### Table 1: Complexity of problems speci#0Ced succinctly. The list is not

in Complexity of Hierarchically and One-dimensional Periodically-specified Problems I: Hardness Results

"... In PAGE 29: ... Our idea then, is to lift the known reduction from 3SAT to problem #05 when the instance is speci#0Ced non-hierarchically, and thus obtain a suitable reduction from L-3SAT to the problem L-#05. This approach helps one to prove easily many more PSPACE-hardness results for hierarchically speci#0Ced instances #28see Table1 #29. We also point out that most of the reductions are quasi-linear size and quasi-linear time reductions and thus provide tightlower bounds on the deterministic time complexityofthe problem #28under standard complexity theoretic assumptions#29.... In PAGE 33: ...heorem 6.2. The following problems are PSPACE-hard for 1-FPN-or L-speci#0Ced planar graphs of O#28log N #29 bandwidth: 3-COLORING, INDEPEN- DENT SET#28IS#29, DOMINATING SET, VERTEX COVER, PARTITION INTO TRIANGLES and HAMILTONIAN PATH#28HP#29. Table1 contains a sample of the results wehave obtained for succinctly speci#0Ced problems. As another example of the applicability of our results we consider the complexity of the monotone circuit value problem for L- and 1-FPN#28BC#29-speci#0Ced inputs.... In PAGE 33: ... We also outlined how our techniques could be used to prove the PSPACE- hardness of several other L-, 1-FPN-, 1-FPN#28BC#29- or 1-PN-speci#0Ced prob- lems. Table1 contains example of the results for L-, 1-FPN-, 1-FPN#28BC#29- or 1-PN-speci#0Ced problems that can be obtained by the techniques presented here. In the following, we explain the various entries in Table 1 and also... In PAGE 33: ... Table 1 contains example of the results for L-, 1-FPN-, 1-FPN#28BC#29- or 1-PN-speci#0Ced problems that can be obtained by the techniques presented here. In the following, we explain the various entries in Table1 and also... In PAGE 34: ...33 1. An entry 1 and 4 in Table1 denotes that the corresponding problem is PSPACE-hard, even when restricted to L- or 1-PN-speci#0Ccations of O#28log N #29 bandwidth bounded instances. These results have pre- viously not appeared in the literature; 4 implies that the result is proved formally in #5BMH+96#5D.... In PAGE 34: ... 2. An entry 2 in Table1 denotes that the problem L-#05 or the problem 1-PN-#05 was shown to be PSPACE-hard in #5BLW92#5Dor#5BOr82a#5D. This can also be shown by the ideas and techniques of this paper, even when restricted to L and 1-PN-speci#0Ccations of O#28log N #29 bandwidth bounded instances.... In PAGE 34: ... 3. An entry 3 in Table1 denotes that the corresponding problem is polynomial time solvable. This is shown in #5BMH+96#5D.... ..."

### Table 2: A sample of complexity results for succinctly specified problems obtained here and in [MH+96a].

1995

"... In PAGE 30: ... For proving easiness results, we reverse the above process and devise reductions from ? to an appropriately chosen -SAT(S) problem. The results obtained using these ideas are summarized in Table2 and also briefly discussed in [MH+96a]. We therefore omit the details.... In PAGE 34: ...esults have previously not appeared in the literature. [LW92]. 2. An entry 2 in Table2 denotes that the problem L- or the problem 1-PN- was shown to be PSPACE-hard in [LW92] or [Or82a]. This can also be shown by the ideas and techniques of this paper, even when restricted to L and 1-PN-specifications of O(log N ) bandwidth bounded instances.... ..."

Cited by 6

### Table 3. Succinct ow logic for the functional fragment.

1998

"... In PAGE 6: ... We express this as follows: (Rd F ; Rc F ; MF ; SF ; WF ) satis es R; M e : S1 ! S2 amp; W Here the proposed solution consists of the ve caches of Table 2 and the entities R; M; S1; S2 and W: R 2 d Env is the environment in which e is to be analysed, M 2 P(Mem) is the set of contexts in which e is to be analysed, S1 2 d Store is the store that is possible immediately before e, S2 2 d Store is the store that is possible immediately after e, and W 2 c Val is the value that e can evaluate to. Since the ve caches of Table 2 remain \constant quot; throughout the veri cation we shall dispense with listing them when de ning the \ quot; relation in Table3 . Note that the clauses are de ned compositionally and hence clearly are well-de ned.... In PAGE 6: ... Given the caches of Example 4 we may verify the following formula for the program of Example 1 [ ]; f g program : [ ] ! [ ] amp; f( ; (y; 3))g re ecting that the initial environment is empty, that the initial context is the empty call string, that the program does not manipulate the store (which hence is empty) and that the nal value is described by f( ; (y; 3))g. The veri cation will amount to a proof using the clauses of Table3 as rules and axioms; if successful, the proof and the caches constitute the analysis of the program. 2 The clause for variables merely demands that the store after x equals the store possible before x and that the value associated with x in the environment equals... In PAGE 8: ... Containments versus equalities. Since the speci cation in Table3 is concerned with verifying whether or not a proposed solution is acceptable it is sensible that the clause for function application employs a containment like takek(l; M) MF ( ) rather than an equality like takek(l; M) = MF ( ). The reason is that there might be other instances of the clause where the label of the application... In PAGE 9: ... In fact it would be incorrect to replace the containment by an equality: if M 6 = ;, k gt; 0 and li1 6 = li2 then it is impossible to obtain takek(li; M) = MF ( ) for all i. Although the clauses in Table3 contain no explicit equalities they do contain a lot of implicit equalities because the same ow variable is used more than once in the same clause. One can avoid this by introducing new variables and then linking them explicitly by containments as illustrated below.... In PAGE 10: ...F ( )dXl dc(M; W1; )e v Rc F( ) ^ takek(l; M) MF( ) for some R1; M1; S11; S12; W1; R2; M2; S21; S22; W2 Clearly there will be proposed solutions that are acceptable according to the modi ed speci cation but that are not acceptable according to Table3 . This motivates being explicit about what we mean by the best solution.... In PAGE 10: ... In other words, we can change containments to equalities if we \collect quot; all terms de ning the same entity. 3 Attribute Grammar Formulations The ow logic of Table3 can be transformed into an attribute grammar. The basic idea behind attribute grammars is as follows.... In PAGE 10: ... We shall now proceed in two stages. First we show that a minor transformation will turn the speci cation of Table3 into an extended attribute grammar with global attributes and side conditions. The second stage will then transform the extended attribute grammar into an attribute grammar using global attributes and de ning the attributes by containments (rather than equalities).... In PAGE 11: ... The global attributes can be used as constants in the construction of terms for the attributes and their values can be further constrained by explicit conditions associated with the syntactic rules. It is now easy to see that Table 4 can be obtained from Table3 and vice versa by simply changing the notation. Hence it should be clear that the two speci cations admit the same acceptable solutions and therefore that the best solution for one equals the best solution for the other.... In PAGE 15: ... In doing so we shall exploit the presence of labels on all subexpressions. We shall write the analysis of an expression tl as (Rd F ; Rc F ; MF ; SF ; WF ; RL; ML; WL; SL) satis es tl and (as in Table3 ) we shall be explicit about the analysis of subexpressions. Allowing minor changes in notation this results in the speci cation of Table 8.... ..."

Cited by 9

### Table 1: Discrete two-dimensional experiments: Scaling in estimation sample size D8. Distributions are bivariate Gaussians discretized on 21 x 21 grid. 1000 repetitions. AM C8

"... In PAGE 5: ... Interestingly, these experiments also show that it is not always best to sample directly from the target distri- bution C8 when the random variable CU has a substantially different structure. We also tested GIS on a simple multi- variate problem, and Table1 shows that GIS handles mul-... ..."

### Table 12 Marginal effects on probability of hospital exit (for-profit and not-for-profit) For-profit exit Not-for-profit exit

"... In PAGE 17: ...Table12 for the marginal effects of county and hospital characteristics on the probability of exit for each group of hospitals. A comparison of coefficient estimates from the two logistic regressions reveals that lower growth of the elderly population increases the probability of exit for both for- profit and not-for-profit hospitals.... In PAGE 17: ... For both groups, the existence of a certificate of need law does not have a significant effect on the probability of exit. It is possible that the difference in for-profit and not-for-profit responsiveness reported in Table12 is due to differences in the distribution of characteristics among hospitals rather than the propensity to exit. To address this, we take the estimates from Table 12 and apply them out of sample to the dataset for the opposite ownership type.... In PAGE 17: ... It is possible that the difference in for-profit and not-for-profit responsiveness reported in Table 12 is due to differences in the distribution of characteristics among hospitals rather than the propensity to exit. To address this, we take the estimates from Table12 and apply them out of sample to the dataset for the opposite ownership type. The results of this exercise are reported in Table 13.... ..."

### Table 1. Parts distribution in diC128erent buC128ers at steady state, the sample parameters, the discrete and the continuous entropies

1999

"... In PAGE 7: ... The initial buC128er levels are set to 10 parts and the demand rate is distrib- uted exponentially with mean of 1=3 parts=time unit . Table1 summarizes the statistics of more than 65 500 data points in diC128erent buC128ers along the line. It presents the distribution of parts in these buC128ers at steady state, the sample parameters, the discrete entropies and the continuous entropies.... In PAGE 8: ...upper bound based on a normal distribution with the indicated sample variance. Note from Table1 that the continuous entropy gives a good estimate for the dis- crete entropy. The normal distribution is selected by the software as the best continuous distribution that fits the data of downstream buC128ers.... ..."