### Table 4 Disaggregation Algorithm

"... In PAGE 23: ... Return I. We show the disaggregation algorithm in Table4 . In the following proposition we summarize its properties.... ..."

### Table 3: Complexity results for propositional counterfactual evaluation

"... In PAGE 15: ... 4 Overview and discussion of results In this section we summarize and discuss our results and indicate possible extensions that we are currently investigating. The main results of this paper are compactly presented in Table3 . Di erent methods according to the di erent change semantics for evaluating a counterfactual p gt; q over a propositional knowledge base correspond to di erent rows in this table.... In PAGE 17: ... The model-based methods, however, all become polynomial under these restrictions and allow counterfactual evaluation even in O(kT k kqk) time in this case. Note that the theorems presented in the following sections often state yet more acute results than the ones shown in Table3 . For instance, we show that most hardness and completeness results hold even in case T is literal base, i.... In PAGE 17: ...ompleteness results hold even in case T is literal base, i.e. T consists of a set (or con- junction) of atoms or negated atoms. All results presented in Table3 are new except the P 2 -completeness of Ginsberg apos;s ap- proach for the general case. The latter has recently been shown by Nebel in [45].... ..."

### Table 5.2: Bounds for symmetric set size multipartite Ramsey numbers mk(n; l) for k; n; l = 1; 2; 3; 4. The known Ramsey numbers and lower bounds are motivated as follows: 1By Proposition 2.4(1). 2By Proposition 2.4(4). 3By Proposition 2.4(3). 4By a theorem of Beineke amp; Schwenk, [2]. 5By Theorem 3.7(1). 6Due to Day et al., [32]. 7By Theorem 3.7(2). 8By Theorem 3.7(3). 9By Theorem 3.7(4). 10By Theorem 3.7(5). The upper bounds are motivated as follows: aBy Theorem 2.5. bBy Proposition 2.6(4). cBy Theorem 4.1(2).

2000

### Table 8.1]. In fading channels, the instantaneous SNR Eb=N0 will be replaced by = , where := Eb=N0 now denotes average bit SNR, and is a random variable with unit mean. Supposing that the PDF of satisfies the conditions of Propositions 1 and 3, we can apply the same technique used in the proof of Proposition 3, with some additional properties of Laplace transforms, to obtain the following expression for the high- SNR average BER (derivation details are omitted due to lack of space):

2003

Cited by 16

### Table 1: Algorithm of Tree Allocation Rule not so, because as you move closer to the root of the tree along a given path, the set of players is increasing, so the resources that appear wasted downstream are not wasted because there would have been no one else to allocate them to. More precisely, if the single resource rule A is e cient, then this algorithm consitutes an allocation rule F which is e cient for the tree. This is expressed formally by the following result. Proposition 3 Let A be a single-resource allocation rule, and F the corre- sponding Tree Allocation Rule. Suppose A is monotone with respect to the e ciency measure m, i.e. 8s; s0 m(s) m(s0) ) m(A(s)) m(A(s0)): (17)

1997

Cited by 34

### Table 5: Comprehensibility of propositional rule-sets. The number of rules includes the positive and negative rules. The number of consequents includes the negated instance of target goals. The fourth column shows the actual number of attributes appeared in the rule-set followed by the corresponding values.

"... In PAGE 6: ... The Comprehensibility of an extracted rule-set is a measure of the number of rules, antecedents, consequents, and actual attributes appearing in the rule-set. The Comprehensibility of the propositional rule-sets is summarised in Table5 . For all data-sets, the number of rules and antecedents are signi cantly lower in the rule-sets for the pruned network as com- pared to the extracted rule-sets before the pruning.... ..."

### Table 4 is straightforward, e.g., the propositions corresponding to the probe are:

"... In PAGE 16: ... Because these stories are so simple, I prefer to refer to them as \episodes. quot; The probe and the \memorized quot; episodes are shown in Table4 . I explain the acronyms for the aspects and types of similarity in the next two subsections.... In PAGE 17: ... 4.2 C lassi cations of sim ila r ity The nal column in Table4 classi es the relationship between each episode and the probe using Gentner, Rattermann and Forbus apos;s [1993] types of similarity,whichhave the following relationships to the presence of the various aspects of similarity: LS (Literal Similarity): all aspects. AN (Analogy, also called True Analogy): FOR, HOR, HOS, and OLI, but not OA or not RFB.... ..."

### Table 1: Fibonacci representations and apos;-expansions of the 15 rst integers Proposition A T apos; is a rational set of J . Indeed, Proposition A appears as the consequence of a much stronger result that, for every integer N, relates its Fibonacci representation and its folded apos;-expansion and which is stated by the following:

"... In PAGE 6: ... Proposition 1). Table1 below gives the apos;-expansion of the rst 15 integers together with their Fibonacci normal representation. The position of the radix point, roughly situated, as Table 1 shows, in the middle of every expansion, suggests that R apos; is not a rational language.... In PAGE 6: ...nite apos;-expansion (cf. Proposition 1). Table 1 below gives the apos;-expansion of the rst 15 integers together with their Fibonacci normal representation. The position of the radix point, roughly situated, as Table1 shows, in the middle of every expansion, suggests that R apos; is not a rational language. It will be eventually shown that R apos; is a linear context-free language9 (see Corollary 4).... In PAGE 6: ...s an element of J will naturally be called the folded apos;-expansion of N; e.g., the folded apos;-expansion of 5 is ( 1 1 0 0 0 0 0 1 ). Table1 gives the folded apos;-expansion of the 15 rst integers as well.Let T apos; be the set of folded apos;-expansions of all positive integers; the announced char- acterization of R apos; then reads : 8It is convenient not to deal with 0.... ..."

### Table 1: The possible transitions A reasonable function f : [a; b] ! lt; is restricted to the following set of possible transitions from one qualitative state to the next. The contents of this table are justi ed by Propositions 3, 4, 6, and 7 in Appendix A. P-Transitions Name QS(f; ti) ) QS(f; ti; ti+1)

1986

"... In PAGE 17: ... Table1 speci es the set of possible transitions that can take place in the qualitative behavior of a single function. The validity of this table is proved... In PAGE 26: .... Select a qualitative state from ACTIVE. 2. For each function, determine (from Table1 ) the set of transitions possible from the current qualitative state. 3.... In PAGE 27: ... The constraints are: DERIV (Y; V ) DERIV (V; A) A(t) = g lt; 0: Figure 5 shows a graphical representation of the constraints and a qual- itative plot of the behavior of Y (t). We start with an active state, t = (t0; t1), whose description is: QS(A; t0; t1) = hg; stdi QS(V; t0; t1) = h(0; 1); deci QS(Y; t0; t1) = h(0; 1); inci For each function, retrieve from Table1 the set of possible qualitative state transitions from the current state of that function. Since the current state represents the time-interval (t0; t1), only I-transitions are applicable.... In PAGE 29: ... 4.4 Function Consistency The possible transitions that a single parameter can take from one quali- tative state to the next are given in Table1 . In Step 2, the current state of each function is used to retrieve the set of applicable transition patterns from Table 1.... In PAGE 29: ...4 Function Consistency The possible transitions that a single parameter can take from one quali- tative state to the next are given in Table 1. In Step 2, the current state of each function is used to retrieve the set of applicable transition patterns from Table1 . Constraints between neighboring functions are not considered until Step 3.... In PAGE 29: ...tage, to eliminate impossible transitions for functions that are (e.g.) always nite or never negative. For any particular qualitative state, Table1 provides at most 4 possible transitions. Thus, if there are n functions in the system, the possible next states are to be found within a product space of at most 4n points.... In PAGE 29: ... At this stage, however, we do not explicitly generate this product space, so we need create at most 4n individual transitions. Appendix A presents the proofs that justify the possible transitions given in Table1 . It also discusses the handling of divergence to 1 and asymptotic approach to limiting values.... In PAGE 36: ... Such a rede nition is not always possible. The structure of QSIM makes it possible to experiment with f+; 0; ?g semantics for qualitative simulation simply by replacing Table1 with an alternate table of legal transitions (Table 2). Figure 6 shows the behavior of the Spring system under the f+; 0; ?g semantics.... In PAGE 37: ...Table 2: Possible transitions under f+; 0; ?g semantics P-Transitions Name QS(f; ti) ) QS(f; ti; ti+1) P1 hlj; stdi hlj; stdi P2 hlj; stdi h(lj; lj+1); inci P3 hlj; stdi h(lj?1; lj); deci P4 hlj; inci h(lj; lj+1); inci P5 h(lj; lj+1); inci h(lj; lj+1); inci P6 hlj; deci h(lj?1; lj); deci P7 h(lj; lj+1); deci h(lj; lj+1); deci Q8 h(lj; lj+1); stdi h(lj; lj+1); stdi Q9 h(lj; lj+1); stdi h(lj; lj+1); inci Q10 h(lj?1; lj); stdi h(lj?1; lj); deci I-Transitions Name QS(f; ti; ti+1) ) QS(f; ti+1) I1 hlj; stdi hlj; stdi I2 h(lj; lj+1); inci hlj+1; stdi I3 h(lj; lj+1); inci hlj+1; inci I4 h(lj; lj+1); inci h(lj; lj+1); inci I5 h(lj; lj+1); deci hlj; stdi I6 h(lj; lj+1); deci hlj; deci I7 h(lj; lj+1); deci h(lj; lj+1); deci J8 h(lj; lj+1); inci h(lj; lj+1); stdi J9 h(lj; lj+1); deci h(lj; lj+1); stdi J10 h(lj; lj+1); stdi h(lj; lj+1); stdi The landmarks are xed as f?1; 0; 1g. The transitions that create new landmarks (I8 and I9 from Table1 ) are eliminated, and new transitions are added (with Q and J names) to permit direction of change std between... In PAGE 41: ... The function u satis es the initial state description QS(F; t0) because it is a qualitative abstraction of the initial conditions to equation (3). Step 2 in QSIM generates all possible qualitative state transitions for the functions in C from a given qualitative state, using Table1 which is justi ed by Propositions 3, 4, 6, and 7. Thus, any change in qualitative state of the system must be included in the possibilities generated.... In PAGE 51: ...A The Qualitative State Transitions This appendix applies the Intermediate-Value and Mean-Value Theorems to prove the validity of the transition rules in Table1 on page 18 that restrict the possible qualitative behaviors of a single function. Let f : [a; b] ! lt; be a reasonable function with distinguished time-points a = t0 lt; lt; tn = b; and landmark values l1 lt; lt; lk: De nition: Where ti is a distinguished time-point, a P-transition of f is a pair of adjacent qualitative states of f, QS(f; ti) ) QS(f; ti; ti+1) whose rst state is the qualitative state at a distinguished time-point.... In PAGE 54: ... Note that the total ordering on the set of landmarks is preserved. Table1 on page 18 collects and names the transitions permitted by Propositions 3, 4, 6, and 7, for use in the QSIM algorithm. A.... ..."

Cited by 382

### Table 2 summarizes the differences between the cases in terms of the five propositions highlighted by Jarvenpaa and Ives (1996).

"... In PAGE 8: ...As indicated in Table2 , the UOX case shares attributes with both cases, while at the same time is unlike any of them. In the following sections the UOX attributes are considered and compared to the quot;American quot; and ROW cases.... ..."