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2 PROCESSES REALIZATION IN
TERMS OF USING A
RULE-BASED SITUATION
FORMALISM OF PROCESSES
SPECIFICATION
Real physical interacting processes almost always
are hybrid ones that is they have discrete and contin-
uous components. A base of the formalism is an
abstract model of hybrid automaton (Henzinger,
1996), in which discrete states are represented by
logical variables. A set of logical variables contains
also a subset of predicates of the continuous states.
The predicates are used to specify interactions be-
tween continuous and discrete-event components of
the processes. The logic variables can determine
both discrete changes of continuous component and
changes of their evolution dynamic. Thus a state of
the processes model may be represented by a set of
real variables
for the continuous components and
a set of logic variables
W for the discrete compo-
nents. The last consists of subset
Q for the discrete
states and subset
G of predicates, that is GQW U
.
To specify processes it is necessary to define transi-
tion functions of the following types:
},{: TrueFalseQW ×→σ — function of dis-
crete state transitions;
XXW →×δ : — function of continuous state
transitions;
},{: TrueFalseGX ×→
— predicate value de-
pendence of continuous process states.
The formalism expressiveness and effectiveness
of the model implementation essentially depend up-
on specific forms of these functions. A definitional
domain of the transition function σ may be repre-
sented with help of logic formulae that makes the
representation rather obvious. In the general case,
the model specification may need whatever logic
formulae. To our mind as the condition it is conven-
ient to use an elementary conjunction of the logic
variables. Such conjunctions may be understandably
interpreted as logic-dynamic situations (Shpakov,
2004). The situation
j
S may be defined as
following:
ni
jjjj
sssS ,...,,...
1
= , where
ii
jj
ws = , or
ii
jj
ws ¬=
,
Ww
i
j
∈
, ||,...1 WNNn
ww
== .
Designating the set of situations
S the type of
the transition function of discrete states may now be
defined as following:
},{: TrueFalseQS ×→
.
This function may be assigned with help of a collec-
tion of production rules “condition → action”. In
the rules the logic-dynamic situation is used as a
condition and the procedure, which assigns defined
values to logic states, is used as an action. At that the
rule takes the following form:
'''
,...,,...,
1 mi
jjjj
rrrS →
,
(1)
where
''
ii
jj
qr = or
''
ii
jj
qr ¬= , Qq
i
j
∈
'
.
To implement an arbitrary logic formula it is
necessary to use several rules with the different con-
ditions and the same executive parts. It is a specifi-
cation of a normal disjunction form and it enables to
realize any logic formula.
The function
must for every mode determine
new values of the process states which correspond
with a current process state, the system dynamic and
also with current states of some processes interacting
with this process. The hybrid process mode may be
naturally represented by a logic situation considered
above. It was proposed to find the new values of the
continuous process states by calculation of their
transitive relations with the current states (Alur,
2000). Algorithms of the transitive relations calcula-
tions for some elementary processes (integration,
differentiation, smoothing and others) are presented
in (Shpakov, 2006). Complicated processes may be
implemented by different connections of the elemen-
tary ones and using arithmetic operations and func-
tional procedures. In this case the transitive function
may be represented with help of a collection of
following rules:
),('
ikkkj
xxxS τ=→
,
(2)
where
SS
j
— the situation, corresponding to the
mode,
k
— transitive relation, Xxx
ik
∈, —
states of the continuous processes.
Since the hybrid process modes very often de-
pend on their states being at some assigned ranges it
is convenient to represent the function
by a col-
lection of following rules:
kjkjjkj
gxbxxax →+
≥ ))()((
4321
,
(3)
where
GgXxxxx
kjjjj
∈∈ ,,,,
4321
,
k
a and
k
b -
constants, corresponding to the range. More compli-
cated predicates may be received by using simple
ones and rules (1).
A computer implementation of the processes
specified by rules (1, 2, 3) is executed with help of
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