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Algebra of Flownomials; Part 1: Binary Flownomials; Basic Theory
"... ' 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. 36 of [Ste91]. The main result is based on a series of papers dealing with the algebra ..."
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' 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. 36 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...
An InitialAlgebra Approach to Directed Acyclic Graphs
, 1995
"... . The initialalgebra approach to modelling datatypes consists of giving constructors for building larger objects of that type from smaller ones, and laws identifying di#erentways of constructing the same object. The recursive decomposition of objects of the datatype leads directly to a recursiv ..."
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. The initialalgebra approach to modelling datatypes consists of giving constructors for building larger objects of that type from smaller ones, and laws identifying di#erentways of constructing the same object. The recursive decomposition of objects of the datatype leads directly to a recursive pattern of computation on those objects, whichisvery helpful for both functional and parallel programming. We showhow to model a particular kind of directed acyclic graph using this initialalgebra approach. Keywords. Graphs, data types, catamorphisms, initial algebras, BirdMeertens Formalism, program derivation. 1 Introduction It is now widely recognized that the traditional adhoc approaches to program construction do not yield reliable software; a more systematic and formal approach is required. One such approach consists of program veri#cationproving after the fact that a given program satis#es its formal speci#cation. This approach turns out to be di#cult to implement, not lea...
A Hierarchy Preserving Hierarchical BottomUp 2Layer Wiring Algorithm With Respect to Via Minimization
, 1993
"... Here, the following hierarchical constrained via minimization problem (HCVM) is examined: "Let Q be a circuit with a given layout hierarchy. Find a 2layer wiring of Q which needs a number of vias minimal with respect to the preservation of hierarchy, i.e., on condition that the description of the ..."
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Here, the following hierarchical constrained via minimization problem (HCVM) is examined: "Let Q be a circuit with a given layout hierarchy. Find a 2layer wiring of Q which needs a number of vias minimal with respect to the preservation of hierarchy, i.e., on condition that the description of the result is (nearly) as short as the description of Q before the 2layer wiring". The problem arises in connection with hierarchical physical synthesis, which is not highly developed yet although absolutely essential for the design of ULSI circuits. The computational complexity of HCVM is unknown till today. This paper presents a hierarchical bottomup algorithm to (a variant of) this problem which is locally optimal, i.e., given the 2layer wirings of the subcircuits of level i, it computes the optimal (partially induced by the wirings of the subcircuits) 2layer wiring of the circuits of level i+1. The algorithm's running time is less than O(n 3 ) where n is the size of the hierarchical ...