the form S = R [ A) where is a preregular order sorted signature [24], and R [ A is a set of equations such that A is a set of sort preserving equational axioms, that is, whenever 7 As we shall see in Section 4.2.1, in the current version of the tool the uni cation and matching algorithms are speci ed equationally, and only for signatures whose operators do not ....

....pair ccp(u; u 0 ; c 0 ) if there exists a substitution such that t =A u, t 0 =A u 0 , and c =A c 0 . Then, given a speci cation S, let MCP(S) denote the set of most general critical pairs between rules in S that, after simplifying both sides of the 8 We denote the least sort [24] of a term t by LS(t) 23 critical pair using the equations in S, are not identical critical pairs modulo A of the form ccp(t; t; c) To determine the overlaps of the lefthand sides of the rules the variables in them have to be renamed in order to get disjoint sets of variables appearing in each ....

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.

....as the term s[f 0g Nat] Nat. Similarly, there is also a binary constructor : for meta representing the lazy membership predicate that does not evaluate the term in question at all, but uses only the syntactic declarations in the module s order sorted signature (that is assumed preregular [21]) to decide whether the least sort of the term is smaller or equal to a given sort. The last declaration for the data 6 Clavel et al. type of terms is a constant error to be used as an error element. 3.2 Representing Modules Functional and system modules are meta represented in a syntax very ....

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.

....The Logical Basis of Maude Maude s functional modules are theories in membership equational logic [49, 6] a Horn logic whose atomic sentences are equalities t = t 0 and membership assertions of the form t : s, stating that a term t has sort s. Such a logic extends order sorted equational logic [31], and supports sorts, subsort relations, subsort polymorphic overloading of operators, and de nition of partial functions with equationally de ned domains. Maude s functional modules are assumed to be Church Rosser; they are executed by the Maude engine according to the rewriting techniques and ....

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

....a supersort of Qid. Sorts not defined by the user, as for example the error sorts added by the system to complete each connected component 5 are in this sort Sort. The function leastSort takes as arguments a module and a term, and computes the least sort of that term in the module (see, e.g. [15]) This function can return an error sort not defined by the user, and that is why it is defined as op leastSort : Module Term Sort . Given a module M with subsort relation M , and sorts s; s 0 2 S, where S is the set of sorts in M , the Boolean function sortLeq(M ;s;s 0 ) is true if and ....

....then give recommendations for its use and some examples. 5.1 Church Rosser Order Sorted Specifications In this section we introduce the notion of Church Rosser order sorted specification. We assume specifications of the form S = Sigma; R[A) where Sigma is a preregular order sorted signature [15], and R [ A is a set of equations such that A is a set of sort preserving equational axioms, that is, whenever t =A t 0 we have LS(t) LS(t 0 ) The equations R will be used as rewrite rules modulo the axioms A. Furthermore, in what follows, and for the purposes of the present tool, the ....

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.

....with whom we have designed and built the Maude language and system. 2 Church Rosser Order Sorted Speci cations In this section we introduce the notion of Church Rosser order sorted speci cation. We assume speci cations of the form S = R [ A) where is a preregular order sorted signature [23], and R [ A is a set of equations such that A is a set of sort preserving equational axioms, that is, whenever t =A t 0 we have 2 LS(t) LS(t 0 ) The equations R will be used as rewrite rules modulo the axioms A. Furthermore, in what follows, and for the purposes of the present tool, the ....

....n )j i p = t i j p . Finally, a term t with its subterm tj p replaced by the term t 0 is denoted t[t 0 ] p . A substitution is a replacement operation uniquely de ned by a mapping from variables to terms, and is written out as fx 1 t 1 ; x n t n g. Given 2 We denote the least sort [23] of a term t by LS(t) 6 a set of axioms A, a substitution is an A uni er of t and t 0 if t =A t 0 , and it is an A match from t to t 0 if t 0 =A t . The set of variables occurring in a term t is denoted vars(t) Then, given a substitution = fx 1 t 1 ; x n t n g, we ....

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

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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.

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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.

No context found.

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.

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.

Goguen,J.A. & Meseguer,J., Order-sorted Algebra I: Equational deduction for multiple inheritance, polymorphism and partial operations. Theor. Comp. Sci. 105 (1992) pp. 217--293.

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