Chapter 10: Design and Analysis of PropositionalLogic RuleBased Systems

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Chapter 10 Design and Analysis of
Propositional-Logic Rule-Based Systems

• Albert M. K. Cheng

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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 ais
• VAL expressions on right-hand side of
  assignment, i.e., the bis
• EC enabling condition

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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

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

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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
  gt', ' 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.

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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.

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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.

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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.

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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.

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

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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