### Table 1. Effector Dynamics and Limits.

### Table 1. Inductive rules for EMPA integrated interleaving semantics

1998

"... In PAGE 9: ...the transitions of the corresponding state and their rates. The formal definition of the integrated interleaving semantics for EMPA is based on the transition relation ???!, which is the least subset of G Act G satisfying the inference rule reported in the first part of Table1 . This rule selects the potential moves having the highest priority level, and then merges together those having the same action type, the same priority level and the same derivative term.... In PAGE 9: ... The first operation is carried out through functions Select : Mu n(Act G) ?! Mu n(Act G) and PL : Act ?! PLevel, which are defined in the third part of Table 1. The second operation is carried out through function Melt : Mu n(Act G) ?! P n(Act G) and partial function Min : (ARate ARate) ?!o ARate, which are defined in the fourth part of Table1 . The name Min should recall the adoption of the race policy: the minimum of a set of random variables has to be computed.... In PAGE 9: ... The rationale behind the use of Melt and Min is thus the possibility of producing standard LTSs as integrated semantic models, without the need to decorate transitions with auxiliary labels like in [16] nor the need to take into account the multiplicity of transitions like in [18]. The multiset PM (E) 2 Mu n(Act G) of potential moves of E 2 G is defined by structural induction in the second part of Table1 . The normalizationof the rates of potential moves resulting from thesynchronization of the sameactive actionwith several independent or alternativepassive actions is carried out through partial function Norm : (AType ARate ARate Mu n(Act G) Mu n(Act G)) ?!o ARate and function Split : (ARate R I ]0;1]) ?! ARate, which are defined in the fifth part of Table 1.... In PAGE 9: ... The multiset PM (E) 2 Mu n(Act G) of potential moves of E 2 G is defined by structural induction in the second part of Table 1. The normalizationof the rates of potential moves resulting from thesynchronization of the sameactive actionwith several independent or alternativepassive actions is carried out through partial function Norm : (AType ARate ARate Mu n(Act G) Mu n(Act G)) ?!o ARate and function Split : (ARate R I ]0;1]) ?! ARate, which are defined in the fifth part of Table1 . Note that Norm(a; ~ 1; ~ 2; PM 1; PM2) is defined if and only if min(~ ; ~ ) = , which is the condition on action rates we have required in Section 2.... In PAGE 23: ... To solve the problem, we follow the proposal of [2] by introducing a priority operator ( ) : priority levels are taken to be potential, and they become actual only inside the scope of the priority operator. We thus consider the language L generated by the following syntax E ::= 0 j lt;a; ~ gt;:E j E=L j E[ apos;] j (E) j E + E j E kS E j A whose semantic rules are those in Table1 except that the rule in the first part is replaced by ( lt;a; ~ gt;; E0) 2 Melt(PM (E)) E a;~ ???! E0 and the following rule for the priority operator is introduced in the second part PM ( (E)) = Select(PM (E)) It is easily seen that EMPA coincides with the set of terms f (E) j E 2 Lg. Definition 5.... ..."

Cited by 74

### Table 1. Inductive rules for EMPA integrated interleaving semantics

1999

"... In PAGE 9: ... The idea is to compute inductively the multiset 3 of all the potential moves of a given term regardless of priority levels and then select those having the highest priority level. The formal definition of the integrated interleaving semantics is based on the transition rela- tion ???!, which is the least subset of G Act G satisfying the inference rule in the first part of Table1 . This rule selects the potential moves that have the highest priority level (or are passive) and then merges together those having the same action type, the same priority level, and the same derivative term.... In PAGE 9: ... This rule selects the potential moves that have the highest priority level (or are passive) and then merges together those having the same action type, the same priority level, and the same derivative term. The first operation is carried out through functions Select : Mu n(PMove) ?! Mu n(PMove) and PL : Act ?! APLev, which are defined in the third part of Table1 . The second operation is carried out through function Melt : Mu n(PMove) ?! P n(PMove) and partial function Aggr : ARate ARate ?! o ARate, which are defined in the fourth part of Table 1.... In PAGE 9: ... We regard Aggr as an associative and commutative operation, thus we take the liberty to apply it to multisets of rates. The multiset PM (E) 2 Mu n(PMove) of potential moves of E 2 G is defined by structural induction in the second part of Table1 according to the intuitive meaning of operators. In order to enforce the bounded capacity assumption, in the rule for the parallel composition operator a nor- malization is required which suitably computes the rates of potential moves resulting from the synchronization of the same active action with several independent or alternative passive actions.... In PAGE 9: ...he multiset obtained by projecting the tuples in multiset M on their i-th component. Thus, e.g., ( 1(PM2))( lt;a; gt;) in the fifth part of Table1 denotes the multiplicity of tuples of PM2 whose first component is lt;a; gt;. UBLCS-99-02... In PAGE 11: ...Table1 . Note that Norm(a; ~ 1; ~ 2; PM 1; PM2) is defined if and only if min(~ ; ~ ) = , which is the condition on action rates in order for a synchronization to be permitted.... ..."

Cited by 1

### Table 1. Inductive rules for EMPAr integrated interleaving semantics

1997

"... In PAGE 5: ... The integrated semantics of EMPAr terms can be de ned by exploiting again the idea of potential move: the multiset 1 of the potential moves of a given term is inductively computed, then those potential moves having the highest priority level are selected and appropriately merged. The formal de nition is based on the transition relation ???!, which is the least subset of Gr Actr Gr satisfying the inference rule reported in the rst part of Table1 . This rule selects the potential moves having the highest priority level, and then merges together those having the same action type, the same priority level and the same 1 We use \fj quot; and \jg quot; as brackets for multisets, \ quot; to denote multiset union, Mu n(S) (P n(S)) to denote the collection of nite multisets (sets) over set S, M(s) to denote the multiplicity of element s in multiset M, and i(M) to denote the multiset obtained by projecting the tuples in multiset M on their i-th component.... In PAGE 5: ...g., ( 1(PM2))( lt;a; ; 0 gt;) in the fth part of Table1 denotes the multiplicity... In PAGE 7: ...Mu n(Actr Gr) ?! Mu n(Actr Gr) and PLr : Actr ?! PLevel, which are de ned in the third part of Table1 . The second operation is carried out through function Meltr : Mu n(Actr Gr) ?! P n(Actr Gr) and partial function Min : (ARate ARate) ?!o ARate, which are de ned in the fourth part of Table 1.... In PAGE 7: ...Mu n(Actr Gr) ?! Mu n(Actr Gr) and PLr : Actr ?! PLevel, which are de ned in the third part of Table 1. The second operation is carried out through function Meltr : Mu n(Actr Gr) ?! P n(Actr Gr) and partial function Min : (ARate ARate) ?!o ARate, which are de ned in the fourth part of Table1 . Observe that function Meltr sums the rewards of the potential moves to merge: this is consistent with the additivity assumption about rewards.... In PAGE 7: ... Observe that function Meltr sums the rewards of the potential moves to merge: this is consistent with the additivity assumption about rewards. The multiset PM r(E) 2 Mu n(Actr Gr) of potential moves of E 2 Gr is de ned by structural induction in the second part of Table1 . The normalization of rates and rewards of potential moves resulting from the synchronization of an action with several independent or alternative passive actions is carried out through partial functions Normr;rate : (AType ARate ARate Mu n(Actr Gr) Mu n(Actr Gr)) ?!o ARate and Normr;reward : (AType AReward AReward Mu n(Actr Gr) Mu n(Actr Gr)) ?! o AReward, and function Split : (ARate R I ]0;1]) ?! ARate, which are de ned in the fth part of Ta- ble 1.... In PAGE 7: ... Such an equivalence was de ned according to the idea of probabilistic bisimulation [8] on the inte- grated semantic model, and we proved that it is necessary to de ne it on the integrated semantic model in order for the congruence property to hold. For the sake of convenience, we can extend EMB to EMPAr since it disregards rewards, provided that like in [1, 2] we introduce a priority operator \ ( ) quot; and we con- sider the language Lr; generated by the following syntax E ::= 0 j lt;a; ~ ; r gt;:E j E=L j E[ apos;] j (E) j E + E j E kS E j A whose semantic rules are those in Table1 except that the rule in the rst part is replaced by ( lt;a; ~ ; r gt;; E0) 2 Meltr(PM r(E)) E a;~ ;r ???! E0... ..."

Cited by 21

### Table 1. Inductive rules for EMPA interleaving semantics

1995

"... In PAGE 11: ... As a consequence, only if we know all the potential moves of the subterms of a given term, we can correctly determine its transitions and their rates. The first relation, denoted with ???! , is the least subset of G Act G satisfying the inference rule reported in the upper part of Table1 . The side condition associated with the inference rule enforces the race policy by selecting among the potentialmoves of a term those having the highest priority level, and then merges together all the potential moves in which actions with the same type and priority as well as the same target term occur.... In PAGE 13: ...3. The second relation, denoted with ` , is defined by structural induction as the least subset of G Mufin(Act G) satisfying the axiom and the inference rules reported in the lower part of Table1 . This relation computes the multiset of all the potential moves of each term, regardless of action priorities.... In PAGE 15: ...e. if we apply exactly the rules reported in Table1 ), then we obtain the LTS reported in Figure 2(c). Notice that the correct rate of both the transition from E2 to E3 and the transition from E2 to E4 is =2 instead of , because in E2 only a timed action with rate occurs hence the exit rate of E2 is .... In PAGE 15: ... Remark 3.10 In the rest of the paper, functions appearing in Table1 will be sometimes abused. However, the way in which they will be used will result clear from the context.... In PAGE 35: ... We thus define the relation ???! as the least subset of Mufin(V) Act Mufin(V) generated by the inference rule reported in the upper part of Table 4, which in turn is based on the relation ` defined as the least subset of Mufin(V) Mufin(Act Mufin(V)) generated by the axioms and the inference rules reported in the lower part of Table 4. These rules are strictly related to the rules reported in Table1 for the operational interleaving semantics of EMPA terms. There are four major differences: Intheoperationalnet semantics functionRaceis unnecessarysincetheracepolicyis inherent in the net firing rule.... In PAGE 37: ...Table1 , then we would get instead the two transitions dec(E) a;~ ???! fj 0k;id; idk; lt;b; ~ gt;:0jg dec(E) b;~ ???! fj lt;a; ~ gt;:0k;id; idk;0jg which are not consistent with the fact that the two subterms of E are independent of each other and, more important, are wrong because they are both enabled at marking dec(E) but the firing of one of them prevents the other one from being executed. Example 6.... ..."

Cited by 23

### Table 4. Inductive rules for EMPA net semantics

1995

"... In PAGE 35: ....2.2 Net transitions The second step in the definition of the operational net semantics consists of introducing an appropriate relation whereby net transitions will be constructed. We thus define the relation ???! as the least subset of Mufin(V) Act Mufin(V) generated by the inference rule reported in the upper part of Table4 , which in turn is based on the relation ` defined as the least subset of Mufin(V) Mufin(Act Mufin(V)) generated by the axioms and the inference rules reported in the lower part of Table 4. These rules are strictly related to the rules reported in Table 1 for the operational interleaving semantics of EMPA terms.... In PAGE 35: ... Example 6.6 Consider term E lt;a; ~ gt;:0k; lt;b; ~ gt;:0 whose decomposition is given by fj lt;a; ~ gt;:0k;id; idk; lt;b; ~ gt;:0jg By applying the rules in Table4 , we get the two transitions fj lt;a; ~ gt;:0k;id jg a;~ ???! fj 0k;id jg fj idk; lt;b; ~ gt;:0jg b;~ ???! fj idk;0 jg as expected. If we replaced the three rules for the parallel composition operator with a single rule UBLCS-95-14... In PAGE 37: ... Example 6.7 Function Race does not appear in the rules of Table4 . Even if we introduced it before applying function Melt in order to rule out lower priority transitions, we would not be able to capture all the cases.... ..."

Cited by 23

### Table 1: Inductive rules for EMPA integrated interleaving semantics

1998

"... In PAGE 8: ..., and i(M) to denote the multiset obtained by projecting the tuples in multiset M on their i-th component. Thus, e.g., ( 1(PM2))( lt;a; gt;) in the fth part of Table1 denotes the multiplicity of tuples of PM2 whose rst component is lt;a; gt;.... In PAGE 9: ... 3(c) is exactly the result of the application to E of the rules in Table 1 equipped with the auxiliary functions mentioned above. The formal de nition of the integrated interleaving semantics for EMPA is based on the transition relation ???!, which is the least subset of G Act G satisfying the inference rule in the rst part of Table1 . This rule selects the potential moves that have the highest priority level (or are passive), and then merges together those having the same action type, the same priority level and the same derivative term.... In PAGE 9: ... The rst operation is carried out through functions Select : Mu n(PMove) ?! Mu n(PMove) and PL : Act ?! APLev, which are de ned in the third part of Table 1. The second operation is carried out through function Melt : Mu n(PMove) ?! P n(PMove) and partial function Min : (ARate ARate) ?! o ARate, which are de ned in the fourth part of Table1 . We recall that function Melt, whose introduction is motivated by the drawback cited in the example above, avoids burdening transitions with auxiliary labels as well as keeping track of the fact that some transitions may have multiplicity greater than one.... In PAGE 11: ...in the second part of Table1 according to the intuitive meaning of operators explained in Sect.... In PAGE 11: ... The normalization operates in such a way that applying Min to the rates of the synchronizations involving the active action gives as a result the rate of the active action itself, and that each synchronization is assigned the same execution probability. This normalization is carried out through partial function Norm : (AType ARate ARate Mu n(PMove) Mu n(PMove)) ?!o ARate and function Split : (ARate R I ]0;1]) ?! ARate, which are de ned in the fth part of Table1 . Note that Norm(a; ~ 1; ~ 2; PM 1; PM 2) is de ned if and only if min(~ ; ~ ) = , which is the condition on action rates we have required in Sect.... In PAGE 27: ... To solve the problem, we follow the proposal of [BBK96] by introducing a priority operator \ ( ) quot;: priority levels are taken to be potential, and they become e ective only within the scope of the priority operator. We thus consider the language L generated by the following syntax E ::= 0 j lt;a; ~ gt;:E j E=L j E[ apos;] j (E) j E + E j E kS E j A whose semantic rules are those in Table1 except that the rule in the rst part is replaced by ( lt;a; ~ gt;; E0) 2 Melt(PM (E)) E a;~ ???! E0 and the following rule for the priority operator is introduced in the second part... In PAGE 33: ...ollowing the guideline of Sect. 3.2, we de ne the transition relation ???! as the least subset of Mu n(V) ActMufin(V) Mu n(V) generated by the inference rule reported in the rst part of Table 2, which in turn is based on the multiset PM (Q) 2 Mu n(ActMufin(V) Mu n(V)) of potential moves of Q 2 Mu n(V) de ned by structural induction in the second part of Table 2. These rules are strictly related to those in Table1 for the integrated interleaving semantics of EMPA terms. The major di erences are listed below and are clari ed by the corresponding upcoming examples: 1.... In PAGE 34: ...6 Consider term E lt;a; ~ gt;:0k; lt;b; ~ gt;:0 whose decomposition is given bydec(E) = fj lt;a; ~ gt;:0 k; id; id k; lt;b; ~ gt;:0jg By applying the rules in Table 2, we get the two independent transitions fj lt;a; ~ gt;:0 k; id jg norm( lt;a;~ gt;; lt;a;~ gt;:0k; id;1) ????????????????????! fj 0k; id jg fj id k; lt;b; ~ gt;:0 jg norm( lt;b;~ gt;;id k; lt;b;~ gt;:0;1) ????????????????????! fj id k; 0 jg as expected. If we replaced the three rules for the parallel composition operator with a single rule similar to that in Table1 , then we would get instead the two alternative transitions dec(E) norm( lt;a;~ gt;; lt;a;~ gt;:0k; id;1) ????????????????????! fj 0k; id; id k; lt;b; ~ gt;:0jg dec(E) norm( lt;b;~ gt;;id k; lt;b;~ gt;:0;1) ????????????????????! fj lt;a; ~ gt;:0k; id; id k; 0 jg which are not consistent with the fact that the two subterms of E are independent, thereby resulting in a violation of the concurrency principle (see Sect. 7:4).... In PAGE 49: ... The tool driver, which is written in C [KR88] and uses Lex [Les75] and YACC [Joh75], includes routines for parsing EMPA speci cations and performing lexical, syntactic, and static semantic (closure, guardedness, niteness) checks on the speci cations. The integrated kernel, which is implemented in C, currently contains only the routines to generate the integrated interleaving semantic model of EMPA speci cations according to the rules of Table1 : this kernel will be extended by implementing a EMB checking algorithm. The functional kernel, which is written in C, is based on a version of CWB-NC [CS96] that was retargeted for EMPA using PAC-NC [CMS95].... ..."

Cited by 25

### Table 3. Summary of qualitative model induction results.

"... In PAGE 29: ...From Forte apos;s point of view, the constraints in the fundamental domain theory are simply predicates that succeed or fail in the course of a proof. Table3 provides a summary of several models Forte induced from behavioral data, ranging from the very simple model of a thrown ball to the much more complex Reaction Control System #28RCS#29 on the space shuttle. As illustrations, we discuss the two cascaded tanks and the RCS below.... ..."

### Table 2: Inductive rules for EMPA integrated location oriented net semantics

1998

"... In PAGE 33: ... 3.2, we de ne the transition relation ???! as the least subset of Mu n(V) ActMufin(V) Mu n(V) generated by the inference rule reported in the rst part of Table2 , which in turn is based on the multiset PM (Q) 2 Mu n(ActMufin(V) Mu n(V)) of potential moves of Q 2 Mu n(V) de ned by structural induction in the second part of Table 2. These rules are strictly related to those in Table 1 for the integrated interleaving semantics of EMPA terms.... In PAGE 33: ...ring rule, as well as di cult to implement, due to the distributed notion of state (see Ex. 7.7). 5. Rate normalization is carried out through function norm : (Act V0 N I +) ?! ActMufin(V) de ned in the third part of Table2 , where V0 is generated by the same syntax as V except that V + V is replaced by V + id and id + V . In order to determine the correct rate of transitions deriving from the synchronization of the same active action with several independent or alternative passive actions of the same type, function norm considers for each transition three parameters: the basic action, the basic place and the passive contribution.... In PAGE 33: ...ynchronization. Again, this is a consequence of the distributed notion of state. 6. Potential move merging is carried out through functions melt1 : Mu n(ActMufin(V) Mu n(V)) ?! P n(ActMufin(V) Mu n(V)) and melt2 : P n(ActMufin(V) Mu n(V)) ?! P n(ActMufin(V) Mu n(V)) de ned in the fourth part of Table2 . Function melt1 merges the potential moves having the same basic action, the same basic place and the same postset by summingtheir passive contributions (see Ex.... In PAGE 34: ...dec(E) = fj ( lt;a; gt;:0 k; id) + lt;c; gt;:0; (id k; lt;b; gt;:0) + lt;c; gt;:0jg By applying the rules in Table2 , we get the following transitions fj ( lt;a; gt;:0k; id) + lt;c; gt;:0jg norm( lt;a; gt;;( lt;a; gt;:0k; id)+id;1) ????????????????????! fj 0k; id jg fj (id k; lt;b; gt;:0) + lt;c; gt;:0jg norm( lt;b; gt;;(id k; lt;b; gt;:0)+id;1) ????????????????????! fj id k; 0 jg dec(E) norm( lt;c; gt;;id+ lt;c; gt;:0;1) ????????????????????! fj 0jg If dec(E) is the current marking then all the transitions above are enabled and ring the rst transition results in marking fj 0k; id; (id k; lt;b; gt;:0)+ lt;c; gt;:0 jg which cannot be the preset of any transition labeled with action type c, because the execution of either lt;a; gt; or lt;b; gt; prevents lt;c; gt; from being executed according to the intended meaning of E.... In PAGE 34: ...dec(E) = fj ( lt;a; gt;:0 k; id) + lt;c; gt;:0; (id k; lt;b; gt;:0) + lt;c; gt;:0jg By applying the rules in Table 2, we get the following transitions fj ( lt;a; gt;:0k; id) + lt;c; gt;:0jg norm( lt;a; gt;;( lt;a; gt;:0k; id)+id;1) ????????????????????! fj 0k; id jg fj (id k; lt;b; gt;:0) + lt;c; gt;:0jg norm( lt;b; gt;;(id k; lt;b; gt;:0)+id;1) ????????????????????! fj id k; 0 jg dec(E) norm( lt;c; gt;;id+ lt;c; gt;:0;1) ????????????????????! fj 0jg If dec(E) is the current marking then all the transitions above are enabled and ring the rst transition results in marking fj 0k; id; (id k; lt;b; gt;:0)+ lt;c; gt;:0 jg which cannot be the preset of any transition labeled with action type c, because the execution of either lt;a; gt; or lt;b; gt; prevents lt;c; gt; from being executed according to the intended meaning of E. This fact is detected by the rules in Table2 , i.e.... In PAGE 34: ...ccording to the intended meaning of E. This fact is detected by the rules in Table 2, i.e. they generate no transition labeled with action type c for the marking above, since the alternative id k; lt;b; gt;:0 of lt;c; gt;:0 is not complete. To understand the presence of Q3 in the rst two rules for the alternative composition operator, let us now slightly modify term E in the following way E0 ( lt;a; gt;: lt;b; gt;:0 kfbg lt;b; gt;:0) + lt;c; gt;:0 where dec(E0) = fj ( lt;a; gt;: lt;b; gt;:0kfbg id) + lt;c; gt;:0; (id kfbg lt;b; gt;:0) + lt;c; gt;:0jg By applying the rules in Table2 , we get the two transitions fj ( lt;a; gt;: lt;b; gt;:0kfbg id) + lt;c; gt;:0jg norm( lt;a; gt;;( lt;a; gt;: lt;b; gt;:0 kfbg id)+id;1) ????????????????????! fj lt;b; gt;:0kfbg id jg dec(E0) norm( lt;c; gt;;id+ lt;c; gt;:0;1) ????????????????????! fj 0jg If dec(E0) is the current marking then all the transitions above are enabled and ring the rst transition results in marking fj lt;b; gt;:0 kfbg id; (id kfbg lt;b; gt;:0)+ lt;c; gt;:0jg which is the preset of the following tran- sitionfj lt;b; gt;:0kfbg id; (id kfbg lt;b; gt;:0) + lt;c; gt;:0jg norm( lt;b; gt;;(id kfbg lt;b; gt;:0)+id;1) ????????????????????! fj 0kfbg id; id kfbg 0jg If Q3 were not taken into account, then the transition above would not be constructed. Example 7.... In PAGE 34: ... Example 7.6 Consider term E lt;a; ~ gt;:0k; lt;b; ~ gt;:0 whose decomposition is given bydec(E) = fj lt;a; ~ gt;:0 k; id; id k; lt;b; ~ gt;:0jg By applying the rules in Table2 , we get the two independent transitions fj lt;a; ~ gt;:0 k; id jg norm( lt;a;~ gt;; lt;a;~ gt;:0k; id;1) ????????????????????! fj 0k; id jg fj id k; lt;b; ~ gt;:0 jg norm( lt;b;~ gt;;id k; lt;b;~ gt;:0;1) ????????????????????! fj id k; 0 jg as expected. If we replaced the three rules for the parallel composition operator with a single rule similar to that in Table 1, then we would get instead the two alternative transitions dec(E) norm( lt;a;~ gt;; lt;a;~ gt;:0k; id;1) ????????????????????! fj 0k; id; id k; lt;b; ~ gt;:0jg dec(E) norm( lt;b;~ gt;;id k; lt;b;~ gt;:0;1) ????????????????????! fj lt;a; ~ gt;:0k; id; id k; 0 jg which are not consistent with the fact that the two subterms of E are independent, thereby resulting in a violation of the concurrency principle (see Sect.... In PAGE 34: ...iolation of the concurrency principle (see Sect. 7:4). Example 7.7 Consider termE ( lt;a; gt;:0 + lt;c; 11;1 gt;:0) kfcg( lt;b; gt;:0 + lt;c; gt;:0) whose decomposition comprises places V1 kfcg id and id kfcg V2 where V1 lt;a; gt;:0 + lt;c; 11;1 gt;:0 V2 lt;b; gt;:0 + lt;c; gt;:0 By applying the rules in Table2 , we get the three transitions... In PAGE 35: ... Example 7.8 Consider term E ( lt;a; gt;:0kfag lt;a; gt;:(0 + 0)) + ( lt;a; gt;:0kfag lt;a; gt;:0) whose decomposition comprises places (V1 kfag id) + (V1 kfag id), (V1 kfag id) + (id kfag V3), (id kfag V2) + (V1 kfag id) and (id kfag V2) + (id kfag V3) whereV1 lt;a; gt;:0 V2 lt;a; gt;:(0 + 0) V3 lt;a; gt;:0 By applying the rules in Table2 , we get the following two transitions dec(E) norm( lt;a; gt;;(V1 kfag id)+id;1) ????????????????????! fj 0kfag id; id kfag(0 + 0) jg dec(E) norm( lt;a; gt;;id+(V1 kfag id);1) ????????????????????! fj 0kfag id; id kfag 0 jg If dec(E) is the current marking then both transitions are enabled and the normalizing factor is 1 for both transitions, as expected. This example motivates the use of V0 instead of V for expressing the basic place: if V were used, then the two transitions above would have the same basic place (beside the same basic action), so they would be given the wrong normalizing factor 1=2 by function norm.... In PAGE 35: ... Example 7.9 Consider termE lt;a; gt;:0kfag(( lt;a; gt;:0 + lt;a; gt;:0) k; lt;a; gt;:0) whose decomposition comprises places V1 kfag id, id kfag(V2 k; id) and id kfag(id k; V3) where V1 lt;a; gt;:0 V2 lt;a; gt;:0 + lt;a; gt;:0 V3 lt;a; gt;:0 By applying the rules in Table2 , we get the following two transitions fj V1 kfag id; id kfag(V2 k; id) jg norm( lt;a; gt;;V1 kfag id;2) ????????????????????! fj 0kfag id; id kfag(0 k; id) jg fj V1 kfag id; id kfag(id k; V3) jg norm( lt;a; gt;;V1 kfag id;1) ????????????????????! fj 0kfag id; id kfag(id k; 0) jg where value 2 for the passive contribution of the rst transition is determined by function melt1. If dec(E) is the current marking then both transitions are enabled and the normalizing factor is 2=3 for the rst transition, and 1=3 for the second transition, as expected.... In PAGE 35: ... Example 7.10 Consider termE lt;a; gt;:0kfag( lt;a; gt;: lt;a; gt;:0 k; lt;a; gt;:0) whose decomposition comprises places V1 kfag id, id kfag(V2 k; id) and id kfag(id k; V3) where V1 lt;a; gt;:0 V2 lt;a; gt;: lt;a; gt;:0 V3 lt;a; gt;:0 By applying the rules in Table2 , we get the following three transitions fj V1 kfag id; id kfag(V2 k; id) jg norm( lt;a; gt;;V1 kfag id;1) ????????????????????! fj 0kfag id; id kfag(V3 k; id) jg fj V1 kfag id; id kfag(id k; V3) jg norm( lt;a; gt;;V1 kfag id;1) ????????????????????! fj 0kfag id; id kfag(id k; 0) jg fj V1 kfag id; id kfag(V3 k; id) jg norm( lt;a; gt;;V1 kfag id;1) ????????????????????! fj 0kfag id; id kfag(0 k; id) jg... In PAGE 36: ... Example 7.11 Consider termE ( lt;a; gt;:0 + lt;a; gt;:0) kfag( lt;a; gt;:0k; lt;a; gt;:0) whose decomposition comprises places V1 kfag id, id kfag(V2 k; id) and id kfag(id k; V2) where V1 lt;a; gt;:0 + lt;a; gt;:0 V2 lt;a; gt;:0 By applying the rules in Table2 , we get the following two transitions fj V1 kfag id; id kfag(V2 k; id) jg norm( lt;a;2 gt;;( lt;a; gt;:0+id)kfag id);1) ????????????????????! fj 0kfag id; id kfag(0 k; id) jg fj V1 kfag id; id kfag(id k; V2) jg norm( lt;a;2 gt;;( lt;a; gt;:0+id)kfag id);1) ????????????????????! fj 0kfag id; id kfag(id k; 0) jg each of which is obtained by applying function melt2 to two potential moves having as a basic place ( lt;a; gt;:0 + id) kfag id and (id + lt;a; gt;:0) kfag id, respectively. If dec(E) is the current marking then both transitions are enabled and the normalizing factor is 1=2 for both transitions, as expected.... ..."

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### Table 2 Representation and integration according to the theory of mental models

2003

"... In PAGE 11: ... over the past two decades, quantified propositions are represented directly in terms of arbitrary individuals. For example, in the Bucciarelli and Johnson-Laird (1999) version of the theory, processing the premisses of AA1A (in non-canonical order) results in the suite of mental models shown in Table2 . Every line in a mental model represents an individual, so for the first premiss we have two individuals, which have the same properties, A and B.... In PAGE 12: ... This is done by trying to refute each of the preliminary conclusions by a counterexample: an extended model in which the premisses are still true but the conclu- sion is false. Such a counterexample can be found for All C are A (as shown in the last row of Table2 ) but not for All A are C , so only the latter survives and is spelled out as the final conclusion. One of the things critics of mental-model theory have complained about is that it is not quite clear what it is, not only because the theory has gone through so many revisions, but because its key tenets remain somewhat underspecified.... In PAGE 12: ... Usually, a version of the mental- model theory comes with one or more computer implementations and a description of what these programs do, but in general this does not suffice to pin down exactly what mental models are. To illustrate, while the first model in Table2 is said to represent the proposition All A are B , we are also told that the model in the third row verifies the proposition All C are A . The former claim suggests that individuals representing the subject term must be enclosed in square brackets, to encode that its representation is exhaustive; the latter suggests that this is not necessary.... In PAGE 12: ... Or consider the sentence Two A are B . How can we represent this in a mental model? One might think that the first model of Table2 is a plausible candidate, but this cannot be right, for two reasons at least. First, this model already represents the interpretation of All A are B , which is patently not synonymous with Two A are B .... ..."

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