## 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... |

537 |
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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 ... |

402 |
Representations of events in nerve nets and finite automata. Automata Studies, .:pp 3–41
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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 ...preted 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 present axiomatic feedback `uparrow' was introduced in [Ste86/90... |

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Citation Context ...R S (st o g xy SS CS) 5 3.4 Short comments and references Matrix theories are classical structures. (See [Elg76b] for their relationship with algebraic theories.) Algebraic theories are introduced 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]. Simil... |

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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... |

47 | Relations and Graphs - Schmidt, Ströhlein - 1993 |

38 | On the algebraic structure of rooted trees - Elgot, Bloom, et al. - 1978 |

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

33 | The lattice of flow diagrams - Scott |

30 | Flow graphs and flow algebras - Milner - 1979 |

23 |
Towards a new algebraic foundation of flowchart scheme theory
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- 1990
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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 ... |

21 | Theory of program structures: schemes, semantics, verification - Greibach - 1975 |

17 | Algebra of Proofs - Szabo - 1978 |

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15 |
Feedback and generalized logic
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- 1976
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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... |

15 | Feedback Theories (A Calculus for Isomorphism Classes of Flowchart Schemes - Ştefănescu - 1986 |

14 | The algebra of stream processing functions - Stefanescu - 1996 |

13 | Regular trees and the free iterative theory - Ginali - 1979 |

11 | Solutions of the iteration equation and extensions of the scalar iteration operation - Bloom, Elgot, et al. - 1980 |

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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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8 |
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Citation Context ...nt, namely the "matrix" with 0 rows and n columns# similarly for n =0. C R 3 CR C C C C S RUC UR S (st o g xy SS CS) 5 3.4 Short comments and references Matrix theories are classical structu=-=res. (See [Elg76b]-=- for their relationship with algebraic theories.) Algebraic theories are introduced in [Law63] and widely used in the study of the semantic of programs, e.g. in [Elg75, Elg76a,WTWG76, GTWR77, ElBT78, ... |

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7 |
diagrams and 2-dimensional rewriting
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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... |

6 | Fixed points in free process algebras - Benson, Tiuryn |

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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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5 | An algebraic theory of flowchart schemes - Stefanescu - 1986 |

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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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3 | Synchronisation flow algebra - Parrow - 1987 |