R. Berghammer and H. Zierer. Relational algebraic semantics of deterministic and nondeterministic programs. Theoretical Computer Science, 43:123--147, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of S. Thus r S codes up the inputoutput relation r S of S and the domain of nontermination of S. An alternative is to take the meaning of S to be a pair hr S ; e S i, where r S is the input output relation and e S is either the domain of nontermination of S (e.g. 5] 41, p. 511] [3]) or, better yet, a binary relation that codes up this domain, such as the set of pairs hoe 3 ; oe 4 i, where oe 3 is the initial state of a nonterminating computation of S and oe 4 is any state whatsoever. The nontermination relation e S is completely determined by its domain, so e S is called a ....

....of A may not be actual binary relations, and may also not behave like them, beyond what is guaranteed by the relationalgebraic axioms. In spite of this generality, much can be proved, as is demonstrated by the extent of this and other papers, starting with [10] followed by [11] 12] 47] 48] [3], and others. See [49] and the references therein. Some material originally done for Re(U ) is generalized here to arbitrary complete relation algebras. Let Stat be the set of statements of a programming language. In Section 6 we define an A interpretation to a pair hr; ei of functions that map ....

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Rudolf Berghammer and H. Zierer, Relational algebraic semantics of deterministic and nondeterministic programs, Theoretical Computer Science 43 (1986), 123--147.

....where inverses arise unexpectedly in specification. Concise as it is, how does one derive an algorithm from it The answer, among many other examples, is to be presented in this thesis. 1.2 Background The use of relations to model programming can be traced to the 80 s. Earlier work (for example, [63, 10]) started with modelling common imperative programming constructs as input output relations. The relational approach was later extended to model datatypes as well as operations on them. Some focused on relations [6, 7, 31, 32] while some took a category theoretical approach [64, 17] Both ....

R. Berghammer and H. Zierer. Relational algebraic semantics of deterministic and nondeterministic programs. Theoretical Computer Science, 43:123--147, 1986. 133

.... on A (of type A A) For every two sets A; B, the relation A B = A B is called the universal relation (of type A B) For every two sets A; B, the relation A B = f(x; y) j false g is called the empty relation (of type A B) For the usual rules of the calculus of relations see [5, 8, 40, 41]. There are many di erent types of relations, however in this paper we shall use only two: total relations, and functions (univalent relations) The relation R A B is total if dom(R A B ) A. The relation R A B is a function (univalent relation) if 8x 2 A:8y; z 2 B: x; y) 2 R (x; z) 2 ....

R. Berghammer, H. Zierer. Relational algebraic semantics of deterministic and nondeterministic programs. Theoretical Computer Science, 43:123-147, 1986.

....complete join semilattice, called a demonic semilattice. The associated operations are demonic join (#) demonic meet (#) and demonic composition (#) Again, we generalize from the case of relation algebra to general KAs. For more details on relational demonic semantics and demonic operators, see [4,5,6,7,9,21]. Definition 4.1. We say that an element a refines an element b [14] denoted by a b, i# #b b . It is easy to show that is indeed a partial ordering. Theorem 4.2. a) The partial order induces a complete upper semilattice, i.e. every subset L K has a least upper bound (wrt L ....

R. Berghammer and H. Zierer. Relational Algebraic semantics of deterministic and nondeterministic programs. Theoret. Comput. Sci., 43:123--147, 1986.

....relation algebras [Orl88a, Orl88b, Orl94] Furthermore, the theory of nonclassical logics is intensively used in program logics. So relation algebras are fundamental to programs. In fact much of the theory of relation algebras was applied to program semantics and program development [BKSS91, BS91, BZ86, BM97, DOR94, DO96, HH86a, HH86b, HH87, Hut93, KM92, SS93] The relational calculus can be applied to not only the theory of programs but also databases [BDJM93, BBG 94, JOB94, OJB94] natural languages [Bot92a, Bot92b, Sup76, Sup79, Sup81] graphs [KM94, Miz93, MiK95, SS93] set theory ....

Berghammer, R. and Zierer, H.: Relational Algebraic Semantics of Deterministic and Nondeterministic Programs, Theoret. Comput. Sci. 43 (1986) 123--147.

....pairing algebras [Mad89, AJN91, SN94] true pairing algebras [Mad81, Mad89, SN94, MSS92] full products [Mad93] and direct products [Mad93] There are also examples of heterogeneous relational specifications of structures that involve projection elements . i.e. heterogeneous) direct products [Sch81, BZ86, Ber91, Gri91, Zie91, SS93, BHSV93] and specification of stacks [Ber91, BS93] In general these examples can be translated to homogeneous algebras with relational reduct in RA and with projection elements . In [BHSV93] a comparison is made between the heterogeneous direct products and fork algebras. Section 3.2 shows a ....

....by [TG87, 4.1(viii) p.97] and 2.14.3 (r; s) Delta (t; q) r; Delta (t; ae) s) Delta (ae; q) rrt) s; Delta (q; ae) rrt) srq) 2 The following relational specification of direct products is presented in [Ber91, Section 4.2, p. 9] See also [SS93] and [BZ86]) Spec PROD j param types m, n rels None laws None target types prod rels : prod m ae : prod n laws T = I ae T ae = I T aeae T = I T ae = L end This specification can be translated into a first order description of a class of algebras. In [BHSV93] this approach to ....

R. Berghammer and H. Zierer. Relational algebraic semantics of deterministic and nondeterministic programs. Theoretical Computer Science, 43:123--147, 1986.

....Demonic Composition Weakest precondition semantics [9] gives an axiomatic description of nondeterministic programs admitting several models. One of these is a relational model with non standard relational composition and union: the so called demonic composition and demonic choice (see e.g. [19, 6]) In this section we motivate and then define demonic composition. In section 6 we define demonic choice. To motivate the definition of demonic composition let us briefly summarise its operational interpretation in the relational model. We consider some set X consisting of program states X ....

....for R ; S. Specifically, van der Woude defined R ; S = R ffi S) u ( ffi R) S [ 38) S [ being the converse of S) whilst Berghammer worked with the formula R ; S = R ffi S) u : ffi R) ffi S) 39) This latter formula was published earlier by Berghammer and Zierer [6]. It is well known that U=V = U ffi V [ so it is clear that these two formula are equivalent. The equivalence of (38) to our own definition of R ; S is left as an exercise. It should become clear after our discussion of monotypes versus vectors below. There are two main differences ....

R. Berghammer and H. Zierer. Relational algebraic semantics of deterministic and nondeterministic programs. Theoretical Computer Science, 43:123--147, 1986.

.... A (of type A A) For every two sets A; B, the relation A B = A B is called the universal relation (of type A B) For every two sets A; B, the relation A B = f(x; y) j false g is called the empty relation (of type A B) For the usual rules of the calculus of relations see [5, 8, 40, 41]. There are many di erent types of relations, however in this paper we shall use only two: total relations, and functions (univalent relations) The relation R A B is total if dom(R A B ) A. The relation R A B is a function (univalent relation) if 8x 2 A:8y; z 2 B: x; y) 2 R (x; z) 2 R ....

R. Berghammer, H. Zierer. Relational algebraic semantics of deterministic and nondeterministic programs. Theoretical Computer Science, 43:123-147, 1986.

.... relational product is not monotonic with respect to refinement, we define a monotonic sequence like operator, which we denote by R 2 R 0 and define as R 2 R 0 = RR 0 RR 0 L: This operator is also known as demonic composition; it has been introduced and discussed by other researchers [1, 3, 2, 8]. We call this operator the monotonic composition to refer to the fact that it is monotonic with respect to the refinement ordering (if A w A 0 then A 2 B refines A 0 2 B, whereas AB does not necessarily refines A 0 B) Other authors [1, 3, 2, 8] refer to it as the demonic composition ....

.... introduced and discussed by other researchers [1, 3, 2, 8] We call this operator the monotonic composition to refer to the fact that it is monotonic with respect to the refinement ordering (if A w A 0 then A 2 B refines A 0 2 B, whereas AB does not necessarily refines A 0 B) Other authors [1, 3, 2, 8] refer to it as the demonic composition because it captures demonic semantics of nondeterminacy: in angelic semantics (where sequence is captured by the relative product RR 0 ) a pair (s; s 0 ) is in RR 0 if and only if there exists (at least one) intermediate state t such that (s; t) 2 R ....

Rudolf Berghammer and Hans Zierer. Relational algebraic semantics of deterministic and nondeterministic programs. TCS, 43:123--147, 1986.

....is not acceptable, because the traditional relational product (represented by ffi) is not monotonic with respect to the refinement ordering; so that if we now refine R 0 by R 0 0 , we have no assurance that R 0 0 ffi R 1 refines R 0 ffi R 1 . Hence we introduce the demonic composition operator [3, 5], which captures the idea of sequential composition and is monotonic with respect to the refinement ordering: P 2 Q = PQ PQL : 6) In [9] it is shown how the definition of demonic composition can be obtained from flow diagrams [22] which are used to provide a general definition of ....

Berghammer, R. and H. Zierer. Relational Algebraic Semantics of Deterministic and Nondeterministic Programs. Theoretical Computer Science 43 (1986) 123--147.

....elements (i.e. pairing algebras [23, 1, 25] true pairing algebras [17, 23, 25, 24] full products [21] and direct products [21] There are also examples of heterogeneous relational specifications of structures that involve projection elements . i.e. heterogeneous) direct products [26, 5, 2, 13, 30, 27, 3] and specification of stacks [2, 4] In general these examples can be translated to homogeneous algebras with relational reduct in RA and with projection elements . 203 L. J. of the IGPL, Vol. 6 No. 2, pp. 203 226 1998 c # Oxford University Press 204 Some Classes Containing a Fork Algebra ....

....#) 1#1 # ) #) # 1 # . Because #and# are quasiprojections by [28, 4.1(viii) p.97] and 2.14.3 (r; s) t; q) r; #) t; #) #; s) #; q) r#t) s#q) The following relational specification of direct products is presented in [2, Section 4.2, p. 9] See also [27] and [5]) Spec PROD # param types m, n target types prod rels # : prod # m and # : prod # n laws # T # = I, # T # = I, ## T # ## T = I and xdiv # T # = L end This specification can be translated into a first order description of a class of algebras. In [3] this approach to products ....

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R. Berghammer and H. Zierer. Relational algebraic semantics of deterministic and nondeterministic programs. Theoretical Computer Science, 43:123--147, 1986.

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R. Berghammer and H. Zierer. Relational algebraic semantics of deterministic and nondeterministic programs. Theoretical Computer Science, 43:123--147, 1986.

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Berghammer R., Zierer H.: Relational algebraic semantics of deterministic and nondeterministic programs. Theoret. Comput. Sci. 43, 123--147 (1986)

....programming. The development starts with the work of de Bakker and de Roever in the early 70 s; see [12, 13] for example. In the following decade, e.g. Hoare and He [17] related the work of Birkhoff on residuals with Dijkstra s weakest precondition approach to programming, a group in Munich (see [23, 6, 32, 31]) constructed semantic domains by relation algebraic means and, thus, was able to treat also languages with higher order functions, and a group in Eindhoven (see [2] developed a theory of data types based on the calculus of relations. At this point also the approach of a group in Rio should be ....

Berghammer R., Zierer H.: Relational algebraic semantics of deterministic and nondeterministic programs. Theoret. Comput. Sci. 43, 123--147 (1986)

....into programs. In the concluding section, we indicate further applications of the ideas underlying this paper. 2 The axiomatic version of the relational calculus was developed by Tarski and his co workers (see, for example, 25, 7] Some applications to computer science can be found in [2, 22, 5, 13, 1]. 2 Boolean Operations, Sequence and Quotients The hardest task in modelling reactive systems is in choosing the right set of possible observations. Rather than fix a particular domain of observations, such as set of pairs as in the relational calculus, as set of words as in the case of regular ....

Berghammer R., Zierer H.: Relational algebraic semantics of deterministic and nondeterministic programs. Theoretical Computer Science 43, 123-147 (1986)

....therefore best avoided. For ease of presentation we have considered only condition event nets in this paper, but other types of Petri nets can be explored in a similar way. In this context it is important to know that the natural numbers can be axiomatized very naturally within relational algebra [1], so that places with multiple tokens can be modelled. 8 ....

Berghammer R., Zierer H.: Relational algebraic semantics of deterministic and nondeterministic programs. Theoret. Comput. Sci. 43, 123-147 (1986)

....fact theorems of the sequential calculus. By carefully choosing the space of all possible observations we can instanti 1 The axiomatic version of the relational calculus was developed by Tarski and his co workers (see, for example, 26,7] Some applications to computer science can be found in [2,23,5,13,1]. ate the sequential calculus to a subset of CSP, which may then be enriched with additional operators and axioms. However, the construction of a design calculus for CSP is a very ambitious project, and a first approach must make some simplifying assumptions. The program calculus of CSP started ....

Berghammer R., Zierer H.: Relational algebraic semantics of deterministic and nondeterministic programs. Theoretical Computer Science 43, 123-147 (1986)

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R. Berghammer and H. Zierer. Relational algebraic semantics of deterministic and nondeterministic programs. Theoretical Computer Science, 43:123--147, 1986.

No context found.

R. Berghammer and H. Zierer. Relational Algebraic semantics of deterministic and nondeterministic programs. Theoretical Computer Science, 43:123--147, 1986.

No context found.

R. Berghammer and H. Zierer, Relational algebraic semantics of deterministic and nondeterministic programs, Theoretical Computer Science43 (1986), 123-- 147.

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Rudolf Berghammer, H. Zierer, Relational algebraic semantics of deterministic and nondeterministic programs, Theoretical Computer Science 43 (1986), 123--147.

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