An Application of Proof-Theory in Answer Set Programming

1 Introduction

The use of proof theory in logic based formalisms for constraint solving is pervasive. For example, in Satisfiability (SAT), proof theoretic methods are used to find 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 Default Logic (and thus stable semantics of logic programs). Bonatti [Bo04] and separately Milnikel [Mi05] devised non-monotonic proof systems to study skeptical consequences of programs and default theories. Lifschitz [Li96] used proof-theoretic methods to approximate well-founded semantics of logic programs. Bondarenko et.al. [BTK93] studied an approach to stable semantics using methods with a clear proof-theoretic flavor. Marek, Nerode, and Remmel in a series of papers, [MNR90a, MNR90b, MNR91, MNR92, MNR94a, MNR94b], developed proof theoretic methods to study what they termed non-monotonic rule systems which have as special cases almost all ASP formalisms that have been seriously studied in the literature. Recently the area of proof systems for ASP (and more generally, nonmonotonic logics) received a lot of attention [GS07, JO07]. It is clear that the community feels that an additional research of this area is necessary. Nevertheless, there is no clear classification of proof systems for nonmonotonic reasoning analogous to that present in classical logic, and SAT in particular.

In this paper, we define a notion of $P$ -proof schemes, which is a kind of a proof system that was previously used by Marek, Nerode, and Remmel to study complexity issues for stable semantics of logic programs [MNR94a]. This proof system abstracts of $M$ -proofs of [MT93] and produces Hilbert-style proofs. The nonmonotonic character of our $P$ -proofs is provided by the presence of guards, called the support of the proof scheme, to insure context-dependence. A different but equivalent, presentation of proof schemes, using a guarded resolution is also possible [MR09].

We shall show that we can use $P$ -proof schemes to find a characterization of stable models via reduced defining equations. While in general these defining equations may be infinite, we study the case of programs for which all these equations are finite. This resulting class of programs, called FSP-programs, turn out to be characterized 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 elucidate the proof-theoretical character of the stable semantics for logic programs, an area with 20 years history [GL88]. The principal results of this paper are:

We show that the Gelfond-Lifschitz operator ${G L}_{P}$ is, in fact a proof-theoretical construct (Proposition 4.3)
As a result of the analysis of the Gelfond-Lifschitz operator we are able to show that the upper-half continuity of that operator is equivalent to finiteness of (propositional) formulas in a certain class associated with the program $P$ (Proposition 4.6)

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

2 Preliminaries

Let $A t$ be a countably infinite set of atoms. We will study programs consisting of clauses built of the atoms from $A t$ . A program clause $C$ is a string of the form

p \leftarrow q_{1}, \dots, q_{m}, \neg r_{1}, \dots, \neg r_{n}

(1)

The integers $m$ or $n$ or both can be $0$ . The atom $p$ will be called the head of $C$ and denoted $h e a d (C)$ . We let $p o s B o d y (C)$ denote the set ${q_{1}, \dots, q_{m}}$ and $n e g B o d y (C)$ denote the set ${r_{1}, \dots, r_{n}}$ . For any set of atoms $X$ , we let $\neg X$ denote the conjunction of negations of atoms from $X$ . Thus, we can write clause (1) as

h e a d (C) \leftarrow p o s B o d y (C), \neg n e g B o d y (C) .

Let us stress that the set $n e g B o d y (C)$ is a set of atoms, not a set of negated atoms as is sometimes used in the literature. A normal propositional program is a set $P$ of such clauses. For any $M \subseteq A t$ , we say that $M$ is model of $C$ if whenever $q_{1}, \dots, q_{m} \in M$ and ${r_{1}, \dots, r_{n}} \cap M = \emptyset$ , then $p \in M$ . We say that $M$ is a model of a program $P$ if $M$ is a model of each clause $C \in P$ . Horn clauses are clauses with no negated literals, i.e. clauses of the form (1) where $n = 0$ . We will denote by $H o r n (P)$ the part of the program $P$ consisting of its Horn clauses. Horn programs are logic programs $P$ consisting entirely of Horn clauses. Thus for a Horn program $P$ , $P = H o r n (P)$ .

Each Horn program $P$ has a least model over the Herbrand base and the least model of $P$ is the least fixed point of a continuous operator $T_{P}$ representing one-step Horn clause logic deduction ([L89]). That is, for any set $I \subseteq A t$ , we let $T_{P} (I)$ equal the set of all $p \in A t$ such that there is a clause $C = p \leftarrow q_{1}, \dots, q_{m}$ in $P$ and $q_{1}, \dots, q_{m} \in I$ . Then $T_{P}$ has a least fixed point $F_{P}$ which is obtained by iterating $T_{P}$ starting at the empty set for $ω$ steps, i.e., $F_{P} = ⋃_{n \in ω} T_{P}^{n} (\emptyset)$ where for any $I \subseteq A t$ , $T_{P}^{0} (I) = I$ and $T_{P}^{n + 1} (I) = T_{P} (T_{P}^{n} (I))$ . Then $F_{P}$ is the least model of $P$ .

The semantics of interest for us is the stable semantics of normal programs, although we will discuss some extensions in Section LABEL:ext. The stable models of a program $P$ are defined as fixed points of the operator $T_{P, M}$ . This operator is defined on the set of all subsets of $A t$ , $P (A t)$ . If $P$ is a program and $M \subseteq A t$ is a subset of the Herbrand base, define operator $T_{P, M} : P (A t) \to P (A t)$ as follows:

T_{P, M} (I) = {p : t h e r e e x i s t a c l a u s e C = p \leftarrow q_{1}, \dots, q_{m}, \neg r_{1}, \dots, \neg r_{n} i n P s u c h t h a t q_{1} \in I, \dots, q_{m} \in I, r_{1} \notin M, \dots, r_{n} \notin M}

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

Proposition 2.1

For every program $P$ and every set $M$ of atoms the operator $T_{P, M}$ is monotone and continuous.

Thus the operator $T_{P, M}$ like all monotonic continuous operators, possesses a least fixed point $F_{P, M}$ .

Given program $P$ and $M \subseteq A t$ , we define the Gelfond-Lifschitz reduct of $P$ , $P_{M}$ , as follows. For every clause $C = p \leftarrow q_{1}, \dots, q_{m}, \neg r_{1}, \dots, \neg r_{n}$ of $P$ , execute the following operations.
(1) If some atom $r_{i}$ , $1 \leq i \leq n$ , belongs to $M$ , then eliminate $C$ altogether.
(2) In the remaining clauses that have not been eliminated by operation (1), eliminate all the negated atoms.
The resulting program $P_{M}$ is a Horn propositional program. The program $P_{M}$ possesses a least Herbrand model. If that least model of $P_{M}$ coincides with $M$ , then $M$ is called a stable model for $P$ . This gives rise to an operator $G L_{P}$ which associates to each $M \subseteq A t$ , the least fixed point of $T_{P, M}$ . We will discuss the operator $G L_{P}$ and its proof-theoretic connections in section 4.2.

3 Proof schemes and reduced defining equations

In this section we recall the notion of a proof scheme as defined in [MNR90a, MT93] and introduce a related notion of defining equations.

Given a propositional logic program $P$ , a proof scheme is defined by induction on its length. Specifically, a proof scheme w.r.t. $P$ (in short $P$ -proof scheme) is a sequence $S = ⟨ ⟨ C_{1}, p_{1} ⟩, \dots, ⟨ C_{n}, p_{n} ⟩, U ⟩$ subject to the following conditions:
(I) when $n = 1$ , $⟨ ⟨ C_{1}, p_{1} ⟩, U ⟩$ is a $P$ -proof scheme if $C_{1} \in P$ , $p_{1} = h e a d (C_{1})$ , $p o s B o d y (C_{1}) = \emptyset$ , and $U = n e g B o d y (C_{1})$ and
(II) when $⟨ ⟨ C_{1}, p_{1} ⟩, \dots, ⟨ C_{n}, p_{n} ⟩, U ⟩$ is a $P$ -proof scheme,
$C = p \leftarrow p o s B o d y (C), \neg n e g B o d y (C)$ is a clause in the program $P$ , and $p o s B o d y (C) \subseteq {p_{1}, \dots, p_{n}}$ , then

⟨ ⟨ C_{1}, p_{1} ⟩, \dots, ⟨ C_{n}, p_{n} ⟩, ⟨ C, p ⟩, U \cup n e g B o d y (C) ⟩

is a $P$ -proof scheme.
When $S = ⟨ ⟨ C_{1}, p_{1} ⟩, \dots, ⟨ C_{n}, p_{n} ⟩, U ⟩$ is a $P$ -proof scheme, then we call (i) the integer $n$ – the length of $S$ , (ii) the set $U$ – the support of $S$ , and (iii) the atom $p_{n}$ – the conclusion of $S$ . We denote $U$ by $s u p p (S)$ .

Example 3.1

Let $P$ be a program consisting of four clauses: $C_{1} = p \leftarrow$ , $C_{2} = q \leftarrow p, \neg r$ , $C_{3} = r \leftarrow \neg q$ , and $C_{4} = s \leftarrow \neg t$ . Then we have the following examples of $P$ -proof schemes:

$⟨ ⟨ C_{1}, p ⟩, \emptyset ⟩$ is a $P$ -proof scheme of length $1$ with conclusion $p$ and empty support.
$⟨ ⟨ C_{1}, p ⟩, ⟨ C_{2}, q ⟩, {r} ⟩$ is a $P$ -proof scheme of length $2$ with conclusion $q$ and support ${r}$ .
$⟨ ⟨ C_{1}, p ⟩, ⟨ C_{3}, r ⟩, {q} ⟩$ is a $P$ -proof scheme of length $2$ with conclusion $r$ and support ${q}$ .
$⟨ ⟨ C_{1}, p ⟩, ⟨ C_{2}, q ⟩, ⟨ C_{3}, r ⟩, {q, r} ⟩$ is a $P$ -proof scheme of length $3$ with conclusion $r$ and support ${q, r}$ .

Proof scheme in (c) is an example of a proof scheme with unnecessary items (the first term). Proof scheme (d) is an example of a proof scheme which is not internally consistent in that $r$ is in the support of its proof scheme and is also its conclusion.

A $P$ -proof scheme carries within itself its own applicability condition. In effect, a $P$ -proof scheme is a conditional proof of its conclusion. It becomes applicable when all the constraints collected in the support are satisfied. Formally, for any set of atoms $M$ , we say that a $P$ -proof scheme $S$ is $M$ -applicable if $M \cap s u p p (S) = \emptyset$ . We also say that $M$ admits $S$ if $S$ is $M$ -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 program $P$ and every set $M$ of atoms, $M$ is a stable model of $P$ if and only if the following conditions hold.

For every $p \in M$ , there is a $P$ -proof scheme $S$ with conclusion $p$ such that $M$ admits $S$ .
For every $p \notin M$ , there is no $P$ -proof scheme $S$ with conclusion $p$ such that $M$ admits $S$ .

Proposition 3.1 says that the presence and absence of the atom $p$ in a stable model depends only on the supports of proof schemes. This fact naturally leads to a characterization of stable models in terms of propositional satisfiability. Given $p \in A t$ , the defining equation for $p$ w.r.t. $P$ is the following propositional formula:

p \Leftrightarrow (\neg U_{1} \lor \neg U_{2} \lor \dots)

(2)

where $⟨ U_{1}, U_{2}, \dots ⟩$ is the list of all supports of $P$ -proof schemes. Here for any finite set $S = {s_{1}, \dots, s_{n}}$ of atoms, $\neg S = \neg s_{1} \land \dots \land \neg s_{n} .$ If $p$ is not the conclusion of any proof scheme, then we set the defining equation of $p$ to be $p \Leftrightarrow ⊥$ . In the case, where all the supports of proof schemes of $p$ are empty, we set the defining equation of $p$ to be $p \Leftrightarrow ⊤$ . Up to a total ordering of the finite sets of atoms such a formula is unique. For example, suppose we fix a total order on $A t$ , $p_{1} < p_{2} < \dots$ . Then given two sets of atoms, $U = {u_{1} < \dots < u_{m}}$ and $V = {v_{1} < \dots < v_{n}}$ , we say that $U ≺ V$ , if either (i) $u_{m} < v_{n}$ , (ii) $u_{m} = v_{n}$ and $m < n$ , or (iii) $u_{m} = v_{n}$ , $n = m$ , and $(u_{1}, \dots, u_{n})$ is lexicographically less than $(v_{1}, \dots, v_{n})$ . We say that (2) is the defining equation for $p$ relative to $P$ if $U_{1} ≺ U_{2} ≺ \dots$ . We will denote the defining equation for $p$ with respect to $P$ by ${E q}_{p}^{P}$ .

For example, if $P$ is a Horn program, then for every atom $p$ , either the support of all its proof schemes are empty or $p$ is not the conclusion of any proof scheme. The first of these alternatives occurs when $p$ belongs to the least model of $P$ , $l m (P)$ . The second alternative occurs when $p \notin l m (P)$ . The defining equations are $p \Leftrightarrow ⊤$ (that is $p$ ) when $p \in l m (P)$ and $p \Leftrightarrow ⊥$ (that is $\neg p$ ) when $p \notin l m (P)$ . When $P$ is a stratified program the defining equations are more complex, but the resulting theory is logically equivalent to

{p : p \in {P e r f}_{P}} \cup {\neg p : p \notin {P e r f}_{P}}

where ${P e r f}_{P}$ is the unique stable model of $P$ .

Let $Φ_{P}$ be the set ${{E q}_{p}^{P} : p \in A t}$ . We then have the following consequence of Proposition 3.1.

Proposition 3.2

Let $P$ be a normal propositional program. Then stable models of $P$ are precisely the propositional models of the theory $Φ_{P}$ .

When $P$ is purely negative, i.e. all clauses $C$ of $P$ have $P o s B o d y (C) = \emptyset$ , the stable and supported models of $P$ coincide [DK89] and the defining equations reduce to Clark’s completion [Cl78] of $P$ .

Let us observe that in general the propositional formulas on the right-hand-side of the defining equations may be infinite.

Example 3.2

Let $P$ be an infinite program consisting of clauses $p \leftarrow \neg p_{i}$ , for all $i \in n$ . In this case, the defining equation for $p$ in $P$ is infinite. That is, it is

p \Leftrightarrow (\neg p_{1} \lor \neg p_{2} \lor \neg p_{3} \lor \dots)

The following observation is quite useful. If $U_{1}, U_{2}$ are two finite sets of propositional atoms then

U1⊆U2 if and only if\  ¬U2⊨¬U1

Here $⊨$ is the propositional consequence relation. The effect of this observation is that not all the supports of proof schemes are important, only the inclusion-minimal ones.

Example 3.3

Let $P$ be an infinite program consisting of clauses $p \leftarrow \neg p_{1}, \dots, \neg p_{i}$ , for all $i \in N$ . The defining equation for $p$ in $P$ is

p \Leftrightarrow [\neg p_{1} \lor (\neg p_{1} \land \neg p_{2}) \lor (\neg p_{1} \land \neg p_{2} \land \neg p_{3}) \lor \dots]

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

p \Leftrightarrow \neg p_{1}

Motivated by the Example 3.3, we define the reduced defining equation for $p$ relative to $P$ to be the formula

p \Leftrightarrow (\neg U_{1} \lor \neg U_{2} \lor \dots)

(3)

where $U_{i}$ range over inclusion-minimal supports of $P$ -proof schemes for the atom $p$ and $U_{1} ≺ U_{2} ≺ \dots$ . Again, if $p$ is not the conclusion of any proof scheme, then we set the defining equation of $p$ to be $p \Leftrightarrow ⊥$ . In the case, where there is a proof scheme of $p$ with empty support, then we set the defining equation of $p$ to be $p \Leftrightarrow ⊤$ . We denote this formula as ${r E q}_{p}^{P}$ , and define $r Φ_{P}$ to be the theory consisting of ${r E q}_{p}^{P}$ for all $p \in A t$ . We then have the following strengthening of Proposition 3.2.

Proposition 3.3

Let $P$ be a normal propositional program. Then stable models of $P$ are precisely the propositional models of the theory $r Φ_{P}$ .

In our example 3.3, the theory $Φ_{P}$ involved formulas with infinite disjunctions, but the theory $r Φ_{P}$ contains only normal finite propositions.

Given a normal propositional program $P$ , we say that $P$ is a finite support program (FSP-program) if all the reduced defining equations for atoms with respect to $P$ are finite propositional formulas. Equivalently, a program $P$ is an FSP-program if for every atom $p$ there is only finitely many inclusion-minimal supports of $P$ -proof schemes for $p$ .

4 Continuity properties of operators and proof schemes

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

4.1 Continuity properties of monotone and antimonotone operators

Let us recall that $P (A t)$ denotes the set of all subsets of $A t$ . We say that any function $O : P (A t) \to P (A t)$ is an operator on the set $A t$ of propositional atoms. An operator $O$ is monotone if for all sets $X, Y \subseteq A t$ , $X \subseteq Y$ implies $O (X) \subseteq O (Y)$ . Likewise an operator $O$ is antimonotone if for all sets $X, Y \subseteq A t$ , $X \subseteq Y$ implies $O (Y) \subseteq O (X)$ . For a sequence $⟨ X_{n} ⟩_{n \in N}$ of sets of atoms, we say that $⟨ X_{n} ⟩_{n \in N}$ is monotonically increasing if for all $i, j \in N$ , $i \leq j$ implies $X_{i} \subseteq X_{j}$ and we say that $⟨ X_{n} ⟩_{n \in N}$ is monotonically decreasing if for all $i, j \in N$ , $i \leq j$ implies $X_{j} \subseteq X_{i}$ .

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 continuous monotone operators and lower-half continuous monotone operators. That is, we say that a monotone operator $O$ is upper-half continuous if for every monotonically increasing sequence $⟨ X_{n} ⟩_{n \in N}$ , $O (⋃_{n \in N} X_{n}) = ⋃_{n \in N} O (X_{n}) .$ We say that a monotone operator $O$ is lower-half continuous if for every monotonically decreasing sequence $⟨ X_{n} ⟩_{n \in N}$ , $O (⋂_{n \in N} X_{n}) = ⋂_{n \in N} O (X_{n}) .$ In the Logic Programming literature the first of these properties is called continuity. The classic result due to van Emden and Kowalski is the following.

Proposition 4.1

For every Horn program $P$ , the operator $T_{P}$ is upper-half continuous.

In general, the operator $T_{P}$ for Horn programs is not lower-half continuous. For example, let $P$ be the program consisting of the clauses $p \leftarrow p_{i}$ for $i \in N$ . Then the operator $T_{P}$ is not lower-half continuous. That is, if $X_{i} = {p_{i}, p_{i + 1}, \dots}$ , then clearly $p \in T_{P} (X_{i})$ for all $i$ . However, $⋂_{i} X_{i} = \emptyset$ and $p \notin T_{P} (\emptyset)$ .

Lower-half continuous monotone operators have appeared in the Logic Programming literature [Do94]. Even more generally, for a monotone operator $O$ , let us define its dual operator $O^{d}$ as follows:

O^{d} (X) = A t ∖ O (A t ∖ X) .

Then an operator $O$ is upper-half continuous if and only if $O^{d}$ is lower-half continuous [JT51]. Therefore, for any Horn program $P$ , the operator $T_{P}^{d}$ is lower-half continuous.

In case of antimonotone operators, we have two additional notions of continuity. We say an antimonotone operator $O$ is upper-half continuous if for every monotonically increasing sequence $⟨ X_{n} ⟩_{n \in N}$ , $O (⋃_{n \in N} X_{n}) = ⋂_{n \in N} O (X_{n}) .$ Similarly, we say an antimonotone operator $O$ is lower-half continuous if for every monotonically decreasing sequence $⟨ X_{n} ⟩_{n \in N}$ , $O (⋂_{n \in N} X_{n}) = ⋃_{n \in N} O (X_{n})$ .

4.2 Gelfond-Lifschitz operator ${G L}_{P}$ and proof-schemes

For the completeness sake, let us recall that the Gelfond-Lifschitz operator for a program $P$ which we denote $G L_{P}$ , assigns to a set of atoms $M$ the least fixpoint of of the operator $T_{P, M}$ or, equivalently, the least model $N_{M}$ of the program $P_{M}$ which is the Gelfond-Lifschitz reduct of $P$ via $M$ [GL88]. The following fact is crucial.

Proposition 4.2 ([Gl88])

The operator $G L$ is antimonotone.

Here is a useful proof-theoretic characterization of the operator ${G L}_{P}$ .

Proposition 4.3

Let $P$ be a normal propositional program and $M$ be a set of atoms. Then

{G L}_{P} (M) = {p : there exists a P -proof scheme S such that M admits S, and p is the conclusion of S}

Proof: Let us assume that $p \in {G L}_{P} (M)$ that is $p \in N_{M}$ . As $N_{M}$ is the least model of the Horn program $P_{M}$ , $N_{M} = ⋃_{n \in N} T_{P_{M}}^{n} (\emptyset)$ . Then it is easy to prove by induction on $n$ , that if $p \in T_{P_{M}}^{n} (\emptyset)$ , then there is a $P$ -proof scheme $S_{p}$ such that $p$ is the conclusion of $S_{p}$ and $S_{p}$ is admitted by $M$ . Conversely, we can show, by induction on the length of the $P$ -proof schemes, that whenever such $P$ -proof scheme $S$ is admitted by $M$ , then $p$ belongs to ${G L}_{P} (M)$ .

4.3 Continuity properties of the operator ${G L}_{P}$

This section will be devoted to proving results on the continuity properties of the operator ${G L}_{P}$ . First, we prove that for every program $P$ , the operator ${G L}_{P}$ is lower-half continuous. We then show that if $f$ is a lower-half continuous antimonotone operator, then $f = {G L}_{P}$ for a suitably chosen program $P$ . Finally, we show that the operator ${G L}_{P}$ is upper-half continuous if and only if $P$ is an FSP-program. That is, ${G L}_{P}$ is upper-half continuous if for all atoms $p$ the reduced defining equation for any $p$ (w.r.t. $P$ ) is finite.

Proposition 4.4

For every normal program $P$ , the operator ${G L}_{P}$ is lower-half continuous.

Proof: We need to prove that for every program $P$ and every monotonically decreasing sequence $⟨ X_{n} ⟩_{n \in N}$ ,

{G L}_{P} (⋂ n \in N X_{n}) = ⋃ n \in N {G L}_{P} (X_{n}) .

Our goal is to prove two inclusions: $\subseteq$ , and $\supseteq$ .
We first show $\supseteq$ . Since

⋂ j \in N X_{j} \subseteq X_{n}

for every $n \in N$ , by antimonotonicity of ${G L}_{P}$ we have

{G L}_{P} (X_{n}) \subseteq {G L}_{P} (⋂ j \in N X_{j}) .

As $n$ is arbitrary,

⋃ n \in N {G L}_{P} (X_{n}) \subseteq {G L}_{P} (⋂ j \in N X_{j}) .

Thus the inclusion $\supseteq$ holds.
Conversely, let $p \in {G L}_{P} (⋂_{n \in N} X_{n})$ . Then, by Proposition 4.3, there must be a proof scheme $S$ with support support $U$ and conclusion $p$ such that

U \cap ⋂ n \in N X_{n} = \emptyset .

But the family $⟨ X_{n} ⟩_{n \in n}$ is monotonically descending and the set $U$ is finite. Thus there is an integer $n_{0}$ so that

U \cap X_{n_{0}} = \emptyset .

This, however, implies that $p \in {G L}_{P} (X_{n_{0}})$ , and thus

p \in ⋃ n \in N {G L}_{P} (X_{n}) .

As $p$ is arbitrary, the inclusion $\subseteq$ holds. Thus ${G L}_{P} (⋂_{n \in N} X_{n}) = ⋃_{n \in N} {G L}_{P} (X_{n})$ .

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

Proposition 4.5

Let $A t$ be a denumerable set of atoms. Let $f$ be an antimonotone and lower-half continuous operator on $P (A t)$ . Then there exists a normal logic program $P$ such that $f = {G L}_{P}$ .

Proof.

We define the program $P = P_{f}$ as follows:

P = {p \leftarrow \neg q_{1}, \dots, \neg q_{i} : p \in f (A t ∖ {q_{1}, \dots, q_{i}})} .

We claim that $f = {G L}_{P}$ , that is, for all $X$ , $f (X) = {G L}_{P} (X)$ .

Let $X \subseteq A t$ be given. We consider two cases.
Case 1: $X$ is cofinite, $X = A t ∖ {q_{1}, \dots, q_{i}}$ . We need to prove two inclusions, (a) $f (X) \subseteq {G L}_{P} (X)$ and (b) ${G L}_{P} (X) \subseteq f (X)$ .
For (a), note that if $p \in f (X)$ , then the clause $p \leftarrow \neg q_{1}, \dots, \neg q_{i}$ belongs to $P$ . Hence $p \leftarrow$ belongs to $P_{X}$ and $p \in {G L}_{P} (X)$ .

For (b), note that if $p \in {G L}_{P} (X)$ , then given the form of the clauses in $P$ , there must be some clause $p \leftarrow \neg q_{i_{1}}, \dots, \neg q_{i_{j}}$ in $P$ where ${q_{i_{1}}, \dots, q_{i_{j}}} \subseteq {q_{1}, \dots, q_{i}}$ . But this means that $p \in f (A t ∖ {q_{i_{1}}, \dots, q_{i_{j}}})$ . Since $f$ is antimonote and $A t ∖ {q_{1}, \dots, q_{i}} \subseteq A t ∖ {q_{i_{1}}, \dots, q_{i_{j}}}$ , we must have

f (A t ∖ {q_{i_{1}}, \dots, q_{i_{j}}}) \subseteq f (A t ∖ {q_{1}, \dots, q_{i}}) = f (X)

and, hence, $p \in f (X)$ . Thus ${G L}_{P} (X) \subseteq f (X)$ .

Case 2: $X$ is not cofinite. Let ${q_{0}, q_{1}, \dots}$ be an enumeration of $A t ∖ X$ . Let $Y_{i} = A t ∖ {q_{0}, \dots, q_{i}}$ . Then, clearly, $X \subseteq Y_{i}$ for all $i \in N$ . Moreover the sequence $⟨ Y_{i} ⟩_{i \in N}$ is monotonically decreasing and $⋂_{i \in N} Y_{i} = X$ . Therefore, by our assumptions on the operator $f$ ,

f (X) = ⋃ i \in N f (Y_{i}) .

Again, we need to prove two inclusions, (a) $f (X) \subseteq {G L}_{P} (X)$ and (b) ${G L}_{P} (X) \subseteq f (X)$ . For (a), note that if $p \in f (X)$ , then for some $i \in N$ , $p \in F (Y_{i})$ . Therefore, for that $i$ , $p \leftarrow \neg q_{0}, \dots, \neg q_{i}$ is a clause in $P$ . But then $X \cap {q_{0}, \dots, q_{i}} = \emptyset$ so that the clause $p \leftarrow$ is in $P_{X}$ and $p \in {G L}_{P} (X)$ .

For the proof of (b), note that if $p \in {G L}_{P} (X)$ , then because of the syntactic form of the clauses in our program there are atoms $r_{0}, \dots, r_{k}$ so that the clause $p \leftarrow \neg r_{0}, \dots, \neg r_{k}$ belongs to the program $P$ , and $r_{0}, \dots, r_{k} \notin X$ . Thus ${r_{0}, \dots, r_{k}} \subseteq {q_{0}, q_{1}, \dots}$ and, hence, for some $i \in N$ , ${r_{0}, \dots, r_{k}} \subseteq {q_{0}, \dots, q_{i}}$ . Now, consider such a $Y_{i}$ . Since $Y_{i}$ is cofinite, it follows from Case 1 that $f (Y_{i}) = {G L}_{P} (Y_{i})$ . Since $X \subseteq Y_{i}$ , $f (Y_{i}) \subseteq f (X)$ by the antimonotonicity of $f$ . But $p \in {G L}_{P} (Y_{i})$ because $r_{0}, \dots, r_{k} \notin Y_{i}$ and, hence, $p \in f (Y_{i})$ . But since $f (Y_{i}) \subseteq f (X)$ , $p \in f (X)$ 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 are equivalent:

$P$ is an FSP-program.
The operator ${G L}_{P}$ is upper-half continuous, i.e.

${G L}_{P} (⋃ n \in N X_{n}) = ⋂ n \in N {G L}_{P} (X_{n})$

for every monotonically increasing sequence $⟨ X_{n} ⟩_{n \in N}$ .

Proof: Two implications need to be proved: $(a) \Rightarrow (b)$ , and $(b) \Rightarrow (a)$ .
Proof of the implication $(a) \Rightarrow (b)$ . Here, assuming $(a)$ , we need to prove two inclusions:
(i) ${G L}_{P} (⋃_{n \in N} X_{n}) \subseteq ⋂_{n \in N} {G L}_{P} (X_{n})$ , and
(ii) $⋂_{n \in N} {G L}_{P} (X_{n}) \subseteq {G L}_{P} (⋃_{n \in N} X_{n})$ .
To prove (i), note that since $X_{n} \subseteq ⋃_{j \in N} X_{j}$ , we have

{G L}_{P} (⋃ j \in N X_{j}) \subseteq {G L}_{P} (X_{n}) .

As $n$ is arbitrary,

{G L}_{P} (⋃ j \in N X_{j}) \subseteq ⋂ n \in N {G L}_{P} (X_{n}) .

This proves (i).
To prove (ii), let $p \in ⋂_{n \in N} {G L}_{P} (X_{n})$ . Then, for every $n \in N$ , $p \in {G L}_{P} (X_{n})$ and so, for every $n \in N$ , there is an inclusion-minimal support $U$ for $p$ such that

U \cap X_{n} = \emptyset .

But by (a) there are only finitely many inclusion-minimal supports for $P$ -proof schemes for $p$ . Therefore there is a support of an inclusion minimal support of a proof scheme of $p$ , $U_{0}$ , such that for infinitely many $n$ ’s

U_{0} \cap X_{n} = \emptyset .

But the sequence $⟨ X_{n} ⟩_{n \in N}$ is monotonically increasing. Therefore for all $n \in N$ , $U_{0} \cap X_{n} = \emptyset$ . But then

U_{0} \cap ⋃ n \in N X_{n} = \emptyset,

so that $p \in G L_{P} (⋃_{n \in N} X_{n})$ . Thus (ii) holds and the implication $(a) \Rightarrow (b)$ follows.

To prove that $(b) \Rightarrow (a)$ , assume that the operator ${G L}_{P}$ is upper-half continuous. We need to show that for every $p$ , the reduced defining equation for $p$ is finite. So let us assume that ${r E q}_{p}^{P}$ is not finite. This means that there is an infinite set $X = {U_{1}, U_{2}, \dots}$ , where $U_{1} ≺ U_{2} ≺ \dots$ , such that

each $U_{i}$ is finite,
the elements of $X$ are pairwise inclusion-incompatible, and
for every set of atoms $M$ , $p \in {G L}_{P} (M)$ if and only if for some $U_{i} \in X$ , $U_{i} \cap M = \emptyset$ .

We will now define two sequences:

a sequence $⟨ K_{n} ⟩_{n \in N}$ of infinite sets of integers and
a sequence $⟨ p_{n} ⟩_{n \in N ∖ {0}}$ of atoms.

We define $K_{0} = N$ , and we define $p_{1}$ as the first element of $U_{1}$ such that

{j : p \notin U_{j}}

is infinite. Clearly, $K_{0}$ is well-defined. We need to show that $p_{1}$ is well-defined. If $p_{1}$ is not well-defined, then for every $p \in U_{1}$ there is an integer $i_{p}$ such that for all $m > i_{p}$ , $p \in U_{m}$ . But $U_{1}$ is finite so taking $n = {max}_{p \in U_{1}} i_{p}$ , we find that for all $m > n$ , $U_{1} \subseteq U_{m}$ - which contradicts the fact that the sets in $X$ are pairwise inclusion-incompatible. Thus $p_{1}$ is well-defined. We now set

K_{1} = {n \in K_{0} : p_{1} \notin U_{n}} = {n \in K_{0} : {p_{1}} \cap U_{n} = \emptyset} .

Clearly. $K_{1}$ is infinite.

Now, let us assume that we already defined $p_{l}$ and $K_{l}$ so that $K_{l} = {n : U_{n} \cap {p_{1}, \dots, p_{l}} = \emptyset}$ is an infinite subset of $N$ . We select $p_{l + 1}$ as the first element $p \in U_{l + 1}$ so that

{j : j \in K_{l} and p \notin U_{j}}

is infinite. Clearly, by an argument as above, there is such $p$ , and so $p_{l + 1}$ is well-defined. We then set

K_{l + 1} = {j \in K_{l} : p_{l + 1} \notin U_{j}} .

Since ${p_{1}, \dots, p_{l}} \cap U_{j} = \emptyset$ for all $j \in K_{l}$ , ${p_{1}, \dots, p_{l + 1}} \cap U_{j} = \emptyset$ for all $j \in K_{l + 1}$ . By construction, the set $K_{l + 1}$ is infinite.

Now, we complete the argument as follows. We set $X_{n} = {p_{1}, \dots, p_{n}}$ . The sequence $⟨ X_{n} ⟩_{n \in N}$ is monotonically increasing. For each $n$ there is $j$ (in fact infinitely many $j$ ’s) so that $X_{n} \cap U_{j} = \emptyset$ . Therefore, for each $n$ , $p \in {G L}_{P} (X_{n})$ . Hence $p \in ⋂_{n \in N} {G L}_{P} (X_{n})$ .

On the other hand, let $X = ⋃_{n \in N} X_{n}$ . Then

X = {p_{1}, p_{2}, . . .} .

By our construction, $p_{n} \in U_{n}$ , and so $U_{n} \cap X \neq \emptyset$ . Therefore $X$ does not admit any $P$ -proof scheme for $p$ . Thus $p \notin {G L}_{P} (X) = {G L}_{P} (⋃_{n \in N} X_{n})$ . But this would contradict our assumption that ${G L}_{P}$ is upper-half continuous. Thus there can be no such $p$ and hence $P$ must be a FSP-program.

5 Extensions to $C C$ -programs

In [SNS02] Niemelä and coauthors defined a significant extension of logic programming with stable semantics which allows for programming with cardinality constraints, and, more generally, with weight constraints. This extension has been further studied in [MR04, MNT07]. To keep things simple, we will limit our discussion to cardinality constraints only, although it is possible to extend our arguments to any class of convex constraints [LT05]. Cardinality constraints are expressions of the form $l X u$ , where $l, u \in N$ , $l \leq u$ and $X$ is a finite set of atoms. The semantics of an atom $l X u$ is that a set of atoms $M$ satisfies $k X l$ if and only if $k \leq | M \cap X |$ . When $l = 0$ , we do not write it, and, likewise, when $u \geq | X |$ , we omit it, too. Thus an atom $p$ has the same meaning as $1 {p}$ while $\neg p$ has the same meaning as ${p} 0$ .

The stable semantics for $C C$ -programs is defined via fixpoints of an analogue of the Gelfond-Lifschitz operator $G L_{P}$ ; see the details in [SNS02] and [MR04]. The operator in question is neither monotone nor antimonotone. But when we limit our attention to the programs $P$ where clauses have the property that the head consists of a single atom (i.e. are of the form $1 {p}$ ), then one can define an operator ${C C G L}_{P}$ which is antimonotone and whose fixpoints are stable models of $P$ . This is done as follows.

Given a clause $C$

p \leftarrow l_{1} X_{1} u_{1}, \dots, l_{m} X_{m} u_{m},

we transform it into the clause

p \leftarrow l_{1} X_{1}, \dots, l_{m} X_{m}, X_{1} u_{1}, \dots, X_{m} u_{m}

(4)

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

p \leftarrow l_{1} X_{1}, \dots, l_{m} X_{m} .

(5)

A $C C$ -Horn program is a $C C$ -program all of whose clauses are of the form (5). If $P$ is a $C C$ -Horn program, we can define the analogue of the one step provability operator $T_{P}$ by defining that for a set of atom $M$ ,

T_{P} (M) = {p : (\exists C = p \leftarrow l_{1} X_{1}, \dots, l_{m} X_{m}) (\forall i \in {1, \dots m}) (| X_{i} \cap M | \geq l_{i})}

(6)

It is easy to see that $T_{P}$ is monotone operator that the least fixed point of $T_{P}$ is given by

l f p (T_{P}) = ⋃ n \geq 0 T_{P}^{n} (\emptyset) .

(7)

We can define the analogue of the Gelfond-Lifschitz reduct of a $C C$ -program, which we call the $N S S$ -reduct of $P$ , as follows. Let $¯ P$ denote the set of all transformed clauses derived from $P$ . Given a set of atoms $M$ , we eliminate from $¯ P$ those clauses where some upper-constraint ( $X_{i} u_{i}$ ) is not satisfied by $M$ , i.e. $| M \cap X_{i} | > u_{i}$ . In the remaining clauses, the constraints of the form $X_{i} u_{i}$ are eliminated altogether. This leaves us with a $C C$ -Horn program $P_{M}$ . We then define ${C C G L}_{P} (M)$ to be the least fixed point of $T_{P_{M}}$ and say that $M$ is a $C C$ -stable model if $M = {C C G L}_{P} (M)$ . The equivalence of this construction and the original construction in [SNS02] for normal $C C$ -programs is shown in [MNT07].

Next we define the analogues of $P$ -proof schemes for normal $C C$ -programs, i.e. programs which consists entirely of clauses of the form (4). This is done by induction as follows. When

C = p \leftarrow X_{1} u_{1}, \dots, X_{k} u_{k}

is a normal $C C$ -clause without the cardinality-constraints of the form $l_{i} X_{i}$ then

⟨ ⟨ C, p ⟩, {X_{1} u_{1}, \dots, X_{k} u_{k}} ⟩

is a $P$ - $C C$ -proof scheme with support ${X_{1} u_{1}, \dots, X_{k} u_{k}}$ . Likewise, when

S = ⟨ ⟨ C_{1}, p_{1} ⟩, \dots, ⟨ C_{n}, p_{n} ⟩, U ⟩

is a $P$ - $C C$ -proof scheme,

p \leftarrow l_{1} X_{1}, \dots, l_{m} X_{m}, X_{1} u_{1}, \dots, X_{m} u_{m}

is a clause in $P$ , and $| X_{1} \cap {p_{1}, \dots, p_{n}} | \geq l_{1}$ , $\dots$ , $| X_{m} \cap {p_{1}, \dots, p_{n}} | \geq l_{m}$ , then

⟨ ⟨ C_{1}, p_{1} ⟩, \dots, ⟨ C_{n}, p_{n} ⟩, ⟨ C, p ⟩, U \cup {X_{1} u_{1}, \dots, X_{m} u_{m}} ⟩

is a $P$ - $C C$ -proof scheme with support $U \cup {X_{1} u_{1}, \dots X_{m} u_{m}}$ . The notion of admittance of a $P$ - $C C$ -proof scheme is similar to the notion of admittance of $P$ -proof scheme for normal programs $P$ . That is, if $S = ⟨ ⟨ C_{1}, p_{1} ⟩, \dots, ⟨ C_{n}, p_{n} ⟩, ⟨ C, p ⟩, U ⟩$ is a $C C$ -proof scheme with support $U = {X_{1} u_{1}, \dots X_{n} u_{n}}$ , then $S$ is admitted by $M$ if for every $X_{i} u_{i} \in U$ , $M ⊨ X_{i} u_{i}$ , i.e. $| M \cap X_{i} | \leq u_{i}$ .

Similarly, we can associate a propositional formula $ϕ_{U}$ so that $M$ admits $S$ if and only if $M ⊨ ϕ_{U}$ as follows:

ϕ_{U} = n ⋀ i = 1 ⋁ W \subseteq X_{i}, | W | = | X_{i} | - u_{i} \neg W .

(8)

Then we can define a partial ordering on the set of possible supports of proof scheme by defining $U_{1} ⪯ U_{2} ⟺ ϕ_{U_{2}} ⊨ ϕ_{U_{1}}$ . For example if $U_{1} = ⟨ {1, 2, 3} 2$ , ${4, 5, 6} 2 ⟩$ and $U_{2} = ⟨ {1, 2, 3, 4, 5, 6}, 4 ⟩$ , then

	$ϕ_{U_{1}}$	$=$	$(\neg 1 \lor \neg 2 \lor \neg 3) \land (\neg 4 \lor \neg 5 \lor \neg 6)$
	$ϕ_{U_{2}}$	$=$	$⋁ 1 \leq i < j \leq 6 (\neg i \land \neg j) .$

Then clearly $ϕ_{U_{1}} ⊨ ϕ_{U_{2}}$ so that $U_{2} ⪯ U_{1}$ . We then define a normal propositional $C C$ -program to be FPS $C C$ -program if for each $p \in A t$ , there are finitely many $⪯$ -minimal supports of $P$ - $C C$ -proof schemes with conclusion $p$ .

We can also define analogue of the defining equation $C C E q_{p}^{P}$ of $p$ relative to a normal $C C$ -program $P$ as

p \Leftrightarrow (ϕ_{U_{1}} \lor ϕ_{U_{2}} \lor \dots)

(9)

where $⟨ U_{1}, U_{2}, \dots ⟩$ is a list of supports of all $P$ - $C C$ -proofs schemes with conclusion $p$ . Again up to a total ordering of possible finite supports, this formula is unique. Let $Φ_{P}$ be the set ${C C E q_{p}^{P} : p \in A t}$ . Similarly, we define the reduced defining equation for $p$ relative to $P$ to be the formula

p \Leftrightarrow (\neg ϕ_{U_{1}} \lor \neg ϕ_{U_{2}} \lor \dots)

(10)

where $U_{i}$ range over $⪯$ -minimal supports of $P$ - $C C$ -proof schemes for the atom $p$ .

Then we have the following analogues of Propositions 3.1 and 3.2.

Proposition 5.1

For every normal propositional $C C$ -program $P$ and every set $M$ of atoms, $M$ is a $C C$ -stable model of $P$ if and only if the following two conditions hold:

for every

An Application of Proof-Theory in Answer Set Programming

Guarded resolution for answer set programming

First-order answer set programming as constructive proof search

onlineSPARC: a Programming Environment for Answer Set Programming

Properties and Applications of Programs with Monotone and Convex Constraints

Positive Dependency Graphs Revisited

An Algebra of Causal Chains

On Elementary Loops of Logic Programs

1 Introduction

1.1 Contributions of the paper

2 Preliminaries

Proposition 2.1

3 Proof schemes and reduced defining equations

Example 3.1

Proposition 3.1

Proposition 3.2

Example 3.2

Example 3.3

Proposition 3.3

4 Continuity properties of operators and proof schemes

4.1 Continuity properties of monotone and antimonotone operators

Proposition 4.1

4.2 Gelfond-Lifschitz operator ${G L}_{P}$ and proof-schemes

Proposition 4.2 ([Gl88])

Proposition 4.3

4.3 Continuity properties of the operator ${G L}_{P}$

Proposition 4.4

Proposition 4.5

Proposition 4.6

5 Extensions to $C C$ -programs

Proposition 5.1

An Application of Proof-Theory in Answer Set Programming

Related Research

Guarded resolution for answer set programming

First-order answer set programming as constructive proof search

onlineSPARC: a Programming Environment for Answer Set Programming

Properties and Applications of Programs with Monotone and Convex Constraints

Positive Dependency Graphs Revisited

An Algebra of Causal Chains

On Elementary Loops of Logic Programs

1 Introduction

1.1 Contributions of the paper

2 Preliminaries

Proposition 2.1

3 Proof schemes and reduced defining equations

Example 3.1

Proposition 3.1

Proposition 3.2

Example 3.2

Example 3.3

Proposition 3.3

4 Continuity properties of operators and proof schemes

4.1 Continuity properties of monotone and antimonotone operators

Proposition 4.1

4.2 Gelfond-Lifschitz operator GLP and proof-schemes

Proposition 4.2 ([Gl88])

Proposition 4.3

4.3 Continuity properties of the operator GLP

Proposition 4.4

Proposition 4.5

Proposition 4.6

5 Extensions to CC-programs

Proposition 5.1

4.2 Gelfond-Lifschitz operator ${G L}_{P}$ and proof-schemes

4.3 Continuity properties of the operator ${G L}_{P}$

5 Extensions to $C C$ -programs