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An Application of Proof-Theory in Answer SetProgramming

V. W. Marek Department of Computer ScienceUniversity of KentuckyLexington, KY 40506

J.B. RemmelDepartments of Computer Science and Mathematics

University of CaliforniaLa Jolla, CA 92093

December 24, 2013

Abstract

Using a characterization of stable models of logic programsP as satis-fying valuations of a suitably chosen propositional theory, called the set ofreduced defining equationsrΦP , we show that the finitary character of thattheory rΦP is equivalent to a certain continuity property of the Gelfond-Lifschitz operatorGLP associated with the programP .We discuss possible extensions of techniques proposed in this paper to thecontext of cardinality constraints.

1 Introduction

The use of proof theory in logic based formalisms for constraint solving is per-vasive. For example, in Satisfiability (SAT), proof theoretic methods are used tofind lower bounds on complexity of various SAT algorithms. However, proof-theoretic methods have not played as prominent role in Answer Set Programming(ASP) formalisms. This is not to say that there were no attempts to apply proof-theoretic methods in ASP. To give a few examples, Marek and Truszczynski in[MT93] used the proof-theoretic methods to characterize Reiter’s extensions in

1

Default Logic (and thus stable semantics of logic programs). Bonatti [Bo04] andseparately Milnikel [Mi05] devised non-monotonic proof systems to study skepti-cal consequences of programs and default theories. Lifschitz [Li96] used proof-theoretic methods to approximate well-founded semantics of logic programs. Bon-darenko et.al. [BTK93] studied an approach to stable semantics using methodswith a clear proof-theoretic flavor. Marek, Nerode, and Remmel in a series ofpapers, [MNR90a, MNR90b, MNR91, MNR92, MNR94a, MNR94b], developedproof theoretic methods to study what they termednon-monotonic rule systemswhich have as special cases almost all ASP formalisms that have been seriouslystudied in the literature. Recently the area of proof systems for ASP (and moregenerally, nonmonotonic logics) received a lot of attention [GS07, JO07]. It isclear that the community feels that an additional research of this area is necessary.Nevertheless, there is no clear classification of proof systems for nonmonotonicreasoning analogous to that present in classical logic, andSAT in particular.In this paper, we define a notion ofP -proof schemes, which is a kind of a proof sys-tem that was previously used by Marek, Nerode, and Remmel to study complexityissues for stable semantics of logic programs [MNR94a]. This proof system ab-stracts ofM -proofs of [MT93] and produces Hilbert-style proofs. The nonmono-tonic character of ourP -proofs is provided by the presence of guards, called thesupportof the proof scheme, to insure context-dependence. A different but equiv-alent, presentation of proof schemes, using a guarded resolution is also possible[MR09].We shall show that we can useP -proof schemes to find a characterization of stablemodels viareduced defining equations. While in general these defining equationsmay be infinite, we study the case of programs for which all these equations arefinite. This resulting class of programs, called FSP-programs, turn out to be char-acterized by a form of continuity of the Gelfond-Lifschitz operator.

1.1 Contributions of the paper

The contributions of this paper consist, primarily of investigations that elucidatethe proof-theoretical character of the stable semantics for logic programs, an areawith 20 years history [GL88]. The principal results of this paper are:

1. We show that the Gelfond-Lifschitz operatorGLP is, in fact a proof-theoreticalconstruct (Proposition 4.3)

2. As a result of the analysis of the Gelfond-Lifschitz operator we are able toshow that the upper-half continuity of that operator is equivalent to finitenessof (propositional) formulas in a certain class associated with the programP(Proposition 4.6)

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We also discuss possible extension of these results to the case of programs withcardinality constraints.

2 Preliminaries

Let At be a countably infinite set of atoms. We will study programs consisting ofclauses built of the atoms fromAt . A program clauseC is a string of the form

p← q1, . . . , qm,¬r1, . . . ,¬rn (1)

The integersm or n or both can be0. The atomp will be called the head ofC and denotedhead(C). We let posBody (C) denote the set{q1, . . . , qm} andnegBody(C) denote the set{r1, . . . , rn}. For any set of atomsX, we let¬Xdenote the conjunction of negations of atoms fromX. Thus, we can write clause(1) as

head(C)← posBody (C),¬negBody(C).

Let us stress that the setnegBody(C) is a set of atoms, not a set of negated atomsas is sometimes used in the literature. A normal propositional program is a setPof such clauses. For anyM ⊆ At , we say thatM is model ofC if wheneverq1, . . . , qm ∈ M and{r1, . . . , rn} ∩M = ∅, thenp ∈ M . We say thatM is amodel of a programP if M is a model of each clauseC ∈ P . Horn clauses areclauses with no negated literals, i.e. clauses of the form (1) wheren = 0. Wewill denote byHorn(P ) the part of the programP consisting of its Horn clauses.Horn programs are logic programsP consisting entirely of Horn clauses. Thus fora Horn programP , P = Horn(P ).Each Horn programP has a least model over the Herbrand base and the least modelof P is the least fixed point of a continuous operatorTP representing one-step Hornclause logic deduction ([L89]). That is, for any setI ⊆ At , we letTP (I) equalthe set of allp ∈ At such that there is a clauseC = p ← q1, . . . , qm in P andq1, . . . , qm ∈ I. ThenTP has a least fixed pointFP which is obtained by iteratingTP starting at the empty set forω steps, i.e.,FP =

⋃n∈ω T

nP (∅) where for any

I ⊆ At , T 0P (I) = I andT n+1

P (I) = TP (TnP (I)). ThenFP is the least model ofP .

The semantics of interest for us is thestable semanticsof normal programs, al-though we will discuss some extensions in Section??. The stable models of aprogramP are defined as fixed points of the operatorTP,M . This operator is de-fined on the set of all subsets ofAt , P(At). If P is a program andM ⊆ At is asubset of the Herbrand base, define operatorTP,M : P(At)→ P(At) as follows:

TP,M (I) = {p : there exist a clause C = p← q1, . . . , qm,¬r1, . . . ,¬rn

in P such that q1 ∈ I, . . . , qm ∈ I, r1 /∈M, . . . , rn /∈M}

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The following is immediate, see [Ap90] for unexplained notions.

Proposition 2.1 For every programP and every setM of atoms the operatorTP,M

is monotone and continuous.

Thus the operatorTP,M like all monotonic continuous operators, possesses a leastfixed pointFP,M .Given programP andM ⊆ At , we define theGelfond-Lifschitz reductof P , PM ,as follows. For every clauseC = p← q1, . . . , qm,¬r1, . . . ,¬rn of P , execute thefollowing operations.(1) If some atomri, 1 ≤ i ≤ n, belongs toM , then eliminateC altogether.(2) In the remaining clauses that have not been eliminated byoperation (1), elimi-nate all the negated atoms.The resulting programPM is a Horn propositional program. The programPM

possesses a least Herbrand model. If that least model ofPM coincides withM ,thenM is called astable modelfor P . This gives rise to an operatorGLP whichassociates to eachM ⊆ At , the least fixed point ofTP,M . We will discuss theoperatorGLP and its proof-theoretic connections in section 4.2.

3 Proof schemes and reduced defining equations

In this section we recall the notion of aproof schemeas defined in [MNR90a,MT93] and introduce a related notion ofdefining equations.Given a propositional logic programP , a proof scheme is defined by induction onits length. Specifically, a proof scheme w.r.t.P (in shortP -proof scheme) is asequenceS = 〈〈C1, p1〉, . . . , 〈Cn, pn〉, U〉 subject to the following conditions:(I) whenn = 1, 〈〈C1, p1〉, U〉 is aP -proof scheme ifC1 ∈ P , p1 = head(C1),posBody (C1) = ∅, andU = negBody(C1) and(II) when 〈〈C1, p1〉, . . . , 〈Cn, pn〉, U〉 is aP -proof scheme,C = p← posBody (C),¬negBody(C) is a clause in the programP , andposBody(C) ⊆{p1, . . . , pn}, then

〈〈C1, p1〉, . . . , 〈Cn, pn〉, 〈C, p〉, U ∪ negBody(C)〉

is aP -proof scheme.WhenS = 〈〈C1, p1〉, . . . , 〈Cn, pn〉, U〉 is aP -proof scheme, then we call (i) theintegern – thelengthof S, (ii) the setU – thesupportof S, and (iii) the atompn– theconclusionof S. We denoteU by supp(S).

Example 3.1 Let P be a program consisting of four clauses:C1 = p ←, C2 =q ← p,¬r, C3 = r ← ¬q, andC4 = s ← ¬t. Then we have the followingexamples ofP -proof schemes:

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(a) 〈〈C1, p〉, ∅〉 is aP -proof scheme of length1 with conclusionp and emptysupport.

(b) 〈〈C1, p〉, 〈C2, q〉, {r}〉 is aP -proof scheme of length2 with conclusionq andsupport{r}.

(c) 〈〈C1, p〉, 〈C3, r〉, {q}〉 is aP -proof scheme of length2 with conclusionr andsupport{q}.

(d) 〈〈C1, p〉, 〈C2, q〉, 〈C3, r〉, {q, r}〉 is aP -proof scheme of length3 with con-clusionr and support{q, r}.

Proof scheme in (c) is an example of a proof scheme with unnecessary items (thefirst term). Proof scheme (d) is an example of a proof scheme which is not in-ternally consistent in thatr is in the support of its proof scheme and is also itsconclusion. ✷

A P -proof scheme carries within itself its own applicability condition. In effect,aP -proof scheme is aconditionalproof of its conclusion. It becomes applicablewhen all the constraints collected in the support are satisfied. Formally, for any setof atomsM , we say that aP -proof schemeS isM -applicableif M∩supp(S) = ∅.We also say thatM admitsS if S isM -applicable.The fundamental connection between proof schemes and stable models [MNR90a,MT93] is given by the following proposition.

Proposition 3.1 For every normal propositional programP and every setM ofatoms,M is a stable model ofP if and only if the following conditions hold.

(i) For everyp ∈ M , there is aP -proof schemeS with conclusionp such thatM admitsS.

(ii) For everyp /∈M , there is noP -proof schemeS with conclusionp such thatM admitsS.

Proposition 3.1 says that the presence and absence of the atom p in a stable modeldependsonly on the supports of proof schemes. This fact naturally leads to acharacterization of stable models in terms of propositional satisfiability. Givenp ∈ At , thedefining equationfor p w.r.t. P is the following propositional formula:

p⇔ (¬U1 ∨ ¬U2 ∨ . . .) (2)

where〈U1, U2, . . .〉 is the list of all supports ofP -proof schemes. Here for anyfinite setS = {s1, . . . , sn} of atoms,¬S = ¬s1 ∧ · · · ∧ ¬sn. If p is not the

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conclusion of any proof scheme, then we set the defining equation of p to bep ⇔⊥. In the case, where all the supports of proof schemes ofp are empty, we setthe defining equation ofp to bep ⇔ ⊤. Up to a total ordering of the finite setsof atoms such a formula is unique. For example, suppose we fix atotal order onAt , p1 < p2 < · · · . Then given two sets of atoms,U = {u1 < · · · < um} andV = {v1 < · · · < vn}, we say thatU ≺ V , if either (i) um < vn, (ii) um = vnandm < n, or (iii) um = vn, n = m, and(u1, . . . , un) is lexicographically lessthan (v1, . . . , vn). We say that (2) is thedefining equationfor p relative toP ifU1 ≺ U2 ≺ · · · . We will denote the defining equation forp with respect toP byEqPp .For example, ifP is a Horn program, then for every atomp, either the support ofall its proof schemes are empty orp is not the conclusion of any proof scheme. Thefirst of these alternatives occurs whenp belongs to the least model ofP , lm(P ).The second alternative occurs whenp /∈ lm(P ). The defining equations arep⇔ ⊤(that isp) whenp ∈ lm(P ) andp ⇔ ⊥ (that is¬p) whenp /∈ lm(P ). WhenPis a stratified program the defining equations are more complex, but the resultingtheory is logically equivalent to

{p : p ∈ Perf P} ∪ {¬p : p /∈ Perf P }

wherePerf P is the unique stable model ofP .Let ΦP be the set{EqPp : p ∈ At}. We then have the following consequence ofProposition 3.1.

Proposition 3.2 LetP be a normal propositional program. Then stable models ofP are precisely the propositional models of the theoryΦP .

WhenP is purely negative, i.e. all clausesC of P havePosBody(C) = ∅, thestable and supported models ofP coincide [DK89] and the defining equationsreduce to Clark’s completion [Cl78] ofP .Let us observe that in general the propositional formulas onthe right-hand-side ofthe defining equations may be infinite.

Example 3.2 Let P be an infinite program consisting of clausesp ← ¬pi, for alli ∈ n. In this case, the defining equation forp in P is infinite. That is, it is

p⇔ (¬p1 ∨ ¬p2 ∨ ¬p3 ∨ . . .)

✷

The following observation is quite useful. IfU1, U2 are two finite sets of proposi-tional atoms then

U1 ⊆ U2 if and only if ¬U2 |= ¬U1

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Here|= is the propositional consequence relation. The effect of this observation isthat not all the supports of proof schemes are important, only the inclusion-minimalones.

Example 3.3 LetP be an infinite program consisting of clausesp← ¬p1, . . . ,¬pi,for all i ∈ N . The defining equation forp in P is

p⇔ [¬p1 ∨ (¬p1 ∧ ¬p2) ∨ (¬p1 ∧ ¬p2 ∧ ¬p3) ∨ . . . ]

which is infinite. But our observation above implies that this formula isequivalentto the formula

p⇔ ¬p1

✷

Motivated by the Example 3.3, we define thereduced defining equationfor p rela-tive toP to be the formula

p⇔ (¬U1 ∨ ¬U2 ∨ . . .) (3)

whereUi range overinclusion-minimalsupports ofP -proof schemes for the atomp andU1 ≺ U2 ≺ · · · . Again, if p is not the conclusion of any proof scheme,then we set the defining equation ofp to bep ⇔ ⊥. In the case, where there isa proof scheme ofp with empty support, then we set the defining equation ofpto bep ⇔ ⊤. We denote this formula asrEqPp , and definerΦP to be the theoryconsisting ofrEqPp for all p ∈ At . We then have the following strengthening ofProposition 3.2.

Proposition 3.3 LetP be a normal propositional program. Then stable models ofP are precisely the propositional models of the theoryrΦP .

In our example 3.3, the theoryΦP involved formulas with infinite disjunctions, butthe theoryrΦP contains only normal finite propositions.Given a normal propositional programP , we say thatP is afinite support program(FSP-program) if all the reduced defining equations for atoms with respect toPare finite propositional formulas. Equivalently, a programP is anFSP-program iffor every atomp there is only finitely many inclusion-minimal supports ofP -proofschemes forp.

4 Continuity properties of operators and proof schemes

In this section we investigate continuity properties of operators and we will see thatone of those properties characterizes the class of FSP programs.

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4.1 Continuity properties of monotone and antimonotone operators

Let us recall thatP(At) denotes the set of all subsets ofAt . We say that anyfunctionO : P(At)→ P(At) is an operator on the setAt of propositional atoms.An operatorO is monotoneif for all setsX,Y ⊆ At , X ⊆ Y impliesO(X) ⊆O(Y ). Likewise an operatorO is antimonotoneif for all setsX,Y ⊆ At , X ⊆ YimpliesO(Y ) ⊆ O(X). For a sequence〈Xn〉n∈N of sets of atoms, we say that〈Xn〉n∈N is monotonically increasingif for all i, j ∈ N , i ≤ j impliesXi ⊆ Xj

and we say that〈Xn〉n∈N is monotonically decreasingif for all i, j ∈ N , i ≤ jimpliesXj ⊆ Xi.There are four distinct classes of operators that we shall consider in this paper.First, we shall consider two types of monotone operators, upper-half continuousmonotone operators and lower-half continuous monotone operators. That is, wesay that a monotone operatorO is upper-half continuousif for every monotoni-cally increasing sequence〈Xn〉n∈N , O(

⋃n∈N Xn) =

⋃n∈N O(Xn). We say that

a monotone operatorO is lower-half continuousif for every monotonically de-creasing sequence〈Xn〉n∈N , O(

⋂n∈N Xn) =

⋂n∈N O(Xn). In the Logic Pro-

gramming literature the first of these properties is calledcontinuity. The classicresult due to van Emden and Kowalski is the following.

Proposition 4.1 For every Horn programP , the operatorTP is upper-half con-tinuous.

In general, the operatorTP for Horn programs isnot lower-half continuous. Forexample, letP be the program consisting of the clausesp ← pi for i ∈ N . Thenthe operatorTP is not lower-half continuous. That is, ifXi = {pi, pi+1, . . .}, thenclearlyp ∈ TP (Xi) for all i. However,

⋂i Xi = ∅ andp 6∈ TP (∅).

Lower-half continuous monotone operators have appeared inthe Logic Program-ming literature [Do94]. Even more generally, for a monotoneoperatorO, let usdefine itsdual operatorOd as follows:

Od(X) = At \O(At \X).

Then an operatorO is upper-half continuous if and only ifOd is lower-half con-tinuous [JT51]. Therefore, for any Horn programP , the operatorT d

P is lower-halfcontinuous.In case of antimonotone operators, we have two additional notions of continuity.We say an antimonotone operatorO is upper-halfcontinuous if for every monoton-ically increasing sequence〈Xn〉n∈N , O(

⋃n∈N Xn) =

⋂n∈N O(Xn). Similarly,

we say an antimonotone operatorO is lower-halfcontinuous if for every monoton-ically decreasing sequence〈Xn〉n∈N , O(

⋂n∈N Xn) =

⋃n∈N O(Xn).

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4.2 Gelfond-Lifschitz operatorGLP and proof-schemes

For the completeness sake, let us recall that the Gelfond-Lifschitz operator for aprogramP which we denoteGLP , assigns to a set of atomsM the least fixpointof of the operatorTP,M or, equivalently, the least modelNM of the programPM

which is the Gelfond-Lifschitz reduct ofP via M [GL88]. The following fact iscrucial.

Proposition 4.2 ([GL88]) The operatorGL is antimonotone.

Here is a useful proof-theoretic characterization of the operatorGLP .

Proposition 4.3 Let P be a normal propositional program andM be a set ofatoms. Then

GLP (M) = {p : there exists aP -proof schemeS such thatM admitsS,

andp is the conclusion ofS}

Proof: Let us assume thatp ∈ GLP (M) that isp ∈ NM . As NM is the leastmodel of the Horn programPM , NM =

⋃n∈N T n

PM(∅). Then it is easy to prove

by induction onn, that if p ∈ T nPM

(∅), then there is aP -proof schemeSp suchthatp is the conclusion ofSp andSp is admitted byM . Conversely, we can show,by induction on the length of theP -proof schemes, that whenever suchP -proofschemeS is admitted byM , thenp belongs toGLP (M). ✷

4.3 Continuity properties of the operatorGLP

This section will be devoted to proving results on the continuity properties of theoperatorGLP . First, we prove that for every programP , the operatorGLP islower-half continuous. We then show that iff is a lower-half continuous anti-monotone operator, thenf = GLP for a suitably chosen programP . Finally, weshow that the operatorGLP is upper-half continuous if and only ifP is anFSP-program. That is,GLP is upper-half continuous if for all atomsp the reduceddefining equation for anyp (w.r.t. P ) is finite.

Proposition 4.4 For every normal programP , the operatorGLP is lower-halfcontinuous.

Proof: We need to prove that for every programP and every monotonically de-creasing sequence〈Xn〉n∈N ,

GLP (⋂

n∈N

Xn) =⋃

n∈N

GLP (Xn).

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Our goal is to prove two inclusions:⊆, and⊇.We first show⊇. Since ⋂

j∈N

Xj ⊆ Xn

for everyn ∈ N , by antimonotonicity ofGLP we have

GLP (Xn) ⊆ GLP (⋂

j∈N

Xj).

As n is arbitrary, ⋃

n∈N

GLP (Xn) ⊆ GLP (⋂

j∈N

Xj).

Thus the inclusion⊇ holds.Conversely, letp ∈ GLP (

⋂n∈N Xn). Then, by Proposition 4.3, there must be a

proof schemeS with support supportU and conclusionp such that

U ∩⋂

n∈N

Xn = ∅.

But the family〈Xn〉n∈n is monotonically descending and the setU is finite. Thusthere is an integern0 so that

U ∩Xn0= ∅.

This, however, implies thatp ∈ GLP (Xn0), and thus

p ∈⋃

n∈N

GLP (Xn).

Asp is arbitrary, the inclusion⊆ holds. ThusGLP (⋂

n∈N Xn) =⋃

n∈N GLP (Xn).✷

The lower-half continuity of antimonotone operators is closely related to programs,as shown in the following result.

Proposition 4.5 LetAt be a denumerable set of atoms. Letf be an antimonotoneand lower-half continuous operator onP(At). Then there exists a normal logicprogramP such thatf = GLP .

Proof.We define the programP = Pf as follows:

P = {p← ¬q1, . . . ,¬qi : p ∈ f(At \ {q1, . . . , qi})}.

We claim thatf = GLP , that is, for allX, f(X) = GLP (X).

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LetX ⊆ At be given. We consider two cases.Case1: X is cofinite,X = At \ {q1, . . . , qi}. We need to prove two inclusions,(a) f(X) ⊆ GLP (X) and (b)GLP (X) ⊆ f(X).For (a), note that ifp ∈ f(X), then the clausep ← ¬q1, . . . ,¬qi belongs toP .Hencep← belongs toPX andp ∈ GLP (X).For (b), note that ifp ∈ GLP (X), then given the form of the clauses inP , theremust be some clausep← ¬qi1 , . . . ,¬qij in P where{qi1 , . . . , qij} ⊆ {q1, . . . , qi}.But this means thatp ∈ f(At \ {qi1 , . . . , qij}). Sincef is antimonote andAt \{q1, . . . , qi} ⊆ At \ {qi1 , . . . , qij}, we must have

f(At \ {qi1 , . . . , qij}) ⊆ f(At \ {q1, . . . , qi}) = f(X)

and, hence,p ∈ f(X). ThusGLP (X) ⊆ f(X).

Case2: X is not cofinite. Let{q0, q1, . . .} be an enumeration ofAt \ X. LetYi = At \ {q0, . . . , qi}. Then, clearly,X ⊆ Yi for all i ∈ N . Moreover the se-quence〈Yi〉i∈N is monotonically decreasing and

⋂i∈N Yi = X. Therefore, by our

assumptions on the operatorf ,

f(X) =⋃

i∈N

f(Yi).

Again, we need to prove two inclusions, (a)f(X) ⊆ GLP (X) and (b)GLP (X) ⊆f(X). For (a), note that ifp ∈ f(X), then for somei ∈ N , p ∈ F (Yi). Therefore,for that i, p ← ¬q0, . . . ,¬qi is a clause inP . But thenX ∩ {q0, . . . , qi} = ∅ sothat the clausep← is in PX andp ∈ GLP (X).

For the proof of (b), note that ifp ∈ GLP (X), then because of the syntactic formof the clauses in our program there are atomsr0, . . . , rk so that the clausep ←¬r0, . . . ,¬rk belongs to the programP , andr0, . . . , rk /∈ X. Thus{r0, . . . , rk} ⊆{q0, q1, . . .} and, hence, for somei ∈ N , {r0, . . . , rk} ⊆ {q0, . . . , qi}. Now, con-sider such aYi. SinceYi is cofinite, it follows from Case 1 thatf(Yi) = GLP (Yi).SinceX ⊆ Yi, f(Yi) ⊆ f(X) by the antimonotonicity off . But p ∈ GLP (Yi) be-causer0, . . . , rk /∈ Yi and, hence,p ∈ f(Yi). But sincef(Yi) ⊆ f(X), p ∈ f(X)as desired. ✷

We are now ready to prove the next result of this paper.

Proposition 4.6 Let P be a normal propositional program. The following areequivalent:

(a) P is anFSP-program.

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(b) The operatorGLP is upper-half continuous, i.e.

GLP (⋃

n∈N

Xn) =⋂

n∈N

GLP (Xn)

for every monotonically increasing sequence〈Xn〉n∈N .

Proof: Two implications need to be proved:(a)⇒ (b), and(b)⇒ (a).Proof of the implication(a) ⇒ (b). Here, assuming(a), we need to prove twoinclusions:(i) GLP (

⋃n∈N Xn) ⊆

⋂n∈N GLP (Xn), and

(ii)⋂

n∈N GLP (Xn) ⊆ GLP (⋃

n∈N Xn).To prove (i), note that sinceXn ⊆

⋃j∈N Xj , we have

GLP (⋃

j∈N

Xj) ⊆ GLP (Xn).

As n is arbitrary,GLP (

⋃

j∈N

Xj) ⊆⋂

n∈N

GLP (Xn).

This proves (i).To prove (ii), letp ∈

⋂n∈N GLP (Xn). Then, for everyn ∈ N , p ∈ GLP (Xn)

and so, for everyn ∈ N , there is an inclusion-minimal supportU for p such that

U ∩Xn = ∅.

But by (a) there are only finitely many inclusion-minimal supports forP -proofschemes forp. Therefore there is a support of an inclusion minimal support of aproof scheme ofp, U0, such that for infinitely manyn’s

U0 ∩Xn = ∅.

But the sequence〈Xn〉n∈N is monotonically increasing. Therefore forall n ∈ N ,U0 ∩Xn = ∅. But then

U0 ∩⋃

n∈N

Xn = ∅,

so thatp ∈ GLP (⋃

n∈N Xn). Thus (ii) holds and the implication(a) ⇒ (b) fol-lows.

To prove that(b) ⇒ (a), assume that the operatorGLP is upper-half continuous.We need to show that for everyp, the reduced defining equation forp is finite.So let us assume thatrEqPp is not finite. This means that there is an infinite setX = {U1, U2, . . .}, whereU1 ≺ U2 ≺ · · · , such that

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1. eachUi is finite,

2. the elements ofX are pairwise inclusion-incompatible, and

3. for every set of atomsM , p ∈ GLP (M) if and only if for someUi ∈ X ,Ui ∩M = ∅.

We will now define two sequences:

1. a sequence〈Kn〉n∈N of infinite sets of integers and

2. a sequence〈pn〉n∈N\{0} of atoms.

We defineK0 = N , and we definep1 as the first element ofU1 such that

{j : p /∈ Uj}

is infinite. Clearly,K0 is well-defined. We need to show thatp1 is well-defined.If p1 is not well-defined, then for everyp ∈ U1 there is an integerip such that forall m > ip, p ∈ Um. But U1 is finite so takingn = maxp∈U1

ip, we find that forall m > n, U1 ⊆ Um - which contradicts the fact that the sets inX are pairwiseinclusion-incompatible. Thusp1 is well-defined. We now set

K1 = {n ∈ K0 : p1 /∈ Un} = {n ∈ K0 : {p1} ∩ Un = ∅}.

Clearly.K1 is infinite.Now, let us assume that we already definedpl andKl so thatKl = {n : Un ∩{p1, . . . , pl} = ∅} is an infinite subset ofN . We selectpl+1 as the first elementp ∈ Ul+1 so that

{j : j ∈ Kl andp /∈ Uj}

is infinite. Clearly, by an argument as above, there is suchp, and sopl+1 is well-defined. We then set

Kl+1 = {j ∈ Kl : pl+1 /∈ Uj}.

Since{p1, . . . , pl} ∩ Uj = ∅ for all j ∈ Kl, {p1, . . . , pl+1} ∩ Uj = ∅ for allj ∈ Kl+1. By construction, the setKl+1 is infinite.Now, we complete the argument as follows. We setXn = {p1, . . . , pn}. Thesequence〈Xn〉n∈N is monotonically increasing. For eachn there isj (in factinfinitely manyj’s) so thatXn ∩ Uj = ∅. Therefore, for eachn, p ∈ GLP (Xn).Hencep ∈

⋂n∈N GLP (Xn).

On the other hand, letX =⋃

n∈N Xn. Then

X = {p1, p2, ...}.

13

By our construction,pn ∈ Un, and soUn ∩X 6= ∅. ThereforeX does not admitany P -proof scheme forp. Thusp /∈ GLP (X) = GLP (

⋃n∈N Xn). But this

would contradict our assumption thatGLP is upper-half continuous. Thus therecan be no suchp and henceP must be aFSP-program. ✷

5 Extensions toCC -programs

In [SNS02] Niemela and coauthors defined a significant extension of logic pro-gramming with stable semantics which allows for programming with cardinalityconstraints, and, more generally, with weight constraints. This extension has beenfurther studied in [MR04, MNT07]. To keep things simple, we will limit our dis-cussion to cardinality constraints only, although it is possible to extend our ar-guments to any class of convex constraints [LT05].Cardinality constraintsareexpressions of the formlXu, wherel, u ∈ N , l ≤ u andX is a finite set of atoms.The semantics of an atomlXu is that a set of atomsM satisfieskXl if and only ifk ≤ |M ∩X|. Whenl = 0, we do not write it, and, likewise, whenu ≥ |X|, weomit it, too. Thus an atomp has the same meaning as1{p} while ¬p has the samemeaning as{p}0.The stable semantics forCC -programs is defined via fixpoints of an analogue ofthe Gelfond-Lifschitz operatorGLP ; see the details in [SNS02] and [MR04]. Theoperator in question is neither monotone nor antimonotone.But when we limitour attention to the programsP where clauses have the property that the head con-sists of a single atom (i.e. are of the form1{p}), then one can define an operatorCCGLP which is antimonotone and whose fixpoints are stable models of P . Thisis done as follows.

Given a clauseCp← l1X1u1, . . . , lmXmum,

we transform it into the clause

p← l1X1, . . . , lmXm,X1u1, . . . ,Xmum (4)

[MNT07]. We say that a clauseC of the form (4) is aCC -Horn clause if it is ofthe form

p← l1X1, . . . , lmXm. (5)

A CC -Horn program is aCC -program all of whose clauses are of the form (5). IfP is aCC -Horn program, we can define the analogue of the one step provabilityoperatorTP by defining that for a set of atomM ,

TP (M) = {p : (∃C = p← l1X1, . . . , lmXm)(∀i ∈ {1, . . . m})(|Xi ∩M | ≥ li)}(6)

14

It is easy to see thatTP is monotone operator that the least fixed point ofTP isgiven by

lfp(TP ) =⋃

n≥0

T nP (∅). (7)

We can define the analogue of the Gelfond-Lifschitz reduct ofa CC -program,which we call theNSS -reduct ofP , as follows. LetP denote the set of all trans-formed clauses derived fromP . Given a set of atomsM , we eliminate fromP those clauses where some upper-constraint (Xiui) is not satisfied byM , i.e.|M ∩ Xi| > ui. In the remaining clauses, the constraints of the formXiui areeliminated altogether. This leaves us with aCC -Horn programPM . We then de-fineCCGLP (M) to be the least fixed point ofTPM

and say thatM is aCC -stablemodel ifM = CCGLP (M). The equivalence of this construction and the originalconstruction in [SNS02] for normalCC -programs is shown in [MNT07].Next we define the analogues ofP -proof schemes for normalCC -programs, i.e.programs which consists entirely of clauses of the form (4).This is done by induc-tion as follows. When

C = p← X1u1, . . . ,Xkuk

is a normalCC -clause without the cardinality-constraints of the formliXi then

〈〈C, p〉, {X1u1, . . . ,Xkuk}〉

is aP -CC -proof scheme with support{X1u1, . . . ,Xkuk}. Likewise, when

S = 〈〈C1, p1〉, . . . , 〈Cn, pn〉, U〉

is aP -CC -proof scheme,

p← l1X1, . . . , lmXm,X1u1, . . . ,Xmum

is a clause inP , and|X1 ∩ {p1, . . . , pn}| ≥ l1, . . ., |Xm ∩ {p1, . . . , pn}| ≥ lm,then

〈〈C1, p1〉, . . . , 〈Cn, pn〉, 〈C, p〉, U ∪ {X1u1, . . . ,Xmum}〉

is aP -CC -proof scheme with supportU ∪ {X1u1, . . . Xmum}. The notion of ad-mittance of aP -CC -proof scheme is similar to the notion of admittance ofP -proofscheme for normal programsP . That is, ifS = 〈〈C1, p1〉, . . . , 〈Cn, pn〉, 〈C, p〉, U〉is aCC -proof scheme with supportU = {X1u1, . . . Xnun}, thenS is admitted byM if for everyXiui ∈ U , M |= Xiui, i.e. |M ∩Xi| ≤ ui.Similarly, we can associate a propositional formulaφU so thatM admitsS if andonly if M |= φU as follows:

φU =n∧

i=1

∨

W⊆Xi,|W |=|Xi|−ui

¬W. (8)

15

Then we can define a partial ordering on the set of possible supports of proofscheme by definingU1 � U2 ⇐⇒ φU2

|= φU1. For example ifU1 = 〈{1, 2, 3}2,

{4, 5, 6}2〉 andU2 = 〈{1, 2, 3, 4, 5, 6}, 4〉, then

φU1= (¬1 ∨ ¬2 ∨ ¬3) ∧ (¬4 ∨ ¬5 ∨ ¬6)

φU2=

∨

1≤i<j≤6

(¬i ∧ ¬j).

Then clearlyφU1|= φU2

so thatU2 � U1. We then define a normal propositionalCC -program to beFPSCC -program if for eachp ∈ At, there are finitely many�-minimal supports ofP -CC -proof schemes with conclusionp.We can also define analogue of the defining equationCCEqPp of p relative to anormalCC -programP as

p⇔ (φU1∨ φU2

∨ · · · ) (9)

where〈U1, U2, . . .〉 is a list of supports of allP -CC -proofs schemes with conclu-sion p. Again up to a total ordering of possible finite supports, this formula isunique. LetΦP be the set{CCEqPp : p ∈ At}. Similarly, we define thereduceddefining equationfor p relative toP to be the formula

p⇔ (¬φU1∨ ¬φU2

∨ . . .) (10)

whereUi range over�-minimalsupports ofP -CC -proof schemes for the atomp.Then we have the following analogues of Propositions 3.1 and3.2.

Proposition 5.1 For every normal propositionalCC -programP and every setMof atoms,M is aCC -stable model ofP if and only if the following two conditionshold:

(i) for everyp ∈ M , there is aP -CC -proof schemeS with conclusionp suchthatM admitsS and

(ii) for everyp /∈ M , there is noP -CC -proof schemeS with conclusionp suchthatM admitsS.

Proposition 5.2 LetP be a normal propositionalCC -program. ThenCC -stablemodels ofP are precisely the propositional models of the theoryΦP .

We also can prove the analogues of Propositions 4.2 and 4.3.

Proposition 5.3 For anyCC-programP , the operatorCCGLP is antimonotone.

16

Proof: It is easy to see that ifM1 ⊆M2, then for any clause

C = p→ l1X1, . . . , lmXm,X1u1, . . . Xmlm,

M2 |= Xiui impliesM1 |= Xiui. Thus it follows thatPM2⊆ PM1

and hencelfp(TPM2

) ⊆ lfp(TPM1). ✷

Proposition 5.4 LetP be a normal propositionalCC -program andM be a set ofatoms. Then

CCGLP (M) = {p : there exists aP -proof schemeS such thatM admitsS,

andp is the conclusion ofS}

Proof: Let us assume thatp ∈ CCGLP (M), i.e. p ∈ lfp(TPM). Sincelfp(TPM

) =⋃n≥1 T

nPM

(∅), we can easily show by induction onn that ifp ∈ T nPM

(∅), then thereis aP -CC -proof schemeSp suchp is the conclusion ofSp andSp is admitted byM .Conversely, we can show, by induction on the length of theP -CC -proof schemes,that whenever there isP -CC -proof schemeS admitted byM , thenp belongs tolfp(TPM

). ✷

Next we prove that analogue of Proposition 4.4.

Proposition 5.5 For every normalCC -programP , the operatorCCGLP is lower-half continuous.

Proof: We need to prove that for every normalCC -programP and every mono-tonically decreasing sequence〈Xn〉n∈N

CCGLP (⋂

n∈N

Xn) =⋃

n∈N

CCGLP (Xn).

We need to prove two inclusions:⊆, and⊇.We first show⊇. Since ⋂

j∈N

Xj ⊆ Xn

for everyn ∈ N , it follows from the antimonotonicity ofCCGLP that we have

CCGLP (Xn) ⊆ GLP (⋂

j∈N

Xj).

As n is arbitrary,⋃

n∈N

CCGLP (Xn) ⊆ CCGLP (⋂

j∈N

Xj).

17

Thus the inclusion⊇ holds.Conversely, letp ∈ CCGLP (

⋂n∈N Xn). Then, by Proposition 5.4, there must be a

CC -proof schemeS with support supportU = {Y1u1, . . . , Ynun} and conclusionp such that

|Yi ∩⋂

n∈N

Xn| ≤ ui for i = 1, . . . , n.

Since the family〈Xn〉n∈n is monotonically descending, it follows that

Yi ∩X1 ⊇ Yi ∩X2 ⊇ · · · .

SinceYi is finite, it is the case that if|Yi ∩⋂

n∈N Xn| ≤ ui, then there is somemi

such that|Yi ∩Xmi| ≤ ui. Hence ifm = max(m1, . . . ,mn), then

|Yi ∩Xm| ≤ ui for i = 1, . . . , n.

This, however, implies thatp ∈ CCGLP (Xm), and thus

p ∈⋃

n∈N

CCGLP (Xn).

Asp is arbitrary, the inclusion⊆ holds. ThusCCGLP (⋂

n∈N Xn) =⋃

n∈N CCGLP (Xn).✷

Next we can prove the analogue of the first half of Proposition4.6.

Proposition 5.6 Let P be a normal propositionalCC -program. Then ifP is anFSP-program, the operatorCCGLP is upper-half continuous, i.e.

CCGLP (⋃

n∈N

Xn) =⋂

n∈N

CCGLP (Xn)

for every monotonically increasing sequence〈Xn〉n∈N .

Proof: Two implications need to be proved:(a)⇒ (b), and(b)⇒ (a).Proof of the implication(a) ⇒ (b). Here, assuming(a) we need to prove twoinclusions:(i) GLP (

⋃n∈N Xn) ⊆

⋂n∈N GLP (Xn), and

(ii)⋂

n∈N GLP (Xn) ⊆ GLP (⋃

n∈N Xn).To prove (i), note that sinceXn ⊆

⋃j∈N Xj , we have

CCGLP (⋃

j∈N

Xj) ⊆ CCGLP (Xn).

18

As n is arbitrary,

CCGLP (⋃

j∈N

Xj) ⊆⋂

n∈N

CCGLP (Xn).

This proves (i).To prove (ii), letp ∈

⋂n∈N CCGLP (Xn). Then, for everyn ∈ N , p ∈ CCGLP (Xn)

and so, for everyn ∈ N , there is a minimal supportUn = {Y(n)1 u

(n)1 , . . . , Y

(n)mn u

(n)Mn}

for p such that|Y

(n)i ∩Xn| ≤ u

(n)i for i = 1, . . . ,mn.

But there are only finitely many�-minimal supports forP -CC -proof schemes forp. Therefore there is a supportU0 = {Z1w1, . . . , Ztwt} for aP -CC -proof schemewith conclusionp such that for infinitely manyn’s

|Zi ∩Xn| ≤ wi for i = 1, . . . , t.

But the sequence〈Xn〉n∈N is monotonically increasing. Therefore forall n ∈ N ,

|Zi ∩Xn| ≤ wi for i = 1, . . . , t.

But since eachZi is finite, then it must be the case that

|Zi ∩⋃

ninN

Xn| ≤ wi for i = 1, . . . , t.

so thatp ∈ CCGLP (⋃

n∈N Xn). ✷

We note that, alternatively, one can easily give a direct reduction of ourCC -programs to normal logic programs using the methods of [FL05] and the distribu-tivity result of [LTT99]. Such reduction, of course, lead toan exponential blow upin the size of the representation.

6 Conclusions

We note that investigations of proof systems in a related area, SAT, play a key rolein establishing lower bounds on the complexity of algorithms for finding the mod-els. We wonder if there are analogous results in ASP. For achieving such a goal, weneed to find and investigate proof systems for ASP. One candidate for such a proofsystem is provided in this paper by usingP -proof schemes. We wonder if sucha proof system can be used to develop a deeper understanding of the complexityissues related to finding stable models.

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Acknowledgments

This research of the first author was supported by the National Science Foundationunder Grant IIS-0325063. This research of the second authorwas supported by theNational Science Foundation under Grant DMS 0654060.

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