Abstract. We present a novel coalgebraic logic for deterministic Mealy machines that is sound, complete and expressive w.r.t. bisimulation. Every finite Mealy machine corresponds to a finite formula in the language. For the converse, we give a compositional synthesis algorithm which transforms every formula into a finite Mealy machine whose behaviour is exactly the set of causal functions satisfying the formula. 1

CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics. CWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms.

In this paper, we describe a symbolic synthesis method which given an algebraic expression that specifies a bitstream function f, constructs a (minimal) Mealy machine that realises f. The synthesis algorithm can be seen as an analogue of Brzozowski’s construction of a finite deterministic automaton from a regular expression. It is based on a coinductive characterisation of the operators of 2-adic arithmetic in terms of stream differential equations. 1

Abstract. Viewing discrete-time causal linear systems as (Mealy) coalgebras, we describe their semantics, minimization and realisation as universal constructions, based on the final coalgebras of streams and causal stream functions. 1

A (binary) Mealy machine is a deterministic automaton which in each step reads an input bit, produces an output bit and moves to a next state. The induced mapping of input streams to output streams is a causal bitstream function, which we call the bitstream function realised by the Mealy machine. In this note, we describe a synthesis method which given an algebraic specification of a bitstream function f, constructs a minimal Mealy machine which realises f. In the design of digital hardware, Mealy machines specify the behaviour of sequential circuits, and there exist algorithms which construct from a (finite) Mealy machine, a sequential circuit which exhibits the specified behaviour. Combining these algorithms with our synthesis algorithm we thus obtain a complete construction from algebraic specification to sequential circuit. The inputs to our synthesis algorithm are called function expressions and they define bitstream functions in the algebra of 2-adic numbers. Here we use that the formal power series representation of a 2-adic integer can be seen as the bitstream of its coefficients. We describe the 2-adic algebra below, but for now such a function

CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics. CWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms.

Vol. 4 (3:9) 2008, pp. 1–22 www.lmcs-online.org