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Figure 10.12 The HLD graph of Program 1.
(*4*) [] d := c IF a > 0
(*5*) [] e := c + 1 IF c ≤ 1 and b > 0
(*6*) [] f := c + 1 | e := c − 1 IF c ≤ 1 and b ≤ 0 (*7*) [] f := c − 1 IF c ≥ 0
(*8*) [] g := 1 | h := 1 IF f > 1 and d > 1 (*9*) [] g := 2 | h := 2 IF f ≤ 1 and e > 1
(*10*) [] x := true IF g = 2 and y = true (*11*) [] x := false IF g = 1
(*12*) [] y := true IF h = 2 and x = false (*13*) [] y := false IF h = 1
(*14*) [] z := true IF x = true and y = true
END.
Applying the HLD graph construction procedure to Program 1, the resulting HLD graph GProgram 1 is shown in Figure 10.12. Rule 1 and rule 2 are assigned higher priorities than rule 3 and rule 4, which in turn are assigned higher priorities than rule 8 and rule 9, and so on. Note that rule 3 and rule 4 are assigned the same priority as rule 5, rule 6, and rule 7.
10.12.5 Program Execution and Response Time
The execution of the n-rule program p is a sequence of recognize–act cycles. Each recognize–act cycle consists of two phases: match and fire.
•During the match phase, a sequence of evaluations are conducted such that rules of p are evaluated in order of their priorities to determine a highest-priority fireable rule. The order of evaluating rules sharing the same priority is nondeterministic or up to the run-time scheduler. Suppose r is checked next. The enabling condition of r is evaluated first to see if r is enabled. If r is not en-
340 DESIGN AND ANALYSIS OF PROPOSITIONAL-LOGIC RULE-BASED SYSTEMS
abled, r is found to be not fireable and another rule is checked if an unchecked rule still exists; otherwise, all of the RHS expressions of r are evaluated in a parallel manner and the resulting values are compared with the values of corresponding LHS variables to see if the firing of r changes the value of a variable. If r is enabled and the firing of r changes the value of a variable, r is found to be fireable. This evaluation continues until either one rule is found to be fireable or all of the rules are found to be not fireable.
•During the fire phase, all of the assignment expressions of the selected fireable rule are carried out in parallel.
The recognize–act cycle continues until none of the rules is fireable. p reaches a fixed point if none of its rules is fireable. The execution of p terminates when p reaches a fixed point.
Since all rules may not have the same number of assignment expressions and the same size of enabling conditions, evaluating and firing different rules may take different amounts of time. However, for each rule r p, it takes only up to a fixed amount of time to evaluate the enabling condition of r, since the size of the enabling condition of r is finite. In addition, the time needed for selecting a rule to evaluate is also finite since only a finite number of rules exist from which to choose. Assume the maximal amount of time needed to select and evaluate a rule is x units of time. Since there are n rules in p, the match phase needs at most n x units of time.
Furthermore, since there are only up to a fixed number, say m, of assignment expressions in each rule, it also takes only up to a fixed amount of time to conduct the fire phase. Assume the maximal amount of time needed to carry out an assignment expression is y units of time. The fire phase needs at most m y units of time. Hence, the time spent during each cycle is at most n x +m y units of time. If the execution takes at most l recognize–act cycles to reach a fixed point, the response time is at most l (n x + m y) units of time. That is, the response time (i.e., execution time) of p is proportional to the number of recognize–act cycles (and thus the number of rule firings) during the execution of p. If we know the maximal amount of actual time needed to select and evaluate a rule and the maximal amount of actual time needed to carry out an assignment expression, then the response time of p can be easily calculated. Since these quantities are machine-dependent, the response time of p in this chapter is measured in terms of the number of rule firings during the execution of p. The program p has a bounded response time if and only if the number of rule firings during the execution of p is bounded by a fixed number.
The exact upper bound on the number of rule firings during the execution of p is the integer i representing the maximal number of rule firings that can possibly happen during the execution of p. An upper bound on the number of rule firings during the execution of p is an integer j, j ≥ i. In the remainder of this chapter, an upper bound for the program p refers to an upper bound on the number of rule firings during the execution of p, unless otherwise stated. Let T p represent an upper bound for the program p.
342 DESIGN AND ANALYSIS OF PROPOSITIONAL-LOGIC RULE-BASED SYSTEMS
However, it has been observed by the authors that sets of syntactic and semantic constraint assertions exist such that if the set of rules S satisfies any of them, the execution of S always terminates in bounded time. A special form is a set of syntactic and semantic constraint assertions on a set of rules. A set of rules satisfying all assertions of special form F is said to be in special form F and is guaranteed to always reach a fixed point in bounded time. Two special forms have been observed and, for each of them, we have developed an algorithm used to calculate response-time upper bounds for programs satisfying it. Hence, if a program (or part of a program) is determined to be in a known special form, the corresponding response-time upper-bound algorithm can be used such that the use of expensive exhaustive state-space graph checks can be avoided (or at least minimized).
10.12.8 Special Form A and Algorithm A
The first set of syntactic and semantic constraint assertions is called Special Form A, which allows only constant expressions to be assigned to non-input variables. In addition, for each rule r in a Special Form A program p, r can be fired at most once during the execution of p.
10.12.9 Special Form A
Special Form A: Let S denote a set of rules. S is in Special Form A if the following conditions hold.
A1. RS = .
A2. For each pair of distinct rules r and r in S, r and r are compatible. A3. L S ∩ TS = .
To check if a set of n rules is in Special Form A, the recognition procedure requires O(n)-time for checking the satisfiability of (A1) and O(k2)-time for checking the satisfiability of (A3), where k is the number of variables in this set of rules. In addition, it takes quadratic time to check the satisfiability of (A2).
Theorem 5. If p is in Special Form A, then the execution of p always terminates in n rule firings, where n is the number of rules in p.
Proof of Theorem 5. The proof can be found in section 10.12.12.
Algorithm A We now improve the upper bound above by exploiting the property of mutual exclusiveness among rules. If r and r are compatible by mutual exclusiveness, then at least one of them, say r, is disabled at any moment during the execution of p. Hence, the rule r cannot be fired throughout the execution of p. This means that the value 1 can be subtracted from the value of n obtained by applying the theorem above to get a tighter (and better) upper bound for p. If a set of m rules exists in
