G. W. Schmidt and T. Strohlein. Relations and Graphs: Discrete Mathematics for Computer Scientists. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....For all morphisms # : X # Y , # : Y # Z and # : X # Z the Dedekind formula ## # # # #(# # # # #) holds. D4. Sub Distributivity] The composition preserves order: If # # # # and # # # # , then ## # # # # # . # The fundamental properties of relational categories is referred to [1, 2, 9, 5]. The following is a basic property of allegories. Proposition 1. Let #, # # : X # Y , #, # # : Y # Z and # : Y # X be morphisms in an allegory A. If ## = id X and ## = id Y , then # = # # . # A morphism # : X # Y is total if id X # ## # (or equivalently, ## Y X = #XX ) A morphism f ....

....e. p : total Note that qi # pi = t(i i)m where t = q#p : GG# GG is the twisting function. # In what follows we assume G = G, m, e, i) is a group in an allegory A. In relational calculus there are a few di#erent ways how to specify subobjects. For example, Schmidt and Strohlein [9] made use of vectors , to represent subobjects in relation algebras. We are going to use relations from a unit I into some object G; these do in fact satisfy the vector equation # II # = #. Definition 6. A nonempty relation # : I # G is a subgroup of G if #i # # and # # # # #. # Note ....

G. Schmidt and T. Strohlein, Relations and graphs -- Discrete Mathematics for Computer Science -- (SpringerVerlag, 1993).

....composition, non deterministic choice and iteration or xed point computation. The eld has been pioneered by Conway [5] in the context of the algebra of regular events. Besides formal languages and automata, Kleene algebras also arise, for instance, in the context of relation algebra (c.f. [20]) and logics, analysis and construction of programs (c.f. 11] We follow Kozen s de nition [13] These structures are called Kozen semirings in [3] For concise proofs of all statements in this text see [24] Some formal proofs with the Isabelle proof checker can be found in [25] A semiring is ....

.... as the re exive transitive closure operation, 0 as the empty, 1 as the identity relation and as set inclusion. The third and fourth statement collects some basic properties of the Kleene star and plus. Most of them are well known also for relation algebra, regular languages or automata (c.f. [5,9,13,20]) Lemma 3. Let A be a Kleene algebra. For all a; b; c 2 A, 1 = 1 ; 12) 1 a ; 13) a a = a ; 14) a a ; 15) a = a ; 16) ac cb ) a c cb ; 17) cb ac ) cb a c; 18) a b) a (ba ) 19) In particular, ab) ....

G. W. Schmidt and T. Strohlein. Relations and Graphs: Discrete Mathematics for Computer Scientists. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1993.

....models iteration or xed point computation. They have been pioneered by Conway [7] in the context the algebra of regular sets over a nite alphabet. Besides formal languages and automata, Kleene algebras or their substructures also arise, for instance, in the eld of relational algebra (c.f. [19]) and logics of programs (c.f. 10] We follow the de nition of Kozen in [14] A semiring is an algebra (A; 0; 1) of a set A, two binary operations of multiplication and addition and two constants 0 and 1 such that (i) A; 0) is a commutative monoid, ii) A; 1) is a monoid, iii) the ....

....Most of them are simple induction free computations that can be formalized using a complete deductive system built from universal Horn logic. The rst lemma presents general properties that are well known for Kleene algebras, regular languages or automata (c.f. 7, 14] or relation algebras (c. f [19]) Lemma 5.1. Let A be a Kleene algebra. For all a; b; c 2 A, 1 = 1 ; 29) 1 a ; 30) a a = a ; 31) a a ; 32) a = a ; 33) ac = cb ) a c = cb ; 34) ab) a = a(ba) 35) a a = aa ; 36) a b) a (ba ) a ....

G. W. Schmidt and T. Strohlein. Relations and Graphs: Discrete Mathematics for Computer Scientists. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1993.

....Introduction In 1941 Tarski [8] proposed a problem, that is, Is every relation algebra isomorphic to an algebra of all Boolean (ordinary) relations on a set . The positive answers of the question, called representation theorem for relation algebras, have been investigated. Schmidt and Strohlein [6, 7] gave a simple proof of the representation theorem for Boolean relation algebras satisfying Tarski rule and a point axiom. A representation theorem for fuzzy relation algebras satisfying a point axiom was proved by Kawahara and Furusawa [1] and categorical representation theorems of fuzzy ....

G. Schmidt and T. Strohlein, Relations and graphs -- Discrete Mathematics for Computer Science -- (Springer-Verlag, Berlin, 1993).

....modern algebraic study of (binary) relations, namely relational calculus, was begun by Tarski; see [12] for details of the history of the study of Boolean relation algebras. In [10] Tarski proposed a formalisation of Boolean relation algebras and their representation problem. Schmidt and Strohlein [8, 9] gave a simple proof of a representation theorem for Boolean relation algebras satisfying the (so called) Tarski rule and a point axiom. Dedekind categories [2] or allegories [3] provide a categorical framework for relational calculus. In relational calculus one calculates with relations in an ....

....of crisp relations different from that in [4, 5, 6] Also the set of all scalar relations is a complete distributive lattice, which is a sublattice of the relation algebra, and scalar relations represent membership values. The concept of point relations was introduced by Schmidt and Strohlein in [8, 9] in the context of applications of (Boolean) relation algebras to theories of graphs and programs, and it played an important role in proofs of representation theorems in [8, 4, 5] In this paper we define a strict point axiom by using our concepts of scalar relations and point relations, and ....

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G. Schmidt and T. Strohlein, Relations and Graphs -- Discrete Mathematics for Computer Science -- (Springer-Verlag, Berlin, 1993).

....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 [Kaw95, TG87] and so on. Orlowska described benefit from ....

.... [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 [Kaw95, TG87] and so on. Orlowska described benefit from applying relational calculus to all sorts of scientific research areas in just one sentence in chapter 6 of [BKS97] so clearly: The main advantages of the relational formalisation are uniformity and modularity. Actually, once ....

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Schmidt, G. and Strohlein, T.: Relations and Graphs -- Discrete Mathematics for Computer Science, Springer, Berlin, 1993.

....Zadeh s invention of the concept of fuzzy sets [19] Goguen [5] generalized the concepts of fuzzy sets and relations to taking values on arbitrary lattices. On the other hand, the theory of relations, namely relational calculus, has been investigated since the middle of the nineteen century, see [13, 16, 17] for more details. Almost all modern formalisations of relation algebras are affected by the work of Tarski [18] Mac Lane [12] and Puppe [15] exposed a categorical basis for the calculus of additive relations. Freyd and Scedrov [2] developed and summarized categorical relational calculus, which ....

G. Schmidt and T. Strohlein, Relations and graphs -- Discrete Mathematics for Computer Science -- (SpringerVerlag, Berlin, 1993).

....Although our key combinatorial technique in Section 4 seems to have quite broad applicability, we have not investigated this aspect of our work in any detail. Further indication of the range of current investigations into relations and relationcalculi may be found in, for example, the books [21] or [5] or the proceedings of the roughly annual RelMiCS conferences. 2 Preliminaries 2.1 De nitions De nition 1. The signature is composed of the binary operations intersection and composition ; usually written as concatenation) two unary operations converse ( and domain dom, and a ....

G. W. Schmidt and T. Strohlein. Relations and Graphs: Discrete Mathematics for Computer Scientists. EATCS Monographs on Theoretical Computer Science. SpringerVerlag, 1993.

....close to certain algebraic theories. Its main advantage is that it evades several difficult issues by using functions on a domain consisting of strings rather than function compositions. Relational semantics has been thoroughly explored by many authors. An excellent text on relational methods is [46]. More discussion of our particular version can be found in [ 32, 44] The internal module documentation model is more than 20 years old and has been reinvented by several researchers. Our effort has been to show how these well understood theories can be applied to the problem of documentation. ....

G. Schmidt and T. Strohlein, Relations and Graphs - Discrete Mathematics for Computer Scientists, Springer-Verlag, 301 pgs, 1993.

....been devoted within the graph transformation community to issues of tool support. A few notable exceptions are the systems GraphEd [4] PROGRES [8] Agg [5] and Treebag [3] Grammatica is a prototype implementation of double pushout algebraic graph transformation [2] based on relation algebra [1, 7]. It has originated in an investigation of the extent to which algebraic graph transformation can be expressed within relation algebra [6] and has been implemented using Mathematica [11] on top of the Combinatorica [9] package. Therefore, it runs on most platforms. The implementation takes the ....

G. Schmidt and T. Strohlein. Relations and Graphs: Discrete Mathematics for Computer Scientists. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1993.

....= find or add unique table (v,T ,E) insert computed table ( F , G, H , R) return R; Fig. 2. The ITE algorithm 4 Representation of Directed Graphs The approach to subgraph isomorphism presented in this paper is based on unlabeled, directed, simple graphs, also called relational graphs [2, 17]. Definition 1. A graph G = V, E) consists of a set V and a relation E # V V . The elements of V are called vertices, and E is called the (arc) relation associated to G. It is said that there is an arc from a node u to a node v if (u, v) # E. A graph G = V, B) is finite if the order V ....

....an isomorphic image of a given graph, called the pattern, in another graph, called the target. The name of this problem comes from the image of the pattern being then a subgraph of the target. Graph homomorphisms are structure preserving relations over the relations associated to the graphs [17, 22, 23]. In particular, a structure preserving relation between the sets of vertices of two graphs is a subgraph isomorphism if it is injective. Let I denote the identity relation. Definition 2. A relation # # V # V is an isomorphism of a graph G # = V # , E # ) into a graph G = V, E) if # T # ....

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G. Schmidt and T. Strohlein. Relations and Graphs: Discrete Mathematics for Computer Scientists. SpringerVerlag, 1993.

....event occurrences are what is really observed. However, one may abstract away from the output values, if states can be unambiguously described by sequences of call event occurrences only. 4 Relational Model of Programs We review the fundamentals of the relational model of programs (e.g. [30]) Data re nement is introduced according to [10] except that, rather than taking relations extended by a bottom element, demonic relational composition and demonic re nement is used. We write S T for the set of all relations between S and T , formally de ned as S T = 2 S T . For ....

G. Schmidt and T. Strohlein. Relations and Graphs: Discrete Mathematics for Computer Scientists. EATCS Monographs in Theoretical Computer Science. Springer-Verlag, Berlin ; New York, 1993.

....Goguen [2] generalized the concepts of fuzzy sets and relations taking values on partially ordered sets. Fuzzy relational equations were initiated by Sanchez [11] and applied to medical models of diagnosis. On the other hand theory of relations, namely relational calculus, has a long history, see [8, 12, 13] for more details on the history. Almost modern formalizations of relation algebras are affected by the work of Tarski [14] Mac Lane [7] and Puppe [10] exposed a categorical basis for calculus of additive relations. Freyd and Scedrov [1] developed and summarized categorical relational calculus, ....

....[4] have been already given by [1] Concerning with applications to the relational theory of graphs and programs, Schmidt and Strohlein [12] gave a simple proof of a representation theorem for Boolean relation algebras satisfying Tarski rule and point axiom. Also they wrote an excellent text book [13] on relations and graphs with many useful examples in computer science. The aim of the paper is to provide an algebraic formalization for fuzzy relations. Fuzzy relations treated here are homogeneous ones on a set X with values in the unit interval [0; 1] that is, functions R : X2X [0; 1] The ....

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G. Schmidt and T. Strohlein, Relations and graphs -- Discrete Mathematics for Computer Science -- (Springer-Verlag, Berlin, 1993).

....parameters to be used in (2) are still work in progress. Locality of Computations A group of computations within an algorithm is said to have a high degree of locality if the algorithm level representation of those computations corresponds to an isolated, strongly connected (sub)graph. [18] More informally, the largest the volume of data being transferred among the nodes belonging to the cluster , in comparison with the volume of data entering and exiting the cluster, the highest will be the degree of locality. 10 By indeed considering such strongly connected sub graphs as ....

G. Schmidt, T. Strohlein. Relations and Graphs - Discrete Mathematics for Computer Scientists. Springer-Verlag, 1993.

....Goguen [2] generalized the concepts of fuzzy sets and relations taking values on partially ordered sets. Fuzzy relational equations were initiated and applied to medical models of diagnosis by Sanchez [12] On the other hand theory of relations, namely relational calculus, has a long history, see [8, 13, 14] for more details on the history. Almost modern formalizations of relation algebras are affected by the work of Tarski [15] Mac Lane [7] and Puppe [11] exposed a categorical basis for calculus of additive relations. Freyd and Scedrov [1] developed and summarized categorical relational calculus, ....

....which they called allegories. Concerning with applications to the relational theory of graphs and programs, Schmidt and Strohlein [13] gave a simple proof of a representation theorem for Boolean relation algebras satisfying Tarski rule and point axiom. Also they wrote an excellent text book [14] on relations and graphs with many useful examples in computer science. Also Kawahara and Mizoguchi [3, 6, 4] developed relational methodology for assertion semantics of programs, theory of graph transformations (or graph grammars) and categorical set theory. A representation theorem for ....

G. Schmidt and T. Strohlein, Relations and graphs -- Discrete Mathematics for Computer Science -- (Springer-Verlag, Berlin, 1993).

....[5] generalized the concepts of fuzzy sets and relations taking values from partially ordered sets. Fuzzy relational equations were initiated and applied to medical models of diagnosis by Sanchez [17] On the other hand, the theory of relations, namely relational calculus, has a long history, see [13, 18, 19] for more details. Almost all modern formalizations of relation algebras are affected by the work of Tarski [20] Mac Lane [12] and Puppe [16] exposed a categorical basis for the calculus of additive relations. Freyd and Scedrov [2] developed and summarized categorical relational calculus, which ....

....which they called allegories. Concerning applications to the relational theory of graphs and programs, Schmidt and Strohlein [18] gave a simple proof of a representation theorem for Boolean relation algebras satisfying the Tarski rule and the point axiom. They also wrote an excellent text book [19] on relations and graphs with many useful examples from computer science. In relational calculus one calculates with relations in an element free style, which makes relational calculus a very useful framework for the study of mathematics [8] and theoretical computer science [1, 7, 11] and also a ....

G. Schmidt and T. Strohlein, Relations and graphs -- Discrete Mathematics for Computer Science -- (Springer-Verlag, Berlin, 1993).

....of set theory and since then theory of relations has been extensively investigated by many mathematicians from the view points of logic, algebra, topology and computer science. For more detailed history of studies on relations the reader refer to R.D. Muddux [14] and G. Schmidt and T. Strohlein [16]. From a view of category theory S. Mac Lane [11, 12] initiated theory of additive relations and D. Puppe [15] established a notion of I categories that was a start point of categorical theory of relations. Peter Freyd [3] investigated theory of allegories as a basis for theory of relations and ....

G. Schmidt and T. Strohlein, Relations and graphs -- Discrete Mathematics for Computer Science --, Springer-Verlag, 1993.

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G. W. Schmidt and T. Strohlein. Relations and Graphs: Discrete Mathematics for Computer Scientists. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1993.

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Schmidt, G., Strohlein, T.: Relations and Graphs; Discrete mathematics for Computer Scientists. Springer-Verlag, Berlin Heidelberg New York (1991)

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G. W. Schmidt and T. Strohlein. Relations and Graphs: Discrete Mathematics for Computer Scientists. EATCS Monographs on Theoretical Computer Science. Springer, 1993.

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G. Schmidt and T. Strohlein, Relations and graphs -- Discrete Mathematics for Computer Science -- (Springer-Verlag, Berlin, 1993).

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G. W. Schmidt and T. Strohlein. Relations and Graphs: Discrete Mathematics for Computer Scientists. EATCS Monographs on Theoretical Computer Science. Springer, 1993. 20

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G. Schmidt and T. Strohlein. Relations and Graphs - Discrete Mathematics for Computer Scientists. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1991.

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