ASP vs. Prolog like programming

ASP vs. Prolog like programming • ASP is adequate for: • NP-complete problems • situation where the whole program is relevant for the problem at hands • If the problem is polynomial, why using such a complex system? • If only part of the program is relevant for the desired query, why computing the whole model?

ASP vs. Prolog • For such problems top-down, goal-driven mechanisms seem more adequate • This type of mechanisms is used by Prolog • Solutions come in variable substitutions rather than in complete models • The system is activated by queries • No global analysis is made: only the relevant part of the program is visited

Problems with Prolog • Prolog declarative semantics is the completion • All the problems of completion are inherited by Prolog • According to SLDNF, termination is not guaranteed, even for Datalog programs (i.e. programs with finite ground version) • A proper semantics is still needed

Well Founded Semantics • Defined in [GRS90], generalizes SMs to 3-valued models. • Note that: • there are programs with no fixpoints of G • but all have fixpoints of G2 P = {a ¬ not a} • G({a}) = {} andG({}) = {a} • There are no stable models • But: G2({}) = {} and G2({a}) = {a}

Partial Stable Models • A 3-valued intr. (T U not F) is a PSM of P iff: • T = GP2(T) • T ÍG(T) • F = HP - G(T) The 2nd condition guarantees that no atom is both true and false: T Ç F = {} P = {a ¬ not a}, has a single PSM: {} This program has 3 PSMs: {}, {a, not b} and {c, b, not a} The 3rd corresponds to the single SM a ¬ not b c ¬ not a b ¬ not a c ¬ not c

WFS definition • [WF Model] Every P has a knowledge ordering (i.e. wrt Í) least PSM, obtainable by the transfinite sequence: • T0 = {} • Ti+1 = G2(Ti) • Td = Ua<d Ta, for limit ordinals d Let T be the least fixpoint obtained. MP = T U not (HP - G(T)) is the well founded model of P.

Well Founded Semantics • Let M be the well founded model of P: • A is true in P iff AÎ M • A is false in P iff not AÎ M • Otherwise (i.e. AÏ M and not AÏ M) A is undefined in P

WFS Properties • Every program is assigned a meaning • Every SM extends one PSM • If WFM is total it coincides with the single SM • It is sound wrt to the SMs semantics • If P has stable models and A is true (resp. false) in the WFM, it is also true (resp. false) in the intersection of SMs • WFM coincides with the perfect model in locally stratified programs (and with the least model in definite programs)

More WFS Properties • The WFM is supported • WFS is cumulative and relevant • Its computation is polynomial (on the number of instantiated rule of P) • There are top-down proof-procedures, and sound implementations • these are mentioned in the sequel

LP and Default Theories • Let DP be the default theory obtained by transforming: H ¬ B1,…,Bn, not C1,…, not Cm into: B1,…,Bn : ¬C1,…, ¬Cm H • There is a one-to-one correspondence between the SMs of P and the default extensions of DP • If LÎ WFM(P) then L belongs to every extension of DP

LPs as defaults • LPs can be viewed as sets of default rules • Default literals are the justification: • can be assumed if it is consistent to do so • are withdrawn if inconsistent • In this reading of LPs, ¬ is not viewed as implication. Instead, LP rules are viewed as inference rules.

LP and Auto-Epistemic Logic • Let TP be the AEL theory obtained by transforming: H ¬ B1,…,Bn, not C1,…, not Cm into: B1 Ù … Ù BnÙ ¬ L C1Ù … Ù ¬ L Cm Þ H • There is a one-to-one correspondence between the SMs of P and the (Moore) expansions of TP • If LÎ WFM(P) then L belongs to every expansion of TP

LPs as AEL theories • LPs can be viewed as theories that refer to their own knowledge • Default negation not A is interpreted as “A is not known” • The LP rule symbol is here viewed as material implication

LP and AEB • Let TP be the AEB theory obtained by transforming: H ¬ B1,…,Bn, not C1,…, not Cm into: B1 Ù … Ù BnÙB ¬C1Ù … ÙB ¬CmÞ H • There is a one-to-one correspondence between the PSMs of P and the AEB expansions of TP • AÎ WFM(P) iff A is in every expansion of TP not AÎ WFM(P) iff B¬A is in all expansions of TP

LPs as AEB theories • LPs can be viewed as theories that refer to their own beliefs • Default negation not A is interpreted as “It is believed that A is false” • The LP rule symbol is also viewed as material implication

SM problems revisited • The mentioned problems of SM are not necessarily problems: • Relevance is not desired when analyzing global problems • If the SMs are equated with the solutions of a problem, then some problems simple have no solution • Some problems are NP-complete. So using an NP-complete language is not a problem. • In case of NP-complete problems, the efficient gains from cumulativity are not really an issue.

SM versus WFM • Yield different forms of programming and of representing knowledge, for usage with different purposes • Usage of WFM: • Closer to that of Prolog • Local reasoning (and relevance) are important • When efficiency is an issue even at the cost of expressivity • Usage of SMs • For representing NP-complete problems • Global reasoning • Different form of programming, not close to that of Prolog • Solutions are models, rather than answer/substitutions

ASP vs. Prolog like programming