Point-Sensitive Circumscription Eyal Amir
Department of Computer Science, Gates Building, 2A wing Stanford University, Stanford, CA 94305-9020, USA
[email protected] December 7, 1997 Abstract
A decade ago, Pointwise Circumscription was proposed as a tool for formalizing common sense. In this paper, we revisit some of its uses and examine some cases in which it does not yield the intended behavior. Speci cally, we explore the cases in which unsatis ability may result from the presence of multiple minimal models (for McCarthy's Circumscription). Then, we present a form of circumscription, called Point-Sensitive Circumscription, that is a generalization of McCarthy's Circumscription and Lifschitz's Generalized Pointwise Circumscription. We illustrate how Point-Sensitive Circumscription handles these cases without losing the control over selective negrained variance of predicates and functions. Last, we compare the two tools and their potential uses in formalizing Theories of Action.
1 Introduction Pointwise Circumscription, devised by Lifschitz in [Lif87] and [Lif89], is a nonmonotonic method de ned along the intuitions of Circumscription [McC80],[McC86] with superior control of the minimization process. Since its debut, it has been used in formalizing Common Sense problems and solutions, such as in [DL94], formalizing some Entailment Classes in the theory of Features and Fluents (cf [San94]), and in [Ami97], using Pointwise Circumscription to formalize the system of [LR94]. It was argued in [Lif87] and [DL94] that Pointwise Circumscription has the power to be a tool for formalizing solutions for the Frame Problem.
1
In this paper, we show that there are cases in which Pointwise Circumscription requires the intended model to be a minimum, rather than merely minimal. We present a form of circumscription, called Point-Sensitive Circumscription, that is a generalization of McCarthy's Circumscription and Lifschitz's Generalized Pointwise Circumscription. Point-Sensitive Circumscription allows us to get a consistent theory where Pointwise Circumscription fails to do so, while maintaining similar control over the minimization process. For Point-Sensitive Circumscription, we allow a varied range of point-wiseness (the removal of elements of the minimized predicate separately), but maintain the same control over selective ne-grained variance of predicates and functions. We give semantics for our approach and demonstrate the di erent behavior using examples from the literature. The paper is organized as follows. Sections 2 and 3 describe Pointwise Circumscription and its semantics and give an example that yields unsatis ability. Sections 4 and 5 describe Point-Sensitive Circumscription and its semantics and reexamine the same example, this time with the new tool. Section 6 closely examines one commonsense application for which Pointwise Circumscription was used and compares the various options for that application. For the rest of the paper, we use the following two conventions: (1) the relation p < q between two propositions refers to the usual ordering of Boolean values (false < true), with p q allowing p = q; and (2) the relation P < Q between two predicates with same arity refers to the strict subset relation (regarding P; Q as the relative sets of elements). For background material on Circumscription and Pointwise Circumscription, the reader is referred to [Lif93] and [Lif87].
2 Pointwise Circumscription
2.1 The Formula
Pointwise Circumscription was rst proposed in [Lif86] and then expanded in [Lif87] and [Lif89]. In [DL94] it was demonstrated that Pointwise Circumscription can be used to formalize various logics of action in the framework of Features and Fluents [San94]. These logics include OCM (Original Chronological Minimization), PCM (Prototypical Chronological Minimization), PCMF (PCM with Filtering), CMON (Chronological Minimization of Occlusion with Nochange Premises) and CMOC (Chronological Minimization of Occlusion and Change). 1
1
Some of the formalizations needed ltering.
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The general idea behind Pointwise Circumscription is to capture the minimization idea of Circumscription [McC80], [McC86], while making the circumscription process more distributed (i.e., making it on a di erent subset of the theory/domain each time). McCarthy's Circumscription formula
Circ[A(P; Z ); P ; Z ] = A(P; Z ) ^ 8p; z (A(p; z) =) :(p < P )) says that in the theory A, with parameter relations and function sequences P; Z , P is a minimal element such that A(P; Z ) is still consistent, when we are allowed to vary Z in order to allow P to become smaller. The basic case for Pointwise Circumscription is the formula
A(P ) ^ 8x:[Px ^ A(y(Py ^ x 6= y))] (where we minimize the subscript P again, varying nothing), intuitively saying that we remove one element at a time. One of the bene ts of such an approach is the rst-orderedness of the circumscription formula. This property disappears, though, in the most general form, which we use below. Let A(S ; :::; Sn) be a sentence in which each Si is a predicate symbol or a function symbol (in particular, it can be a 0-ary function symbol, i.e., an object constant). We want to minimize one of the predicate symbols from this list, Si0 (Thus, Si0 corresponds to P , and the other members of the list correspond to Z in the notation used above). Let us write EQV (P; Q) (\P and Q are equal outside V ") for 1
EQV (P; Q) def= 8x(:V x =) (Px Qx))
(1)
for P; Q predicates or functions, and V a predicate, all with same arity. The Pointwise circumscription of Si0 in A with Si allowed to vary on Vi is, by de nition,
A(S ) ^ 8xs:[Si0 x ^ :si0 x ^
n ^ i=1
EQVi x(si; Si) ^ A(s)]
(2)
Here, S stands for S ; :::; Sn, s is a list s ; :::; sn of predicate and function variables corresponding to the predicate and function constants S , and xuVi(x; u) (i = 1; :::; n) is a predicate without parameters that does not contain S ; :::; Sn and whose arity is the arity of Si0 plus the arity of Si. We denote (2) by CPW [A; Si0 ; S =V ; :::; Sn=Vn]. 1
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2.1.1 An Intuitive Explanation of Pointwise Circumscription The Pointwise Circumscription formula (2) has the property that we remove x from Si0 while varying S (recall that S is the list of predicates/functions S ; :::; Sn) in each of the Vix separately, xing all the other parts of the domain, at that time. In other words, the bracketed part of the formula says that, in the process of removing x from Si0 , we allow only some parts of S to change. These parts are determined by Vix by saying that only where Vix is TRUE si may be di erent from Si (recall that EQVi x(si; Si) means that si is equal to Si outside u(Vi(x; u))). 1
2.2 Semantics for Pointwise Circumscription
In [Lif87], Pointwise Circumscription was given semantic interpretation. Take, without loss of generality, i = 1. For a model M of A(S ), let jMj be the associated universe, and for every term, function or predicate a, a is the realization of a in M. De nition 2.1 Let M ; M have the same universe U , and let 2 U k , where k is the arity of S . We say that M M i 1. K 1 = K 2 for every function or predicate constant K that is not in S, 2. for any i = 1; :::; n, Si 1 and Si 2 coincide on f j :Vi 1 (; )g 3. S 1 ( ) =) S 2 ( ). 0
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Proposition 2.2 ([Lif87]) Let M 2 [ A(S )]]. M j= CPW [A; S ; S =V ; :::; Sn=Vn] () 8M 2 [ A(S )]] 8 2 jMjk :(M M ^ M 6 M ) Other words, M is a model of CPW [A; S ; S =V ; :::; Sn=Vn] i for each 2 jMjk, M is minimal relative to . 1
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3 A Counter-intuitive Example In this section, we discuss only one simple example.
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Let A(P ) = (P (X ) _ P (Y )) ^ X 6= Y . For this simple language (the only predicate is P , and there are exactly two object constant symbols X; Y ), we get the Pointwise Circumscription formula
A(P ) ^ 8xp:[Px ^ :px ^ EQVP x(p; P ) ^ A(p)]: Recall that VP limits the di erences between p and P , according to the point x. Assume that for all x, (VP x) True, so that p is allowed to di er from P with no limitations. We get the further simpli ed Pointwise Circumscription formula
A(P ) ^ 8xp:[Px ^ :px ^ A(p)]: For A(P ) = (P (X ) _ P (Y )) ^ X 6= Y , we get the following result: Proposition 3.1 CPW [A(P ); P ; P=x:True] j= FALSE . Proof CPW [A(P ); P ; P=x:True] A(P ) ^ 8xp:[P (x) ^ :p(x) ^ A(p)] A(P ) ^ 8xp:[P (x) ^ :p(x) ^ (p(X ) _ p(Y )) ^ X 6= Y ] =) A(P ) ^ 8p: PP ((YX)) ^^ ::pp((YX))^^((pp((XX))__pp((YY))))^^XX 6=6= YY _ P ( X ) ^ : p ( X ) ^ p ( Y ) ^ X = 6 Y _ (3) A(P ) ^ 8p: P (Y ) ^ :p(Y ) ^ p(X ) ^ X 6= Y Let px = fX g; py = fY g be two possible values of p. Then, from the formula on the last line, we get P ( X ) ^ : p ( X ) ^ p ( Y ) ^ X = 6 Y _ A(P ) ^ 8p: P (Y ) ^ :p(Y ) ^ p(X ) ^ X 6= Y =) 3 2 P (X ) ^ :px(X ) ^ px(Y ) ^ X 6= Y _ 6 P (Y ) ^ :px (Y ) ^ px (X ) ^ X 6= Y _ 7 A(P ) ^ : 64 P (X ) ^ :p (X ) ^ p (Y ) ^ X 6= Y _ 75 y y P ( Y ) ^ : p ( Y ) ^ p (X ) ^ X 6= Y y y 3 2
FALSE _ P (Y ) ^ (TRUE () X 6= Y ) ^ X 6= Y _ 77 P (X ) ^ (TRUE () X 6= Y ) ^ X 6= Y _ 5 FALSE A(P ) ^ :[(P (Y ) _ P (X )) ^ X 6= Y ] P (X ) _ P (Y ) ^ X 6= Y ^ :[(P (Y ) _ P (X )) ^ X 6= Y ] FALSE
6 A(P ) ^ : 64
5
This proposition reveals a limitation of Pointwise Circumscription, i.e., the requirement, in some cases, that P be a minimum rather than minimal . This conclusion is supported by the semantics given by Lifschitz (proposition 2.2 above). Let U = fx; yg be the set of elements in the universe. Let MX , MY , MXY , M be the models with universe U , with X; Y interpreted to x; y, respectively, and the following interpretations for the predicate P : P MX = fX g. P MY = fY g. P MXY = fXY g. P M = ;. PSfrag replacements Figure 1 below displays the di erent models for A(P ). x 2
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x ,y between the models. In order to check these partial orders, we rst
notice that the rst two conditions of de nition 2.1 are met by any pair of models from the four examined above. Using the third condition, we get the obvious M y MX y MXY and MXY 6y MX . M x MY x MXY and MXY 6x MY . but also the somewhat less obvious M x MY x MX and MX 6x MY . M y MX y MY and MY 6y MX . ;
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Examining the conditions of proposition 2.2 for each one of the models, we reveal the following: M is not a model of C = CPW [A(P ); P ; P=VP ] because it is not a model of A(P ). MX is not a model of C because MY j= A(P ) and MY x MX and MX 6x MY . MY is not a model of C because MX j= A(P ) and MX y MY and MY 6y MX . MXY is not a model of C because MX j= A(P ) and MX x MXY and MXY 6y MX . Thus, again, there is no model that satis es C . Figure 2 below displays the two orders on models. ;
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Figure 2: The orders x and y . The di erent colors represent the di erent orders.
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4 Point-Sensitive Circumscription
4.1 The formula
We present Point-Sensitive Circumscription, a modi ed version of Pointwise Circumscription in which the minimized predicate is minimized according to a minimization domain. This minimization domain may be a point and may be the complete set of elements. We preserve the ability to select/vary parts of the theory/domain dynamically. It is important to notice at this point that Lifschitz already proposed a global circumscription that has some control over the varied domain (see formula (14) in [Lif87]). We shall compare that proposal with Pointwise Circumscription and Point-Sensitive Circumscription in section 6. We use similar notations to those used in section 2. Let A(S ; :::; Sn) be a sentence in which each Si is a predicate symbol or a function symbol (in particular, it can be a 0-ary function symbol, i.e., an object constant). We want to minimize one of the predicate symbols from this list, Si0 . In addition the de nition of EQV in (1), let us write LSR (P; Q) (\P is smaller than Q in the region R", or P \ R $ Q \ R) for 1
LSR (P; Q) def= 8x(:Rx =) (Px =) Qx)) ^ 9x(Rx ^ :Px ^ Qx) (4) for P; Q predicates or functions, and R a predicate, all with same arity. The Point-Sensitive Circumscription of Si0 in A with Si allowed to vary on Vi, and Si0 minimized using R, denoted CPS [A; Si0 =R; S =V ; :::; Sn=Vn], is, by de nition, 1
A(S ) ^ 8xs:[LSRx (si0 ; Si0 ) ^
n ^ i=1
1
EQVix(si; Si) ^ A(s)]
(5)
where S stands for S ; :::; Sn, s is a list s ; :::; sn of predicate and function variables corresponding to the predicate and function constants S , and xuR(x; u), xuVi(x; u) (i = 1; :::; n) are predicates without parameters that do not contain S ; :::; Sn and whose arity is the arity of Si0 plus the arity of Si0 and Si , respectively (here Rx = uR(x; u), Vix = uV (x; u)). 1
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4.1.1 An Intuitive Explanation The bracketed part of the Point-Sensitive Circumscription formula (5) says that, in the process of making Si0 smaller, we use R to de ne what \smaller" means and we allow only some parts of S to change. These parts are determined by Rx and Vix by saying that si may be di erent from Si only where 8
Vix is TRUE (recall that EQVi x(si; Si) means that si is equal to Si outside u(Vi(x; u))) and that we actually care if si0 is smaller only where Rx is TRUE. Notice that the only di erence between the Point-Sensitive Circumscription formula (5) and the Pointwise Circumscription formula (2) is in the rst component. We changed Si0 x ^ :si0 x (in (2)) to LSRx(si0 ; Si0 ) (in (5)). It is also important to notice that even for the cases where R is uniformly TRUE (in these cases, x is used only in the second component in the formula in parentheses), the second component is actually a conjunction of formulas, all using x. Therefore, although we do not require that x be removed from the predicate Si0 , we do require that the variance of the di erent predicates/functions (including Si0 ) be correlated with x as an index.
4.2 Semantics
We follow the lines of [Lif87] with regard to the semantics for our PointSensitive Circumscription. Take, without loss of generality, i = 1. For a model M of A(S ), let jMj be the associated universe and, for every term, function or predicate a, a is the realization of a in M. We write the corresponding de nition to 2.1. 0
M
De nition 4.1 Let M ; M have the same universe U , and let 2 U k , where k is the arity of S . We say that M M (a strict partial order) 1
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3. LSR() (S1 1 ; S1 2 ). M
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Let [ A(S )]] be the set of models of A(S ). The following proposition says that every model of the circumscription formula for A(S ) is minimal in [ A(S )]] according to all of the orders , and vice versa. Proposition 4.2 (Semantics for Point-Sensitive Circumscription) Let M 2 [ A(S )]].
M j= CPS [A; S =R; S =V ; :::; Sn=Vn] () 8M 2 [ A(S )]] 8 2 jMjk :(M M) 1
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Proof For the forward direction, let M j= CPS [A; S =R; S =V ; :::; Sn=Vn]. Assume M 2 [ A(S )]] M M for some 2 jMjk . Then, LSR (S ; S ), by de nition 4.1 (requirement 3). Take s to be S , the sequence of predicates/functions interpreting S (the sequence of predicate/function constant symbols) in M . For x = ( picked above), we get that in the model M, LSR (s (x); S (x)), because s = (s ; :::; sn) (that is our notation), and the way we picked (s.t., s = S , and S ( ) < S ( )), and A(s), because s = S and M j= A(S ), and Vni EQVi x(si; Si), because of de nition 4.1 (requirement 2). This contradicts M j= CPS [A; S =R; S =V ; :::; Sn=Vn], because we found x and s that satisfy the bracketed part of formula (5). The reverse direction works by the same method. If M is -minimal for every 2 jMjk, but Vdoes not satisfy formula (5), then there are s; x such that LSR (s ; S ) ^ ni EQVi x(si; Si) ^ A(s). But then we can build a model M with the same universe, taking all the constants other than those in S to be the same, and the constants of S take the values of s. For M , we show now that M x M. We follow the conditions of de nition 4.1 one by one. 1. is satis ed by the construction of M . 2. is satis ed because s satis es the second component of the bracketed part of the CPS formula. 3. is satis ed because s satis es the rst component of the bracketed part of the CPS formula. We have found a model M with M x M. Contradiction. 1
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Lemma 4.3 For 2 jMjk, is a strict partial order over models of L. Proof Irre exivity is simply because LSR (P; Q) is irre exive (R \ P $ Q \ R). It is now enough to prove transitivity. Let M ; M ; M with same domain, and 2 jM jk . Assume M M and M M . We prove the conditions for M M one by one. 1
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2. For i 2 f1; :::; ng, Si 1 ; Si 2 coincide on f j :Vi 1 (; )g, and Si 2 ; Si 3 coincide on f j :Vi 2 (; )g. Since Vi 2 = Vi 1 (recall that xuVi(x; u) is a formula that contains no element from S ), Si 2 ; Si 3 coincide on f j :Vi 1 (; )g. Thus Si 1 ; Si 3 coincide on f j :Vi 1 (; )g. 3. LSR (S 1 ; S 2 ) and LSR (S 2 ; S 3 ), and thus (transitivity of $) LSR (S 1 ; S 3 ). Thus we proved all three requirements for M M and transitivity is proved. Assymetry follows from transitivity and irre exivity. In the case of R uniformly true (LSRx (P; Q) P $ Q), we get the following proposition. Proposition 4.4 Let A be a satis able theory with nitely many models, and let Rx = Ry for all x; y 2 jMjk . Then CPS [A; S =R; S =V ; :::; Sn =Vn] is satis able. Proof Since [ A] is nite and is a partial order, there must be at least one minimal element M for from the models of A. If a minimal model M of is not a minimal element according to then has a smaller model M . Since there are nitely many models, if we continue such a chain of models, we will eventually end up with a cycle. Without loss of generality assume that the cycle is M ; M ; :::; Mn; M . But then we get LSR (S 1 ; S n ); :::LSR (S 2 ; S 1 ), and thus S 1 $ S n $ ::: $ :::S 2 $ S 1 , and thus S 1 $ S 1 . Contradiction. Last, we nd out the following proposition. Proposition 4.5 1. Point-Sensitive Circumscription with R TRUE and V TRUE is equivalent to McCarthy's Circumscription. 2. Point-Sensitive Circumscription with Rx x (x = x ) is equivalent to Lifschitz's Generalized Pointwise Circumscription. M
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5 Examples using Point-Sensitive Circumscription
Let us rst recall the example from section 3. Let A(P ) = (P (X ) _ P (Y )) ^ X 6= Y . Let (VP x) TRUE for all x, so that p is allowed to di er from P with no limitations. Let R TRUE , so that LSRx (P; Q) (P $ Q). We get the simpli ed Point-Sensitive Circumscription formula A(P ) ^ 8p:[p < P ^ A(p)]: 11
For A(P ) = (P (X ) _ P (Y )) ^ X 6= Y , we get the following result (contrary to proposition 3.1 for the Pointwise Circumscription case): Proposition 5.1 CPS [A(P ); P=R; P=x:True] (8x(P (x) () x = X )) _ (8x(P (x) () x = Y )) ^ X 6= Y .
Proof CPS [A(P ); P ; P=x:True] A(P ) ^ 8p:[p < P ^ A(p)] At this point, all we need to notice is that we got McCarthy's Circumscription which is known to have the required equivalence. Let us now examine the situation as seen by the semantics of proposition 4.2. Let U = fx; yg be the set of elements in the universe. We explore the same models discussed in section 3, MX , MY , MXY and M . Figure 1 above displays the di erent models for A(P ). Figure 2 above then describes the two orders x, y between the models. Now we examine the semantics of proposition 4.2. First, we notice that both X ; Y exactly correspond to the $ relation for the respective P 's. Therefore, we get M x MX x MXY and MXY 6 x MX and MX 6 x MY . M x MY x MXY and MXY 6 x MY and MY 6 x MX . Examining the conditions of proposition 4.2 for each one of the models, we reveal the following: M is not a model of C = CPS [A(P ); P ; P=VP ] because it is not a model of A(P ). MXY is not a model of C because MX j= A(P ) and MX MXY . MX is a model of C because MX j= A(P ) and MXY 6 MX and MY 6 MX . MY is a model of C because MY j= A(P ) and MXY 6 MY and MX 6 MY . Thus, again, there are exactly two models that satis es C , one in which P = fX g and the other in which P = fY g. Figure 3 below displays the two orders on models. ;
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Figure 3: The orders X and Y .
6 Applying Point-Sensitive to Theories of Action We examine one application of Pointwise Circumscription for the formalization of Prototypical Chronological Minimization (PCM), which is one known solution to the Frame Problem (for some ontological classes ). Finally, we compare it with the proposed Point-Sensitive Circumscription and with Generalized Global Circumscription (as proposed in [Lif87] (formula (14))). 3
6.1 PCM Using Pointwise Circumscription
Doherty and Lukaszewicz [DL94] examined various solutions to the Frame Problem, implementing them using either McCarthy's Circumscription or Lifschitz's Generalized Pointwise Circumscription (formula 2 above). We brie y review some of the theory of Features and Fluents [San94] and one of the solutions that was examined in [DL94], namely, Prototypical Chronological Minimization (PCM), for which Pointwise Circumscription was used. 3
See [San94].
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6.1.1 The general theory for PCM The language L(FL) is a sorted rst-order language with equality. There are two domain independent sorts, T for time points (t; s are variables of that sort) and F for propositional uents (f; g are variables of that sort), which
are functions from time to truth values. We include the predicate symbols Holds; Clip of type T F and the predicate symbols
