### Tableau Calculi for Answer Set Programming

2006

Cited by 6

### Tableaux Calculi for Answer Set Programming

### Tableaux Calculi for Answer Set Programming 3

### Table 1: Complexity of answer set checking for various fragments of DL, prop. case.

"... In PAGE 7: ...3 Previous Results As mentioned in the Introduction, previous work on the complexity of ASP mostly considered the case of propositional programs. Table1 , which is taken from [17], provides a complete overview... ..."

### Table 1. De nition Introduction Phase

2002

"... In PAGE 9: ...nnermost (i.e., ti does not contain operation-rooted subterms) such that each one of them shares at least a common variable with at least one more subterm in T 2. Apply the DEFINITION INTRODUCTION RULE to generate : Rdef = (fnew(x) ! T) where fnew is a new function symbol not appearing R, and x is the set of variables of T OUTPUT: Definition Rule (Eureka) Rdef In order to avoid these risks, our approach generates eureka de nition following a very simple strategy ( Table1 ) that obtains similar levels of generality than pre- vious approaches and covers most practical cases. The main advantages are that the analysis is terminating, easy and quickly, and does not perform redundant calculus (like unfolding steps, instantiations, etc.... In PAGE 9: ...) that properly corresponds to subsequent phases. As illustrated by step 1 in Example 2, our eureka generator proceeds as the algorithm in Table1 shows. Observe that the input of the algorithm is the original program R and a selected rule R 2 R which de nition is intended to be 4 This fact is observed during the so-called quot;program extraction phase quot; in [20].... In PAGE 13: ...contrast with the previous case, where an optimal de nition of fnew is obtained, now the process tries with rules in R in order to replace and reuse as much as possible the optimized de nition of fnew into the original program R. In particular, we know that there exists at least a rule R = (l ! r) 2 R (the one considered in Table1 to generate de eureka Rdef) that veri es TEST(R; Rdef)=2. Hence, similarly as done before, we can apply an abstraction and folding steps to it, obtaining the new rule: l ! r[zj]Pj where hz1; : : : ; zni = fnew(x): where the call to fnew enhances the nal de nition of the old function symbol that roots l.... ..."

### Table 1. Complexity of answer set checking for various fragments of DL, prop. case.

2004

"... In PAGE 6: ...3 Previous Results As mentioned in the Introduction, previous work on the complexity of ASP mostly con- sidered the case of propositional programs. Table1 , which is taken from [17], provides a complete overview for those fragments of the language we also consider in this paper. The rows specify the form of disjunction allowed (in particular, a77 a78 = no disjunction).... ..."

Cited by 2

### Table 2: Experiments computing all answer sets.

2007

"... In PAGE 6: ... Only cmodels and smodelscc never time out, but they are much slower. Table2 shows results for computing all answer sets, or for determining that no answer sets exist (0 #sol). Given that as- sat cannot enumerate answer sets, we only include it on un- satisfiable programs.... ..."

Cited by 11

### Table 2. Neuro-fuzzy system answers

"... In PAGE 2: ...ig. 1. Voltage peaks within 10 min. periods (maximal values). Experiment results are shown in Table2 . The neuro-fuzzy system was trained with 2000 training vectors consisting of 5 random values describing the disturbances within a 10 min.... In PAGE 2: ... The lower the output value the smaller the probability of a malfunction. Test input vectors and respective outputs are summarized in Table2 . In case 2 swells were severe, so the system output is close to 1.... ..."

### Table 4. Neuro-fuzzy system answers

"... In PAGE 3: ... On the contrary, if THD is high and H19, H21, overvoltages are allowed - normal operation is expected. Investigation results are shown in Table4 . As before, most cases were correctly recognized, accordingly to the learning patterns (Table 4, cases 1.... In PAGE 3: ... Investigation results are shown in Table 4. As before, most cases were correctly recognized, accordingly to the learning patterns ( Table4 , cases 1.... ..."

### Table 1: Complexity of deciding strong / weak answer set existence for dl-programs KB (completeness results)

2004

Cited by 90