### Table 1 Our complexity results are summarized in Table 1, in which each column refers to a minimality problem, and each row refers to a condition on the set S. The rst three rows correspond to cases ; ; , respectively, as de ned in Section 2. The fourth row corresponds to the complementary case, i.e. in which S contains at least one relation which is not 0-valid, at least one which is not 1-valid, ..., and at least one which is not a ne. The symbol P means polynomiality, while the symbol coNP means co-NP-completeness. We want to point out that we have provided a detailed picture of the tractability threshold of the problem. Moreover we gave a negative answer to Kolaitis and Papadimitriou apos;s question [2, Open Problem 4], since model checking problem is co-NP-complete even for a subclass of propositional circumscriptive formulae.

1992

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### Table 1. Tractability of symmetry breaking and dominance detection

"... In PAGE 35: ... Unfortunately, efficient symmetry breaking by such dominance-detection schemes has its limits, as we have identified some CSP classes where dominance detection is intractable. Table1 summarizes our main results, where P (Thm i) means that breaking all the symmetries mentioned in the corresponding row is feasible with a poly- nomial overhead with respect to both time and space at every node explored for the corresponding (set-) CSP in the column, as proved in Theorem i. Some of these positive tractability results, namely the ones marked P (from Thm i) , are trivially derivable as consequences from Theorem i.... ..."

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### TABLE 1: Fixed-parameter tractability of LCS.

### Table 1: The maximal tractable subalgebras of RCC-5.

"... In PAGE 6: ...XfDRgY i X \ Y = ? XfPOgY i 9a; b; c : a 2 X; a 62 Y; b 2 X; b 2 Y; c 62 X; c 2 Y XfPPgY i X Y XfPPIgY i X Y XfEQgY i X = Y Table1 : The ve basic relations of RCC-5. Y X X Y X Y Y X X Y DR(X; Y ) PO(X; Y ) PP(X; Y ) PPI(X; Y ) EQ(X; Y ) Figure 1: Pictorial example of the relations in RCC-5.... ..."

### Table 2: The maximal tractable subalgebras of RCC-5.

### Table 2. Subclasses of Reduction and Scan

1997

"... In PAGE 6: ....3.3. Subclasses of Reduction and Scan. Functions in the classes Reduction and Scan have a local and a global in- put. Thus, the function g which determines the subclassifi- cation is exactly the binary function preS: g (a; b) = preS (a; b) Table2 lists different forms of function g with their cor- responding subclasses of Reduction and Scan. 3.... In PAGE 6: ...1. Reduction For all functions in the subclasses of Reduction which are listed in Table2 , we use a tree-like processor network with n processors which computes the result with time(n) = O(log n), costBrent(n) = O(n) and pipe(n) = O(1). The distributed input (as in Figure 1) determines the val- ues at the leaves; at each inner node, a function computes the value from the values received from the children; the result... ..."

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### Table 1: Information categories and their subclasses

1996

"... In PAGE 3: ... 2 Information Categories In interpreting the data, the first step was to determine a set of information categories into which the contents of participants apos; protocols could be fit. Table1 shows the four major categories and their subclasses. We derived the four categories from theoretical discussions and historical evidence on how external representations convey meanings and concepts, from past literature on design processes that suggest what architects generally think of in design process, and from intensive study of the protocols.... ..."

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### Table 3. Subclasses of Sequential and Broadcast

1997

"... In PAGE 6: ... Thus, the function which determines the subclassification is: g b = preS ((); b) Despite the fact that this function is not binary, associa- tivity again plays a key ro le in the subclassification of the classes Sequential and Broadcast. Table3 lists different forms of function g with their corresponding subclasses of Sequential and Broadcast. 4.... In PAGE 8: ... Remember that n, the depth of the recursion, is still determined by the list of local inputs as = [(); n times : : : ; ()]. For all subclasses of Sequential, shown in Table3 , ex- cept Identity, we use a processor network with log n pro- cessors in a row (repeat) which computes the result with time(n) = O(log n), costBrent(n) = O(log n) and pipe(n) = O(1). We assume n to be a power of 2.... In PAGE 9: ....3. Broadcast In this class we apply function g to the global input n times and return a list of intermediate results and the final re- sult whose implementation is known from the previous sec- tion. For all subclasses shown in Table3 , we use a tree-like processornetworkwith n processorswhich computes the re- sult with time(n) = O(log2 n), costBrent(n) = O(n) and pipe(n) = O(1). At each node, a function is applied which receives the input from its parent on the left side and pro- vides two outputs to its children on the right.... ..."

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