H. Ehrig and B. Mahr. Fundamentals of algebraic speci cation 1. equations and initial semantics. EATCS Monographs on Theoretical Computer Science, Vol.6, Springer-Verlag, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....well known examples of first order structures in mathematics and in theoretical computer science. For instance, relational) databases can be represented by first order structures with respect to a first order language without function symbols [8] Algebraic specification of abstract data type [6, 15] form another possible field of applications. 1 Preliminaries For basic notions on first order structures, we refer the readers to Burris and Sankappanavar [5] and Keisler [9] Note that first order structures are also known as algebraic systems (Mal cev [10] 1.1 L structures, morphisms and ....

....S i . For each relation symbol r, we define r on S i by (x 1 ; x n ) 2 r ( 9y 1 2 (x 1 ) 9y n 2 (x n ) y 1 ; y n ) 2 r (where n is the arity of r) Then the ultraproduct S i is an L structure and F is a morphism of L structures. More details can be found in [5, 6]. We say that a class of L structures is a quasivariety if it is closed under taking substructures, direct products and ultraproducts. It is a variety if it is closed under taking substructures, homomorphic images and direct products. Finally, a class of finite L structures is called a ....

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H. Ehrig and B. Mahr. Fundamentals of algebraic specification 1, EATCS Monographs on Theoretical Computer Science 6, Springer, Berlin, 1985.

....simulation, compilation and prototype generation is described in [20] 2.3. Full LOTOS with TM Full LOTOS introduces value communication and data types to the processalgebraic concepts of Basic LOTOS. The original data language for Full LOTOS is based on the ACT ONE theory of abstract data types [21]. For LOTOS TM, we use TM as the data language instead. The constructs of Full LOTOS that involve value expressions are briefly introduced below. In Full LOTOS, an event (or gate) name may be followed by value and variable declarations, together also called event attributes. An event name ....

H. Ehrig and B. Mahr, Fundamentals of Algebraic Specification 1, EATCS Monographs on Theoretical Computer Science, 6, Springer-Verlag, Berlin, 1985.

....that Eq Eq. Definition 3 (Equational specification) Let SigEq be an equational signature. A SigEq Specification SpecEq is a pair (Sigq, E) where E is a set of conditional equations over Sigq. 2. 1 Equational abstract data specification Abstract data types (ADT s) can be specified equationally [11, 12, 19, 44]. Our intended semantics of equational ADT specifications is the initial algebra semantics explained in the references above. Basically, in this semantics, a data element is a closed term modulo equality. This means that each data elements of an ADT is named by at least one closed term, and that ....

H. Ehrig and B. Mahr. Fundamentals of Algebraic Specification 1. Equa- tions and Initial Semantics. Springer, 1985. EATCS Monographs on Theoretical Computer Science, Vol. 6.

....x COMPANY ) DATE, which represents the date an employee is hired by a company. This is really a relationship attribute in ER terms. 3.1.2 Formalization To formalizc this, we need the concepts of conservative enrichment of a specification and rcduct of a model. These are defined By Ehrig and Mahr [18]. A signature Sig = S, F) is an extension of Sigo = So, o) Fo) if So C S, o C , and Fo C F. It is an enrichment if So = S and o= Spec = S, F, E) is an extension of Speco = So, o) Fo, Eo) if the signature of Spec is an extension of the signature of Speco and Eo C E. Let ....

H. Ehrig and B. Mahr. Fundamentals of Algebraic Specification 1. Equations and Initial Semantics. Springer, 1985. EATCS Monographs on Theoretical Computer Science, Vol. 6.

....that we can use integer arithmetic. Although we have left this open until now, we can simply assume that first order logic with equality is part of our language, and that an equational theory of the integers has been specified in this language. The theory of how this can be done is standard [EM85] Now, let b be a variable whose value is the balance of an account, w(m) be the action withdraw at least m from the account , and d(m) be the action deposit at least m on the account , and let n and m be constants. In terms of our theory of actions, we take as atomic action names the set A = ....

H. Ehrig and B. Mahr. Fundamentals of Algebraic Specification 1. Equations and Initial Semantics. Springer, 1985. EATCS Monographs on Theoretical Computer Science, Vol. 6.

....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 ....

H. Ehrig and B. Mahr, Fundamentals of Algebraic Specification 1. Equations and Initial Semantics, Springer (1985). EATCS Monographs on Theoretical Computer Science, Vol. 6.

....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 ....

....to exist as well. The initial state is specified by a finite set of equations. An example CM specification 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 ....

H. Ehrig and B. Mahr. Fundamentals of Algebraic Specification 1. Equations and Initial Semantics. Springer, 1985. EATCS Monographs on Theoretical Computer Science, Vol. 6.

....(Language Of Temporal Ordering Speci cations) see [40, 28] is a wide spectrum language, which is suitable for specifying systems at various levels of abstraction. Consequently, it can be used at both ends of the software development spectrum. Its natural division into ADT part (based on ACT ONE [16]) and process algebra part (similar to CSP [26] and CCS [37] is advantageous since it provides the exibility of two di erent semantic models for expressing behaviour, whilst managing to integrate them in a relatively coherent fashion. LOTOS provides an elegant way to specify services and to ....

H. Ehrig and Mahr B. Fundamentals of Algebraic Specication I. Springer-Verlag, Berlin, 1985. EATCS Monographs on Theoretical Computer Science (6).

....we consider classes of algebras (denoted by A) which satisfy particular properties; e.g. the class of all monoids or the class of all commutative monoids. Now, suppose we have some set X and we want to have an algebra of A which at least contains the elements of X and which is typical (see [EM85] for details) for class A. This can be achieved by the free algebra. 4 Let A be a class of algebras and let X be some set. An algebra A = A; OA ) 2 A along with some mapping u : X A is called free over X in A if for each algebra B = B; O B ) 2 A and each mapping h : X B there exists a ....

Hartmut Ehrig and Bernd Mahr. Fundamentals of Algebraic Specifications 1, Equations and Initial Semantics, EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1985.

....simulation, compilation and prototype generation is described in [20] 2.3. Full LOTOS with TM Full LOTOS introduces value communication and data types to the processalgebraic concepts of Basic LOTOS. The original data language for Full LOTOS is based on the ACT ONE theory of abstract data types [21]. For LOTOS TM, we use TM as the data language instead. The constructs of Full LOTOS that involve value expressions are briefly introduced below. In Full LOTOS, an event (or gate) name may be followed by value and variable declarations, together also called event attributes . An event name ....

H. Ehrig and B. Mahr, Fundamentals of Algebraic Specification 1, EATCS Monographs on Theoretical Computer Science, 6 , Springer-Verlag, Berlin, 1985.

....value has a type and comes into existence if and when its type is defined, typically as part of a structure definition. This may be at language design time (for built in types) or at program design time (for user defined types) A specification corresponds to its use in algebraic specification [EM85] in object oriented terms it can be roughly equated to the (public) interface of a class along with its invariants and assertions (see, e.g. Eiffel [Mey88] In algebraic terms a structure is a valid model of a specification, and the types in a structure correspond to the carrier sets of such a ....

H. Ehrig and B. Mahr. Algebraic Specification, volume 1/2 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1985.

....development. LOTOS is used only to make the object oriented models more concrete, as development approaches implementation. 1.2. 5 LOTOS LOTOS is chosen as the object oriented requirements capture and design language because: ffl Its natural division into ADT part (based on ACT ONE 5 [48]) and process algebra part (similar to CSP [65] and CCS [87] suits our need for semantic continuity from analysis to design: the ACT ONE requirements model can be incorporated within the full LOTOS design model. ffl LOTOS has already been the subject of research with regard to its suitability ....

H. Ehrig and Mahr B. Fundamentals of Algebraic Specification I. Springer-Verlag, Berlin, 1985. EATCS Monographs on Theoretical Computer Science (6).

....Of Temporal Ordering Specifications) see [11, 48, 28] is a wide spectrum language, which is suitable for specifying systems at various levels of abstraction. Consequently, it can be used at both ends of the software development spectrum. Its natural division into ADT part (based on ACT ONE [21]) and process algebra part (similar to CSP [27] and CCS [37] is advantageous sinceit provides the flexibility of two different semantic models for expressing behaviour, whilst managing to integrate them in a relatively coherent fashion. 2.1 Syntax and Semantics LOTOS is a language using a ....

H. Ehrig and Mahr B. Fundamentals of Algebraic Specification I. SpringerVerlag, Berlin, 1985. EATCS Monographs on Theoretical Computer Science (6).

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H. Ehrig and B. Mahr. Fundamentals of algebraic speci cation 1. equations and initial semantics. EATCS Monographs on Theoretical Computer Science, Vol.6, Springer-Verlag, 1985.

No context found.

Ehrig, H.; Mahr, B.: Fundamentals of Algebraic Specification 2. Module Specifications and Constraints. EATCS Monographs on Theoretical Computer Science, Vol. 21, Springer-Verlag (1990)

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Ehrig, H., Mahr, B.: Fundamentals of Algebraic Specification 1. Equations and Initial Semantics. EATCS Monographs on Theoretical Computer Science, Vol. 6, Springer-Verlag (1985)

....formalisms constitute a net data model, and are used for de ning the syntax of the system, in other terms to capture its structure. The net model, which is a kind of advanced Petri net [20] is used to describe the process architecture of the system; the data model, which is an algebraic formalism [15], is used for specifying the data structures of the system. The third formalism, which is a rewriting logic [23] is used for de ning the semantics of the system, or in other words to describe its behaviour. According to this logic, the system behaviour may be explained by formal reasoning. ....

E. Ehrig and B. Mahr. Fundamentals of Algebraic Specications 1. EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1985.

....of states by firing of transitions. In layer 3 we have net processes, which can be defined for high level nets similar to those in the low level case. In the system architecture level 4 we have well known parameterization and modularization concepts for algebraic specifications [EM 85, EM 90] net transformations defined by rule based transformation of nets, and the concepts of union and fusion to build up larger nets from smaller components. However, a suitable modularization concept including data type and net parts is still missing. 4.1.3 Attributed Graph Transformation Similar ....

Ehrig, H.; Mahr, B.: Fundamentals of Algebraic Specification 2. Module Specifications and Constraints. EATCS Monographs on Theoretical Computer Science, Vol. 21, Springer-Verlag (1990)

....formalisms, where the axioms are ranging from classical equations, via conditional equations and Universal Horn to first and higher order logical formulas. As typical representatives of this class we consider the (classical) algebraic approach based on algebraic specifications in the sense of [EM 85] and extensions of algebraic specifications, where states are defined by algebras as proposed in [DG 94] AZ 94] and [GR 95, 97, 98] or processes are specified algebraically [Ehr et al. 90] 3.1.2 State Model Oriented Approaches This class of state model oriented specification formalisms ....

....the transformation of states by firing of transitions. In layer 3 we have net processes, which can be defined for high level nets similar to those in the low level case. In the system architecture level 4 we have well known parameterization and modularization concepts for algebraic specifications [EM 85, EM 90] net transformations defined by rule based transformation of nets, and the concepts of union and fusion to build up larger nets from smaller components. However, a suitable modularization concept including data type and net parts is still missing. 4.1.3 Attributed Graph Transformation ....

Ehrig, H., Mahr, B.: Fundamentals of Algebraic Specification 1. Equations and Initial Semantics. EATCS Monographs on Theoretical Computer Science, Vol. 6, SpringerVerlag (1985)

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Ehrig, H. and Mahr, B. : Fundamentals of Algebraic Specification 1, EATCS Monographs on Theoretical Computer Science, Vol. 6, Springer-Verlag (1985).

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Ehrig, H. and Mahr, B. : Fundamentals of Algebraic Specification 1, EATCS Monographs on Theoretical Computer Science, Vol. 6, Springer-Verlag (1985).

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H. Ehrig and B. Mahr. Fundamentals of Algebraic Specification 1, Equations and Initial Semantics. EATCS Monographs on Theoretical Computer Science. Springer Verlag, w. brauer and r. rozenberg and a. salomaa edition, 1985.

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Ehrig, H., and Mahr, B. Fundamentals of Algebraic Speci cation 1, Equations and Initial Semantics. EATCS Monographs on Theoretical Computer Science, vol. 6, Springer-Verlag, 1985.

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H. Ehrig, B. Mahr, Fundamentals of Algebraic Specification 1. EATCS Monographs on Theoretical Computer Science. Berlin: Springer 1985.

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H. Ehrig, B. Mahr, Fundamentals of Algebraic Specification 1. EATCS Monographs on Theoretical Computer Science. Berlin: Springer 1985.

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