1. Chapter 10: Design and Analysis of Propositional-Logic Rule- Based Systems Albert M. K. Cheng

2. Analysis of Rule-Based Systems The RULES section is composed of a finite set of rules each of which is of the form: a1 := b1 ! a2 := b2 ! … ! am := bm IF enabling condition VAR = set of variables on left-hand side of the assignment, i.e., the ai’s VAL = expressions on right-hand side of assignment, i.e., the bi’s EC = enabling condition

3. Simple Rule-Based Program (* 1 *) object_detected := true IF sensor_a = 1 AND sensor_a_status = good (* 2 *) []object_detected := true IF sensor_b = 1 AND sensor_b_status = good (* 3 *) []object_detected := false IF sensor_a = 0 AND sensor_a_status = good (* 4 *) []object_detected := false IF sensor_b = 0 AND sensor_b_status = good

4. State Space Representation A D E F G H FP1 FP2 C L J K B I M N P FP3

5. Problem Complexity In general, the analysis problem is undecidable if the program variables can have infinite domains, i.e., there is no general procedure for answering all instances of the decision problem.

6. Proof Outline Any two-counter machine can be encoded by an equational rule-based program that uses only `+' and `-' as operations on integer variables and `>', `=' as atomic predicates such that a two-counter machine accepts an input if and only if the corresponding equational rule-based program can reach a fixed point from an initial condition determined by the input to the two-counter machine.

7. Analysis Problem is Solvable for some Cases All the variables of an equational rule- based program range over finite domains. Set of variables in VAR and set of variables in VAL and EC are disjoint. Enabling conditions are mutually exclusive. Only constants are assigned to variables in VAR.

8. Compatibility of Rules Let L_x denote the set of variables appearing in LHS of rule x. Two rules a and b are said to be compatible iff at least one of the following conditions holds: (CR1) Test a and test b are mutually exclusive. (CR2) L_a and L_b are disjoint. (CR3) Suppose L_a and L_b are not disjoint. Then for every common variable v in L_a and L_b, the same expression must be assigned to v in both rule a and b.

9. Special Form A Let L and T be sets of variables in VAR and EC of rules. A set of rules are in special form A if the following conditions hold: (1) Constant terms are assigned to all the variables in L. (2) All of the rules are compatible pairwise. (3) L and T are disjoint.

10. Example 1. a1 := true IF b = true AND c = true 2. [] a1 := true IF b = true AND c = false 3. [] a2 := false IF c = true Rules 1 and 2 are compatible by conditions CR1 and CR3. Rules 1 and 3 are compatible by condition CR2. Rules 2 and 3 are compatible by condition CR2.

11. General Analysis Strategy Rule-based program Simpler programs Rule rewriter State-space analyzer Special form recognizer No independent ruleset in special form Independent rulsets in special form(s)

12. Example input: read(b, c) 1. a1 := true IF b = true AND c = true 2.[]a1 := true IF b = true AND c = false 3.[]a2 := false IF c = true 4.[]a3 := true IF a1 = true AND a2 = false 5.[]a4 := true IF a1 = false AND a2 = false 6.[]a4 := false IF a1 = false AND a2 = true input: read(b, c) 1. a1 := true IF b = true AND c = true 2.[]a1 := true IF b = true AND c = false 3.[]a2 := false IF c = true

13. Applications of Analysis Tools Cryogenic Hydrogen Pressure Malfunction Procedure in the Pressure Control System of the Space Shuttle Vehicle Integrated Status Assessment Expert System Fuel Cell Expert System Orbital Maneuvering and Reaction Control System