From Default and Autoepistemic Logics to Disjunctive Answer Set Programs via the Logic of GK
Jianmin Ji
1and Hannes Strass
2Abstract. We show how the pure logic of GK can be embedded into disjunctive logic programming. The translation we present is polyno- mial, but not modular, and introduces new variables. The result can then be used to compute the extension/expansion semantics of de- fault and autoepistemic logics using disjunctive ASP solvers.
1 Introduction
Lin and Shoham [6] proposed a logic with two modal operatorsK and A, standing for knowledge and assumption, respectively. The idea is that one starts with a set of assumptions (those true under the modal operatorA), computes the minimal knowledge under this set of assumptions, and then checks to see if the assumptions were justified in that they agree with the resulting minimal knowledge.
In this paper, for the first time, we consider computing models of GK theories by disjunctive logic programs. We shall propose a poly- nomial translation from a (pure) GK theory to a disjunctive logic program such that there is a one-to-one correspondence between GK models of the GK theory and answer sets of the resulting disjunc- tive logic program. The result can then be used to compute the ex- tension/expansion semantics of default logic [9] and autoepistemic logic [8]. To substantiate this claim, we have implemented the trans- lation into a working prototype gk2dlp.3 A longer version of this paper with more details is available as a workshop paper [5].
2 Main Result: From Pure GK to Disjunctive ASP
Syntactically, the logic of GK is a propositional modal language with modalitiesAandK;pureGK formulas contain no nested modalities.
For space reasons, we cannot present background and refer to [6, 2].
Before presenting the translation, we introduce some notations.
LetF be a pure GK formula, we usetrp(F)to denote the propo- sitional formula obtained fromF by replacing each occurrence of a formula Kφ(called aK-atom) bykφ and each occurrence of a formulaAψ(anA-atom) byaψ, wherekφandaψare new atoms with respect toφandψrespectively. For a pure GK theoryT (a set of pure GK formulas), we definetrp(T) = V
F∈Ttrp(F). Intu- itively, the new atomkφwill be used to encodeφ ∈ K(M)for a GK (Kripke) modelMforT, that is,φis known inM. Likewise,aφ
encodesφ ∈A(M), which means thatφis assumed inM. Given a propositional formulaφand an atoma, we useφato denote the propositional formula obtained fromφby replacing each occurrence of an atompwith a new atompawith respect toa. Intuitively, such new atoms will be used to guarantee the existence of certain interpre- tations witnessing various technical properties.
1 School of Computer Science and Technology, University of Science and Technology of China, Hefei, China
2Computer Science Institute, Leipzig University, Leipzig, Germany
3http://informatik.uni-leipzig.de/˜strass/gk2dlp/
We now stepwise work our way towards the main result. We start out with a result that relates a pure GK theoryT to a propositional formula. In what follows,AtomK(T)andAtomA(T)denote the sets ofK-atoms andA-atoms occurring inT, respectively.
Proposition 1 LetT be a pure GK theory. A Kripke interpretation Mis a model ofTif and only if there exists a modelI∗of the propo- sitional formulaΦT =trp(T)∧Φsnd∧ΦKwit∧ΦAwitwith
Φsnd= ^
φ∈AtomK(T)
(kφ⊃φk)∧ ^
φ∈AtomA(T)
(aφ⊃φa)
ΦKwit= ^
ψ∈AtomK(T)
¬kψ ⊃
¬ψkψ∧ ^
φ∈AtomK(T)
(kφ⊃φkψ)
ΦAwit= ^
ψ∈AtomA(T)
¬aψ⊃
¬ψaψ∧ ^
φ∈AtomA(T)
(aφ⊃φaψ)
• K(M)∩AtomK(T) ={φ|φ∈AtomK(T)andI∗|=kφ};
• A(M)∩AtomA(T) ={φ|φ∈AtomA(T)andI∗|=aφ}.
The proposition examines the relationship between models of a pure GK theory and particular models of the propositional for- mulaΦT. The first conjuncttrp(T) of the formulaΦT indicates that thek-atoms anda-atoms in it can be interpreted in accordance withK(M)andA(M)such thatI∗ |= trp(T)iffM is a model ofT. The soundness formulaΦsndachieves that the sets{φ|φ ∈ AtomK(T)andI∗ |=kφ}and{φ|φ∈AtomA(T)andI∗ |=aφ} are consistent. The witness formulasΦwitindicate that, ifI∗|=¬kψ
for someψ∈AtomK(T)(resp.ψ∈AtomA(T)) then there exists a modelI0ofK(M)(resp.A(M)) such thatI0 |=¬ψ, whereI0is explicitly indicated by newly introducedpkψ(resp.paψ) atoms.
While Proposition 1 aligns Krikpe models and propositional mod- els of the translation, there is yet no mention of GK’s typical mini- mization step. This is the task of the next result, which extends the above relationship to GK models.
Proposition 2 LetT be a pure GK theory. A Kripke interpretation M is a GK model ofT if and only if there exists a modelI∗of the propositional formulaΦTsuch that
• K(M) =A(M) =Th({φ|φ∈AtomK(T)andI∗|=kφ});
• for eachψ∈AtomA(T),
I∗|=aψiffψ∈Th({φ|φ∈AtomK(T)andI∗|=kφ})
• there does not exist another modelI∗0such that
– I∗0∩ {aφ|φ∈AtomA(T)}=I∗∩ {aφ|φ∈AtomA(T)}, – I∗0∩ {kφ|φ∈AtomK(T)}(I∗∩ {kφ|φ∈AtomK(T)}.
In Proposition 2, we only need to consider Kripke interpretations Msuch thatA(M)∪K(M)is consistent. This means that formula ΦTcan be modified toΨT =trp(T)∧Ψsnd∧Ψwitwith
Ψsnd= ^
φ∈AtomK(T)
(kφ⊃φ)∧ ^
φ∈AtomA(T)
(aφ⊃φ)
Ψwit = ^
ψ∈AtomK(T)
¬kψ⊃ΨKψ
!
∧ ^
ψ∈AtomA(T)
¬aψ⊃ΨAψ
!
ΨKψ =¬ψkψ∧ ^
φ∈AtomK(T)
(kφ⊃φkψ)∧ ^
φ∈AtomA(T)
(aφ⊃φkψ)
ΨAψ =¬ψaψ∧ ^
φ∈AtomK(T)
(kφ⊃φaψ)∧ ^
φ∈AtomA(T)
(aφ⊃φaψ)
Using this new formula, the result of Proposition 2 can be restated.
Proposition 3 LetT be a pure GK theory. A Kripke interpretation M is a GK model ofT if and only if there exists a modelI∗of the propositional formulaΨTsuch that
• K(M) =A(M) =Th({φ|φ∈AtomK(T)andI∗|=kφ});
• for eachψ∈AtomA(T),
ifI∗|=aψthenψ∈Th({φ|φ∈AtomK(T)andI∗|=kφ})
• there does not exist another modelI∗0ofΦT such that
– I∗0∩ {aφ|φ∈AtomA(T)}=I∗∩ {aφ|φ∈AtomA(T)}, – I∗0∩ {kφ|φ∈AtomK(T)}(I∗∩ {kφ|φ∈AtomK(T)}.
We are now ready for our main result, translating a pure GK theory to a disjunctive logic program. First, we introduce some notations.
LetTbe a pure GK theory, we usetrne(ΨT)to denote the nested ex- pression obtained fromΨT by first converting it to negation normal form, then replacing “∧” by “,” and “∨” by “;”. For a propositional formulaφ, we usetrc(φ)to denote the set of rules obtained from the conjunctive normal form ofφ(possibly containing new variables) by replacing each clause(p1∨ · · · ∨pl∨ ¬pl+1∨ · · · ∨ ¬pm)by a rule p1;. . .;pl ←pl+1, . . . , pm. We useφbto denote the propositional formula obtained fromφby replacing each occurrence of an atom pby a new atomp. We useˆ Φ∗T to denote the propositional formula obtained fromΦTby replacing each occurrence of an atomp(except atoms of the formaφfor someφ∈AtomA(T)) by a new atomp∗. Intuitively, by Proposition 3,trne(ΨT)is used to restrict inter- pretations for introduced k-atoms and a-atoms so that these in- terpretations serve as candidates M for GK models. By Proposi- tion 1,Φ∗T constructs possible modelsM0 of the GK theory (with A(M0) =A(M)) that are used to test whetherM is a GK model.
Inspired by the linear translation from parallel circumscription into disjunctive logic programs in [3], we have the following theorem.
Theorem 1 LetT be a pure GK theory. A Kripke interpretationM is a GK model ofT if and only if there exists an answer setSof the logic programtrlp(T):
(1) ⊥ ←nottrne(ΨT)
(2) p0;¬p0← > (for each atomp0occurring intrne(ΨT)) (3) u;A←B (for each ruleA←Bintrc(Φ∗T)) (4) u;cφ1;· · ·;cφm ← > ({φ1, . . . , φm}=AtomK(T)) (5) u←cφ, not kφ (for eachφ∈AtomK(T)) (6) u←kφ∗, not kφ (for eachφ∈AtomK(T)) (7) u←cφ, k∗φ, not¬kφ (for eachφ∈AtomK(T)) (8) u;cφ;kφ∗←not¬kφ (for eachφ∈AtomK(T))
(9) p∗←u (for each new atomp∗intrc(Φ∗T))
(10) cφ←u (for eachφ∈AtomK(T))
(11) ⊥ ←not u (12) v;A←B
(for each ruleA←Bin thetrc(·)translation of
^
φ∈AtomK(T)
(kφ⊃φ)b ∧ ¬ ^
φ∈AtomA(T)
(aφ⊃φ))b
(13) pˆ←v
(for each atompˆexceptk-atoms anda-atoms intrc(·)of
^
φ∈AtomK(T)
(kφ⊃φ)b ∧ ¬ ^
φ∈AtomA(T)
(aφ⊃φ))b
(14) ⊥ ←not v
whereu,v, andcφ(for eachφ∈ AtomK(T)) are new atoms, such thatK(M) =A(M) =Th({φ|φ∈AtomK(T)andkφ∈S}).
Intuitively, rules (1) and (2) intrlp(T)guarantee that each answer set is a model of the formulaΨT. Rules (3) to (8) then create model candidates that violate the minimal knowledge condition; rules (9) to (11) eliminate answer sets for which such models exist. Finally, rules (12) to (14) check whether assumptions and knowledge coincide.
Due to the results of Eiter and Gottlob [2] and Lin and Zhou [7], our Theorem 1 yields a complexity result for the pure logic of GK.
Proposition 4 LetT be a pure GK theory. The problem of deciding whetherT has a GK model isΣP2-complete.
3 Discussion
We have presented the first translation of pure formulas of the logic of GK to disjunctive answer set programming. Among other things, this directly leads to implementations of default and autoepistemic logics under different semantics. The translation presented in this paper is a generalization of the one presented for Turner’s logic of universal causation by Ji and Lin [4]. In recent related work, Chen et al. [1]
presented the dl2asp system that implements propositional default logic by translating default theories to (non-disjunctive) ASPs. For their translation, the size of the translated logic program might grow exponentially in the size of the input default theory. In contrast, the size increase of our translation via the logic of GK is polynomial.
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