New Semantics for Epistemic Specifications
Michael Gelfond
Texas Tech University
michael.gelfond@ttu.edu
Abstract. This note presents a new version of the language of epistemic
specifications. The semantics of the new language is arguably closer to the
intuitive meaning of epistemic operators. It eliminates some unintended
interpretations which exist under the old definition. The author hopes
that the new language will be better suited for the design of intelligent
agents capable of introspective reasoning with incomplete information.
1
Introduction
The language of epistemic specifications [1,4] is an extension of the language of
disjunctive logic programs [3] by modal operators K and M, where KF stands
for F is known to be true and MF stands for F may be believed to be true.
The stated purpose of this work was to allow for the correct representation of
incomplete information in the presence of multiple belief sets. To illustrate the
need for the extension the authors considered the following simple example.
Example 1. [Scholarship Eligibility]
Consider a collection of rules
1.
2.
3.
4.
eligible(X) ← highGP A(X)
eligible(X) ← minority(X), f airGP A(X)
¬eligible(X) ← ¬f airGP A(X), ¬highGP A(X)
interview(X) ← not eligible(X),
not ¬eligible(X)
used by a certain college for awarding scholarships to its students. The first three
rules are self explanatory (we assume that variable X ranges over a given set of
students) while the fourth rule can be viewed as a formalization of the statement:
(*) The students whose eligibility is not determined by the college rules should
be interviewed by the scholarship committee.
We assume that this program is to be used in conjunction with a database DB
consisting of literals specifying values of the predicates minority, highGP A, and
f airGP A. Consider, for instance, DB consisting of the following two facts about
one of the students:
5. f airGP A(ann)
6. ¬highGP A(ann)
J. Delgrande and W. Faber (Eds.): LPNMR 2011, LNAI 6645, pp. 260–265, 2011.
c Springer-Verlag Berlin Heidelberg 2011
New Semantics for Epistemic Specifications
261
(Notice that DB contains no information about the minority status of Ann.) It
is easy to see that rules (1)–(6) allow us to conclude neither eligible(ann) nor
¬eligible(ann), therefore eligibility of Ann for the scholarship is undetermined
and, by rule (4), she must be interviewed. Formally this argument is reflected
by the fact that program T1 consisting of rules (1)–(6) has exactly one answer
set:
{f airGP A(ann), ¬highGP A(ann), interview(ann)}.
The situation changes significantly if disjunctive information about students is
allowed to be represented in the database. Suppose, for instance, that we need to
augment rules (1)–(3) by the following information:
(**) Mike’s GPA is fair or high.
The corresponding program T2 consists of rules (1)–(3) augmented by the disjunction
7. f airGP A(mike) or highGP A(mike)
T2 has two answer sets:
A1 = {highGP A(mike), eligible(mike)}
and
A2 = {f airGP A(mike)},
and therefore the reasoner associated with T2 does not have enough information to establish Mike’s eligibility for the scholarship (i.e. his answer to query
? eligible(mike) will be unknown). If we now expand this theory by (*) we expect the new theory T3 to be able to answer yes to a query interview(mike). It
is easy to see however that if (*) is represented by (4) this goal is not achieved.
The resulting theory T3 consisting of (1)–(4) and (7) has two answer sets
A3 = {highGP A(mike), eligible(mike)}
A4 = {f airGP A(mike), interview(mike)}
and therefore the answer to query interview(mike) is unknown. The reason of
course is that (4) is too weak to represent (*). The informal argument we are
trying to capture goes something like this: theory T3 answers neither yes nor no to
the query eligible(mike). Therefore, the answer to this question is undetermined,
and, by (*), Mike should be interviewed. To formalize this argument our system
should have a more powerful introspective ability than the one captured by the
notion of answer sets from [1].
To remedy this problem [1,4] introduced the language of epistemic specifications.
Literals of this language were divided into
– objective – expressions of the form p(t̄), ¬p(t̄), and
– subjective – expressions of the form Kl, ¬Kl, Ml, ¬Ml where l is an objective
literal.
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M. Gelfond
Epistemic specifications were defined as collections of rules of the form:
l1 or . . . or lk ← gk+1 , . . . , gm , not lm+1 , . . . , not ln
(1)
where the l’s are objective literals and the g’s are subjective or objective literals1 .
The semantics of an epistemic specification T has been given via the notion of
a world view of T - a collection of simple theories about the world which can
be built by a rational reasoner on the instructions from T . For a program T not
containing operators K and M the world view of T coincided with the collection
of all the answer sets of T . The precise definition went as follows:
Let T be a ground epistemic specifications and S be a collection of sets of ground
objective literals in the language of T ; S entails K l (S |= K l) if for every W ∈ S,
l ∈ W . Otherwise S |= ¬K l). Similarly for M.
A disjunctive logic program T S was obtained from T by:
1. removing from T all rules containing subjective literals not entailed by S.
2. removing from rules in T all other occurrences of subjective literals.
T S was referred to as the reduct of T with respect to S. A set S was called a
world view of T if S were the collection of all answer sets of T S . Elements of S
were called belief sets of T . Epistemic specifications with variables were viewed
as shorthands for the collection of their ground instances.
The following example shows that the language of epistemic specifications provided the way to deal with the problem outlined in example 1.
Example 2. [Example 1 revisited]
The statement (*) above could be naturally expressed in the language of epistemic specifications by the rule:
interview(X) ← not Keligible(X),
not K¬eligible(X)
which corresponds closely to the intuitive meaning of (*). The epistemic specification T consisting of this rule together with the rules (1) – (3) and (7) from
Example 1 has the world view A = {A1 , A2 } where
A1 = {highGP A(mike), eligible(mike), interview(mike)}
A2 = {f airGP A(mike), interview(mike)}
Therefore T answers unknown to the query eligible(mike) and yes to the query
interview(mike) which is the intended behavior of the system.
Unfortunately, as was first noticed by Teodor Przimusinski, world views of epistemic specifications do not always correspond to those intended by the authors.
Consider for instance the following example:
1
The actual language defined in these papers is substantially more general but its
simple version we present here is sufficient for our purpose.
New Semantics for Epistemic Specifications
263
Example 3. [Unsupported Beliefs]
Consider epistemic specification T1 consisting of the rule
p ← Kp
It is easy to check that it has two world views: A1 = {∅} and A2 = {{p}}.
Clearly the second one is unintended. A rational agent will not have a belief p
which is supported only by Kp.
Even though some attempts to remedy the situation were made in [2] we have
never been able to obtain a fully satisfactory solution to the problem of unintended world views. This paper is another attempt to the solution. The work is
of course preliminary but the author thought that it may be worth publishing
since there seems to be some renewed interest in epistemic specifications (see,
for instance, [6],[7].
2
The New Definition of Epistemic Specifications
The new definition of epistemic specifications suggests changes in both, syntax
and semantics of the language. Objective literals are defined as before; the subjective literals have the form K l where l is an objective literal possibly preceded
by the default negation no. Expression K not p, which was not allowed in the
old version becomes a subjective literal of the new language. According to the
new definition epistemic specification is a collection of rules of the form:
l1 or . . . or lk ← gk+1 , . . . , gm , not lm+1 , . . . , not ln
(2)
where the l’s are objective literals and the g’s are subjective or objective literals.
The new syntax allows modal operator M to be expressed in terms of K.
M l =def ¬K not l
As before, programs with variables are viewed as shorthands for their ground
instantiations. The second, more substantial, change is in the definition of the
notion of the reduct.
Definition 1. [New Reduct]
Let T be an epistemic specification and S be a collection of sets of ground
literals in the language of T . By T S we will denote the disjunctive logic program
obtained from T by:
1. removing all rules containing subjective literals g such that S |= g,
2. removing all other occurrences of subjective literals of the form ¬K l,
3. replacing remaining occurrences of literals of the form K l by l.
The definition of world view of T remains unchanged: a set S is called a world
view of T if S is the collection of all answer sets of T S .
The new definition deals correctly with the eligibility examples. According to
the new definition specification T from Example 2 has exactly the world views it
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M. Gelfond
had under the old definition. This is not surprising, since the rules of T contain
no positive occurrences of K and hence the old reduct coincides with the new
one.
Let us now see how the new definition deals with unintended world views.
Example 4. [Example 3 revisited]
Consider
T1 = {p ← Kp}
from Example 3. It is easy to see that, under the new semantics, A1 = {∅}
remains a world view of T1 . However, A2 = {{p}} is not a world view of T1
according to the new definition; T1A2 = {p ← p}. Clearly its answer set, ∅, is not
equal to {p}. The new definition of reduct helps to eliminate the unsupported
beliefs.
The next example shows how the new language can be used for an alternative
formalization of the closed world assumption (CWA) [5]. The assumption, which
says that p(X) should be assumed to be false if there is no evidence to the
contrary, is normally expressed by an ASP rule
¬p(X) ← not p(X)
(3)
An interesting alternative representation of CWA may be given by epistemic rule
¬p(X) ← ¬Mp(X)
To better understand this formalization let us consider
Example 5. [Closed World Assumption]
Let T consist of the rules
1.
2.
3.
4.
p(a) or p(b).
p(c).
q(d).
¬p(X) ← ¬Mp(X)
According to the new definition the specification has one world view,
A = {{q(d), p(a), p(c), ¬p(d)}, {q(d), p(b), p(c), ¬p(d)}}.
To see that it is enough to recall that the rule
¬p(X) ← ¬Mp(X)
is a shorthand for
¬p(X) ← K not p(X)
and hence the corresponding reduct is
(4)
New Semantics for Epistemic Specifications
1.
2.
3.
4.
265
p(a) or p(b).
p(c).
q(d).
¬p(d) ← not p(d)
If, however, we replace the last rule by the rule 3 the world view will change.
Now it would be
B = {{q(d), p(a), ¬p(b), p(c), ¬p(d)}, {q(d), p(b), ¬p(a), p(c), ¬p(d)}}.
Hence the first program will answer unknown to a query ¬p(a) or ¬p(b) while
the second will answer yes. To have the semantics of M in which program T
above will have exactly one world view, A, was part of the original goal of
[1,4]. Unfortunately, however, the goal was not achieved. According to the old
definition specification T above has three world views: the world view A above
and
A1 = {{q(d), p(c), p(a), ¬p(b), ¬p(d)}},
A2 = {{q(d), p(c), p(b), ¬p(a), ¬p(d)}},
The new definition of reduct allows us to get rid of the two unintended world
views.
Obviously the work presented in this note is preliminary. The first next step is
to see if the known results establishing properties of epistemic specifications and
the corresponding reasoning algorithms can be adopted to the new language.
There are many possible applications. Most immediate ones are to investigate
the use of the language for conformant planning and, when suitably expanded,
for probabilistic reasoning.
References
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5. Reiter, R.: On Closed World Data Bases. In: Logic and Data Bases, pp. 119–140.
Plenum Press, New York (1978)
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(2009)