1 Introduction
Craig’s interpolation theorem [Craig1957] is an important theorem known for propositional logic and firstorder logic (FOL). It says that if are two logical formulae and , then there is a formula such that and (“” is the classical logical deduction relation; is the language of (the set of formulae built with the nonlogical symbols of , )). Such interpolation theorems allow us to break inference into pieces associated with sublanguages of the language of that theory [McIlraith and Amir2001], for those formal systems in which they hold. In AI, these properties have been used to speed up inference for constraint satisfaction systems (CSPs), propositional logic and FOL (e.g., [Dechter and Pearl1988, Darwiche1998, McIlraith and Amir2001, Dechter and Rish1994, Darwiche1997, Amir and McIlraith2000, Dechter1999] and to build structured representations [Darwiche1998, Amir2000, Darwiche1997]
In this paper we present interpolation theorems for three nonmonotonic systems: circumscription [McCarthy1980], default logic [Reiter1980] and logic programs with the Answer Set semantics [Gelfond and Lifschitz1991, Gelfond and Lifschitz1988]. In the nonmonotonic setup there are several interpolation theorems for each system, with different conditions for applicability and different form of interpolation. This stands in contrast to classical logic, where Craig’s interpolation theorem always holds. Our theorems allow us to use methods for the decomposition of reasoning (ala [Amir and McIlraith2000, McIlraith and Amir2001]) under some circumstances for these systems, possibly increasing their applicability and tractability for structured theories. We list the main theorems that we show in this paper below, omitting some of their conditions for simplicity.
For circumscription we show that, under some conditions, iff there is some set of formulae such that and . For example, to answer , we can compute this formula from without applying circumscription, and then solve (where may be significantly smaller than ).
For default logic, letting mean that every extension of entails (cautious entailment), we show that, under some conditions, if , then there is a formula such that and . For logic programs we show that if are two logic programs and such that , then there is such that and (here is the brave entailment for logic programs).
This paper focuses on the form of the interpolation theorems that hold for those nonmonotonic logics. We do not address the possible application of these results to the problem of automated reasoning with those logics. Nonetheless, we mention that direct application of those results is possible along the lines already explored for propositional logic and FOL in
[Amir and McIlraith2000, McIlraith and Amir2001].No interpolation theorems were shown for nonmonotonic reasoning systems before this paper. Nonetheless, some of our theorems for default logic and logic programs are close to the splitting theorems of [Lifschitz and Turner1994, Turner1996], which have already been used to decompose reasoning for those logics. The main difference between our theorems and those splitting theorems is that the latter change some of the defaults/rules involved to provide the corresponding entailment. Also, they do not talk about an interpolant , but rather discuss combining extensions.
Since its debut, the nonmonotonic reasoning line of work has expanded and several textbooks now exist that give a fair view of nonmonotonic reasoning and its uses (e.g., [Gabbay et al.1993]). The reader is referred to those books for background and further details.
2 Logical Preliminaries
In this paper, we use the notion of logical theory for every set of axioms in FOL or propositional logic, regardless of whether the set of axioms is deductively closed or not. We use to denote the signature of , i.e., the set of nonlogical symbols. denotes the language of , i.e., the set of formulae built with . is the set of logical consequences of (i.e., those formulae that are valid consequences of in FOL). For a firstorder structure, , in , we write for the universe of elements of . For every symbol, , in , we write for the interpretation of in .
Finally, we note Craig’s Interpolation Theorem.
Theorem 2.1 ([Craig1957])
Let be sentences such that . Then there is a formula involving only nonlogical symbols common to both and , such that and .
3 Circumscription
3.1 McCarthy’s Circumscription: Overview
McCarthy’s circumscription [McCarthy1980, McCarthy1986] is a nonmonotonic reasoning system in which inference from a set of axioms, , is performed by minimizing the extent of some predicate symbols , while allowing some other nonlogical symbols, to vary.
Formally, McCarthy’s circumscription formula
(1) 
says that in the theory
, with parameter relations and function vectors (sequence of symbols)
, is a minimal element such that is still consistent, when we are allowed to vary in order to allow to become smaller.Take for example the following simple theory:
Then, the circumscription of in , varying nothing, is
Roughly, this means that is a minimal predicate satisfying . Computing circumscription is discussed in length in [Lifschitz1993] and others, and we do not expand on it here. Using known techniques we can conclude
This means that there are no other blocks in the world other than those mentioned in the original theory .
We give the preferential semantics for circumscription that was given by [Lifschitz1985, McCarthy1986, Etherington1986] in the following definition.
Definition 3.1 ([Lifschitz1985])
For any two models and of a theory we write if the models differ only in how they interpret predicates from and and if the extension of every predicate from in is a subset of its extension in . We write if for at least one predicate in the extension in is a strict subset of its extension in .
We say that a model of is minimal if there is no model such that .
Theorem 3.2 ([Lifschitz1985]: Circumscript. Semantics)
Let be a finite set of sentences. A structure is a model of iff is a minimal model of .
This theorem allows us to extend the definition of circumscription to set of infinite number of sentences. In those cases, is defined as the set of sentences that hold in all the minimal models of . Theorem 3.2 implies that this extended definition is equivalent to the syntactic characterization of the original definition (equation (1)) if is a finite set of sentences. In the rest of this paper, we refer to this extended definition of circumscription, if is an infinite set of FOL sentences (we will note those cases when we encounter them).
Circumscription satisfied Left Logical Equivalence (LLE): implies that . It also satisfies Right Weakening (RW): and implies that ).
3.2 Model Theory
Definition 3.3
Let be structures, for FOL signature and language . We say that is an elementary extension of (or is an elementary substructure of ), written , if and for every and vector of elements of , iff .
is an elementary embedding if is an injective (onetoone) homomorphism from to and for every and vector of elements from (i.e., ), iff .
For FOL signatures , and for an structure, we say that is the reduct of to , the structure with the same universe of elements as , and the same interpretation as for those symbols from that are in (there is no interpretation for symbols not in ). For A theory in a language of , let be the set of all consequences of in the language of .
The following theorem is a modeltheoretic property that is analogous to Craig’s interpolation theorem (Theorem 2.1).
Theorem 3.4 (See [Hodges1997] p.148)
Let be FOL signatures with and a theory in the language of . Let be an structure. Then, if and only if for some model of , ( is an elementary substructure of the reduct of to ).
3.3 Interpolation in Circumscription
In this section we present two interpolation theorems for circumscription. Those theorems hold for both FOL and propositional logic. Roughly speaking, the first (Theorem 3.8) says that if nonmonotonically entails (here this means ), then there is such that classically entails () and nonmonotonically entails (). In the FOL case this can be an infinite set of sentences, and we use the extended definition of Circumscription for infinite sets of axioms for this statement.
The second theorem (Theorem 3.11) is similar to the first, with two main differences. First, it requires that . Second, it guarantees that as above (and some other restrictions) exists iff nonmonotonically entails . This is in contrast to the first theorem that guarantees only that if part. The actual technical details are more fine than those rough statements, so the reader should refer to the actual theorem statements.
In addition to these two theorems, we present another theorem that addresses the case of reasoning from the union of theories (Theorem 3.10). Before we state and prove those theorems, we prove several useful lemmas.
Our first lemma says that if we are given two theories , and we know the set of sentences that follow from in the intersection of their languages, then every model of this set of sentences together with can be extended to a model of .
Lemma 3.5
Let be two theories, with signatures in , respectively. Let be a set of sentences logically equivalent to . For every structure, , that satisfies there is a structure, , that is a model of such that .
Proof Let be a structure that is a model of . Then . Noticing that is logically equivalent to (by definition of ), we get that . Consequently, because .
Now we use Theorem 3.4 with , , and . We know that . Thus, . To use Theorem 3.4 we need to show that . We use Craig’s interpolation theorem (Theorem 2.1) to show this is indeed the case.
First notice that is true because . We show that . Take . By definition, and . The deduction theorem for FOL implies that , for some finite subset . Craig’s interpolation theorem for FOL implies that there is such that and . Thus, . Consequently, . Using the deduction theorem again we get that , implying that .
Thus, we showed that . From and we get that .
Finally, the conditions of Theorem 3.4 for , , and hold. We conclude that there is a structure, , that is a model of such that .
Our second lemma says that every minimal model of that is also a model of is a minimal model of .
Lemma 3.6
Let be a theory and vectors of nonlogical symbols. If and , then .
Proof Let be a model of such that . If there is such that , then and . Contradiction. Thus, there is no such and .
The following theorem is central to the rest of our results in this section. It says that when we circumscribe in we can replace by its consequences in , for some purposes and under some assumptions.
Theorem 3.7
Let be two theories and two vectors of symbols from such that . Let a set of sentences logically equivalent to . Then, for all , if , then .
Proof We show that for every model of there is a model of whose reduct to is an elementary extension of the reduct of the first model to .
Let be a structure that is a model of . Then, . From Lemma 3.5 we know that there is a structure, , that is a model of such that .
Thus, is a minimal model of . To see this, assume otherwise. Then, there is a model for the signature such that and . Take such that the interpretation of all the symbols in is exactly the same as that of and such that the interpretation of all symbols in is exactly the same as that of . Then, because . Also, , for because and agree on the interpretation of symbols in (). Thus, , since agree on all the interpretation of all symbols in . This contradicts , so is a minimal model of .
Thus, , and . From Lemma 3.6 we get that .
Now, let such that . Then every model of satisfies . Let be a model of in the language . Then there is as above, i.e., and . Thus, . Since , . Thus every model of is a model of .
Theorem 3.8 (Interpolation for Circumscription 1)
Let be a theory, vectors of symbols, and a formula. If , then there is such that
Furthermore, this can be logically equivalent to the consequences of in .
Proof We use Theorem 3.7 to find this . For as in the statement of the theorem we define as follows. We choose such that and : Let for a set of tautologies such that . We choose such that it includes and has a rich enough vocabulary so that . Let , for a set of tautologies such that . Let , .
Theorem 3.7 guarantees that if then from that theorem satisfies for every . This implies that for every , . In particular, , and this satisfies our current theorem.
This theorem does not hold if we require instead of . For example, take , , where are propositional symbols. . However, every logical consequence of in is a tautology. Thus, if the theorem was correct with our changed requirement, would be equivalent to and .
Theorem 3.9
Let be two theories, two vectors of symbols from such that and . Let be a set of sentences logically equivalent to . Then, for all , if , then .
Proof We show that every model of is also a model of . Let be a structure that is a model of . Then , implying that also .
Assume that there is such that . From Lemma 3.5, there is such that and . Since agree on all the symbols of , we get that (because ). Finally, we get that , contradicting the assumption of being minimal satisfying . Thus, is a model of .
Now, let such that . Then every model of satisfies . Let be a model of in the language . Then, and . Thus, .
Theorem 3.10 (Interpolation Between Theories)
Let be two theories, vectors of symbols in such that and . Let be a set of sentences logically equivalent to . Then, for every ,
Theorem 3.11 (Interpolation for Circumscription 2)
Let be a theory, vectors of symbols such that . Let be a set of nonlogical symbols. Then, there is such that and for all ,
Furthermore, this can be logically equivalent to the consequences of in .
Proof Let be a set of tautologies such that . Also, let , for a set of tautologies such that . Let , . Theorem 3.10 guarantees that from that theorem satisfies for every .
The theorems we presented are for parallel circumscription, where we minimize all the minimized predicates in parallel without priorities. The case of prioritized circumscription is outside the scope of this paper.
4 Default Logic
In this section we present interpolation theorems for propositional default logic. We also assume that the signature of our propositional default theories is finite (this also implies that our theories are finite).
4.1 Reiter’s Default Logic: Overview
In Reiter’s default logic [Reiter1980] one has a set of facts (in either propositional or FOL) and a set of defaults (in a corresponding language). Defaults in are of the form with the intuition that if is proved, and are consistent (throughout the proof), then is proved. is called the prerequisite, ; are the justifications, and is the consequent, . We use similar notation for sets of defaults (e.g., ). Notice that the justifications are checked for consistency one at a time (and not conjoined).
Take, for example, the following default theory :
(2) 
Intuitively, this theory says that birds normally fly and that is a bird.
An extension of is a set of sentences that satisfies , follows the defaults in , and is minimal. More formally, is an extension if it is minimal (as a set) such that , where we define to be , a minimal set of sentences such that

; .

For all if and , then .
The following theorem provides an equivalent definition that was shown in [Marek and Truszczyński1993, Risch and Schwind1994, Baader and Hollunder1995]. A set of defaults, is grounded in a set of formulae iff for all , , where .
Theorem 4.1 (Extensions in Terms of Generating Defaults)
A set of formulae is an extension of a default theory iff for a minimal set of defaults such that

is grounded in and

for all :
Every minimal set of defaults as mentioned in this theorem is said to be a set of generating defaults.
Normal defaults are defaults of the form . These defaults are interesting because they are fairly intuitive in nature (if we proved then is proved unless previously proved inconsistent). We say that a default theory is normal, if all of its defaults are normal.
We define as cautious entailment sanctioned by the defaults in , i.e., follows from every extension of . We define as brave entailment sanctioned by the defaults in , i.e., follows from at least one extension of .
4.2 Interpolation in Default Logic
In this section we present several flavors of interpolation theorems, most of which are stated for cautious entailment.
Theorem 4.2 (Interpolation for Cautious DL 1)
Let be a propositional default theory and a propositional formula. If , then there are such that , and all the following hold:
Proof Let be the set of consequences of in . Let be the set of extensions of and the set of extensions of . We show that every extension has an extension such that . This will show that is as needed.
Take and define . We assume that because otherwise we can take a logically equivalent extension whose sentences are in . We show that satisfies the conditions for extensions of :

,

For all , if and , then .
The first condition holds by definition of . The second condition holds because every default that is consistent with is also consistent with and vice versa. We detail the second condition below.
For the first direction (every default that is consistent with is also consistent with ), let be such that . We show that .
By definition, . implies that because . Using the deduction theorem for propositional logic we get (taking here to be a finite set of sentences that is logically equivalent to in (there is such a finite set because we assume that is finite)). Using Craig’s interpolation theorem for propositional logic, there is such that and . However, this means that , by the way we chose . Thus . Since we get that . Since we get that .
The case is similar for : if then by the same argument as given above for . Finally, if then because .
The opposite direction (every default that is consistent with is also consistent with ) is similar to the first one.
Thus, satisfies those two conditions. However, it is possible that is not a minimal such set of formulae. If so, Theorem 4.1 implies that there is a strict subset of the generating defaults of that generate a different extension. However, we can apply this new set of defaults to generate an extension that is smaller than , contradicting the fact that is an extension of .
Now, if logically follows in all the extensions of then it must also follow from every extension of together with . Let , for the (finite) set of (logically nonequivalent) extensions of (we have a finite set of those because is finite). Then, . Take such that and , as guaranteed by Craig’s interpolation theorem (Theorem 2.1). These are those promised by the current theorem: , , , and .
Theorem 4.3 (Interpolation for Cautious DL 2)
Let be a propositional default theory and a propositional formula. If , then there are , and all the following hold:
The proof is similar to the one for Theorem 4.2.
Corollary 4.4
Let be a default theory and a formula. If , then there is a set of formulae, such that and .
Proof Follows immediately from Theorem 4.2 with there corresponding to our needed .
It is interesting to note that we do not get stronger interpolation theorems for prerequisitefree normal default theories. [Imielinski1987] provided a modular translation of normal default theories with no prerequisites into circumscription, but Theorem 3.8 does not lead to better results. In particular, the counter example that we presented after that theorem can be massaged to apply here too.
Theorem 4.5 (Interpolation Between Default Extensions)
Let be default theories such that . Let be a formula such that . If there is an extension of in which holds, then there is a formula , an extension of such that , and an extension of such that .
Proof Let be the set of generating defaults of that belong to . Notice that these defaults are grounded in because there is no information that may have come from applying the rest of the generating defaults in (we required that
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