Validating dynamic properties of rule-based systems

rules , and generates the set of all paths in the rule base . Path Hunter found 512 abstract rules in the Blackbox Expert’s rule base , which formed 516 paths . A single path found by Path Hunter is shown in Figure 1 . The square nodes represent abstract rules , the ‘‘round’’ nodes represent predicates , and the directed arcs represent data dependencies between the rules . The logical completion that is asserted by this path is GMAP – B ∧ CERTAIN – BALLS ∧ GMAP – CERT – B . The semantics for this path are as follows : upon examining the evidence (beam entry and exit points) currently available , select the actions of placing a ball on the grid and marking the ball as certain ; and the outcome is , a ball is placed and this ball is marked as certain . 2 . 2 . DATA REQUIREMENTS The data requirements of a rule-based system can be determined using our formal model for capturing rule-base structure . We recall that a path is a set of rules that advance the state of the problem being solved from one goal state to another . The facts required by the rules in a path to fire are the precedence constraints for advancing the state of the problem being solved . There are two issues in using the path model to determine the data requirements of a rule-based system : determining the precedence constraints required for a single rule in a path to fire , and determining when all the rules in a path will be able to fire . The initial set of rules that must fire in a path are called start rules . Start rules will then assert facts enabling the other rules in the path . The start rules of a path P k , denoted by SR k , is the set of rules r i P F where the templates in the LHS of r i do not depend upon any other rule r j P F ; SR k 5 h r i u r i P F , ( ; r j P F , r i a / r i ) j . For start rules , W 5 [ . The start set of a path P k is the set of facts required by the start rules of that path to become enabled to fire . A start set of a path is identified by the set of start templates , denoted by ST k , which is the union of the set of templates present on the LHS of each of the start rules in the path ; ST k 5 < r i P SR k ( r i . The set of predicates A . D . PREECE , C . GROSSNER & T . RADHAKRISHNAN 152 used in the start templates for a path is given by SP k 5 h l j u ; k L i , L i l P ST k , 9 (( L i , L i )) 5 l j j . The completion set of a path is the set of facts required for all the rules in that path , except start rules , to become enabled to fire . A completion set of a path is identified by the set of completion templates , and is denoted by CT k . The completion templates is the set of all templates present on the LHS of all the rules of the path , except start rules , which specify a fact that is not asserted by any rule in the path . Formally , CT k 5 ! r i P ( F 2 SR k ) ( ( r i 2 PT i ,k ) . PT i ,k is the set of templates from ( r i that are matched by the facts asserted when the rules that enable r i fire . PT i ,k 5 hk L i , L i l u ( ; r j P W , W › r i )( ' d , k L i , L i l ? d 5 k l i , l i l , k l i , l i l P ! r j ) j . The set of predicates used in the completion templates for a path is given by CP k 5 h l j u ; k L i , L i l P CT k , 9 ( k L i , L i l ) 5 l j j . Now , let us consider the precedence constraints for a rule in a path . Given a path P k , r i P F , r i ̧ SR k , and W › r i , then the set of completion templates for r i is given by CT i ,k 5 ( ( r i 2 PT i ,k ) . Lemma 1 . If facts matching the completion templates CT i ,k of rule r i in a path P k are present in WM , r i ̧ SR k , and the ( r j P W , W › r i ) fire , then r i can fire . Proof . After the r j P W fire , WM contains the union of all facts asserted by the r j P W , in addition to the facts matching CT i ,k . These facts match the templates CT i ,k < PT i ,k by definition . This simplifies to ( r i (by putting CT i ,k 5 ( ( r i 2 PT i ,k )) ; thus , the facts in WM match all the LHS templates of r i , and hence r i can fire . Now that we have understood the conditions required for an individual rule in a path to fire , we can determine the conditions for all the rules in a path to become to fire-able . Theorem 1 . Gi y en a path P k , if facts matching the completion templates CT k and start templates ST k are present in WM , then all the rules in the path can fire . Proof . The proof of the theorem follows directly from Lemma 1 by induction . According to Theorem 1 , ST k and CT k identify the data items will be required by the rule-based system to achieve its goals . In Theorem 1 , we have shown that the precedence constraints for each path , ST k and CT k , indicate the facts required for all the rules in that path to fire , achieving a goal ; the goal achieved is identified by the logical completion asserted by the rules in the path . Our Path Hunter tool provides the data requirements for each path ( ST k and CT k ) . For example , the data requirements for our earlier example path are shown in Figure 2 . The set of start rules for the example path shown in Figure 1 is h RA-14-Left%1 j . The data items required by this path to achieve the goal represented by the logical completion it asserts are given by its start predicates h GMAP , SHOT-RECORD , GRIDSIZE j , and its completion predicates h GMAP – CERT , CERTAIN – BALLS j . The start predicates for this path indicate that this path will access facts the indicate the contents of the grid squares , the beams that have been fired . The completion predicates indicate that this path requires access to facts that indicate the certainty of the hypothesis for the contents of the grid squares that have been identified . DYNAMIC VALIDATION OF RULE-BASED SYSTEMS 153