### Table 10: Inference Rules

"... In PAGE 18: ...out(t):X = out(t):nil j X OUT2 out(t):nilbX [] out(t):nilbY = out(t):nil b (X Y ) OUT3 out(t):nilbX out(t):nilbY = out(t):nil b (X Y ) EVAL eval(X):Y = Xj Y READ t ( t0; i 2 if(t) =) t0 i = var(ti) read(t):X = in(t):(out(t0):nil j X) IN1 t0 ( + t in(t):X[]in(t0):Y = in(t0):if eq(t0; t) then X Y else Y IN2 if(t) \ ia(t0) 6 = ;; if(t0) \ ia(t) 6 = ;; t (* + ) t0 in(t):X[]in(t0):Y = in(t):if eq(t; t0) then X (Y (t0; t)) else X [] in(t0):if eq(t0; t) then X (t; t0) Y else Y IN3 t (* + ) t0 in(t):X in(t0):Y = in(t):if eq(t; t0) then X (Y (t0; t)) else X in(t0):if eq(t0; t) then X (t; t0) Y else Y Table 9: Linda Laws The rules of the proof system are in Table10 . Most of them are borrowed from [30] and should be self{explanatory.... In PAGE 18: ... The soundness and completeness proof proceeds in two steps: rst a reduced proof system is considered and its soundness and completeness for nite processes is proven; then, the inference rules are used to establish soundness and completeness of CP. Let RP be the proof system obtained from CP by deleting rules IV(a) and VI in Table10 , dealing with possibly in nite terms. We shall write E vRP F to indicate that E v F can be derived within RP.... In PAGE 39: ...AL and their coincidence (Theorem 5.13) can be proven again. A proof system for IPAL is obtained by adding a speci c law for assignment AS x := e:X = X[e=x] to the proof system for PAL. Since stores can only be investigated by communication or by conditional choice, we do not introduce any additional inference rule for ensuring substitutivity of value expressions (rules VII and VIII in Table10 are su cient). All of the results related to the equational semantics of PAL can be proven for IPAL as well.... ..."

### Table 2. Inference rules for 6`.

1995

"... In PAGE 6: ... The premise of the rule assumes the existence of a diagnostic trace for #5Bl 0 ;r#28g ^ D#29#5D : Inv#28 apos;#29 and the side-condition of the rule provides information as to howonemay extend this trace #28obviously with an a-transition#29 in order to obtain a diagnostic trace for #5Bl;D#5D:Inv#28 apos;#29. The rules in Table2 are sound and complete in the following sense: Theorem 7. Let A be a timed transition system with initial node l 0 .... In PAGE 6: ...3 Obtaining an Algorithm Given a symbolic state #5Bl;D#5D of the automata A and a property apos; it is decidable whether there exists a diagnostic trace #1B suchthat#1B 6` #5Bl;D#5D: apos;.We obtain an algorithm by using the rules in Table2 in two phases, In Phase 1 a goal directed search, starting in the symbolic state #5Bl;D#5D, searching for a violating symbolic state, is performed by using the inference rules in Table 2. Wehave the following two termination criteria for the symbolic state #5Bl n ;D n #5D and the property apos; n : #7B #28Success#29 c or p axiom can be applied, #7B #28Fail#29 for some i, l n = l i , D n #12 D i and apos; n = apos; i .... In PAGE 6: ...3 Obtaining an Algorithm Given a symbolic state #5Bl;D#5D of the automata A and a property apos; it is decidable whether there exists a diagnostic trace #1B suchthat#1B 6` #5Bl;D#5D: apos;.We obtain an algorithm by using the rules in Table 2 in two phases, In Phase 1 a goal directed search, starting in the symbolic state #5Bl;D#5D, searching for a violating symbolic state, is performed by using the inference rules in Table2 . Wehave the following two termination criteria for the symbolic state #5Bl n ;D n #5D and the property apos; n : #7B #28Success#29 c or p axiom can be applied, #7B #28Fail#29 for some i, l n = l i , D n #12 D i and apos; n = apos; i .... In PAGE 7: ...However, if Phase 1 terminates on the Success criterion it follows that #1B 6` #5Bl;D#5D : apos;. The rules in Table2 provide a way to synthesize the diagnostic trace of the conclusion from a diagnostic trace of the premise, constituting Phase 2. If the search in Phase 1 is performed using a breadth-#0Crst strategy, a resulting trace will be a shortest diagnostic trace.... ..."

Cited by 4

### Table 7: Partially Compressed Decision Table Rules

1997

"... In PAGE 25: ... As an example of the difference, consider a cost minimization problem where the knowledge source is a decision table. For illustration purposes, let us use the simple decision table depicted in Table7 . A joint approach such as [MM78] can use this decision table to find the optimal solution.... In PAGE 25: ... A separate approach requires that decision table reveal all possible compressed rules. Table7 is only partially compressed because it is missing a compressed rule1: I1=F, I2=F, I3=- (dash). The search space is larger for partially compressed decision tables because missing or implied rules must be discovered in the search process, but the knowledge source is easier to generate.... ..."

Cited by 4

### Table 2: Inference Rules of Many Sorted Monadic Equational Logic

1991

"... In PAGE 7: ...omplex assertions, e.g. formulas of rst order logic, then they should be interpreted by subobjects; in particular equality = : A should be interpreted by the diagonal [[A]]. The formal consequence relation on the set of equations is generated by the inference rules for equivalences ((re ), (simm) and (trans)), congruence and substitutivity (see Table2 ). This formal consequence relation is sound and complete w.... In PAGE 12: ...7 Given a signature for the programming language, let be the signature for the metalanguage with the same base types and a function p: 1 ! T 2 for each command p: 1 * 2 in . The translation from programs over to terms over is de ned by induction on raw programs: x [x]T (let x1(e1 in e2) (letT x1(e1 in e2 ) p(e1) (letT x(e1 in p(x)) [e] [e ]T (e) (letT x(e in x) The inference rules for deriving equivalence and existence assertions of the simple programming language can be partitioned as follows: general rules (see Table 6) for terms denoting computations, but with variables ranging over values; these rules replace those of Table2 for many sorted monadic equational logic rules capturing the properties of type- and term-constructors (see Table 7) after interpretation of the programming language; these rules replace the additional rules for the metalanguage given in Table 4.... ..."

Cited by 585

### Table 1. Domain model maintenance rules (partial)

1998

"... In PAGE 6: ... Finally, a page relocation results in updating the repos- itory to include the new URL for each instance where the obsolescent URL was mentioned. Table1 provides a partial set of rules for maintaining the domain model (as represented in the repository) consistent with respect to the Web site. Rules r1 -r5 handle the page in- sertion scenario.... ..."

Cited by 6

### Table 1. Domain model maintenance rules (partial)

1998

"... In PAGE 6: ...ual, i.e. having no correspondents in the Web pages. Finally, a page relocation results in updating the repos- itory to include the new URL for each instance where the obsolescent URL was mentioned. Table1 provides a partial set of rules for maintaining the domain model (as represented in the repository) consistent with respect to the Web site. Rules r1 -r5 handle the page in- sertion scenario.... ..."

Cited by 6

### Table 2: Rules extracted from network \Iris-Norm quot; by Partial-RE technique. Rule Rule Iris Certainty Soundness Completeness False-Alarm

1999

"... In PAGE 12: ... Final rules are written in the format: \If Xi i And Xg g cf ?! Consequentj quot;. See Table2 and 6 for examples. Partial-RE can be used e ciently in applications where the main objective of extracting rules from trained neural networks is to study the main parameters that cause speci c output decisions to be taken.... In PAGE 23: ... Also, it shows the remarkable performance of the rules extracted from network \Iris-Cont quot; by Full-RE. Note that: (i) The numeric values compared with input features Iis in the rules extracted by both BIO-RE and Partial-RE represent the mean ( is) of these input feature (see the rule bodies in Table 1 and Table2... In PAGE 24: ...g., rule 4 in Table2 ), completeness may be much less than soundness, because some instances where these rules would re correctly have already been covered by other preceding rules. (iv) Full-RE leads to three simple rules that classify the iris data set very well.... In PAGE 28: ... Also, they were extracted to cover all training and testing data sets and hence increase the completeness of the extracted set of rules. Examples of such rules are: R1 and R4 of Table 1, R4 of Table2 , and R2-R5 of Table 5. 5 Performance Evaluation Since both iris and breast cancer problems have continuous input features, Full-RE is the best technique to be used to extract rules from both \Iris-Cont quot; and \Cancer-Cont quot; networks which were trained with the original continuous input features.... ..."

Cited by 29

### Table 6: Rules extracted from network \Cancer-Norm quot; by Partial-RE technique. Rule Rule B-Cancer Certainty Soundness Completeness False-Alarm

1999

Cited by 29

### Table 3. Control rules in the Adaptive Place Advisor that handle user responses.

1999

"... In PAGE 7: ... Finally, if the query returns only a few candidates, the last rule (Q4) selects one of these items and recommends it to the user. Table3 presents another set of control rules that are responsible for dealing more directly with user responses. Many of the conditions here refer to user- applied operators, which we assume another part of the Place Advisor infers from the user apos;s behavior.... ..."

Cited by 12

### Table 3. Control rules in the Adaptive Place Advisor that handle user responses.

1999

"... In PAGE 7: ... Finally, if the query returns only a few candidates, the last rule #28Q4#29 selects one of these items and recommends it to the user. Table3 presents another set of control rules that are responsible for dealing more directly with user responses. Many of the conditions here refer to user- applied operators, which we assume another part of the Place Advisor infers from the user apos;s behavior.... ..."

Cited by 12