Guarded resolution for answer set programming

1 Introduction

In this note, we introduce a rule of proof, called guarded unit resolution. Guarded unit resolution is a generalization of a special case of the resolution rule of proof, namely, positive unit resolution. In positive unit resolution, one of the inputs is an atom unit clause. Positive unit resolution is complete for Horn clauses, specifically, given a consistent Horn theory $T$ and an atom $p$ , the atom $p$ belongs to the least model of $T$ , $l m (T)$ , if and only if there is a positive unit resolution proof of $p$ from $T$ [Dowling and Gallier (1984)].

The modification we introduce in this note concerns guarded atoms and guarded Horn clauses. Guarded atoms are strings of the form: $p : {r_{1}, \dots, r_{m}}$ where $p, r_{1}, \dots, r_{m}$ are propositional atoms. Guarded Horn clauses are strings of the form $p \leftarrow q_{1}, \dots, q_{n} : {r_{1}, \dots, r_{m}}$ again with $p, q_{1}, \dots, q_{n}, r_{1}, \dots, r_{m}$ propositional atoms.

These guarded atoms and guarded rules will be used to obtain a characterization of stable models of normal logic programs. There are many characterizations of stable models of logic programs. In fact, in [Lifschitz (2008)], Lifschitz lists twelve different characterizations of stable models of logic programs. The characterization of stable models that we present in this paper has a distinctly proof-theoretic flavor and makes easy to prove some basic results on Answer Set Programming such as Fages’ Theorem [Fages (1994)], Erdem-Lifschitz Theorem [Erdem and Lifschitz (2003)], and Dung’s Theorem [Dung and Kanchansut (1989)].

It should be observed that in [Dung and Kanchansut (1989)] Dung and Kanchansut consider so-called quasi-interpretations which, in the formalism of our paper, can be viewed as collections of guarded atoms. The difference between our approach and that of [Dung and Kanchansut (1989)] is that we elucidate the proof theoretic content of the Gelfond-Lifschitz operator and show how this technique allows for uniform proof of various results in the theory of stable models of programs.

The outline of this paper is as follows. First, we introduce the definition of the guarded resolution rule of proof and then derive its connections with the Gelfond-Lifschitz operator [Gelfond and Lifschitz (1988)]. Once we do this, we will obtain the desired lifting of the classical result on the completeness of positive unit resolution for Horn theories [Dowling and Gallier (1984)] to the context of the stable semantics of logic programs. In Section 3, we show how guarded resolution proofs can be used to prove various standard results in the theory of stable models of propositional programs. Finally, in Section 4, we show how the theory developed in this paper can be used to obtain an algorithm for computation of stable models that does not use loop formulas and runs in polynomial space in the size of the program.

2 Guarded resolution and Stable Semantics

By a logic program clause, we mean a string of the form

C = p \leftarrow q_{1}, \dots,, q_{n}, n o t r_{1}, \dots, n o t r_{m} .

(1)

A program $P$ is a set of logic program clauses.

We will interpret program clause $C$ given in (1) as a guarded Horn clause:

g (C) = p \leftarrow q_{1}, \dots,, q_{n} : {r_{1}, \dots, r_{m}} .

We define $g (P) = {g (C) : C \in P}$ . Observe that when we interpret a logic program clause as a guarded Horn clause, the polarity of atoms appearing negatively in the body of the programming clause changes in its representation as the guarded Horn clause. That is, they occurred negatively in the body of clause and they now appear positively in the guard. By convention, we think of a propositional atom as a guarded atom with an empty guard.

We now introduce our guarded resolution rule as follows. It has two arguments: the first is a guarded Horn clause and the second is a guarded atom $q : {r_{1}, \dots, r_{n}}$ . The guarded atom $q$ must occur in the body of the guarded Horn clause. The result of the application of the rule is a guarded Horn clause whose body is the body of the original guarded Horn clause minus the atom $q$ . The guard of the resulting guarded Horn clause is the union of the guard of the guarded atom and the guard of the original guarded Horn clause. Formally, our guarded resolution rule has the following form:

\frac{p \leftarrow q_{1}, \dots,, q_{n} : {r_{1}, \dots, r_{m}} q_{j} : {s_{1}, \dots, s_{h}}}{p \leftarrow q_{1}, \dots, q_{j - 1}, q_{j + 1}, \dots, q_{n} : {r_{1}, \dots, r_{m}, s_{1}, \dots, s_{h}}} .

Next, we discuss the Gelfond-Lifschitz operator associated with a normal propositional program. Given a set of atoms $M$ and a normal logic program $P$ , we first define the Gelfond-Lifschitz reduct $P^{M}$ of $P$ . $P^{M}$ is constructed according to the following two step process. First, if

C = p \leftarrow q_{1}, \dots,, q_{n}, n o t r_{1}, \dots, n o t r_{m}

is a clause in $P$ and $r_{j} \in M$ for some $1 \leq j \leq m$ , then we eliminate $C$ . Second, if $C$ is not eliminated after step 1, then we replace $C$ by

p \leftarrow q_{1}, \dots, q_{n} .

Clearly, $P^{M}$ is a Horn program. Thus $P^{M}$ has a least model $N_{M}$ . The Gelfond-Lifschitz operator assigns to $M$ the set of atoms $N_{M}$ .

Our guarded unit resolution rule naturally leads to the notion of a guarded resolution proof $P$ of a guarded atom $p : S$ from the program $P$ . Here $S$ is a, possibly empty, set of atoms. That is, a guarded resolution proof of $p : S$ is a labeled tree such that every node that is not a leaf has two parents, one labeled with a guarded Horn clause and the other labeled with a guarded atom, where the label of the node is the result of executing the guarded unit resolution rule on the labels of the parents. Each leaf is either a guarded Horn clause $p \leftarrow q_{1}, \dots,, q_{n} : {r_{1}, \dots, r_{m}}$ such that $p \leftarrow q_{1}, \dots,, q_{n}, n o t r_{1}, \dots, n o t r_{m}$ is in $P$ or a guarded atom $q : {r_{1}, \dots, r_{m}}$ such that $q \leftarrow n o t r_{1}, \dots, n o t r_{m}$ is in $P$ . In the special case where the tree consist of a single node, we assume that the node is labeled with a guarded atom $q : {r_{1}, \dots, r_{m}}$ where $q \leftarrow n o t r_{1}, \dots, n o t r_{m}$ is in $P$ . Note that in a guarded resolution proof, guards only grow as we proceed down the tree. That is, as we resolve, the guards are summed up. For this reason, the guard of the root of the proof contains the guards of every label in the tree.

We say that a set of atoms $M$ admits a guarded atom $p : S$ , if $M \cap S = \emptyset$ and that $M$ admits a guarded resolution proof $P$ if it admits the label of the root of $P$ . The following statement follows from the containment properties of guards in a guarded resolution proof.

Lemma 2.1

If $M$ admits the guarded resolution proof $P$ , then $M$ admits every guarded atom occurring as a label in $P$ and $M$ is disjoint from the guard of every guarded clause in $P$ .

We then have the following proposition.

Proposition 2.1

Let $P$ be a propositional logic program and let $M$ be a set of atoms. Then ${G L}_{P} (M)$ consists exactly of atoms $p$ such that there exists a set of atoms $Z$ where the guarded atom $p : Z$ is the conclusion of a guarded resolution proof $P$ from $g (P)$ admitted by $M$ .

Proof: Let $Q = P_{M}$ and assume that $p \in {G L}_{P} (M)$ . Then by definition, $p \in T_{Q}^{ω}$ where $T_{Q}$ is the standard one-step provability operator for $Q$ . We claim that we can prove by induction on $n \in N$ that whenever $p \in T_{Q}^{n}$ , then there exists a set of atoms $Z$ such that $p : Z$ possesses a guarded resolution proof from $g (P)$ admitted by $M$ . If $n = 1$ , then it must be the case that the $p \leftarrow$ belongs to $Q$ . But then, for some $r_{1}, \dots, r_{m}$ ,

p \leftarrow n o t r_{1}, \dots, n o t r_{m}

belongs to $P$ and ${r_{1}, \dots, r_{m}} \cap M = \emptyset$ . Therefore the guarded atom $p : {r_{1}, \dots, r_{m}}$ is admitted by $M$ and it possesses a guarded resolution proof from $g (P)$ , namely, the one that consists of the root labeled by $p : {r_{1}, \dots, r_{m}}$ . Now, let us assume $p \in T_{Q}^{n + 1}$ . Then there is a clause $C = p \leftarrow q_{1}, \dots, q_{s}$ in $Q$ such that $q_{i} \in T_{Q}^{n}$ for $i = 1, \dots, s$ . Thus by induction, there are sets of atoms $S_{i}$ , $1 \leq i \leq n$ , such that $q_{i} : S_{i}$ possesses a guarded resolution proof from $g (P)$ admitted by $M$ . As $p \leftarrow q_{1}, \dots, q_{n}$ belongs to $Q$ , there must exist atoms $r_{1}, \dots, r_{m} \notin M$ such that

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

is a clause in $P$ . It is easy to combine the guarded resolution proofs of $q_{i} : S_{i}$ , $1 \leq i \leq n$ and the guarded clause $p \leftarrow q_{1}, \dots, q_{n} : {r_{1}, \dots, r_{m}}$ to obtain a guarded resolution proof from $g (P)$ of the following guarded atom:

p : S_{1} \cup \dots \cup S_{n} \cup {r_{1}, \dots, r_{m}} .

As all the sets occurring in the guard of this guarded atom are disjoint from $M$ , the resulting guarded resolution proof is admitted by $M$ . This shows the inclusion $\subseteq$ .

Conversely, suppose $p : S$ has a guarded resolution proof $P$ from $g (P)$ admitted by $M$ . By the lemma, all the guards occurring in $P$ are disjoint from $M$ . We can then prove by induction on the height of the tree $P$ that $p \in {G L}_{P} (M)$ . If the height of $P$ is $0$ , then it must be the case that

p \leftarrow n o t r_{1}, \dots, n o t r_{m}

belongs to $P$ where $S = {r_{1}, \dots, r_{m}}$ . But since $M \cap S = \emptyset$ , the clause $p \leftarrow$ belongs to $Q$ . Hence $p \in {G L}_{P} (M)$ .

Now, for the inductive step, assume that whenever $q : S$ has a guarded resolution proof from $g (P)$ that is admitted by $M$ of height less than or equal to $n$ , then $q \in {G L}_{P} (M)$ . We now show that the same property holds for all guarded atoms $p : U$ which have a guarded resolution proof from $G (P)$ that is admitted by $M$ of the height $n + 1$ . What does such a guarded resolution proof look like? First the root must be the result of a guarded unit resolution of the form

\frac{p \leftarrow q : Z_{1} q : S_{0}}{p : Z_{1} \cup S_{0}} .

As $(Z_{1} \cup S_{0}) \cap M = \emptyset$ , $Z_{1} \cap M = \emptyset$ and $S_{0} \cap M = \emptyset$ . Now, $q : S_{0}$ has a guarded resolution proof from $g (P)$ that is admitted by $M$ of height at most $n$ and, hence, $q \in {G L}_{P} (M)$ by our inductive assumption. Since as we progress down the proof tree, the body of guarded clauses only get smaller and the guards of guarded clauses only get bigger, there must exist a path starting at the guarded clause $p \leftarrow q : Z_{1}$ which consists of guarded clauses of the form

	$p \leftarrow q, q_{1}, \dots, q_{t} : Z_{t + 1}$
	$⋮$
	$p \leftarrow q, q_{1} : Z_{2}$
	$p \leftarrow q : Z_{1}$

where $Z_{t + 1} \subseteq Z_{t} \subseteq \dots \subseteq Z_{1}$ and for each $i$ , there is a node in the tree of the form $q_{i} : S_{i}$ such that the resolution of $p \leftarrow q, q_{1}, \dots, q_{i} : Z_{i + 1}$ and $q_{i} : S_{i}$ results in the clause $p \leftarrow q, q_{1}, \dots, q_{i - 1} : Z_{i}$ . Now each $q_{i} : S_{i}$ is the root of a guarded resolution proof from $g (P)$ that is admitted by $M$ of height less than or equal to $n$ and, hence, $q_{i}$ is in $G L_{P} (M)$ for $i = 1, \dots, t$ .

Since $p \leftarrow q, q_{1}, \dots, q_{p} : Z_{t + 1}$ is a leaf, there must be a clause

p \leftarrow q, q_{1}, \dots, q_{t}, n o t r_{1}, \dots, n o t r_{m}

in $P$ where $Z_{t + 1} = {r_{1}, \dots, r_{m}}$ . Since $M$ admits the proof tree, it must be the case that ${r_{1}, \dots, r_{m}} \cap M = \emptyset$ and, hence, $p \leftarrow q, q_{1}, \dots, q_{t}$ is in $Q$ . But then, since $q, q_{1}, \dots, q_{t}$ are in $G L_{P} (M)$ , it follows that $p \in G L_{P} (M)$ .

Proposition 2.1 tells us that the Gelfond-Lifschitz operator $G L$ is, essentially, a proof-theoretic construct. Here is one consequence, this time a semantic one.

Corollary 2.1

Let $P$ be a propositional logic program and let $M$ be a set of atoms. Then $M$ is a stable model of $P$ if and only if

for every $p \in M$ , there is a set of atoms $S$ such that there is a guarded resolution proof of $p : S$ from $g (P)$ admitted by $M$ and
for every $p \notin M$ , there is no set of atoms $S$ such that there is a guarded resolution proof of $p : S$ from $g (P)$ admitted by $M$ .

When $P$ is a Horn program Corollary 2.1 reduces to the classical fact [Dowling and Gallier (1984)] that the elements of the least model of the Horn programs are precisely those that can be proved out of clausal representation of $P$ using positive unit resolution.

Given a finite set of atoms $S$ , we write $\neg S$ for the conjunction $⋀_{q \in S} \neg q$ . Next, let us call $S$ such that $p : S$ has a guarded resolution proof from $g (P)$ a support of $p$ with respect to $P$ . We can then form an equation for $p$ with respect to $P$ , ${e q}_{P} (p)$ , as follows:

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

where $S_{1}, S_{2}, \dots$ is the list of all supports of $p$ with respect to $P$ . If the only support of $p$ is the empty set, then we let ${e q}_{P} (p) = p$ and if there are no supports for $p$ , then we let ${e q}_{P} (p) = \neg p$ . Next, let $E_{P}$ be the propositional theory consisting of ${e q}_{P} (p)$ for all $p \in A t$ . We then have the following theorem resembling Clark’s completion theorem, except we get it for stable models, not supported models.

Proposition 2.2

Let $P$ be a propositional program and let $M$ be a set of atoms. Then $M$ is a stable model of $P$ if and only if $M ⊨ E_{P}$ .

Proof: First, assume that $M$ is a stable model of $P$ . Then if $p \in M$ , it follows from Corollary 2.1 that there is an $S$ such that $p : S$ has a guarded resolution proof admitted by $M$ . Hence $M \cap S = \emptyset$ and $M ⊨ \neg S$ . Thus $M$ satisfies both $p$ and one of the disjuncts on the right-hand side of ${e q}_{P} (p)$ . Hence $M ⊨ {e q}_{P} (p)$ . Next assume that $p \notin M$ . Then there is no $Z$ such that $p : Z$ has a guarded resolution proof admitted by $M$ . It follows that either ${e q}_{P} (p)$ equals $\neg p$ or $M$ satisfies both negation of $p$ and of the negation of every disjunct on the right-hand-side of ${e q}_{P} (p)$ . Thus again $M ⊨ {e q}_{P} (p)$ . This shows “if” part of the theorem.

For the other direction, suppose that $M ⊨ {e q}_{P} (p)$ . Then if $p \in M$ , either ${e q}_{P} (p) = p$ or ${e q}_{P} (p) = p \Leftrightarrow (\neg S_{1} \lor \neg S_{2} \lor \dots)$ . In the former case, this means that the tree consisting of a single node $p : \emptyset$ is a guarded resolution proof and, hence, $p \leftarrow$ is a clause in $P$ . Thus $p$ must be in $M$ . In the latter case, there must be some $S_{i}$ such that $M ⊨ \neg S_{i}$ . But by definition, $p : S_{i}$ is the root of some guarded resolution proof $P$ for $g (P)$ and since every guard in such a guarded resolution proof is contained in $S_{i}$ , it must be the case that $M$ admits $P$ . But then we have shown that $p \in G L_{P} (M)$ . Thus $M \subseteq G L_{P} (M)$ .

On the other hand, if $p \notin M$ , then either ${e q}_{P} (p) = \neg p$ or ${e q}_{P} (p) = p \Leftrightarrow (\neg S_{1} \lor \neg S_{2} \lor \dots)$ . In the former case, there must be be no guarded resolution proofs of $p$ and, hence, $p$ is not in $M$ . In the latter case, it must be that $M$ does not satisfy any $\neg S_{i}$ . This means that there is no guarded resolution proof from $g (P)$ whose root is of the form $p : S$ such that $M$ admits $p$ and, hence, $p \notin G L_{M} (P)$ . This implies $G L_{P} (M) \subseteq M$ and, hence, $G L_{p} (M) = M$ . Thus $M$ is a stable model of $P$ .

If we look carefully at the structure of any resolution proof tree of a guarded atom $p : S$ , we see that $S$ collects a set atoms which guarantee that $p \in N_{M}$ whenever $M \cap S = \emptyset$ . Thus in defining ${e q}_{P} (p)$ , we essentially unfold the atoms in $S$ to conjunctions of negated atoms $\neg S$ (cf. [Brass and Dix (1999)]).

We observe that, in principle, when the program $P$ is infinite, the theory $E_{P}$ may be infinitary. Specifically, the formulas ${e q}_{P} (p)$ may be infinitary formulas, since the disjunction on the right-hand-side of the equivalence may be over an infinite set of finite conjunctions. But the semantics for infinite propositional logic is well-known [Karp (1964)] and can be applied here. The authors studied the necessary and sufficient conditions for $E_{P}$ to be finitary in [Marek and Remmel (2010)].

3 Some applications

In this section we will use the results of Section 2 to derive the result of Erdem and Lifschitz [Erdem and Lifschitz (2003)]. This result generalizes a theorem by Fages [Fages (1994)] which is useful as a preprocessing step for the computation of stable models of programs. We will also prove a result of Dung [Dung and Kanchansut (1989)] on stable models of purely negative programs.

We say that a set of atoms $M$ has levels with respect to a program $P$ if

$M$ is a model of $P$ , and
There is a function $r k : M \to O r d$ such that, for every $p \in M$ , there is a clause $C$ such that
1. $p = h e a d (C)$ ,
2. $M ⊨ b o d y (C)$ , and
3. For all $q \in p o s B o d y (C)$ , $r k (q) < r k (p)$ .

Clearly, the least model of a Horn program has levels since the function which assigns to an atom $p \in M$ , the least integer $n$ such that $p \in T_{P}^{n} (\emptyset)$ is the desired rank function for condition (2).

We now prove the following proposition.

Proposition 3.1

Let $P$ be a propositional logic program and $M⊆\and$ . Then $M$ is a stable model of $P$ if and only if $M$ has levels with respect to $P$ .

Proof: Clearly, when $M$ is a stable model of $P$ , then $M$ has levels with respect to $P$ . Namely, the rank function whose existence is postulated in (2) is the rank function inherited from the Horn program $P_{M}$ .

Converse implication can be proved in a variety of ways. Our proof, in the spirit of the proof-theoretic approach of this paper, uses guarded resolution. That is, assume that $M$ has levels with respect to $P$ where $r k$ is the rank function in condition (2). Our goal is to prove that $M = {G L}_{P} (M)$ . First, let us observe that since $M ⊨ P$ , ${G L}_{P} (M) \subseteq M$ . Thus, all we need to show is that $M \subseteq {G L}_{P} (M)$ . By Corollary 2.1, we need only show that for given any $p \in M$ , there is a $Z$ such that $p : Z$ possesses a guarded resolution proof from $g (P)$ that is admitted by $M$ . We construct the desired set $Z$ and guarded resolution proof by using the rank function $r k$ . Let $S = {r k (p) : p \in M}$ , i.e. $S$ is the range of rank function. We proceed by transfinite induction on the elements of $S$ . Let $p$ be an atom in $M$ such that $r k (p)$ is the least element of $S$ . By assumption, there must exist a clause $C$ in $P$ such that $M ⊨ b o d y (C)$ , $p = h e a d (C)$ and for all $q \in p o s B o d y (C)$ , $r k (q) < r k (p)$ . Since $M ⊨ b o d y (C)$ , there can be no $q$ ’s in positive part of the body of $C$ because any such $q$ must be in $M$ and have rank strictly less than $p$ . Thus the clause $C$ has the following form:

p \leftarrow n o t r_{1}, \dots n o t r_{m} .

As $M ⊨ b o d y (C)$ , $r_{1}, \dots, r_{m} \notin M$ . But then $p : {r_{1}, \dots, r_{m}}$ is a guarded atom admitted by $M$ and so $p : {r_{1}, \dots, r_{m}}$ has a guarded resolution proof from $g (P)$ which consists of a single node labeled with $p : {r_{1}, \dots, r_{m}}$ .

Now, assume that whenever $β \in S$ , $β < α$ , and $r k (q) = β$ , then there is a guarded resolution proof of $q : S$ from $g (P)$ admitted by $M$ for some set $S$ . Let us assume that $p \in M$ and $r k (p) = α$ . By our assumption, there is a clause $C$

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

with $M ⊨ b o d y (C)$ and $r k (q_{1}) < r k (p), \dots, r k (q_{n}) < r k (p)$ . By inductive assumption, for every $q_{i}$ , $1 \leq i \leq n$ , there is a finite set of atoms $Z_{i}$ such that there is guarded resolution proof $D_{i}$ from $g (P)$ of $q_{i} : Z_{i}$ admitted by $M$ . In particular $Z_{i} \cap M = \emptyset$ . We can then easily combine the guarded resolution proofs for $q_{i} : Z_{i}$ with $n$ applications of guarded unit resolution starting with the leaf

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

to produce a guarded resolution proof of

p : Z

from $g (P)$ where $Z = Z_{1} \cup \dots \cup Z_{n} \cup {r_{1}, \dots, r_{n}}$ . Since all $Z_{i}$ s are disjoint from $M$ and $M \cap {r_{1}, \dots, r_{m}} = \emptyset$ , it follows that $M \cap Z = \emptyset$ . Thus the resulting resolution proof is admitted by $M$ . This completes the inductive argument and thus the proof of the Proposition.

We observe that, in fact, it is sufficient to limit the functions $r k$ in the definition of $M$ having levels respect to $P$ to those rank functions that take values in $N$ , the set of natural numbers.

We get, as promised, several corollaries. One of these is the result of Erdem and Lifschitz [Erdem and Lifschitz (2003)]. Following [Erdem and Lifschitz (2003)], we say that a program $P$ is tight on a set of atoms $M$ if there is a rank function $r k$ defined on $M$ such that whenever $C$ is a clause in $P$ and $h e a d (C) \in M$ , then for all $q \in p o s B o d y (C)$ , $r k (q) < r k (h e a d (C)) .$ Here is the result of Erdem and Lifschitz.

Corollary 3.1 (Erdem and Lifschitz)

If $P$ is tight on $M$ and $M$ is a supported model of $P$ , then $M$ is a stable model of $P$ .

Proof: Indeed, tightness on $M$ requires that for any $p \in M$ , there is a support for $p$ and that all clauses $C$ that provide the support for the presence of $p$ in $M$ have the property that the atoms in the positive part of the body of $C$ have smaller rank. In Proposition 3.1, we showed that it is enough to have just one such clause. Since tightness on $M$ implies existence of such a supported clause, the corollary follows.

Since all stable models are supported [Gelfond and Lifschitz (1988)], one can express Erdem-Lifschitz Theorem in a more succinct way.

Corollary 3.2 (Erdem-Lifschitz)

If for every supported model $M$ of a program $P$ , $P$ is tight on $M$ , then the classes of supported and stable models of $P$ coincide.

Fages Theorem [Fages (1994)] is a weaker version of Corollary 3.1 (but with assumptions that are easier to test). Specifically, we say that a program $P$ is tight if there is a rank function $r k$ such that for every clause $C$ of $P$ , the ranks of the atoms occurring in the positive part of the body of $C$ are smaller than the rank of the head of $C$ . Clearly, if $P$ is tight, then $P$ is tight on any of its models $M$ since $r k$ will also witness that $P$ is tight on $M$ . Thus one has the following corollary.

Corollary 3.3 (Fages)

If $P$ is tight, then the classes of supported and stable models of $P$ coincide.

Tightness is a syntactic property that can be checked in polynomial time by inspection of the positive call-graph of $P$ . This is not the case for the stronger assumptions of Proposition 3.1 and Corollary 3.2.

Let $S t a b (P)$ be the set of all stable models of $P$ . We say that programs $P$ , $P^{'}$ are equivalent if $S t a b (P) = S t a b (P^{'})$ . This notion and its strengthenings were studied by ASP community [Lifschitz, Pearce and Valverde (2001)], [Truszczynski (2006)]. We have the following fact.

Lemma 3.1

If $P, P^{'}$ prove the same guarded atoms, then $P$ and $P^{'}$ are equivalent.

As a corollary we get the following result due to Dung [Dung and Kanchansut (1989)]

Corollary 3.4 (Dung)

For every program $P$ , there is purely negative program $P^{'}$ such that $P$ , $P^{'}$ are equivalent.

The program $P^{'}$ is quite easy to construct. That is, for each atom $p$ , if

{e q}_{P} (p) = p \Leftrightarrow (\neg S_{1} \lor \neg S_{2} \lor \dots),

then we add to $P^{'}$ , all clauses of the form

p \leftarrow n o t r_{i, 1}, \dots, n o t r_{i, m_{i}}

where $S_{i} = {r_{i, 1}, \dots, r_{i, m_{i}}}$ . If ${e q}_{P} (p) = p$ , then we add $p \leftarrow$ to $P^{'}$ . Finally, if ${e q}_{P} (p) = \neg p$ , then we add nothing to $P^{'}$ . It is then easy to see that $E_{P} = E_{P^{'}}$ and hence $P$ and $P^{'}$ are equivalent.

4 Computing stable models via satisfiability, but without loop formulas or defining equations

Proposition 2.2 characterized the stable models of a propositional program in terms of the collection of all propositional valuations of the underlying set of atoms. In this section, we give an alternative characterization in terms of the models of suitably chosen propositional theories. The proof of this characterization uses Proposition 2.2, but relates stable models of finite propositional programs $P$ to models of theories of size $O (| P |)$ . This is in contrast to Proposition 2.2 since the set of defining equations is, in general, of size exponential in $| P |$ .

A subequation for an atom $p$ is either a formula $\neg p$ or a formula

p \Leftrightarrow \neg S

where $S$ is a support of a guarded resolution proof of $p$ from $P$ . Here if $S = \emptyset$ , then by convention we interpret $p \Leftrightarrow \neg S$ to be simply the atom $p$ . The idea is that a subequation either asserts absence of the atom $p$ in the putative stable model or provides the reason for the presence of $p$ in the putative stable model. A candidate theory for program $P$ is the union of $P$ and a set of subequations, one for each $p∈\and$ . By $C_{P}$ we denote the set of candidate theories for $P$ .

Proposition 4.1

Let $T$ be a candidate theory for $P$ (i.e. $T \in C_{P}$ ). If $T$ is consistent, then every propositional model of $T$ is a stable model for $P$ .
For every stable model $M$ of $P$ , there is theory $T \in C_{P}$ such that $M$ is a model for $T$ .

Proof: (1) Let $T$ be a candidate theory for $P$ and suppose that $M ⊨ T$ . We need to show that $M$ is a stable model for $P$ . In other words, we need to show that

{G L}_{P} (M) = M .

The inclusion ${G L}_{P} (M) \subseteq M$ follows from the fact that $M$ is a model of $P$ . That is, since $M$ is a model of $P$ , it immediately follows that $M$ is a model of the Gelfond-Lifschitz transform of $P$ , $P_{M}$ . Since ${G L}_{P} (M)$ is the unique minimal model of $P_{M}$ , it automatically follows that ${G L}_{P} (M) \subseteq M$ .

To show that $M \subseteq {G L}_{P} (M)$ , suppose that $p \in M$ . Then the subequation for $p$ that is in $T$ can not be $\neg p$ . Therefore it is of the form

p \Leftrightarrow \neg U_{p}

where there is some guarded resolution proof $P$ of $p : U_{p}$ from $g (P)$ . Since $M ⊨ T$ and $p \in M$ , $M ⊨ \neg U_{p}$ . But then $M \cap U_{p} = \emptyset$ so that $M$ admits $P$ . Hence by Corollary 2.1, $p \in {G L}_{P} (M)$ .

(2) Let $M$ be a stable model of $P$ . We construct a candidate theory $T$ so that $M$ is a model of $T$ . To this end, for each $p \notin M$ , we select $\neg p$ as a subequation for $p$ . For each $p \in M$ , we select a guarded resolution proof of some $p : U_{p}$ from $g (P)$ that admitted by $M$ . We then add to $T$ the formula

p \Leftrightarrow \neg U_{p} .

Clearly, $T$ is a candidate theory, and $M ⊨ T$ , as desired.

Next we give an example of our approach to reducing the computation of of stable models to satisfiability of propositional theories. It will be clear from this example that our approach avoids having to compute the completion of the program and thus significantly reduces the size of the input theories.

Example 4.1

Let $P$ be a propositional program as follows:

	$p \leftarrow t, \neg q$
	$p \leftarrow \neg r$
	$q \leftarrow \neg s$
	$t \leftarrow$

Let us observe that the atom $p$ has two inclusion-minimal supports, namely ${q}$ and ${r}$ . The atom $q$ has just one support, namely ${s}$ , and the atom $t$ also has just one support, namely $\emptyset$ . The atoms $r$ and $s$ have no support at all.

Thus there are three subequations for $p$ :

	$p \Leftrightarrow \neg q$
	$p \Leftrightarrow \neg r$
	$\neg p$

Now, $q$ has only two subequations: $q \Leftrightarrow \neg s$ , and $\neg q$ . Similarly, $t$ has only two subequations, $t$ and $\neg t$ , but the second one automatically leads to contradiction whenever it is chosen. Finally each of $r$ and $s$ have just one defining equation, $\neg r$ , and $\neg s$ , respectively.

First let us choose for $p$ , the subequation $\neg p$ , and for $q$ , the subequation $q \Leftrightarrow \neg s$ . The remaining subequations are forced to $t$ , $\neg r$ , and $\neg s$ . The resulting theory has nine clauses, when we write our program in propositional form:

S = {\neg p, \neg r, \neg s, t, q \Leftrightarrow \neg s} \cup {\neg t \lor p \lor q, r \lor p, s \lor q, t} .

It is quite obvious that this theory is inconsistent. However, if we choose for $p$ , the subequation $p \Leftrightarrow \neg r$ and for $q$ , the subequation $q \Leftrightarrow \neg s$ , then the resulting theory written out in propositional form is

S = {p \Leftrightarrow \neg r, \neg r, \neg s, t, q \Leftrightarrow \neg s} \cup {\neg t \lor p \lor q, r \lor p, s \lor q, t} .

In this case, ${p, q, t}$ is a model of $S$ and hence, ${p, q, t}$ is a stable model of $P$ .

It should be clear that Proposition 4.1 implies an algorithm for computation of stable models. Namely, we generate candidate theories and find their propositional models.

Let us observe that the algorithm described above can be implemented as a two-tier backtracking search, with the on-line computation of supports of guarded resolution proofs, and the usual backtracking scheme of DPLL. This second backtracking can be implemented using any DPLL-based SAT-solver. Proposition 4.1 implies that the algorithm we outlined is both sound and complete. Indeed, if the SAT solver returns a model $M$ of theory $T$ , then, by Proposition 4.1(1), $M$ is a stable model for $P$ . Otherwise we generate another candidate theory and loop through this process until one satisfying assignment is found. Proposition 4.1(2) guarantees the completeness of our algorithm.

Our algorithm is not using loop formulas like the algorithm of Lin and Zhao [Lin and Zhang (2002)] but systematically searches for supports of proof schemes, thus providing supports for atoms in the putative model. It also differs from the modified loop formulas approach of Ferraris, Lee and Lifschitz [Ferraris, Lee and Lifschitz (2006)] in that we do not consider loops of the call-graph of $P$ at all. Instead, we compute systematically proof schemes and their supports for atoms. While the time-complexity of our algorithm is significant because there may be exponentially many supports for any given atom $p$ , the space complexity is $| P |$ . This is the effect of not looking at loop formulas at all ([Lifschitz and Razborov (2006)]).

5 Conclusions and Further Work

We have shown that guarded unit resolution, a proof system that is a nonmonotonic version of unit resolution, is adequate for description of the Gelfond-Lifschitz operator ${G L}_{P}$ and its fixpoints. That is, we can characterize stable models of logic programs in terms of guarded resolution.

There are several natural questions concerning extensions of guarded resolution in the context of Answer Set Programming. For example:

(1) Is there an analogous proof system for the disjunctive version of logic programming?
or
(2) Are there analogous proof systems for logic programming with constraints?

We believe that availability of such proof systems could help with finding a variety of results on the complexity of reasoning in nonmonotonic logics. An interesting case is that of propositional Default Logic. We will show in a subsequent paper that that we can develop a similar guarded resolution proof scheme for propositional Default Logic. The main difference is that we must allow proof trees where the leaves can be tautologies rather than just guarded atoms or guarded clauses that are derived from the given program $P$ as in this paper.

References

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Guarded resolution for answer set programming

Typed SLD-Resolution: Dynamic Typing for Logic Programming

An Application of Proof-Theory in Answer Set Programming

A Brief History of Updates of Answer-Set Programs

A Functional View of Strong Negation in Answer Set Programming

A Denotational Semantics for First-Order Logic

Semantics for Possibilistic Disjunctive Programs

Smodels: A System for Answer Set Programming

1 Introduction

2 Guarded resolution and Stable Semantics

Lemma 2.1

Proposition 2.1

Corollary 2.1

Proposition 2.2

3 Some applications

Proposition 3.1

Corollary 3.1 (Erdem and Lifschitz)

Corollary 3.2 (Erdem-Lifschitz)

Corollary 3.3 (Fages)

Lemma 3.1

Corollary 3.4 (Dung)

4 Computing stable models via satisfiability, but without loop formulas or defining equations

Proposition 4.1

Example 4.1

5 Conclusions and Further Work

References

Guarded resolution for answer set programming

Related Research

Typed SLD-Resolution: Dynamic Typing for Logic Programming

An Application of Proof-Theory in Answer Set Programming

A Brief History of Updates of Answer-Set Programs

A Functional View of Strong Negation in Answer Set Programming

A Denotational Semantics for First-Order Logic

Semantics for Possibilistic Disjunctive Programs

Smodels: A System for Answer Set Programming

1 Introduction

2 Guarded resolution and Stable Semantics

Lemma 2.1

Proposition 2.1

Corollary 2.1

Proposition 2.2

3 Some applications

Proposition 3.1

Corollary 3.1 (Erdem and Lifschitz)

Corollary 3.2 (Erdem-Lifschitz)

Corollary 3.3 (Fages)

Lemma 3.1

Corollary 3.4 (Dung)

4 Computing stable models via satisfiability, but without loop formulas or defining equations

Proposition 4.1

Example 4.1

5 Conclusions and Further Work

References