## Algebra of Flownomials; Part 1: Binary Flownomials; Basic Theory

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@MISC{Stefanescu_algebraof,

author = {Gheorghe Stefanescu},

title = {Algebra of Flownomials; Part 1: Binary Flownomials; Basic Theory},

year = {}

}

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### Abstract

' morphism for connecting flowgraphs are used in [CaU82] and in all of our subsequent papers on flowchart schemes and flownomials, see [Ste87a, Ste87b, CaS88a, CaS90a, CaS92]. This chapter folows Chapter B, sec. 3--6 of [Ste91]. The main result is based on a series of papers dealing with the algebraization of flowchart schemes, including [CaU82, BlEs85, Ste86/90, Bar87a, CaS88a, CaS90b]. With different sets of operators various algebras for flowgraphs appear in [Mil79, Parr87, CaS90b, CaS88b]. In the classical algebraic calculus for regular languages it is often the case that certain abstract semirings are used instead of the Boolean f0; 1g semiring, e.g. by using formal series with such coefficients. 5 This property is similar to the universal property of the polynomials over a ring. Chapter 6 Graph isomorphism with various constants In this chapter we extend the axiomatistion for flowgraphs modulo isomorphism to the case where more constants for generating relations are present i...

### Citations

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Citation Context ...es modulo bisimulation. 2 113 C R9 O R S C O W V OUR 9.2 Short comments and references Bisimulation is a standard equivalnce used in the algebraic studies on concurent processes. It was introduced in =-=[Park80]-=- in connection with Milner's work on concurrency [Mil80, Mil89]. See also [BeK84, BaW90, BenT89, BlET93]. The main result of this chapter is based on [BeS93, BeS94a] where it is shown that bisimulatio... |

478 |
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Citation Context ...uced in [Law63] and widely used in the study of the semantic of programs, e.g. in [Elg75, Elg76a,WTWG76, GTWR77, ElBT78, ArM80, Man92, BlEs93a]. The term symmetric strict monoidal category is used in =-=[MacL71]-=-. Similar strucutres, called x-categories were used by Hotz, see [Hot65, Mol88]. In the field of flowchart theories ssms'c are used is [ElS82, Ste86, Bar87a, CaS88a, CaS90a]. In logic ssmc's are used ... |

374 |
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Citation Context ...i S , then they specify the same morphism when they are interpreted in an arbitrary aff-flow. 4.5 Short comments and references The first axiomatic looping operation was `Kleene's star' introduced in =-=[Kle56]-=- and used as a key operation of regular algebras, see [Con71]. Another axiomatic looping operation is `Elgot dagger' introduced in [Elg75] as the key operation of iterative algebraic theories. The pre... |

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Citation Context ...axiomatic looping operation was `Kleene's star' introduced in [Kle56] and used as a key operation of regular algebras, see [Con71]. Another axiomatic looping operation is `Elgot dagger' introduced in =-=[Elg75]-=- as the key operation of iterative algebraic theories. The present axiomatic feedback `uparrow' was introduced in [Ste86/90]. [Looping operations similar to our feedback may be found in various places... |

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35 | Functional specification of time-sensitive communicating systems - Broy - 1993 |

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29 | The lattice of flow diagrams - Scott - 1971 |

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Towards a new algebraic foundation of flowchart scheme theory
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Citation Context ... used, then the possibility to get an algebra for flowgraphs themselves is lost.] The normal form flownomial expressions are introduced in [Ste86, Ste86/90]. In the general form they are presented in =-=[CaS90a]-=-. The presentation from this chapter follows the one in Chapter B, sec. 1--2 of [Ste91]. Chapter5 Algebra of flowgraphs (aff--flow) In this chapter we justify our claim that aff-flownomials represent ... |

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Citation Context ...heories. The present axiomatic feedback `uparrow' was introduced in [Ste86/90]. [Looping operations similar to our feedback may be found in various places, e.g. in [Con71] (the `linear mechanism') or =-=[Bai76]-=-, but not in an axiomatic setting using ssmc's. If stronger structures then ssmc's are used, then the possibility to get an algebra for flowgraphs themselves is lost.] The normal form flownomial expre... |

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Classes of finite relations as initial abstract data types
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- 1994
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Citation Context ...g and y 2 fff# fi# fl# ffig the algebra xy-IRel S of Ssorted xy-relations is an initial object in the category xy-SSMC Ss. 2 2.7 Short comments and references The results of this chapter are based on =-=[CaS91]-=-. The presentation we have given follows the slightly improved version presented in Chapter A of [Ste91]. In [Laf92] a confluent and terminating rewriting system is given for affi-terms (functions) . ... |

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7 |
diagrams and 2-dimensional rewriting
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- 1992
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Citation Context ...MC Ss. 2 2.7 Short comments and references The results of this chapter are based on [CaS91]. The presentation we have given follows the slightly improved version presented in Chapter A of [Ste91]. In =-=[Laf92]-=- a confluent and terminating rewriting system is given for affi-terms (functions) . Chapter 3 Critical acyclic structures (strong xy-ssmc's) In this chapter we introduce three critical structures that... |

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Partially additive categories and flow-diagram semantics
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Citation Context ...he dual one. 2 C R 7 OWGR V OURS# G R C S 0 7.4 Short comments and references This chapter is a technical one. It is based on Chapter C of [Ste91]. The enzymatic rule via general functions is used in =-=[ArM80]-=- under the name `functoriality rule', a name which was also used in our previous papers. In an implicit way the enzymatic rule for functions appear in the axiomatization of iteration theories in [Esi8... |

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Citation Context ...pter D, sec. 1--3 of [Ste91]. In the particular case of flowgraphs connected by partial functions a stronger result is provided by the equational axiomatisation of regular trees obtained by Esik, see =-=[Esi80]-=-. Esik's ideea was to replace the enzymatic axiom for functions by an weaker equational version. A problem with Esik's axiom is the fact that it is a difficult, global one and up to now there is no be... |

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Structured programming with and without go to statements
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Citation Context ...c \Phi? d ) (I a \Phi? b ) \Delta f \Delta (? c \Phi I d ) (? a \Phi I b ) \Delta f \Delta (I c \Phi? d ) (? a \Phi I b ) \Delta f \Delta (? c \Phi I d ) ! : We mention here the following result (see =-=[Elg76a]-=-). Let B be a matrix theory over (IN# +# 0). On the set B(1# 1) we introduce the operations f [ g =s1 \Delta (f \Phi g) \Delta 1# f \Delta g = f \Delta g# 0=? 1 \Delta? 1# and 1=I 1 : So we get a semi... |

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On flowchart theories II: The nondeterministic case
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- 1987
(Show Context)
Citation Context ...n this first part we present the kernel of the calculus of binary flownomials. Complete proofs are given for all the results, except for some commuting lemmas used in Chapter 11 which may be found in =-=[Ste87b]-=- and for which I have no better proofs. A few references are included. More refernces and connections with related topics, applications, particular models and equivalent presentations for various alge... |