## Synthesis of mealy machines using derivatives (2006)

Venue: | In: Proceedings of CMCS 2006. ENTCS |

Citations: | 9 - 5 self |

### BibTeX

@INPROCEEDINGS{Hansen06synthesisof,

author = {Helle Hvid Hansen and David Costa},

title = {Synthesis of mealy machines using derivatives},

booktitle = {In: Proceedings of CMCS 2006. ENTCS},

year = {2006},

pages = {2006},

publisher = {Elsevier}

}

### OpenURL

### Abstract

In Rutten [13] the theoretical basis was given for the synthesis of binary Mealy machines from specifications in 2-adic arithmetic. This construction is based on the symbolic computation of the coalgebraic notion of stream function derivative, a generalisation of the Brzozowski derivative of regular expressions. In this paper we complete the construction of Mealy machines from specifications in both 2-adic and modulo-2 arithmetic by describing how we decide equivalence of expressions via reduction to normal forms; we present a Haskell implementation of this Mealy synthesis algorithm; and a theoretical result which characterises the (number of) states in Mealy machines constructed from rational 2-adic specifications.

### Citations

334 |
On the Synthesis of a Reactive Module
- Pnueli, Rosner
- 1989
(Show Context)
Citation Context ...ng derivatives of ω-regular expressions, albeit in a much more complex setting. Synthesis of Mealy (or Moore) type automata from logic specifications has a long and well-established history, see e.g. =-=[3,10,15,7]-=-. Themainideahereisthata logic formula ϕ specifies a relation Rϕ between input and output streams, and from ϕ one can construct an automaton Aϕ which essentially accepts Mealy machines whose stream fu... |

281 |
Switching and finite automata theory
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(Show Context)
Citation Context ...r.com/locate/entcs Mealy machines are finite state transducers used in the modelling and specification of systems performing synchronous, ongoing computations such as sequential digital circuits (cf. =-=[9]-=-), and more generally, reactive systems (see e.g. [14]). Synthesis 1 Email: hhhansen@few.vu.nl 2 Email: costa@cwi.nl. Supported by FCT grant 13762 – 2003, Portugal. 3 Email: janr@cwi.nl 1571-0661/$ – ... |

209 | Derivatives of regular expressions
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- 1964
(Show Context)
Citation Context ... Theoretical Computer Science 164 (2006) 27–45 41 5 The Construction Our method for constructing a Mealy machine from a given bitstream specification θ can be seen as a generalisation of Brzozowski’s =-=[2]-=- method for constructing deterministic finite automata from regular expressions. Starting from the specification θ, we compute for each bit a ∈ 2, the transitions corresponding with input a, and itera... |

137 |
Solving sequential conditions by finite-state strategies
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(Show Context)
Citation Context ...ng derivatives of ω-regular expressions, albeit in a much more complex setting. Synthesis of Mealy (or Moore) type automata from logic specifications has a long and well-established history, see e.g. =-=[3,10,15,7]-=-. Themainideahereisthata logic formula ϕ specifies a relation Rϕ between input and output streams, and from ϕ one can construct an automaton Aϕ which essentially accepts Mealy machines whose stream fu... |

59 | Partial derivatives of regular expressions and finite automata constructions
- Antimirov
- 1995
(Show Context)
Citation Context ...d over a single bitstream variable σ by an arithmetic signature which contains binary function symbols for addition, multiplication and division, a unary function symbol for minus, and constants [0], =-=[1]-=-, X n for n ∈ N. The constants are in both cases interpreted as the following bitstreams: [0] = (0, 0, 0,...), [1] = (1, 0, 0,...), X n =(0,...,0, 1, 0, 0,...). � �� � n times Thus as is standard, X 0... |

58 |
Automata Languages and Machines, Vol
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(Show Context)
Citation Context ... Mealy machine is an initialised Mealy coalgebra (Q, φ, q0) in which the set of states Q, and the input/output sets A and B are finite. Mealy machines are also referred to as sequential machines (cf. =-=[4]-=-), and rather than being a language recognition device, a Mealy machine is a so-called deterministic transducer, i.e., it transform input streams to output streams in a deterministic manner. This inpu... |

50 | Behavioural differential equations: a coinductive calculus of streams, automata, and power series, Theoret
- Rutten
- 2003
(Show Context)
Citation Context ...results on rational 2-adic functions were conjectured based on data generated by our Haskell program. In section 2 we introduce basic notions together with the coalgebraic view on Mealy machines (cf. =-=[12,13]-=-), and in section 3 we present the bitstream algebras that we use as specification languages. Parts of these preliminary sections are already contained in [13], but in the current presentation we are ... |

35 | An automata-theoretic approach to fair realizability and synthesis
- Vardi
- 1995
(Show Context)
Citation Context ...ng derivatives of ω-regular expressions, albeit in a much more complex setting. Synthesis of Mealy (or Moore) type automata from logic specifications has a long and well-established history, see e.g. =-=[3,10,15,7]-=-. Themainideahereisthata logic formula ϕ specifies a relation Rϕ between input and output streams, and from ϕ one can construct an automaton Aϕ which essentially accepts Mealy machines whose stream fu... |

23 | On circuits and numbers
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(Show Context)
Citation Context ...this structure is its relevance for sequential binary arithmetic and digital circuits. Not much literature seems to be available on this subject, with the exception of the work by Vuillemin, see e.g. =-=[16]-=-. The other bitstream algebra is based on addition modulo-2, and it is also motivated by its connection to digital circuits, and switching theory [9], in particular, to the theory and design of linear... |

22 |
p-adic Numbers
- Gouvêa
- 1993
(Show Context)
Citation Context ...the set {0, 1} as well as the integer 2. The context should make clear which reading is intended. The first bitstream algebra to be represented is based on the arithmetic operations on 2-adic numbers =-=[5]-=-. The motivation for studying this structure is its relevance for sequential binary arithmetic and digital circuits. Not much literature seems to be available on this subject, with the exception of th... |

19 | Algebraic specification and coalgebraic synthesis of mealy automata, in: L.B. Zhiming Liu (Ed
- Rutten
(Show Context)
Citation Context ...079, NL–1090 GB Amsterdam, Netherlands Jan Rutten 3 Centrum voor Wiskunde en Informatica (CWI) and Free University Amsterdam (VUA) P.O. Box 94079, NL–1090 GB Amsterdam, Netherlands Abstract In Rutten =-=[13]-=- the theoretical basis was given for the synthesis of binary Mealy machines from specifications in 2-adic arithmetic. This construction is based on the symbolic computation of the coalgebraic notion o... |

9 | Synthesis of uninitialized systems
- Henzinger, Krishnan, et al.
- 2002
(Show Context)
Citation Context |

8 | Final semantics for eventpattern reactive programs
- Sánchez, Sipma, et al.
- 2005
(Show Context)
Citation Context ...r, and it would be interesting to see if this technique can be generalised to other specification languages and automaton types than the ones already mentioned. Of particular interest, we mention PAR =-=[14]-=- which is a recently introduced declarative language for the specification of event-pattern reactive programs, a certain type of Mealy machines. The behaviour of PAR programs is defined corecursively,... |

4 | Using proofs by coinduction to find ”traditional” proofs - Grabmayer - 2005 |

3 |
Construction of a deterministic ω-automaton using derivatives
- Redziejowski
- 1999
(Show Context)
Citation Context ...s from regular expressions. Other related work includes Antimirov [1] in which partial derivatives are used in constructing nondeterministic finite automata from regular expressions, and Redziejowski =-=[11]-=- who constructs ω-automata using derivatives of ω-regular expressions, albeit in a much more complex setting. Synthesis of Mealy (or Moore) type automata from logic specifications has a long and well-... |