Slides01-10 - Relationships Among Semantics If a program EDB has a strati ed or perfect locally strati ed model then that is the unique stable

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Unformatted text preview: Relationships Among Semantics If a program + EDB has a strati ed or perfect locally strati ed model, then that is the unique stable model. A program + EDB can have a unique stable model even if there is no perfect model. Example p :- NOT q q :- NOT p p :- NOT p Only fpg is a stable model. Note that without the 3rd rule, both fpg and fqg are stable. Why Strati ed Models are Stable Intuition to prove strati ed model M is stable: Divide M into strata M0 ; M1; : : : M0 includes the EDB. LFP of stratum 0 does not involve negation, so its instantiated rules survive to the inference step of the GL transform, and exactly M0 is inferred. Thus, M0 , and nothing else for stratum 0, is in GLM . Consider what happens computing the strati ed model at stratum 1. 3 Negated subgoals are resolved according to M0 . 3 Instantiated rules with a negated member of M0 e ectively disappear, and those with a negated nonmember of M0 e ectively lose that subgoal. Thus, the LFP for stratum 1 looks just like the GL transform for the relevant instantiated rules. Example Let's revisit the locally strati ed Win" example: 1 2 3 whose relevant instantiated rules were: 1 r1: win1 :- move1,2 & NOT win2 r2: win1 :- move1,3 & NOT win3 r3: win2 :- move2,3 & NOT win3 Recall stratum 0 = win3 + EDB facts; stratum 1 = win2; stratum 2 = win1. Also recall: the strati ed model is fwin1; win2g + the EDB; we must show this model is also stable. Stratum 0: No relevant rules, so win3 is false. Likewise, win3 is not inferred in the GL procedure. Stratum 1: Since win3 is false, the rules for stratum 1 become: r3: win2 :win2 is inferred, both in the strati ed and GL procedures. Stratum 2: Since win2 is true and win3 is false, the rule for stratum 2 becomes: r2: win1 :Again, the same thing happens to this rule in the GL procedure, so we infer win1 in both strati ed and stable approaches. Well-Founded Model 3-valued model: true, false, unknown. WF model has positive facts like p1 and negative facts like :p1. IDB ground atoms not mentioned are assumed to be unknown." EDB ground atoms not mentioned are assumed false. Two Modes of Inference 1. If body is true, infer head. 2. Look for unfounded sets : If, after instantiating rules in all possible ways and eliminating those with a known false subgoal, there is a set U of positive ground atoms such that every rule with a member of U in the head has a member of U as one of its subgoals, then U is unfounded. 3 We cannot prove any member of U , because we would have to prove another 2 member rst. 3 In WF semantics, we infer the negation of all members of U . Repeat inference modes until no new inferences of either type are possible. Example Win" rule with EDB 1 ! 2, 2 ! 1, 2 ! 3 ! 4 ! 5 ! 6. 1 2 3 4 5 6 Start by instantiating the Win rule winX :- moveX,Y & NOT winY in all possible ways. 3 Eliminate rules with false bodies. 3 Also eliminate true subgoals from remaining bodies. 3 For convenience, eliminate rules whose head has already been inferred. win1 win2 win2 win3 win4 win5 ::::::- NOT NOT NOT NOT NOT NOT win2 win1 win3 win4 win5 win6 Round 1: No positive inferences. Largest unfounded set = fwin6g. Infer :win6. Round 2: Infer win5. Delete last two rules. One now has a false subgoal, the other an alreadyinferred head. win1 win2 win2 win3 ::::- NOT NOT NOT NOT win2 win1 win3 win4 Largest unfounded set = fwin4g. Infer :win4. Round 3: Infer win3, delete last two rules. win1 :- NOT win2 win2 :- NOT win1 Now, no unfounded sets, so done. WF model is fwin3; win5; :win4; :win6g 3 Truth value of win1 and win2 is unknown." Example p q r s ::::- q p p&q NOT p & NOT q fp; qg is an unfounded set. But fp; q; rg is the largest unfounded set. 3 Note: union of unfounded sets is unfounded, so there is always a largest. WF model: f:p; :q; :r; sg. Alternating Fixed Point 1. Instantiate rules in all possible ways. 2. Eliminate rules with false EDB or arithmetic subgoal; eliminate true EDB and arithmetic subgoals from remaining rules for convenience. 3. Initialize all IDB ground atoms to false. 4. Repeatedly evaluate IDB subgoals by applying the GL transform to the model consisting of the EDB + an IDB based on the previous round's true false decisions. In the limit, IDB ground atoms that converge to true are true. Those that converge to false are false. Those that oscillate are unknown. Example Repeating above Win" example: 1 2 3 4 Rules processed by 1 and 2: win1 win2 win2 win3 win4 win5 ::::::- NOT NOT NOT NOT NOT NOT win2 win1 win3 win4 win5 win6 Using alternating xed point: 4 5 6 Round win1 win2 win3 win4 win5 win6 0 0 0 0 0 0 0 1 1 1 1 1 1 0 2 0 0 0 0 1 0 3 1 1 1 0 1 0 4 0 0 1 0 1 0 5 1 1 1 0 1 0 Another Example p :- q; q :- NOT p Round p q 012345 010101 010101 Round 0: Both are 0 as always. Round 1: Rules simplify to p :- q; q :-. Infer both q and p. Round 2: Rules simplify to p :- q. No inference possible! Round 3 and later: Repeats. Conclude both p and q are unknown." Relationships Among Semantics If there is a 2-valued WF model, it is the unique stable model. If there is a perfect model i.e., program + EDB is locally strati ed, then this model is also the stable and WF model, and obviously is 2-valued. There can be a 3-valued WF model when there is no stable semantics i.e., no unique stable model. Example Win program with EDB move1,2, move2,1. Two stable models, fwin1g and fwin2g. Thus, a stable semantics does not exist. However, the WF model exists and makes both win1 and win2 unknown." There can be a unique stable model 6= WF model. 5 Example p :- NOT q q :- NOT p p :- NOT p WF model makes p, q unknown." fpg is the only stable model. There can be a 2-valued WF model when there is no locally strati ed model. Example winX :- moveX,Y & NOT winY with move de ned by 1 2 3 4 There is a cycle among 1, 2, and 3, so this program and EDB is not locally strati ed. However, the WF model is 2-valued, intuitively because the cycle is not followed from board 3 on best play. win1 win2 win3 win3 ::::- NOT NOT NOT NOT Round win1 win2 win3 win4 win2 win3 win1 win4 0 0 0 0 0 1 1 1 1 0 2 0 0 1 0 3 1 0 1 0 4 1 0 1 0 Comparisons Between Stable and WF Approaches In win" program, true = forced win; false = forced loss; unknown = draw with best play. However, in cafeteria" example to follow, the rules are the same but the intuitive semantics favors the stable approach. 6 Example Consider a collection of buildings: Buildings have either lounges or cafeterias, not both. No adjacent buildings both have cafeterias. If a building does not have a cafeteria, then an adjacent building must have one. loungeX :- adjX,Y & cafeteriaY cafeteriaX :- NOT loungeX If we get rid of cafeteria": loungeX :- adjX,Y & NOT loungeY Looks just like win." Problem is really: nd a maximal independent set of buildings in which to put cafeterias. The stable models are the maximal independent sets. But the WF model makes cafeteriaX true i X is in every maximal independent set. Comparison of Complexity for Stable Versus WF It is NP-hard to tell whether a propositional logic program has a stable semantics, i.e., a unique stable model. It is polynomial to construct the WF model. Same comments hold for rst-order logic, but with complexity measured in terms of the EDB size rather than the number of propositions. Modularly Strati ed Semantics Motivation: largest known class of Datalogwith-negation programs for which magic-sets query optimization technique to be discussed works. Must be able to partition the rules into modules," such that 1. All recursion is within a module. 7 2. All modules have locally strati ed semantics with respect to the EDB and the previously computed models for any lower modules. 3 I.e., treat all true facts belonging to lower modules as if they were EDB facts. Modularly strati ed semantics = what we get by computing locally strati ed semantics for modules, bottom up. Note modules are partially ordered by dependence among their predicates, because all recursion must take place within a single module. Example Consider: winX :- moveX,Y & NOT winY with move relation fmove1; 2g. We might appear to have a cycle in the instantiated rules r1: win1 :- move1,2 & NOT win2 r2: win2 :- move2,1 & NOT win1 But the fact that move2; 1 is false removes r2. The only dependence is win1 ! win2, and this program + EDB is locally strati ed. Example Continued Next, suppose we add an IDB predicate move1 to be identical to move: winX :- move1X,Y & NOT winY move1X,Y :- moveX,Y Now, with the same EDB, we instantiate the win rule as: r1: win1 :- move11,2 & NOT win2 r2: win2 :- move12,1 & NOT win1 It is not apparent that win2 does not depend on win1, so it looks like we have a cycle in the dependency graph. 3 The di erence is that move1 is IDB, while move is EDB. This program + EDB is not locally strati ed. 8 However, the program + EDB is modularly strati ed, because we can group move and move1 into a module and win into a higher module. Module for move and move1: We rst compute the locally strati ed model for move1, which is fmove11; 2g. Module for win: We discover that r2 can be removed, because move12; 1 is known to be false. Thus, there is no cycle, and the win module is also locally strati ed. Thus, the whole program + EDB is modularly strati ed. Summary of Semantics De nition A above de nition B means that every program + EDB that has a semantics in B 1. Has a semantics according to A, and 2. The meaning chosen model is the same for both A and B . Stable semantics Well-founded semantics Two-valued well-founded semantics Modularly strati ed semantics Locally strati ed negation Strati ed negation No negation 9 ...
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This document was uploaded on 01/06/2012.

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