[
Abstract
The definition of stable models for propositional formulas with infinite conjunctions and disjunctions can be used to describe the semantics of answer set programming languages. In this note, we enhance that definition by introducing a distinction between intensional and extensional atoms. The symmetric splitting theorem for firstorder formulas is then extended to infinitary formulas and used to reason about infinitary definitions. This note is under consideration for publication in Theory and Practice of Logic Programming.
Infinitary Formulas with Extensional Atoms]Stable Models for Infinitary Formulas
with Extensional Atoms
A. Harrison and V. Lifschitz]AMELIA HARRISON, VLADIMIR LIFSCHITZ
University of Texas, Austin, Texas, USA
1 Introduction
The original definition of a stable model (Gelfond and Lifschitz, 1988) was applicable only to quantifierfree formulas of a restricted syntax. Stable models for arbitrary firstorder sentences were defined by Ferraris et al. (2007) using the stable model operator SM. This definition can be used to define the semantics of some rules with aggregate expressions. For instance, the following rule, written in the input language of the ASP system clingo,^{1}^{1}1http://potassco.sourceforge.net
q : #count{X:p(X)} = 0  (1) 
can be identified with the firstorder sentence
(2) 
In Ferraris et al. (2011), that definition was generalized to allow a distinction between “extensional” and “intensional” predicate symbols. (Under the original definition all predicate symbols are treated as intensional.) Intuitively, an intensional predicate is one whose extent is defined by the program, while all other, extensional, predicates are defined externally. Similar distinctions have been proposed many times: Gelfond and Przymusinska (1996) distinguish between input and output predicates in their “lpfunctions”, Oikarinen and Janhunen (2008) distinguish between input and output atoms, and Lierler and Truszczynski (2011) between input and noninput atoms. These distinctions are useful because they allow for a modular view of logic programs. For example, in the splitting theorem from Ferraris et al. (2009), the authors showed that stable models for a program can sometimes be computed by breaking the program into parts and computing the stable models of each part separately using different sets of intensional predicates.
Using the approach proposed by Ferraris (2005), Truszczynski (2012) extended the definition of a stable model in a different direction: he showed how to apply this concept to infinitary propositional formulas. He also showed that the definition of firstorder stable models in terms of the 2007 definition of the operator SM could be reduced to the definition of infinitary stable models. Infinitary stable models were used in that paper as a tool for relating firstorder stable models to the semantics of firstorder logic with inductive definitions. Infinitary stable models are important also because they provide an alternative understanding of the semantics of aggregates. For instance, rule (1) can be identified with the infinitary formula
(3) 
where the conjunction in the antecedent is understood as ranging over all ground terms not containing arithmetic operations. The advantage of this approach over the use of firstorder formulas is that it is more flexible. For example, it is applicable to aggregates involving #sum. In recent work, Gebser et al. (2015) use this idea to define a precise semantics for a large class of ASP programs, including programs with local variables and aggregate expressions.
However, Truszczynski’s definition of stable models for infinitary formulas does not allow a distinction between extensional and intensional atoms. It treats all atoms as intensional. In this note, we generalize the definition of stable models for infinitary formulas to accommodate both intensional and extensional atoms, and we study properties of this definition. As might be expected, the definition of firstorder stable models with extensional predicates can be reduced to the definition proposed in this note. We use this definition to generalize the results on firstorder splitting from Ferraris et al. (2009). In particular, we look at the splitting lemma from Ferraris et al. (2009), which showed that under certain conditions the stable models of a formula can be computed by computing the stable models of the same formula with respect to smaller sets of intensional predicates. We find that a straightforward infinitary counterpart to the splitting lemma does not hold, and show how the lemma needs to be modified for the infinitary case. The situation is similar for the splitting theorem discussed above. The infinitary splitting theorem is used to generalize the lemma on explicit definitions due to Ferraris (2005), which describes how adding explicit definitions to a program affects its stable models. In the version presented in this note, the program can include infinitary formulas and the definition can be recursive.
2 Review: Infinitary Formulas and their Stable Models
This review follows Truszczynski (2012), Harrison et al. (2015). Let be a propositional signature, that is, a set of propositional atoms. For every nonnegative integer , (infinitary propositional) formulas (over ) of rank are defined recursively, as follows:

every atom from is a formula of rank 0,

if is a set of formulas, and is the smallest nonnegative integer that is greater than the ranks of all elements of , then and are formulas of rank ,

if and are formulas, and is the smallest nonnegative integer that is greater than the ranks of and , then is a formula of rank .
We will write as , and as . The symbols and will be understood as abbreviations for and respectively; stands for , and stands for . These conventions allow us to view finite propositional formulas over as a special case of infinitary formulas.
A set or family of formulas is bounded if the ranks of its members are bounded from above. For any bounded family of formulas, the formula will be denoted by , and similarly for disjunctions.
Subsets of a signature will be also called interpretations of . The satisfaction relation between an interpretation and a formula is defined recursively, as follows:

For every atom from , if .

if for every formula in , .

if there is a formula in such that .

if or .
An infinitary formula is tautological if it is satisfied by all interpretations. Two infinitary formulas are equivalent if they are satisfied by the same interpretations.
The reduct of a formula w.r.t. an interpretation is defined recursively, as follows:

For every atom from , is if , and otherwise.

is .

is .

is if , and otherwise.
If is a set of infinitary formulas then the reduct is the set . An interpretation is a stable model of a set of formulas if it is minimal w.r.t. set inclusion among the interpretations satisfying the reduct .
3 stable Models
Following Ferraris et al. (2011), we will assume that some atoms in a program are designated “intensional” while all others are regarded as “extensional”.
Recall that denotes a propositional signature. Let be a (possibly infinite) set of atoms. The partial order is defined as follows: for any sets , we say that if and . (Intuitively, if the atoms in are treated as intensional and all other atoms from are treated as extensional, the relation holds if and agree on all extensional atoms.) An interpretation is called an (infinitary) stable model of a formula if it is a minimal model of w.r.t. .
Observe that if then stable models of a formula are the same as stable models. If then stable models are all models of . Truszczynski observed that an interpretation satisfies iff satisfies (Truszczynski, 2012, Proposition 1). It follows that all stable models of also satisfy .
Example (continued)
To illustrate the definition of stability, let’s find
all stable
models^{2}^{2}2 Here, we understand as implicitly defined to be the
set containing and all atoms of the form where is a ground
term. of (3).
The stable model of (3) is stable as well, because
it is a minimal model of (4) w.r.t. . On the
other hand, any nonempty set of atoms of the form is
stable too. Indeed, the reduct of (3) w.r.t.
such a set is an implication whose antecedent has as one of its
conjunctive terms. Such a formula is tautological so that it is satisfied by
. Furthermore, is a minimal model w.r.t.
since any subset of will disagree with it on extensional atoms.
The fact that all stable models of (3) are also stable is an instance of a more general fact: If is an stable model of and is a subset of then is also a stable model of . This follows directly from the definition of stability.
The following proposition provides two alternative definitions for stability.
Proposition 1
The following three conditions are equivalent:

is an stable model of ;

is a minimal model (w.r.t. set inclusion) of
(5) 
is a stable model of
(6)
We first establish that conditions (i) and (ii) are equivalent: is an stable model of
iff  
iff  
iff  
iff  
iff  
iff  
iff 
4 Relating Infinitary and FirstOrder Stable Models
As mentioned in the introduction, Truszczynski (2012) showed that infinitary stable models can be viewed as a generalization of firstorder stable models in the sense of Ferraris et al. (2011). In this section, we will show that the corresponding result holds for stable models as well.^{3}^{3}3The definition of stable models, where is a list of distinct predicate symbols, can be found in Ferraris et al. (2011), Section 2.3. First, we review Truszczynski’s results.
Let be a firstorder signature, and be an interpretation of with nonempty domain . For each element of , by we denote a new object constant, called the name of . By we denote the signature obtained by adding the names of all elements of to . An interpretation is identified with its extension to in which for each in , . By we denote the set of all atomic sentences over built with relation symbols from and names of elements in , and by we denote the subset of that describes in the obvious way the extents of the relations in . Let be a formula over signature . Then the grounding of w.r.t. , gr is defined recursively, as follows:

gr is ;

gr is ;

gr is if and otherwise;

gr is gr gr, where ;

gr is {gr;

gr is {gr.
(By we denote the result of substituting for all free occurrences of in .) It is clear that for any firstorder sentence over signature , gr is an infinitary formula over the signature .
Example (continued)
If consists of the unary predicate
and the propositional symbol , and is an
interpretation of such that the domain is the set of all ground
terms , then the grounding of (2) w.r.t.
is (3). (To simplify notation we identify the name of each term
with .)
According to Theorem 5 from Truszczynski (2012), if is a firstorder sentence and is an interpretation, then is a firstorder stable model of iff is an infinitary stable model of gr. The proposition below generalizes this result to the case of stable models. By we denote the atomic formulas in built with predicates from .
Example (continued)
If is then is the set
of all atoms of the form .
Proposition 2
For any firstorder sentence over and any tuple of distinct predicate symbols from , an interpretation is a stable model of iff is a stable model of gr.
Example (continued)
Let be the interpretation that interprets as
identically false and assigns the value to . Then is .
Let be an interpretation that satisfies at least one atomic formula
and assigns the value to . Then is (the same as from the previous section). We saw in the
previous section that stable models of (3) are
and any nonempty set of atoms of the form . In accordance with the
proposition above, and are stable models of (2).
[Proof of Proposition 2] Consider a firstorder sentence and list of distinct predicate symbols Let be the set of all predicates occurring in but not in . Consider an interpretation of the signature of . By Theorem 2 from Ferraris et al. (2011), is a stable model of iff it is a stable model of
where is a list of distinct object variables the same length as the arity of . By Theorem 5 from Truszczynski (2012), is a stable model of the formula above iff is a stable model of the grounding of this formula w.r.t. . The grounding of the formula above w.r.t. is
(7) 
By Proposition 1, is a stable model of (7) iff it is a stable model of gr.
5 Review: FirstOrder Splitting Lemma
The lemma presented in the next section of this note is a generalization of the splitting lemma from Ferraris et al. (2009).
In order to state that lemma, we first review the definition of the predicate dependency graph given in that paper. We say that an occurrence of a predicate symbol or a subformula in a firstorder formula is positive if it occurs in the antecedent of an even number of implications and strictly positive if it occurs in the antecedent of no implication. An occurrence of a predicate constant is said to be negated if it belongs to a subformula of the form , and nonnegated otherwise. A rule of a firstorder formula is a strictly positive occurrence of an implication in . The (positive) predicate dependency graph of a firstorder formula w.r.t. a list of distinct predicates, denoted DG is the directed graph that

has all predicate symbols in as its vertices, and

has an edge from to if, for some rule of ,

has a strictly positive occurrence in , and

has a positive nonnegated occurrence in .

We say that a partition^{4}^{4}4We understand a partition of to be a set of disjoint subsets (possibly empty) that cover . of the vertices in a graph is separable (on ) if every strongly connected component of is a subset of either or . (Here, we identify the list with the set of its members.)
The following assertion is a reformulation of Version 1 of the splitting lemma from Ferraris et al. (2009).
Splitting Lemma
If is
a firstorder sentence and are lists of distinct predicate symbols
such that the partition is separable on DG
then
is a stable model of iff it is
both
a stable model and a stable
model of .
6 Infinitary Splitting Lemma
The statement of the infinitary splitting lemma refers to the positive dependency graph of an infinitary formula. As we will see, the vertices of this graph correspond to intensional atoms. This definition is similar to the definition of a predicate dependency graph given in Ferraris (2007) and Ferraris et al. (2009) and reviewed in the previous section. The concepts necessary to define the dependency graph of an infinitary formula are all straightforward extensions of the concepts used in the previous section to define the predicate dependency graph in the firstorder case. However, because infinitary formulas are not syntactic structures, we have to define these concepts recursively.
We define the set of strictly positive atoms of an infinitary formula , denoted P(), recursively, as follows:

For every atom , P() is ;

P() is P(), and so is P();

P() is P().
The set of positive nonnegated atoms and the set of negative nonnegated atoms of an infinitary formula , denoted Pnn() and Nnn() respectively, were introduced in Lifschitz and Yang (2012). These sets are defined recursively as well:

For every atom , Pnn() is ;

Pnn() is Pnn(), and so is Pnn();

Pnn() is if is and Nnn() Pnn() otherwise.
and

For every atom , Nnn() is ;

Nnn() is Nnn(), and so is Nnn();

Nnn() is if is and Pnn() Nnn() otherwise.
The set of rules of an infinitary formula is defined as follows:

The rules of are and all rules of ;

The rules of and are the rules of all formulas in .
Example (continued)
The set of positive nonnegated atoms in formula (3) is
the same as the set of strictly positive atoms: . The only rule of
formula (3) is the formula itself.
For any infinitary formula the (positive) dependency graph of (relative to a set of atoms , denoted DG, is the directed graph, that

has all atoms in as its vertices, and

has an edge from to if, for some rule of ,

is an element of P(), and

is an element of Pnn().

The following statement appears to be a plausible counterpart to the splitting lemma reproduced in Section 5 for infinitary formulas:
If is an infinitary formula and are sets of atoms such that the partition is separable on DG then is a stable model of iff it is both a stable model and a stable model of .  () 
But this statement does not hold; in the case of infinitary formulas separability is not a sufficient condition to ensure splittability. Let be the infinitary conjunction
where the conjunction extends over all integers . Let be the set of all atoms . Let be the set , and be the set Then the partition is separable on DG (shown in Figure 1). Indeed, the strongly connected components of this graph are singletons. If is the set of all atoms then the reduct of w.r.t. is itself. It is easy to check that is a stable model as well as a stable model of , but is not stable. This counterexample shows that () is incorrect.
In order to extend the splitting lemma to infinitary formulas, we will need a stronger notion of separability. An infinite walk of a directed graph is an infinite sequence of vertices occurring in , such that each pair in corresponds to an edge in . A partition of the vertices in will be called infinitely separable (on ) if every infinite walk of visits either or finitely many times, that is either or is finite.
Proposition 3
For any graph ,

every infinitely separable partition of is separable, and

if has finitely many strongly connected components and partition is separable on then it is infinitely separable on .
(i) We will prove the contrapositive: if is a partition that
is not separable on , then there is some strongly connected component of
that contains at least one vertex from and at least one vertex
from . Let’s call these vertices and , respectively.
Since and are in the same strongly connected component, each vertex is
reachable from the other. Then there is an infinite walk that
visits each of these vertices (and therefore both and )
infinitely many times, so that the partition is not infinitely separable on
.
(ii) Again we prove the contrapositive:
if is a partition that is not infinitely separable on ,
then there is some infinite walk
of that visits both and infinitely
many times. Since there are only finitely many strongly connected components
in , at least one strongly connected component of and
at least one strongly connected component of must be visited
infinitely many times. Call these strongly connected components and
respectively; then must be reachable from and vice versa. Then so that the partition is not separable on .
Claim () will become correct if we require the partition to be infinitely separable:
Infinitary Splitting Lemma
If is
an infinitary formula and are sets of atoms such
that the partition is infinitely separable on
DG then
is a
stable model of iff it is
both
a stable model and a stable
model of .
The splitting lemma reproduced in Section 5 is a consequence of the infinitary splitting lemma in view of Theorem 2 and the following fact:
Proposition 4
For any firstorder sentence and tuple of distinct predicate symbols, if is a partition of that is separable on DG, then for any interpretation , is infinitely separable on DG.
If is a partition of that is separable on DG, then for any interpretation , the partition is separable on the atomic dependency graph of gr with respect to . Furthermore, it is easy to see that DG must have finitely many strongly connected components, so that must be infinitely separable on it.
7 Proof of the Infinitary Splitting Lemma
The following two lemmas can be easily proved by induction on the rank of .
Lemma 1
If does not satisfy then the reduct is equivalent to .
Lemma 2
If the set is disjoint from P and satisfies , then satisfies .
In particular, if satisfies then satisfies . (This is the direction lefttoright of Proposition 1 from Truszczynski (2012).)
Lemma 3
For any disjoint sets of atoms , interpretation , and formula ,

If is disjoint from Pnn and satisfies then satisfies .

If is disjoint from Nnn and satisfies then satisfies .
Both parts of the lemma are proved simultaneously by induction on the rank of . Here, we show only the most interesting case when is of the form . (i) If does not satisfy the reduct is equivalent to so that the proposition is trivially true. Assume that satisfies and that is disjoint from Pnn. Then either is or is disjoint from both Nnn and Pnn. If is then the set P is empty, so that is disjoint from it. Then by Lemma 2, if satisfies then satisfies . If, on the other hand, is disjoint from both Nnn and Pnn, then by part (i) of the induction hypothesis we may conclude that
(8) 
and by part (ii) of the induction hypothesis we may conclude that
(9) 
Assume that satisfies . Then by (9), satisfies . Then, since satisfies , that interpretation must satisfy . Then by (8) we can conclude that satisfies . It follows that that satisfies . (ii) Similar to Part (i).
Lemma 4
Let be disjoint sets of atoms and let be an infinitary formula such that there are no edges from to in DG. If satisfies then so does .
The proof is by induction on the rank of . Again we show only the most interesting case when is of the form . Assume that satisfies . We need to show that also satisfies . If is disjoint from P, then by Lemma 2, satisfies , and therefore satisfies . If, on the other hand, is not disjoint from P then must be disjoint from Pnn, because there are no edges from to in DG. Then by Lemma 3(i), satisfies . Since we assumed that satisfies , it follows that satisfies . Since every edge in DG occurs in DG there is no edge from to in DG. Then by the induction hypothesis, satisfies and therefore satisfies .
Lemma 5
For any nonempty graph and any infinitely separable partition on , there exists a nonempty subset of the vertices in such that

is either a subset of or a subset of , and

there are no edges from to vertices not in .
Since is infinitely separable on , there is some vertex such that the set of vertices reachable from is either a subset of or a subset of . (If no such existed then would be reachable from every vertex in and vice versa, and we could construct an infinite walk visiting both elements of the partition infinitely many times.) It is easy to see that the set of all vertices reachable from satisfies both (i) and (ii). {proof}[Proof of the Infinitary Splitting Lemma] Let be an infinitary formula such that the partition is infinitely separable on DG. We need to show that is an stable model of iff it is an stable model and an stable model of . The direction lefttoright is obvious. To establish the direction righttoleft, assume that is both an stable model and an stable model of . By Proposition 1 it is sufficient to show that is a minimal model of
(10) 
Clearly, satisfies this formula. It remains to show that is minimal. Assume there is some nonempty subset of such that satisfies (10). Then satisfies the second conjunctive term of (10), so . Consequently, . Consider the sets and . Since and are infinitely separable on DG, the sets and must be infinitely separable on DG. Then by Lemma 5, there is some nonempty set that is either a subset of or a subset of and such that there are no edges from to . We will show that satisfies
(11) 
which contradicts the assumption that is an stable model of . Since satisfies the first conjunctive term of (11), by Lemma 4 so does . Assume, for instance, that is a subset of . Then is a subset of , so that is a subset of . We may conclude that satisfies the second conjunction term of (11) as well.
8 Infinitary Splitting Theorem
The infinitary splitting lemma can be used to prove the following theorem, which is similar to the splitting theorem from Ferraris et al. (2009).
Infinitary Splitting Theorem
Let be infinitary formulas, and
be disjoint
sets of atoms such that
the partition is infinitely separable on
DG. If
is disjoint from P, and
is disjoint from P,
then for any interpretation , is an stable model of
iff it is both an stable model of and an stable
model of .
Example (continued)
Consider the conjunction of (3) with the formula
where is as before some nonempty set of atoms of the form
. We saw previously that
and all nonempty sets of atoms of the form are
stable models of (3).
It is easy to check that stable models of
are and . In accordance with the
splitting theorem, is the only stable model of this formula.
The following lemma, analogous to Theorem 3 from Ferraris et al. (2011), is used to prove the infinitary splitting theorem.
Lemma 6
For any infinitary formulas , if is disjoint from P then is an stable model of iff it is an stable model of and satisfies .
: Assume is an stable model of and satisfies . Since satisfies it satisfies . Since is an stable model of , it is a minimal w.r.t. among the models of , and consequently among the models of .
: Assume is an stable model of . Then is a minimal model of w.r.t. . So satisfies and therefore satisfies . It remains to show that there is no proper subset of such that and satisfies . Assume that there is some such . Then must not satisfy . (If it did, then would not be minimal with respect to among the models of .) Let denote . Since is disjoint from P, so is . So by Lemma 2, must satisfy . Contradiction.
[Proof of the Infinitary Splitting Theorem] Let be infinitary formulas and let be disjoint sets of atoms such that the partition is infinitely separable on DG and the other conditions of the infinitary splitting theorem hold. By the infinitary splitting lemma, is an stable model of iff it is both an stable model and an stable model of . Since is disjoint from P, by Lemma 6, is an stable model of iff it is an stable model of and it satisfies . Similarly, is an stable model of iff it is an stable model of and it satisfies . It remains to observe that if is an stable model of then it satisfies , and similarly if is an stable model of .
9 Application: Infinitary Definitions
About a formula and a set of atoms we will say that is a definition for if it is a conjunction of a set of formulas of the form , where is an atom in , is a subset of (possibly empty), and no atoms from occur in .^{5}^{5}5The relation occurs in is defined recursively in a straightforward way.
A simple special case is “explicit definitions”: conjunctions of formulas such that atoms from don’t occur in any . For example, (3) is an explicit definition of . The conjunction of the formulas
for all from some set of indices, which represents the usual recursive definition of transitive closure, is a definition in our sense as well. On the other hand, the formula is not a definition.
The following theorem shows that all definitions are “conservative”.
Theorem on Infinitary Definitions
For any infinitary formula , any set of
atoms that do not occur in , and any definition for , the map
is a 11 correspondence between the stable models
of and the stable models of .
This theorem generalizes the lemma on explicit definitions due to Ferraris (2005) in two ways: it applies to infinitary formulas, and it allows definitions to be recursive.
Lemma 7
If all atoms that occur in belong to then, for any interpretation , is an stable model of iff is a stable model of .
Lemma 8
Let be a definition for a set of atoms, and let be a model of . For any subset of such that , satisfies iff satisfies .
Proof.
We can show that satisfies a conjunctive term
of iff satisfies its reduct
as follows:
iff  
iff  
iff  
iff  
iff  
iff  
iff 
Lemma 9
Let be a definition for a set of atoms. For any set of atoms disjoint from there exists a unique stable model of such that .
Let be the intersection of all models of such that . We will show first that satisfies . Assume otherwise, and take a conjunctive term of that is not satisfied by . Then satisfies , , and . By the choice of , it follows that there is a model of such that and . On the other hand, since satisfies and does not differ from on atoms occurring in , satisfies . Since , satisfies . Hence does not satisfy one of the conjunctive terms of , which is a contradiction. Thus is a model of , and consequently a model of . To prove that it is stable, consider any model of such that . By Lemma 8, is also a model . By the choice of , it follows that . Consequently .
It remains to show that is unique. Let be a stable model of such that . It is easy to see that . Furthermore, satisfies and satisfies , so by Lemma 8, satisfies . Since , it follows that .
[Proof of Theorem on Infinitary Definitions] Let denote the set of all atoms occurring in . Since atoms from do not occur in and P, there are no edges from to in DG Consequently the partition is infinitely separable on this graph. By the splitting theorem for infinitary formulas, an interpretation is a stable model of iff it is a stable model of and a stable model of . Consider a stable model of . We have seen that is a stable model of . By Lemma 7, it follows that is a stable model of . Consider now a stable model of , and let be the set of all interpretations such that . We will show that contains exactly one stable model of , or equivalently, that there is exactly one interpretation that is a stable model of and a stable model of in . By Lemma 7, any interpretation in is a stable model of . By Lemma 9, contains exactly one stable model of .
10 Conclusion
In this note, we defined and studied stable models for infinitary propositional formulas with extensional atoms. The use of extensional atoms facilitates a more modular view of logic programs, as evidenced by the Theorem on Infinitary Definitions. The proof of this theorem relies on the Splitting Theorem, and the proof of that theorem makes critical use of the distinction between intensional and extensional atoms.
Acknowledgements
Many thanks to Yuliya Lierler, Dhananjay Raju, and the anonymous referees for useful comments. Both authors were partially supported by the National Science Foundation under Grant IIS1422455.
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