### Table 1: Complexity of subsumption in term-forming languages

1990

"... In PAGE 9: ...erators can be speci ed as follows: (C t C0)I = CI [ C0I (:C)I = D n CI (9 nR: C)I = fx 2 Dj kRI(x) \ CIk ng (9R: C)I = (9 1R: C)I = fx 2 Dj RI(x) \ CI 6 = ;g (9R)I = (9R: gt;)I = fx 2 Dj RI(x) 6 = ;g (R # R0)I = fx 2 Dj RI(x) = R0I(x)g Similar to concept-forming operators, role-forming operators are conceivable, and have indeed been used in di erent term-forming languages.8 The symbol S will be used to denote atomic roles, while R will be used to denote role-descriptions: R ! S atomic role j R u R0 role conjunction j :R role negation j RjC range restriction j R R0 role chain The formal meaning of these operators can be speci ed as follows: (R u R0)I = RI \ R0I (:R)I = (D D) n RI (RjC)I = RI \ (D CI) (R R0)I = fhx; yij 9z: z 2 RI(x) ^ y 2 R0I(z)g As can be seen in Table1 , when trying to extend the term-forming part of STL, one inevitably ends up with languages for which subsumption determination is intractable or, more precisely, NP-hard. Further- more, introducing role-chains and agreements on roles or role-negation and conjunction, the result is even worse, subsumption determination becomes undecid- able.... In PAGE 10: ... It may well be the case that it is possible to nd algorithms that are well-behaved in all normal cases. Furthermore, it seems promising to reevaluate the results shown in Table1 in this light. In our case, a reasonable restriction on the form of terminologies, which can be considered as a \normal case, quot; is that the \depth of a terminology quot; over roles and de nitions is logarithmicly bounded by the size of the terminology [38].... In PAGE 10: ... But there are some special cases, in which the computation of instance relationships is easy. For example, in the back system, which uses an incomplete subsumption algorithm because subsump- tion determination is intractable (see Table1 ), the re- alization algorithm described in [41] and [37, Ch. 4] turns out to be complete in the important special case when the world description contains enough informa- tion in order to decide whether a role relationship be- tween two objects holds or not|when the world de- scription is role-closed.... ..."

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### Table 3. Query plan operators

1995

"... In PAGE 21: ...2#29 rather than sets of tuples. Table3 lists some of the opera- tors. The operators for grouping and for returning complex objects in a query result are more complicated and are not shown.... In PAGE 23: ... Figure 5 shows one plan for Query #281#29 of Section 5.1 using the operators in Table3 . The arrows between operators are annotated with the sets of object assignments resulting from each operator, with an object assignment represented as a set of #5Cpath expression pre#0Cx :#5Bidenti#0Cer#5D quot; pairs.... ..."

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### Table 3. Query plan operators

"... In PAGE 21: ...2) rather than sets of tuples. Table3 lists some of the opera- tors. The operators for grouping and for returning complex objects in a query result are more complicated and are not shown.... In PAGE 23: ... Figure 5 shows one plan for Query (1) of Section 5.1 using the operators in Table3 . The arrows between operators are annotated with the sets of object assignments resulting from each operator, with an object assignment represented as a set of \path expression pre x : [identi er] quot; pairs.... ..."

### Table 2 Complexity of domain-independent planning. Language How the Allow Allow ne- plan existence plan length restrictions operators delete gated pre- (telling if a plan (if there is a plan

1995

"... In PAGE 9: ... 4 Complexity Results Based on various syntactic criteria on what planning operators are allowed to look like, we have developed a comprehensive theory of the complexity of planning. The results are summarized in Table2 ; for details see [10]. When there are no function symbols and only nitely many constant symbols (so that planning is decidable), the computational complexity varies from constant time to expspace-complete, depending on a wide variety of conditions: { whether or not delete lists are allowed; { whether or not negative preconditions are allowed; { whether or not the predicates are restricted to be propositional (i.... In PAGE 11: ... Our results on planning with a xed set of operators reveal that for any given planning domain that can be described with STRIPS operators, the complexity of planning is at most in pspace, and that there exists such domains for which planning is pspace-complete. Examination of Table2 reveals several interesting properties: (i) If the planning operators are extended to allow conditional e ects, this does not a ect our results. This contradicts a widespread belief that planning with conditional operators is harder than planning with regu- lar STRIPS operators.... In PAGE 13: ...1 Planning Bylander [3,4] has done several studies on the complexity of propositional planning. We have stated some of his results in Table2 . More recently, he has studied the complexity of propositional planning extended to allow a limited amount of inference in the domain theory [4].... ..."

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### Table 2 Complexity of domain-independent planning. Language How the Allow Allow ne- plan existence plan length restrictions operators delete gated pre- (telling if a plan (if there is a plan

1995

"... In PAGE 9: ... 4 Complexity Results Based on various syntactic criteria on what planning operators are allowed to look like, we have developed a comprehensive theory of the complexity of planning. The results are summarized in Table2 ; for details see [10]. When there are no function symbols and only nitely many constant symbols (so that planning is decidable), the computational complexity varies from constant time to expspace-complete, depending on a wide variety of conditions: { whether or not delete lists are allowed; { whether or not negative preconditions are allowed; { whether or not the predicates are restricted to be propositional (i.... In PAGE 11: ... Our results on planning with a xed set of operators reveal that for any given planning domain that can be described with STRIPS operators, the complexity of planning is at most in pspace, and that there exists such domains for which planning is pspace-complete. Examination of Table2 reveals several interesting properties: (i) If the planning operators are extended to allow conditional e ects, this does not a ect our results. This contradicts a widespread belief that planning with conditional operators is harder than planning with regu- lar STRIPS operators.... In PAGE 13: ...1 Planning Bylander [3,4] has done several studies on the complexity of propositional planning. We have stated some of his results in Table2 . More recently, he has studied the complexity of propositional planning extended to allow a limited amount of inference in the domain theory [4].... ..."

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### Table 1: Complexity of unsatis ability and subsumption in AL-languages. Notes with a star refer to partial results.

1991

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### Table 2: Complexity results for at representations. plan evaluation plan existence

1997

"... In PAGE 4: ...6 SUMMARY OF RESULTS Table 1 summarizes our results, which are explained in more detail in later sections. Table2 summarizes a set of results for at domains; these are described in our extended technical report (Goldsmith, Littman, amp; Mundhenk 1997). 2 ACYCLIC PLANS Given a planning domain M = hS; s0; A; t; Gi, a plan P = hQ; q0; ; ; ; !i is an acyclic plan where Q and are nite sets of plan steps and e ects labels, respectively,... ..."

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### Table 2: Complexity results for at representations. plan evaluation plan existence

1997

"... In PAGE 4: ...6 SUMMARY OF RESULTS Table 1 summarizes our results, which are explained in more detail in later sections. Table2 summarizes a set of results for at domains; these are described in our extended technical report (Goldsmith, Littman, amp; Mundhenk 1997). 2 ACYCLIC PLANS Given a planning domain M = hS; s0; A; t; Gi, a plan P = hQ; q0; ; ; ; !i is an acyclic plan where Q and are nite sets of plan steps and e ects labels, respectively,... ..."

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### Table X. Numbers of operators in plans

2005

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