## Algebraic specification and coalgebraic synthesis of Mealy machines (2006)

Venue: | In: Proceedings of FACS 2005. ENTCS |

Citations: | 19 - 7 self |

### BibTeX

@TECHREPORT{Rutten06algebraicspecification,

author = {J. J. M. M. Rutten},

title = {Algebraic specification and coalgebraic synthesis of Mealy machines},

institution = {In: Proceedings of FACS 2005. ENTCS},

year = {2006}

}

### Years of Citing Articles

### OpenURL

### Abstract

We introduce the notion of functional stream derivative, generalising the notion of input derivative of rational expressions (Brzozowski 1964) to the case of stream functions over arbitrary input and output alphabets. We show how to construct Mealy automata from algebraically specified stream functions by the symbolic computation of functional stream derivatives. We illustrate this construction in full detail for various bitstream functions specified in the algebraic calculus of the 2-adic numbers. This work is part of a larger ongoing effort to specify and model component connector circuits in terms of (functions and relations on) streams.

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Citation Context ...ebras, for instance as used in the context of functional programming languages. Digital circuits: It is well-known how to construct a digital circuit out of a binary Mealy automaton (see for instance =-=[14]-=- for a classical reference). Thus one can start with an algebraic specification of a binary stream function, then use (a suitable implementation of) our construction to obtain a Mealy automaton, and f... |

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Citation Context ...or circuits in terms of (functions and relations on) streams. Keywords: Stream function, functional stream derivative, Mealy automaton, bitstream, binary arithmetic, 2-adic integer. 1 Introduction In =-=[7]-=-, Brzozowski showed how to construct a deterministic finite automaton for a rational expression by computing its finitely many derivatives, herewith lifting the well-known fact that rational languages... |

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Citation Context ...on. (Other types of operators will be mentioned in the conclusions.) Let 2 ω be the set of bitstreams: infinite sequences of 0’s and 1’s. Before introducing the 2-adic operators, we first recall (cf. =-=[13,10]-=-) that bitstreams are also known as 2-adic numbers, since they can be viewed as the binary representations of ordinary numbers. For rational numbers with odd denominator, this works as follows. We def... |

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Citation Context ...gital circuits. Here we want to point out that the present work is part of a broader ongoing research effort to develop models and specification formalisms for component connector circuits. In [5] and=-=[4]-=-, we used relations on streams and so-called constraint automata as models of Arbab’s [2] component connector calculus Reo. There we showed how to go from connector circuits to (relations on) streams ... |

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Citation Context ...n of digital circuits. Here we want to point out that the present work is part of a broader ongoing research effort to develop models and specification formalisms for component connector circuits. In =-=[5]-=- and[4], we used relations on streams and so-called constraint automata as models of Arbab’s [2] component connector calculus Reo. There we showed how to go from connector circuits to (relations on) s... |

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Citation Context ...ns. Since then, various applications and generalisations have been studied. In [1], Antimirov introduced the notion of partial derivative and used it to construct nondeterministic finite automata. In =-=[19,20]-=-, we reformulated Brzozowski’s original approach in coalgebraic terms and generalised it to formal power series over arbitrary semirings, providing at the same time a generalisation of Antimirov’s res... |

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Citation Context ...on. (Other types of operators will be mentioned in the conclusions.) Let 2 ω be the set of bitstreams: infinite sequences of 0’s and 1’s. Before introducing the 2-adic operators, we first recall (cf. =-=[13,10]-=-) that bitstreams are also known as 2-adic numbers, since they can be viewed as the binary representations of ordinary numbers. For rational numbers with odd denominator, this works as follows. We def... |

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Citation Context ... minimal automaton. In this way, one can use, for instance, Theorem 5.1 to compute lowerbounds for the numbers of registers needed to implement arithmetic functions (not much is known about this, cf. =-=[12,22]-=-). Implementation: For algebras such as that of the 2-adic numbers, the construction of Mealy automata from algebraic expressions can in many cases be automated. For this, one has to be more precise a... |

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Citation Context ...here does not seem to exist much literature on the construction of Mealy automata from algebraically specified stream functions. Most approaches use logical specification formalisms (see for instance =-=[6,11]-=-). There one starts with a logically specified relation (sometimes called stream requirement) on input and output streams, and the goal is to find at least one Mealy automaton (out of many possible on... |

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Citation Context ...e same time a generalisation of Antimirov’s results. A similar generalisation to formal power series and rational expressions with multiplicities was found, independently, by Lombardy and Sakarovitch =-=[15,16]-=-. Here we present yet another generalisation of Brozowski’s construction. We look at deterministic Mealy automata [9], with inputs and outputs over arbitrary alpha1 Email: janr@cwi.nl Electronic Notes... |

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Citation Context ...here does not seem to exist much literature on the construction of Mealy automata from algebraically specified stream functions. Most approaches use logical specification formalisms (see for instance =-=[6,11]-=-). There one starts with a logically specified relation (sometimes called stream requirement) on input and output streams, and the goal is to find at least one Mealy automaton (out of many possible on... |

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7 |
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Citation Context ...uce the (semantic) notion of functional stream derivative (Section 2), which can be used to construct for any stream function a minimal Mealy automaton. Functional stream derivatives already occur in =-=[17]-=- under the name of state (of a sequential function). Also minimisation of Mealy automata is well known [9]. The main contribution of the present paper is the insight that functional stream derivatives... |

7 |
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Citation Context ...ns. Since then, various applications and generalisations have been studied. In [1], Antimirov introduced the notion of partial derivative and used it to construct nondeterministic finite automata. In =-=[19,20]-=-, we reformulated Brzozowski’s original approach in coalgebraic terms and generalised it to formal power series over arbitrary semirings, providing at the same time a generalisation of Antimirov’s res... |

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Citation Context ... = 1 inverse As usual, we shall write τ/σ for τ × (1/σ); note that 1/σ is defined only for σ with σ(0) = 1. It is not difficult to prove that this system uniquely determines these four operators (see =-=[21]-=- for details), and that they are causal. We briefly explain the form of these equations. The sum σ + τ is computed by elementwise addition but with the proviso that ‘overflow’ bits are carried over to... |

1 |
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Citation Context ...or circuits to automata. In addition, initial ideas how to construct circuits from automata (which amounts to a non-trivial generalisation of well-known techniques from logic design) are described in =-=[3]-=-. We intend to generalise the techniques of the present paper to the specification and symbolic construction of (constraint) automata for component connector circuits. 2 Mealy automata We give the bas... |

1 |
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Citation Context ...ions with multiplicities was found, independently, by Lombardy and Sakarovitch [15,16]. Here we present yet another generalisation of Brozowski’s construction. We look at deterministic Mealy automata =-=[9]-=-, with inputs and outputs over arbitrary alpha1 Email: janr@cwi.nl Electronic Notes in Theoretical Computer Science 160 (2006) 305–319 www.elsevier.com/locate/entcs 1571-0661/$ – see front matter © 20... |