Semantics of most knowledge representation languages are defined as collections of interpretations or possible-world structures. The sets of interpretations and possible-world structures, with some natural orderings, form complete lattices. Logic programs, and default and autoepistemic theories determine operators on these lattices. In many cases, semantics of programs and theories are given as fixpoints of these operators. Consequently, an abstract framework of lattices, operators on lattices and their fixpoints has emerged as a powerful tool in investigations of semantics of these logics. Studying semantics of nonmonotonic reasoning systems within an algebraic framework allows us to eliminate inessential details specific to a particular logic, simplify arguments and find common principles underlying different nonmonotonic formalisms.
The roots of this algebraic approach can be traced back to studies of semantics of logic programs [vEK76, AvE82, Fit85, Prz90] and of applications of lattices and bilattices in knowledge representation [Gin88]. Exploiting the concept of a bilattice and relying on some general properties of operators on lattices and bilattices, Fitting proposed an elegant algebraic treatment of all major 2-, 3- and 4-valued semantics of logic programs [Fit01], that is, the supported-model semantics [Cla78], stable-model semantics [GL88], Kripke-Kleene semantics [Fit85, Kun87] and well-founded semantics [VRS91].
In [DMT00a], we extended Fitting’s work to a more abstract setting of the study of fixpoints of lattice operators. Central to our approach is the concept of an approximation of a lattice operator . An approximation is an operator defined on a certain bilattice (the product of the lattice by itself, with two appropriately defined lattice orderings). Using purely algebraic techniques, for an approximation operator for we introduced the notion of the stable operator and the concepts of the Kripke-Kleene, well-founded and stable fixpoints, and showed how they provide information about fixpoints of the operator . In [DMT00a] we noted that our approach generalizes the results described in [Fit01]. We observed that the 4-valued immediate consequence operator is an approximation operator for the 2-valued immediate consequence operator and showed that all the semantics considered by Fitting can be derived from by means of the general algebraic constructions that apply to arbitrary approximation operators.
In [DMT00b], we applied our algebraic approach to default and autoepistemic logics. Autoepistemic logic was defined by Moore [Moo84] to formalize the knowledge of a rational agent with full introspection capabilities. In Moore’s approach, an autoepistemic theory defines a characteristic operator on the lattice of all possible-world structures. Fixpoints of (or, to be precise, their theories) are known as expansions. In [DMT00b], we proposed for an approximation operator, , defined on a bilattice of belief pairs (pairs of possible-world structures). Complete fixpoints of correspond to expansions of (fixpoints of ), the least fixpoint of provides a constructive approximation to all expansions (by analogy with logic programming, we called it the Kripke-Kleene fixpoint). Using general techniques introduced in [DMT00a] we derived from its stable counterpart, the operator . Complete fixpoints of yield a new semantics of extensions for autoepistemic logic. Finally, the least fixpoint of the stable operator results in yet another new semantics, the well-founded semantics for autoepistemic logic (again, called so due to analogies to the well-founded semantics in logic programming), which approximates all extensions.
The same picture emerged in the case of default logic [DMT00b]. For a default theory we defined an operator and characterized all major semantics for default logic in terms of fixpoints of . In particular, the standard semantics of extensions [Rei80] is determined by complete fixpoints of the stable operator derived from . Our results on autoepistemic and default logics obtained in [DMT00b] allowed us to clarify the issue of their mutual relationship and provided insights into fundamental constructive principles underlying these two modes of nonmonotonic reasoning.
These result prove that the algebraic framework developed in [DMT00a] is an effective tool in studies of semantics of knowledge representation formalisms. It allowed us to establish a comprehensive semantic treatment for nonmonotonic logics and demonstrated that major nonmonotonic systems are closely related. However, the approach, as it was developed, is not entirely satisfactory. It provides no criteria that would allow us to prefer one approximation over another when attempting to define the concept of a stable fixpoint or when approximating fixpoints by means of the Kripke-Kleene or well-founded fixpoints. It does not give us any general indications how to obtain approximations and which approximation to pick. Thus, our theory leaves out a key link in the process of defining and approximating fixpoints of operators on lattices.
In particular, when defining semantics of nonmonotonic formalisms, we select an approximation operator, rather then derive it in a principled way. The approximations used, the bilattice operators , and , are not algebraically determined by their corresponding lattice operators , and , respectively. Consequently, some programs or theories with the same basic operators have different Kripke-Kleene, well-founded or stable fixpoints associated with them.
We address this problem here. We extend our theory of approximations and introduce the notion of the precision of an approximation. We show that each lattice operator has a unique most precise approximation which we call the ultimate approximation of . Since the ultimate approximation is determined by , it is well suited for investigations of fixpoints of . As a result we obtain concepts of ultimate stable fixpoints, the ultimate Kripke-Kleene fixpoint and the ultimate well-founded fixpoint that depend on only and not on a (possibly arbitrarily) selected approximation to .
We apply our theory to logic programming, default logic and autoepistemic logic (only the first system is discussed here, due to space limitations). We compare ultimate semantics with the corresponding “standard” semantics of logic programs. In particular, we show that the ultimate Kripke-Kleene and the ultimate well-founded semantics are more precise then the standard Kripke-Kleene and well-founded semantics. This better accuracy comes, however, at a cost. We show that ultimate semantics are in general computationally more complex. On the other hand, we show that for wide classes of theories, including theories likely to occur in practice, the complexity remains the same. Thus, our new semantics may prove useful in computing stable models and default extensions.
The ultimate semantics have also properties that are attractive from the logic perspective. In particular, two programs or theories determining the same basic 2-valued operator have the same ultimate semantics. This property, as we noted, is not true in the standard case.
In summary, our contributions are as follows. We extend the algebraic theory of approximations by providing a principled way of deriving an approximation to a lattice operator. In this way, we obtain concepts of Kripke-Kleene fixpoint, well-founded fixpoint and stable fixpoints that are determined by the operator and not by the choice of an approximation. In specific contexts of most commonly used nonmonotonic systems we obtain new semantics with desirable logical properties and possible computational applications.
Let be a poset and let be an operator on . A poset is chain-complete if it contains the least element and if every chain of elements of has a least upper bound () in . An element of is a pre-fixpoint of if ; is a fixpoint of if .
Let be a monotone operator on a chain-complete poset . Let us define a sequence of elements of by transfinite induction as follows: (1) ; (2) ; (3) , for a limit ordinal . One can show that this sequence is well defined, that is has in its least upper bound and that this least upper bound is the least fixpoint of (, in symbols). One can also show that the least fixpoint of a monotone operator on a chain-complete poset is the least pre-fixpoint of . That is, we have . Monotone operators on chain-complete posets and their fixpoints and pre-fixpoints are discussed in [Mar76].
A lattice is a poset such that and every pair of elements has a unique greatest lower bound and least upper bound. A lattice is complete if its every subset has a greatest lower bound and a least upper bound. In particular, a complete lattice has a least and a greatest element denoted by and , respectively.
For any two elements , we define . If is a complete lattice and , then is a complete lattice, too.
Let be a complete lattice. By the product bilattice [Gin88] of we mean the set with the following two orderings and :
Both orderings are complete lattice orderings for . However, in this paper we are mostly concerned with the ordering .
An element is consistent if . We can think of a consistent element as an approximation to every such that . With this interpretation in mind, the ordering , when restricted to consistent elements, can be viewed as a precision ordering. Consistent pairs that are “higher” in the ordering provide tighter approximations. Maximal consistent elements with respect to are pairs of the form . We call approximations of the form — exact.
We denote the set of all consistent pairs in by . The set is not a lattice. It is, however, chaincomplete. Indeed, the element is the least element in and the following result shows that every chain in has (in ) the least upper bound.
Let be a complete lattice. If is a chain of elements in then and .
It follows that every -monotone operator on has a least fixpoint.
3 Partial Approximations
For an operator , we denote by and its projections to the first and second coordinates, respectively. Thus, for every , we have . An operator is a partial approximation operator if it is -monotone and if for every , . We denote the set of all partial approximation operators on by . Let . Since is -monotone and is chain-complete, has a least fixpoint, called the Kripke-Kleene fixpoint of (, in symbols). Directly from the definition, it follows that approximates all fixpoints of .
If and is an operator on such that then we say that is a partial approximation of . We denote the set of all partial approximations of by . If is a partial approximation of then is a fixpoint of if and only if is a fixpoint of . Thus, for every fixpoint of , we have or, equivalently, , where and are the two components of the pair .
Operators from describe ways to revise consistent approximations. Of particular interest are those situations when the revision of an approximation leads to another one that is at least as accurate. Let be an operator on . We call an approximation -reliable if .
Let be a complete lattice and . If is -reliable then, for every , and, for every , .
Proof: Let . Then . By the -monotonicity of ,
The last inequality follows from the fact that is -reliable. The second part of the assertion can be proved in a similar manner.
This proposition implies that for every -reliable pair , the restrictions of to and to are in fact operators on and , respectively. Moreover, they are -monotone operators on the posets and . Since and are complete lattices, the operators and have least fixpoints in the lattices and , respectively. We define:
We call the mapping , defined on the set of -reliable elements of , the stable revision operator for . When is clear from the context, we will drop the reference to from the notation.
Directly from the definition of the stable revision operator it follows that for every -reliable pair, and .
The stable revision operator for is crucial. It allows us to distinguish an important subclass of the class of all fixpoints of . Let be a complete lattice and let . We say that is a stable fixpoint of if is -reliable and is a fixpoint of the stable revision operator (that is, and ). By the -reliability of , the second requirement is well defined.
Stable fixpoints of an operator are, in particular, its fixpoints.
Let be a complete lattice and let . If is a stable fixpoint of then is a fixpoint of .
Proof: Since is stable, . In particular, . Similarly, .
Let be an operator on a complete lattice and let . We say that is an -stable fixpoint of if is a stable fixpoint of . The notation is justified. Indeed, it follows from Proposition 3.2 and our earlier remarks that every stable fixpoint of is, in particular, a fixpoint of .
The notion of -reliability is not strong enough to guarantee desirable properties of the stable revision operator. In particular, if is -reliable, it is not true in general that is consistent nor that . There is, however, a class of -reliable pairs for which both properties hold. An -reliable approximation is -prudent if . We note that every stable fixpoint of is -prudent. We will now prove several basic properties of -prudent approximations.
Let be a complete lattice, and be -prudent. Then, is consistent, -reliable and -prudent and .
Proof: By the definition of and we have that and . Moreover, since is -prudent, it follows that .
Next, since is -reliable, it follows that and . Thus, is a pre-fixpoint of . Consequently, (as is the least fixpoint of ). Hence, .
By the -monotonicity of we obtain:
It follows that is a pre-fixpoint of the operator . Thus, and so, is consistent.
Let us now observe that . Similarly, . Thus, the pair is reliable.
Lastly, we note that for every , (the last inequality follows by the -reliability of ). Hence, and, consequently, is -prudent.
Let us observe that an -reliable pair is revised by an operator into a more accurate approximation . An -prudent pair can be revised “even more”. Namely, it is easy to see that and . Thus, . In other words, is indeed at least as precise revision of as is.
The stable revision operator satisfies a certain monotonicity property.
Let be a complete lattice and let . If is -reliable, is -prudent and if , then .
Proof: Clearly, we have . By the -monotonicity of , it follows that . Thus, is a pre-fixpoint of . Since is the least fixpoint of , it follows that .
It remains to prove that . Let . By Proposition 3.3, . Since , it follows that . Further, by the -reliability of and , we have and . Thus, and . Consequently,
It follows that . In particular, is a pre-fixpoint of . Since is the least fixpoint of , . Hence, .
We now have (the first inequality follows from the assumption , the second one follows by Proposition 3.3 from the assumption that is -prudent). Thus, and the -monotonicity of implies
Hence, is a pre-fixpoint of . Since is the least fixpoint of , it follows that .
Since stable fixpoints are prudent, we obtain the following corollary.
Let be a complete lattice, and let be a stable fixpoint of . If is -reliable and then .
The next result states that the limit of a chain of -prudent pairs is -prudent.
Let be a complete lattice, and let be a chain of -prudent pairs from . Then, is -prudent.
Proof: Let us set and . By Proposition 2.1, is consistent and . Let us now observe that, by -reliability of and -monotonicity of , we have Thus, It follows that is -reliable.
The -reliability of implies, in particular, that for every , . Thus, by -monotonicity of , for every
Hence, pre-fixpoints of are prefixpoints of and, consequently,
Since is -prudent, we have that . Thus, for arbitrary , and, consequently, . It follows that is -prudent.
We will now prove that the set of all stable fixpoints of an operator has a least element (in particular, it is not empty). To this end, we define a sequence of elements of by transfinite induction:
If , we define and
If is a limit ordinal, we define .
The sequence is well defined, -monotone and its limit is the least stable fixpoint of a partial approximation operator .
Proof: It is obvious that is -prudent. Thus, by the transfinite induction it follows that each element in the sequence is well defined and -prudent (Propositions 3.3 and 3.6 settle the cases of successor ordinals and limit ordinals, respectively). In the same way, one can establish the -monotonicity of the sequence.
Let . By Proposition 3.6, is -prudent. Thus, is -reliable. Moreover, we have and . Thus, is a stable fixpoint of . Further, it is easy to see by transfinite induction and Corollary 3.5 that approximates all stable fixpoints of . Thus, it is the least stable fixpoint of .
We call this least stable fixpoint the well-founded fixpoint of and denote it by . The well-founded fixpoint approximates all stable fixpoints of . In particular, it approximates all -stable fixpoints of the operator . That is, for every -stable fixpoint of , or, equivalently, , where and are the two components of the pair . Moreover, the well-founded fixpoint is more precise than the Kripke-Kleene fixpoint: for , .
In [DMT00b, DMT00a], we showed that when applied to appropriately chosen approximation operators in logic programming, default logic and autoepistemic logic, these algebraic concepts of fixpoints, stable fixpoints, the Kripke-Kleene fixpoint and the well-founded fixpoint provide all major semantics for these nonmonotonic systems and allow us to understand their interrelations.
We need to emphasize that the concept of a partial approximation introduced here is different from the concept of approximation introduced in [DMT00a]. The latter notion is defined as an operator of the whole bilattice . That choice was motivated by our search for generality and potential applications of inconsistent fixpoints in situations when we admit a possibility of some statements being overdefined. While different, both approaches are very closely related111We will include a detailed discussion of the relationship between the two approaches in the full version of the paper..
4 Ultimate Approximations
Partial approximations in can be ordered. Let . We say that is less precise than (, in symbols) if for each pair , . It is easy to see that if then there is an operator on the lattice such that .
Let be a complete lattice and . If and is -prudent then is -prudent and .
Proof: Clearly, . Thus, is -reliable.
For each pre-fixpoint of , . Consequently, is a prefixpoint of . It follows that . Since , . Thus is -prudent.
Likewise, we can prove that any pre-fixpoint of is a prefixpoint of , and consequently, . Since also , it follows that .
More precise approximation have more precise Kripke-Kleene and well-founded fixpoints.
Let be an operator on a complete lattice . Let . If then and .
Proof: Let us denote by the sequence of elements of obtained by iterating the operator over . The sequence is defined in the same way. Since , it follows by an easy induction that for every ordinal , . Since is the limit of the sequence and is the limit of the sequence , it follows that .
To prove the second part of the assertion, we will now assume that the sequences and denote the sequences used in the definition of the well-founded fixpoints of and , respectively. To prove the assertion we will now show that for every ordinal , .
Clearly, . Let us assume that and that . Since is -prudent, Lemma 4.1 entails that it is -prudent and
By Proposition 3.4,
The case of the limit ordinal is straightforward.
Since and are the limits of the sequences and , respectively, the second part of the assertion follows.
The next result shows that as the precision of an approximation grows, all exact fixpoints and exact stable fixpoints are preserved.
Let be an operator on a complete lattice . Let . If then every exact fixpoint of is an exact fixpoint of , and every exact stable fixpoint of (that is, an -stable fixpoint of ) is also an exact stable fixpoint of (that is, a -stable fixpoint of ).
Proof: Since for every , , the first part of the assertion follows. Let us now assume that is an exact stable fixpoint of . In particular, it follows that is a fixpoint of and is -prudent. By Lemma 4.1, is -prudent and . The latter pair is consistent (Proposition 3.3). Consequently, is and hence is an exact stable fixpoint of .
Non-exact fixpoints are not preserved, in general. Let us consider two partial approximations and such that . Let us also assume that (that is, has a strictly less precise well-founded fixpoint than ). Then, clearly, is no longer a stable fixpoint of . Thus, fixpoints of may disappear when we move on to a more precise approximation .
More precise approximations of a non-monotone operator yield more precise well-founded fixpoints and additional exact stable fixpoints. The natural question is whether there exists an ultimate approximation of , that is, a partial approximation most precise with respect to the ordering . Such approximation would have a most precise Kripke-Kleene and well-founded fixpoint and a largest set of exact stable fixpoints. We will show that the answer to this key question is positive. Such ultimate approximation, being a distinguished object in the collection of all approximations can be viewed as determined by . Consequently, fixpoints of the ultimate approximation of (including stable, Kripke-Kleene and well-founded fixpoints) can be regarded as determined by and can be associated with it.
We start by providing a non-constructive argument for the existence of ultimate approximations. Let us note that the set is not empty. Indeed, let us define , if , and , otherwise. It is easy to see that and that it is the least precise element in . Next, we observe that with the ordering is a complete lattice, as the set is closed under the operations of taking greatest lower bounds and least upper bounds. It follows that has a greatest element (most precise approximation). We call this partial approximation the ultimate approximation of and denote it by .
We call the Kripke-Kleene and the well-founded fixpoints of , the ultimate Kripke-Kleene and the ultimate well-founded fixpoint of . We denote them by and , respectively. We call a stable fixpoint of an ultimate partial stable fixpoint of . We refer to an exact stable fixpoint of as an ultimate stable fixpoint of . Exact fixpoints of all partial approximations are the same and correspond to fixpoints of . Thus, there is no need to introduce the concept of an ultimate exact fixpoint of . We have the following corollary to Theorems 4.2 and 4.3.
Let be an operator on a complete lattice . For every , , and every -stable fixpoint of is an ultimate stable fixpoint of .
We will now provide a constructive characterization of the notion. To state the result, for every such that , we define .
Let be an operator on a complete lattice . Then, for every , .
Proof: We define an operator by setting
First, let us notice that since , the operator maps into . Moreover, it is easy to see that is -monotone. Lastly, since ,
and, consequently, . Thus, it follows that is a partial approximation of . Since is the most precise approximation, we have .
On the other hand, for every . Therefore for all and thus . Similarly, . Since are arbitrary, , as desired.
With this result we obtain an explicit characterization of ultimate stable fixpoints of an operator .
Let be a complete lattice. An element is an ultimate stable fixpoint of an operator if and only if is the least fixpoint of the operator regarded as an operator on .
We conclude this section by describing ultimate approximations for monotone and antimonotone operators on .
If is a monotone operator on a complete lattice then for every , . If is antimonotone then for every , .
Proof: By Theorem 4.5,
Now, it is easy to see that if is monotone, then and . If is antimonotone, then and . The proposition follows.
Let be an operator on a complete lattice . If is monotone, then the least fixpoint of is the ultimate well-founded fixpoint of and the unique ultimate stable fixpoint of . If is antimonotone, then and every fixpoint of is an ultimate stable fixpoint of .
5 Ultimate Semantics for Logic Programming
The basic operator in logic programming is the one-step provability operator introduced in [vEK76]. It is defined on the lattice of all interpretations. This lattice consists of subsets of the set of all atoms appearing in and is ordered by inclusion (we identify truth assignments with subsets of atoms that are assigned the value t).
Let be a logic program. We denote by the ultimate approximation operator for the operator . By specializing Theorem 4.5 to the operator we obtain that for every two interpretations ,
Replacing the ultimate approximation operator in the definitions of ultimate Kripke-Kleene, well-founded and stable fixpoints with results in the corresponding notions of ultimate Kripke-Kleene, well-founded and stable models (semantics) of a program .
We are now in a position to discuss commonsense reasoning intuitions underlying abstract algebraic concepts of ultimate approximation and its fixpoints. Let us consider two interpretations and such that . We interpret as a current lower bound and as a current upper bound on the set of atoms that are true (under ). Thus, specifies atoms that are definitely true, while specifies atoms that are possibly true. Arguably, if an atom is derived by applying the operator to every interpretation , it can safely be assumed to be true (in the context of the knowledge represented by and ). Thus, the set can be viewed as a revision of .
Similarly, since every interpretation must be regarded as possible according to the pair
of conservative and liberal estimates, an atom might possibly be true if it can be derived by the operatorfrom at least one interpretation in . Thus, the set , consisting of all such atoms, can be regarded as a revision of . Clearly, and, consequently, can be viewed as a way to revise our knowledge about the logical values of atoms as determined by a program from to .
By iterating starting at , we obtain the ultimate Kripke-Kleene model of as an approximation that cannot be further improved by applying . The ultimate Kripke Kleene model of approximates all fixpoints of and, in particular, all supported models of . Often, however, the Kripke-Kleene model is too weak as we are commonly interested in those (partial) models of that satisfy some minimality or groundedness conditions. These requirements are satisfied by ultimate stable models and, in particular, by the ultimate well-founded model of .
When constructing the ultimate well-founded model, we start by assuming no knowledge about the status of atoms: no atom is known true and all atoms are assumed possible. Our goal is to improve on these bounds.
To improve on the lower bound, we proceed as follows. Our current knowledge does not preclude any interpretation and all of them (the whole segment ) need to be taken into account. If some atom can be derived by applying the operator to each element of then, arguably, could be accepted as definitely true. The set of all these atoms is exactly . So, this set, say , can be taken as a safe new lower bound, giving a smaller interval of possible interpretations. We now repeat the same process and obtain a new lower bound, say , consisting of those atoms that can be derived from every interpretation in . It is given by . Clearly, improves on