CONTENTS

ALGEBRAIC MODEL FOR THE SYNCHRONOUS SR-FLIP-FLOP BEHAVIOUR 237
3.2. The case of synchronous SR flip-flop. From the structural point of view,
in order to obtain the synchronous SR flip-flop we consider the asynchronous circuit
and we add an extra level of AND gates for involving the clock signal.
We consider two levels for specifying the synchronous SR flip-flop behaviour, a
specification and an implementation, and we conclude with the equivalence result for
these models.
We propose the higher-level specification for the synchronous SR flip-flop behaviour
based on the agents:
(3) SpecSRs(m, n, c) = (SpecInput(c) × SpecSR(m, n)) ↾ E SRmn(c)
where the set of external actions is E SRmn(c) = {CLKc, σ, ρ, γ, δ} for a specific
value of c ∈ {0, 1}. Note that the parameters m and n have complemented values, by
definition of the flip-flop.
We also propose the lower-level specification for the synchronous SR flip-flop
behaviour based on the agents:
(4) ImpSRs(m, n, c) = (ImpInput(c) × ImpSR(m, n)) ↾ E SRmn(c)
It is for the interest of this paper results to prove the next equivalence.
Proposition 1. The previous agents SpecSRs(m, n, c) and ImpSRs(m, n, c) for (m, n) ∈
{(0, 1), (1, 0)} and c ∈ {0, 1} are bisimulation equivalent.
The previous agents SpecSRs(m, n, c) and ImpSRs(m, n, c) are defined for all of
the binary combinations on the input lines S and R of the flip-flop. In order to extend
the algebraic model of the flip-flops behaviour to the algebraic model of the more complex
sequential logic circuits based on flip-flops, it is useful to have the projection of
such an agent on only one input binary combination (S, R). For achieving this target,
we have to project each of the generic agents SpecSRs(m, n, c) and ImpSRs(m, n, c) on
each of the binary combinations (S, R) ∈ {(0, 0), (0, 1), (1, 0), (1, 1)}. The final agents
are SpecCBBSRs(m, n, S, R, c) and ImpCBBSRs(m, n, S, R, c) for each current state
(m, n), input binary combination (S, R) and clock signal c. These agents are also
bisimulation equivalent.
3.3. Automatic verification. In order to have an automatic verification for the
bisimilarities between the corresponding specification and implementation agents already
defined in the previous subsections, we have developed the SCCS files with
respect to the CWB-NC platform syntax. For each of the Spec-Impl pair of agents,
we have verified the appropriate bisimilarity. The corresponding CWB-NC answer is
TRUE for all of these tests. We have also obtained successfull CWB-NC automatic
verification for the corresponding implementation agents and for the bisimilarities
tests.
4. Conclusions
Based on ideas from [3, 5], in this paper we have considered the internal structure
of specific memory cells, namely SR flip-flops. As original contribution we have
defined specific agents for modelling the concrete flip-flops behaviour for both the