An introduction to idempotency

Gunawardena, Jeremy

doi:10.1017/cbo9780511662508.003

Cited by 45 publications

(42 citation statements)

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“…For instance, in control theory, one looks for a controller C such that its parallel composition with a plant P satisfies a given specification S, that is, P C ⊆ S. It might be helpful to rewrite this into C ⊆ P −1 S. (4) Many more examples of specifications by behavioural differential equations and the corresponding implementations are needed, in order to get a better understanding of their usefulness. Notably, other tropical and idempotent semirings, as described in [Gun98], should be investigated from the present perspective. Another class of examples is to do with probabilistic nd-transition systems and Markov chains.…”

Section: Discussionmentioning

confidence: 99%

Automata, Power Series, and Coinduction: Taking Input Derivatives Seriously (Extended Abstract)

Rütten

1999

Automata, Languages and Programming

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REPORTRAPPORT AbstractFormal power series, which are functions from the set of words over an alphabet A to a semiring k, are viewed coalgebraically. In summary, this amounts to supplying the set of all power series with a deterministic automaton structure, which has the universal property of being final. Finality then forms the basis for both definitions and proofs by coinduction, the coalgebraic counterpart of induction. Coinductive definitions of operators on power series take the shape of what we have called behavioural differential equations, after Brzozowski's notion of input derivative, and include many classical differential equations for analytic functions. The use of behavioural differential equations leads, amongst others, to easy definitions of and proofs about both existing and new operators on power series, as well as to the construction of finite (syntactic) nondeterministic automata, implementing them.

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Section: Discussionmentioning

confidence: 99%

Automata, Power Series, and Coinduction: Taking Input Derivatives Seriously (Extended Abstract)

Rütten

1999

Automata, Languages and Programming

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“…For one thing, one can look at other fields (such as the complex numbers) or even arbitrary semirings (such as the Booleans or so-called tropical semirings (Gunawardena 1998)). The role of inverse will then be replaced by the…”

Section: ))mentioning

confidence: 99%

A coinductive calculus of streams

Rütten¹

2005

Math. Struct. Comp. Sci.

115

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We develop a coinductive calculus of streams based on the presence of a final coalgebra structure on the set of streams (infinite sequences of real numbers). The main ingredient is the notion of stream derivative, which can be used to formulate both coinductive proofs and definitions. In close analogy to classical analysis, the latter are presented as behavioural differential equations. A number of applications of the calculus are presented, including difference equations, analytical differential equations, continued fractions, and some problems from discrete mathematics and combinatorics.

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“…Topical functions are intensively studied (see [2,3]), and they have many applications in various parts of applied mathematics in particular in the modeling of discrete event system (see [2,3]). Topical functions are also interesting from a different point of view, namely as a tool in the study of the so-called downward sets.…”

Section: Introductionmentioning

confidence: 99%