From: AAAI Technical Report SS-95-07. Compilation copyright © 1995, AAAI (www.aaai.org). All rights reserved.
Relating
Formalizations
of Actions
Tom Costello*
Dept. of Computer Science,
Stanford University,
Stanford,
CA 94305
cost clio@sail,
stanford,
edu
Abstract
possible
moveconcurrently.
We relatethishighlycomplexdomainto a muchsimplerdomain,wherewe have
onlya finitenumberof possible
actions
at anypoint-wherewe onlyconsider
serialactions--where
alltheeffectsof actions
arcimmediate.
In thispaperwe showa general
method
of relating
two
formalizations
so thatwe cansaytheybothderivethe
samesentences,
modulo
translation,
in a certain
class.
In
thecaseof ourexample
we willconsider
thattwoknots
canbe madeequalby tying/untying.
Reasoning about action is central to muchof intelligent behavior. Muchof the reasoning that
is needed consists not of reasoning about long
or complicated sets of actions, but of thinking of a problem at various different levels of
abstraction. Problems that in one representation seem enormous can be reduced to a simple
exercise in a more suitable representation. A
transformation that is especially useful is one
from continuous to discrete change. The transformations necessary to move from a complex
continuous system to a simple discrete systems
involve both introducing new properties that
change with time--fluent names--or no longer
modeling some other fluent names, and considering new composite actions.
1
1.1
1.2 Outlineof Paper
We firstsketch
theessential
properties
of a formalizationof actionthatwe needin orderto relate
it to anotherformalism.
In the following
section
we describe
how we formalize
the relation
betweentwoformalizations,a translation,
anda classof sentences,
and we
discuss
whatrelationships
pairsoftheories
standin.
Finally
we comment
on thegeneral
applicability
of this
method.We relateourformalisms
to otherformalisms
inthefield.
Introduction
Aims of this
Paper
The goal of this paper is to show how complex reasoning
about continuous concurrent systems in complex space,
can sometimes be replaced by much simpler reasoning.
Weshall show how to state relations between various
formalizations, and show that we can precisely say when
one formalization captures a part of another more complicated formalization.
Ourbasictoolwillbe versions
of the situation
calculus,
developed
by McCarthy
and
Hayes in [McCH69].
In an extended version of this paper we consider the
domain of tying and untying (tame 1) knots. This involves arbitrary curves2 in 3 space. The changes possible
are explained in terms of the notion of a subsection of
string movingalong a path, while other pieces of string
*This work was partly supported by ARPA(ONR)grant
N00014-94-1-0775 and was partly done while the author
was visiting the University of Toronto. The author thanks
Ray Reiter, Anna Patterson, and John McCarthyfor useful
comments.
1A tame knot is a knot that has only a finite numberof
loops that intersect.
2Weconsider just piecewise polynomials for expository
reason.
45
2
Requirements
2.1 Discrete Theories
In this paper we consider discrete theories of action
that have a notion of situations sitthat are identified
with sequences of actions evts, and properties or fluent names flt that hold at these situations.
A model
of a discrete theory is therefore an assignment of fluent
names to the sequences of actions, including the empty
sequence which we call the initial situation or sO.
Wewill also briefly mention axiomatizations of discrete theories. These will consist of sentences of three
types:
¯ That a fluent holds at a situation.
¯ That an action causes a fluent to hold in the next
situation if certain fluents hold.
¯ That a domain constraint is always satisfied
Axiomatizations of discrete action such as Kartha and
Lifschitz’s AR, and extension of Gelfond and Lifschitz’s
A [GeLi91], or Lin and Shoham’s"Provably Correct Theories of Action" [LiSh91], Reiter’s monotonicversion in
[Rei91], or the author’s SSC[Cos95b], are representative of these. Weuse ~d~,c to represent the relation
of consequence in these theories. Domainconstraints,
or ramifications, are rules that govern how fluents may
change. Thus the initial situation must obey the domain
constraint, and every action must preserve the domain
constraints. This can be expressed as saying that the
effects of an action are closed under logical consequence,
or the rule K of modal logic. This solution to the ramification problem was proposed by the author in [Cos91].
The original solution to this was proposed by Baker in
[Bak91], but is more complicated.
The discrete theory we will consider as our motivating
example is an axiomatization of manipulating knots. In
this we represent a knot as a sequence of crossings. Each
crossing is either under or over. Wehave three types of
actions, the Reidemeister moves [Reid32]. Wecan make
or destroy a simple loop, cross two pieces that are next
to each other, or movea piece over a crossing if it is
above/below both pieces in the crossing.
Wedo not give an axiomatization here for reasons of
space. The axiomatization is straight forward enough,
save for the use of an ezists fluent name function, following McCarthy in [MeC76] to represent what crossings exist. Thus we create and destroy objects--these
crossings--as we go along.
2.2 Continuous Theories
Continuous theories have as their "backbone" not sequences of actions, but rather an association of events
with a timeline, i.e. a timeline of the events that occurred up to a momentin time identifies a situation in
a given timeline. This can alternately be seen as a sequence of events, with an associated function that returns the time between each event. Weassume that situations are dense along timelines. Properties that hold
at these situations are again called fluents. In general
a property will follow a curve, trajectory, or fluxion between events. At events the curve or graph that the property follows over time may change. It should be noted
that events may be stated externally to the system, or
may be caused by internal confluences of processes. A
sink overflowing is perhaps the canonical example of an
event that is caused by a confluence of the effects of past
events.
Wecan either state explicitly what time-lines or histories we wish to consider, or we can consider all possible
histories. Wenote that we now say that a fluent holds
at a situation in a given history. Whenthe history is
obvious will we sometimes revert to the usual notation.
Axiomatizations of continuous theories, will then need
statement of four types,
¯ That a fluent holds at a situation.
¯ That an event causes a fluent to begin following a
fluxion if certain fluents hold.
¯ That a domainconstraint is always satisfied
¯ That a event occurs in a history
Shanahan’s cireumscriptive theories of events[Shun94],
Reiter and Levesque’s Golog, and the author’s system F
[Cos95b] are examples of continuous systems with the
above properties. Weuse ~cont to describe the consequences of a theory of continuous change.
Our example is the manipulating of knots. Werepresent these as piecewise polynomials. Weallow only a
finite number of pieces, thus we exclude wild knots. We
use abstract syntax to describe piecewise polynomials,
and we represent our string by these. Weallow arbitrary
deformations along piecewise polynomials. If a deformation breaks the string, or passes two pieces through each
other we trigger an fail event.
Wedo not give the axiomatization for reasons of space
again, there are no major difficulties in generating it once
a theory of piece-wise polynomials is in place.
3
Relating Theories
Wehave sketched two theories that might on the surface
seem very different. One is clearly much stronger and
more expressive than the other. What we now show is
that for a certain class of sentences, the sentences that
state whether one knot can be manipulated so that it is
equal to another, the two systems are equivalent.
Wefurthermore generalize the notion of relating theories, to any two theories, a translation, and a set of
sentences. Wethen consider its applicability to reasoning about events, and show reasons why we should be
hopeful that complicated domains may be reduced to
simpler ones.
3.1 Removing Fluents
Wefirst consider a relation between two theories, where
one theory has a subset of the others fluent names. We
ask, when is it possible to keep the same results, yet
remove certain fluent names. In general we will find
that we can remove all fluent names save those that are
true exactly when a particular event occurs. Wewill
show that every other statement about fluent names is
provably equivalent to a statement about these fluent
names.
Wefirst need to define what it meansfor a set of fluent names to be sufficient to represent a theory. A set
will be deemedsufficient if all sentences involving fluent
names outside that set are equivalent to sentences involving only fluent names inside that set, or facts about
the initial situation.
Definition: 1 A set of fluent names F is sufficient to
represent a theory T if every sentence q~ in an axiomatization of T is provably equivalent to sentence ~b where
~b does not contain any fluent name f name outside F
that is not in a expression of the form holds(f, sO).
Example: 1 If we have a set of fluent name constants
F, and we have functions A, V, -1, that map fluent name
(pairs) to the conjunction of the fluent names, then the
fluent nameconstants are sufficient to represent a theory
involving the closure of F under these functions.
holds(f1 A f2, s) =_. holds(f1, s) A holds(f2,
Given the above sentence, we can show that every sentence in terms of fl ^ f~ can be written as a sentence
in terms of fl and f2, without using the function A on
fluent names.
Wenote that we allow statements about the initial
situation to use other fluent names, as long as they are
not needed elsewhere. This is analogous to using initial
conditions in solving a differential equation.
Example: 2 If we wish to describe the change in position of a body that moves with constant, in a straight
line, we can either describe its position as a function of
time, or else describe its velocity as a function of time,
and give the bodies initial position. It is often simpler
to describe systems by using their derivative rather than
the values themselves.
If vel is the velocity of the object, and Ioc(z) it location, and time returns the time associated with a situation then its position can be stated as:
holds(loc(z),
s) - time(s) - time(sO) = 10 * vel
Or more simply as
holds(loc( O), sO) A holds( velocity( vel
Wherevelocity is the velcity of the particle, and we apply
inertia to it.
Here we have given an example where velocity is a
frame for a theory, in the presence of an initial condition,
the value of loe at sO.
We nowdefinewhatit meansto changea theoryon a
I.
setof fluent
names
F toa different
sufficient
setF
Definition: 2 The theory N(F’, T), is the set of sentences dp not involving fluent names not in FI, that are
in the theory T
The other sentences, those involving facts about the
initial state we call init(F’, T).
Theorem: 1 If T is a domain theory, and FI a sufficient set of fluent names, then
T ~,ont¢ 3¢N(F’, T) ~cont ¢ and init(f’,
T)A ¢ ~
Weomit proof of this and the following theorems for
space reasons. The proofs present no special difficulties,
and can be proved by induction on each formula in the
theory.
Theorem: 2 If T is a domain theory, and F’ a sufficient set of fluent names, then
T ~,on, ¢ -3¢Y(f’, T) ~cont ¢ and init(F’, T) A ¢ ~
Wenote that the theorems above show that for continuous and discrete theories, we can change the fluent
names that we work with. Wenote there are large computational advantages in being able to work with a much
smaller set of fluent names. More importantly for our
purposes, we can change one set of fluent names into
another, yet still describe the same underlying domain.
3.2 l~’ames
Wenow define a fluent frame. A frame of fluents for a
set of fluent namesF is a set of fluent names,and a set of
sentences, that has the property that every fluent name
in the original set is definable from the fluent names
in the frame, and that this is derivable from the set of
sentences.
Definition: 3 If F is a set of fluent names, and F’, S
a frame for that set, then .for every .f 6 F there is a
sentence
Vs.holds(.f,
s) - ¢ suchthat
S ~ Vs.holds(.f, s) -and no fluent names but the fluent names in F’ occur in
¢
Thus a frame may introduce an entirely new set of
fluent names, and relate them to the old set.
Example: 3 If we want to describe continuous circular
motion we can do so by writing sentences describing how
the z and y co-ordinates of a particle change with time.
holds(z ~_ sin(d), s) =_ time(s) - time(sO)
holds(z ~_ cos(d), s) - time(s) - time(sO)
Alternately we can use a circular co-ordinate scheme,
where we have r as the distance from the origin and 0
as the angle. If we denote the first derivative of 8 with
respect to time as 8’ then
holds(z ~_ a, s) =_holds(r ~_ s)A
holds(O ~_ c) A a = b * eos8
Wealso note that to describe continuous circular motion
we need only give the initial value orS, and the values of
r and 8’. These do not change over time. Thus r and 8’,
with the inital value .for 0, are a frame for this example
Wenow show that we can reason with a translation
of our old theory into these new fluents, and get the
correct results. First we must define what a translation
of a theory looks like.
Definition: 4 The translation of a theory T into a new
frame F’, S, is the set of sentences that result from replacing the fluent-names in the sentences of T by fluent
names in F’ subject to the following rules. Wesay that
i.fVs.holds(.f, s) = ¢ then f is to be translated by ¢. For
a given sentence X involving .f, we write the translation
of that sentence to remove f as Tr(x(.f), ¢) Weexplain
how this is done by structural induction on ¢
1. Tr(x(.f), holds(.f’, s)) = X(.f’)
L Tr(x(.f), z = y) = (z
3. Tr(x(f),
¢ A ¢) = Tr(x(.f),
¢) A T~(x(.f)
, ¢)
~. Tr(x(f),-~¢)=-~Tr(x(f),¢)
5. Tr(X(f), Vz.f(z)) Vz.Tr(x(f), ¢(z))
This gives us a method of removing one fluent-name from
a sentence. Repeating this process will thus remove all
fluent-names not in F’.
Wenow need to show that the order in which we removefluent names by the above process does not matter
nor by which statement we remove them, and that reasoning in the new system gives the same results.
Theorem: 3 If F’, S is a frame .for T, then
3¢.Tr(T, S) ~,~t ¢ and S A ¢ ~
The proof goes by induction on the sentences in the
theory.
Theorem:4 If F’, S is a frame for T, then
T ~dite¢ =-
3¢.Tr(T,
S)~d.o¢ andS ^ ¢ ~ ¢
The proofs go by induction on the sentences in the theory.
Wefinally introduce the notion of modeling a domain
with a coarser set of fluents. These are a set of fluents which do not have the properties above, that all
other fluents can be reproduced from them, and possible the initial values of fluents. Rather these fluents are
coarser, in that though we cannot find expression that
exactly match, we can find expressions that are strictly
weaker/stronger in the effects/preconditions of actions.
In this way we can find a new domainwhere a state will
be reachable in the original domainif it is reachable in
the new domain, but not necessarily vice-versa. If we can
reach a particular situation just using actions that obey
these stronger preconditions, then we reach it with the
original actions. Wenotice that the analogous operation
of weakeningthe effects has the same result.
Finally we notice that the opposite of this, weakening
preconditions, and/or strengthening effects, is useful is
we wish to showthe impossibility of an situation resulting. This is commonlydone when using induction, that
is we prove something stronger than we actually need,
as it is simpler to prove.
Thus we define the notion of a weak frame as follows:
Definition: 5 If F is a set of fluents, and F’, S a weak
frame for that set, then F’ is a new set of fluent names,
and S consists of sentences of the form:
Vs.holds(f , s) ¢
where f is a fluent name in F’ and dp contains only
fluent-names from F.
This is considerably weaker than the previous definitions but we knowsee that there are occasions when we
can use this notion.
The idea here is that although the weak frame does
not capture all of the theory, it captures all of the theory
that it is able to describe.
Definition: 6 The theory EX(T, F’, S) is the theory
that uses only the fluent names in F’, such that when
we consider the models of it union the sentences S, then
the parts of those modelsthat describe objects in the original theory T are substructures of some model of T, and
further more are all such substructures.
Example: 4 Assume that in the blocksworld, we have
three locations on the table lot1, loc2, and lot3. We
introduce a new fluent name function ontable, defining
it as:
holds( ontable( z ), -- holds(on(z, locl ), s)V
holds(on(z, lot2), s) V holds(on(z, Ioc3),
If we have a sentence
holds(on(A,
locl ),
we translate this to
holds( ontable( A ),
Rather than it being necessary for a location to be clear
to move a block, it now is the case that there must be
or fewer blocks on the table.
The model that just uses ontable(z) rather than the
individual locations is a weaker model. The fluent name
function ontable(z), with the above sentence is a weak
frame for this blocksworld theory.
In particular cases this function is effectively calculuable given a particular theory. This is the ease in the
systems F. This transaltion can be done by a recursive
procedure that replaces the original fluent names, by the
new names.
Theorem:5 If F’, S is a weak frame for T, and q~ contains only fluents in F’,and fluent name variables ranging over FI, then
T ~oo,. ¢-3¢.EX(T,F’, S) ~o~, ¢ andS ^ ¢ ~
3.3 Composite Actions
Wehave shown how various manipulations on fluents can
be used to simplify reasoning, we now look at what we
can do with actions.
Changing actions is much more subtle than changing
fluents. For when we change actions we can decide to
make out theory stronger or weaker. Wesay one theory
is weaker than another if it proves less sentences. If
we remove actions from a theory, it becomes weaker in
that we can "reach" less situations. Thus the notions
of weaker and stronger can be relativized to a set of
sentences, so that a theory is said to be weaker with
respect to a set of sentences, than another theory, if it
proves less sentences in that set.
It is often useful to consider stronger theories in this
respect. Sometimes it is easier to prove that a certain
situation is not reachable by considering stronger actions
than you actually have. If you can prove this from the
stronger actions, then it clearly is unreachable by the
weaker actions.
Thus the sets of sentences that we will consider will be
of the form that particular situations are reachable/nonreachable.
A composite actions can be thought of as a disjunction
of actions, where all we know is that one action in the
set occurred. This however is not general enough, we
need to be able to parameterize the set by the state of
the world. Thus we have a set A(s), the actions that
the composite action is made up of at a given situation.
Thus a composite action has as its result, the common
results of all actions from the set of actions A(s).
This will give us extra flexibility. Howeverwe will see
that this is not enough, we will want to be able to consider composite actions as sets, again parameterized by
a situation, not of other actions, but of other sequences
of actions, or partial time-lines. Wefind that this generality will allow us to state the relationship between our
two formalizations of knots.
Wefirst define some notions of composite action.
Definition: 7 A (set of} composite action(s) is defined
as a (set of) function(s) from situations to sets of
tions. Wecan consider a situation to be characterized
by the fluents holding at it in this case. Thus we associate a function a : ~lt ---r 2errs, with each composite
action. The effects of a composite actions are exactly the
fluents that holds after every action in the composite action. The preconditions for a given effect of a composite
action are exactly the preconditions that satisfy all of the
actions it contains.
Exanaple: 5 If we imagine that we are in a room, we
can describe our position by what quarter of the room we
are in, or, alternately, where we are exactly in terms of
a finer co-ordinate system that measures in feet from the
center. Assume we have the actions of moving a certain
distance in a certain direction. Then we can make the
composite action, moving to the next quarter, by assigning to each situation, those actions that would movefrom
that situation to one where we were in the next quarter.
Wenote that the set of actions that we associate with
each composite action varies from situation to situation.
If we were in the middle, the perhaps all movesof greater
than 10 feet would be in the composite action of moving
to the next quarter. If we were near the edge then perhaps
all moves of one foot or more might count.
Wenow define the translation
a set of composite actions.
of a domain theory by
Definition: 8 The translation of a domain theory T by
a set of composite actions A, TR(T, A) is defined
the theory, where what holds after a sequence of composite actions, or partial time-line, is the disjunction of
all possible sequences, or partial time-lines, of choices
of actions out of the sets associated with the composite
actions.
This can be generated effectively from a theory in
SSC.
By reachable(f, h) we mean there is a sequence of actions that result in f holding at a situation in a given
history h. Defining this is closely related to an induction principle over situations. In the continuous case it
is not purely induction, as we can have infinite sequences
of later and later situations, that do not exhaust all of
time. It is necessary to consider whether the elapsed
time between all the situations converges in the limit.
Wenow give a theorem relating composite actions of
the first type, to the situations that are reachable.
Theorem: 6 If T is a domain theory, and A a set of
composite actions, and S the translation ofT by A, then
T ~t reachable(f,
T ~cont -,reachable(f,
h) =_ S ~ont reachable(f,
h) - S ~cont -,reachable(f,
Here we state the theorem for continuous models, an
analogous theorem holds for for the discrete case. We
now define the second notion of composite actions,
namely composite actions of the second type.
Definition: 9 A composite action of the second type is a
function from partial histories to events, where a partial
history is defined by the fluents that hold at the beginning
of that time interval, and what fluxions they are following, and a set of events duration pairs that occur form
then on in that history, until a duration d has passed.
Werepresent the first situation by a fluent. We represent this as A2 : 2flt×flux × dur x 2evts×dur --+ evts
Example: 6 In the last example we thought of being in
a room, and having a single parameterized move action.
We now imagine that the size of the areas we can be
in can change over time--perhaps they are marked by
a spotlight. Nowa certain action might send us into a
region in one situation. The same action in a situation
that is exactly equivalent might not send us into a lit
region in another case, because the lit region might move
in the second case. We see that the events that might
send us into a new region, need to be parameterized by
the entire partial history, not just the previous events.
Weneed to take into account that the success of an event
might depend on other factors, not just on the situation
it is attempted in.
Wenowneed to specify a translation of a theory T by a
composite action of the second type. Wenotice that this
translation will be from a continuous formalization, to a
discrete one e.g. a theory in ~" to the simple situation
calculus.
Definition: 10 The translation of a domain theory T
by a set of composite actions of the second type A,
TR2(T, A) is defined as follows: It is a theory in a discrete language, with events, the events that are mapped
to, by A2, and fluents the same as the theory T. A fluent holds after a sequence of actions in a model of this
theory iff, there is a model of T, such that that fluent
holds at a situation in T, and the partial histories that
lead to that situations are mappedto the events that lead
to the situation in the discrete model.
Wenow extend the last theorem to composite actions
of the second type.
Let reachable(f) be there is a possible history where
reachable(f, h) in the continuous ease and in the discrete
ease the exhibition of holds(f, s) for somes.
Theorem: 7 If T is a domain theory, and A a set of
composite actions of the second type, and S the translation of T by A, then
T ~,~t reachable(f)
=_ S ~d~,, reachable(f)
T ~,o~t ~reachable(f) = S ~d~,¢ -~reachable(f)
Wenote that this theorem does not have an analogous
diserete theorem, as in the discrete ease there is no difference between a partial history and the last situation.
Nothing happens between actions in the discrete ease.
4 Knots
This allows us to state the relation between the two forrealizations of knots. This theorem is thus the generalization of the theorem in topology that states that if
two knots have the same topology, then the two representations of the knots can be made identical by a finite
sequence of Reidemeister moves.
A knot is usually specified by a projection. If we represent our knot as a curve in three space, removing any
of the dimensions, gives a projection. Wewould like to
get a projection that had the property that it had a small
numberof multiple points. A multiple point of a projection is a point in the plane that is the image of more
than one point in the knot. Wewant to get a projection
with just double points, and where every double point
represents a true crossing, not a tangent. A theorem
of topology[Crow63]tells us that "every knot is equivalent under an arbitrarily small rotation to a knot whose
projection has only true regular points".
Thus every knot can be represented as a projection
in two space. Once we have a projection in two space,
we can further reduce this to a sequence of crossings by
choosing any point on the curve and following it around,
numbering the crossings encountered, and then telling
which number goes under at each point.
Moving from a representation as a curve in 3 space
to this representation is an example of what we called
a weak frame. While it throws away much information
it retains the topology, and thus whether the knot is
equivalent to another. Thus the set of fluents that tell
what the knots crossings are, and whether they go over
or under is a weak frame for the full theory, whenpaired
with the sentences that relate this to the full theory.
Whenwe consider the three types of Reidemeister
moves, we notice that these are exampleof composite actions of the second type. They depend on the transaltion
of the fluents given above. This is because they mapeach
partial history of the knot, to the set of events that would
transform it into a knot that wouldproject to a different
set of crossings.
Thus the classic theorem of topology is equivalent to
the statement that the Reidemeister moves are a set of
composite actions for the translation of a curve in three
space with arbitrary deformations, by the weak frame
given by the above translation.
5
Usefulness
of Relations
Wehave exhibited that there are manydifferent relations
that can be made between theories. In particular we
have examined five.
¯ Firstly we have shownthat all the fluents that are
in the theory, are "not necessary". They can be
eliminated in favor of a muchsmaller set, the fluents
that correspond to actions.
¯ Secondly we showed how a formalization can be related to a formalization using an entirely different
set of fluents. This allows the changing of language
from one formalization to another.
¯ Thirdly we have shown that we can reason in a set
of fluents that are insufficient to represent the complexities of the theory, yet are sufficient to represent
the parts that we care about.
¯ Fourthly we saw how a formalization can be related
to another formalization by changing the actions, or
events.
¯ Lastly we saw a generalization of this that allowed
partial time-lines to be mappedto events in another
formalization. Wesaw how this allowed reasoning in
concurrent continuous domains, as it related them
to domains where reasoning is well understood.
5O
Theserealtions
between
theories
allowreasoning
to be
simplified
by recasting
it in a simpler
domain.
Thiswork
hasbeenmainlymeta-theoretic--thcse
relations
needto
be expressed
in a objectlevellanguage
to be fullyexploited
by a reasoning
machine.
References
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[Cos91]
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Second International
Symposium on Commonsense Reasoning
[Cos95b] Costello, T (1995) ’The Method Of Fluxions’, to appear
[Crow63] Crowell, R. H., Fox, R. H. (1963) Introduction to Knot Theory, Boston
[Kar94]
Kartha, G. N., Lifschitz,
V. (1994) ’Actions with Indirect Effects’, Proceedingof the
Fourth International
Conference on Knowledge Representation and Reasoning, Morgan
Kaufmann
[GeLi91] Gelfond, M., Lifschitz, V. (1992) ’Representing Actions in Extended Logic Programming’ in Proc. Joint Int’l Conf. and Syrup. on
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