Circumscription and Generic Mathematical Objects - CiteSeerX
Circumscription and Generic Mathematical Objects Leopoldo E. Bertossi
Ponti cia Universidad Catolica de Chile Escuela de Ingenieria Departamento de Ciencia de la Computacion Casilla 306, Santiago 22, Chile e-mail: [email protected]
Raymond Reiter
Department of Computer Science University of Toronto Toronto, Canada M5S 1A4 and The Canadian Institute for Advanced Research e-mail: [email protected] Abstract
We investigate the possibility of using circumscription for characterizing the concept of a generic object in the context of a formalized mathematical theory. We show that conventional circumscriptive policies do not give the intuitively expected results for elementary geometry, and that there is a common explanation for this failure and the failure of circumscription in some standard instances of commonsense reasoning. It turns out, however, that scoped circumscription does provide the right mechanism for this task.
1
1 Generic Mathematical Objects One of the motivations for the development of nonmonotonic logical formalisms was to characterize the concept of a generic or prototypical object as it arises in natural language and commonsense reasoning (see Reiter [13]). However, we also nd this concept in mathematics. The best example is provided by set theory, more speci cally in connection with the method of forcing, where the concept of generic set is a well de ned notion in the theory. Generic sets help us prove certain propositions without using any of their \speci c" properties (see [1]). Nevertheless, the concept of generic object also appears in its intuitive, non-formalized version in the context of a mathematical theory. In geometry, for example, a triangle without any special properties, e.g. right-angled or isosceles, etc. or lines without special positions with respect to a coordinate system, would be considered generic. Some computational systems for mechanical theorem proving in geometry use auxiliary diagrams to guide the proofs of theorems. These diagrams are constructed with genericity in mind. As P.C. Gilmore puts it in his logical reconstruction of Gelernter's Geometry Machine [4]: \One more quali cation is generally added in order not to admit as a diagram a set of points with special relationships that are not part of the premises ..." ([5, page 179]). Our concern in this paper is with formalizing the concept of a generic mathematical object. Speci cally, we explore the possibility of using circumscription (McCarthy [8, 9]) for this purpose. We will restrict ourselves to the case of elementary plane geometry. For this theory, we will show that conventional forms of circumscription are inadequate for de ning the property of a generic triangle. We will show, on the other hand, that scoped circumscription (Etherington, Kraus and Perlis [3]) is exactly what is required.
2 Elementary Geometry and Generic Triangles Let L be a rst-order language for plane geometry. It is well known (see [21], [17] or [11]) that geometry can be nitely axiomatized starting with only two primitive predicates (:; :; :) and (:; :; :; :). Here, (x; y; z) means that point x lies between points y and z; (x; y; u; v) means that the distance between points x and y equals that between points u and v. Let G0 be such a nite 2
axiomatization, say, any of the equivalent axiomatizations 2 in [21] or in [17]. We can de ne in L the predicates O; S1; S2 for \triangle", \isosceles triangle", \right-angled triangle", respectively. Let us denote by DO ; D1; D2 the corresponding axioms. Then, DO is E
O(x; y; z)
00
P
( (x; y; z) (y; z; x) (z; x; y));
:
_
(1)
_
D1 is S1(x; y; z) O(x; y; z) (x; y; y; z);
^
and D2 is
S2(x; y; z) (x; y; z) (y; x; z) (z; x; y);
_
_
where (x; y; z) means \(x; y; z) is a triangle with right angle at x" and can be de ned by the following axiom [17]:
(x; y; z) ( u; v)[u = y u = z v = y v = z (x; y; x; v) (x; z; x; u) (y; z; u; v) (y; z; z; v) (2) (y; z; y; u)]: The objects of interest are the triangles. Singular triangles are the rightangled and isosceles triangles. Our focus is on the \generic" triangles, that is, in the triangles outside S1 S2. Let G be the conjunction of G0 with the de nitions DO , D1, D2 , and let a = (a; b; c) be a triple of constants of L (corresponding to points) that do not appear in G0. We would like a to be a \generic" triangle. The natural rst approach is to circumscribe the singular properties of triangles, that is, to compute
9
6
^
^
^
6
^
6
^
6
^
^
^
[
Circum(G O(a); S1; S2):
(3)
^
Unfortunately, this circumscription does not entail S1(a) S2(a), so that we cannot conclude that a is a generic triangle. In fact, models = (M; : : : ; OM ; S1M ; S2M ; aM ) of G O(a) with aM S1M (or S2M ) are <S1 ;S2 minimal. This is so since the theory contains de nitions of the singular predicates, and the extensions in the model of these predicates are completely :
^:
M
^
2
3
determined by the extension of M and the extensions of the basic predicates. In particular, if 0 is a model of G O(a) and 0 S1 ;S2 , then 0 = . This can be seen syntactically as well since the following is valid: Circum(G O(a); S1; S2) G O(a): This circumscription does not prevent a from belonging to S1 S2. We have seen that there are <S1 ;S2 -minimal models where a is a rightangled triangle (or an isosceles triangle) and others where this is not the case. a is xed in the circumscription (3) above. The possibility of allowing a to vary in the circumscription process (so that it can be accommodated outside the minimized Si's), i.e. considering the circumscription Circum(T ; S1; S2; a); (4) with variable a, does not help either. It can be proved that the minimization in (4) does not depend on a; the same argument preceding this remark applies in this case. Notice that in the circumscriptions above the basic predicates and are xed, and this could be the reason for not obtaining the desired conclusion. The natural question is whether we can do better by allowing and to vary in the circumscription. Since O is de ned in terms of and , we must allow this predicate to vary also. Therefore, we consider the circumscription: M
^
M
^
M
M
M
^
[
Circum(G O(a); S1; S2; ; ; O) (5) We now show that circumscription with variable predicates does not give the desired result either. We will use the predicate (x; y; z) whose intended meaning is \(x; y; z) determine a triangle with right angle at x". As we have seen, this predicate can be de ned in terms of and (formula (2)). Let D be this axiom that de nes . Instead of (5), we will consider the apparently simpler circumscription: Circum(G0 D DO O(a); ; ; ; O) (6) That is, our axioms consist of the usual axioms G0 for geometry, the axiom de ning a class of singular triangles, and the axiom DO de ning \trianglehood" (axiom (1)). By means of circumscription (6) we would like to conclude that (a; b; c) does not form a right-angled triangle, i.e. (a; b; c), (c; a; b) and (b; c; a). This conclusion is impossible: ^
^
^
^
:
:
:
4
Theorem 1 Circum(G0 D DO O(a; b; c); ; ; ; O) = (a; b; c). Proof: There is a model = (M; M ; M ; M ; OM ; aM ; bM ; cM ) of G00 := ^
^
^
6j
:
M
G0 D DO O(a; b; c) with (aM ; bM ; cM ) M . We show that this model is
Ponti cia Universidad Catolica de Chile Escuela de Ingenieria Departamento de Ciencia de la Computacion Casilla 306, Santiago 22, Chile e-mail: [email protected]
Raymond Reiter
Department of Computer Science University of Toronto Toronto, Canada M5S 1A4 and The Canadian Institute for Advanced Research e-mail: [email protected] Abstract
We investigate the possibility of using circumscription for characterizing the concept of a generic object in the context of a formalized mathematical theory. We show that conventional circumscriptive policies do not give the intuitively expected results for elementary geometry, and that there is a common explanation for this failure and the failure of circumscription in some standard instances of commonsense reasoning. It turns out, however, that scoped circumscription does provide the right mechanism for this task.
1
1 Generic Mathematical Objects One of the motivations for the development of nonmonotonic logical formalisms was to characterize the concept of a generic or prototypical object as it arises in natural language and commonsense reasoning (see Reiter [13]). However, we also nd this concept in mathematics. The best example is provided by set theory, more speci cally in connection with the method of forcing, where the concept of generic set is a well de ned notion in the theory. Generic sets help us prove certain propositions without using any of their \speci c" properties (see [1]). Nevertheless, the concept of generic object also appears in its intuitive, non-formalized version in the context of a mathematical theory. In geometry, for example, a triangle without any special properties, e.g. right-angled or isosceles, etc. or lines without special positions with respect to a coordinate system, would be considered generic. Some computational systems for mechanical theorem proving in geometry use auxiliary diagrams to guide the proofs of theorems. These diagrams are constructed with genericity in mind. As P.C. Gilmore puts it in his logical reconstruction of Gelernter's Geometry Machine [4]: \One more quali cation is generally added in order not to admit as a diagram a set of points with special relationships that are not part of the premises ..." ([5, page 179]). Our concern in this paper is with formalizing the concept of a generic mathematical object. Speci cally, we explore the possibility of using circumscription (McCarthy [8, 9]) for this purpose. We will restrict ourselves to the case of elementary plane geometry. For this theory, we will show that conventional forms of circumscription are inadequate for de ning the property of a generic triangle. We will show, on the other hand, that scoped circumscription (Etherington, Kraus and Perlis [3]) is exactly what is required.
2 Elementary Geometry and Generic Triangles Let L be a rst-order language for plane geometry. It is well known (see [21], [17] or [11]) that geometry can be nitely axiomatized starting with only two primitive predicates (:; :; :) and (:; :; :; :). Here, (x; y; z) means that point x lies between points y and z; (x; y; u; v) means that the distance between points x and y equals that between points u and v. Let G0 be such a nite 2
axiomatization, say, any of the equivalent axiomatizations 2 in [21] or in [17]. We can de ne in L the predicates O; S1; S2 for \triangle", \isosceles triangle", \right-angled triangle", respectively. Let us denote by DO ; D1; D2 the corresponding axioms. Then, DO is E
O(x; y; z)
00
P
( (x; y; z) (y; z; x) (z; x; y));
:
_
(1)
_
D1 is S1(x; y; z) O(x; y; z) (x; y; y; z);
^
and D2 is
S2(x; y; z) (x; y; z) (y; x; z) (z; x; y);
_
_
where (x; y; z) means \(x; y; z) is a triangle with right angle at x" and can be de ned by the following axiom [17]:
(x; y; z) ( u; v)[u = y u = z v = y v = z (x; y; x; v) (x; z; x; u) (y; z; u; v) (y; z; z; v) (2) (y; z; y; u)]: The objects of interest are the triangles. Singular triangles are the rightangled and isosceles triangles. Our focus is on the \generic" triangles, that is, in the triangles outside S1 S2. Let G be the conjunction of G0 with the de nitions DO , D1, D2 , and let a = (a; b; c) be a triple of constants of L (corresponding to points) that do not appear in G0. We would like a to be a \generic" triangle. The natural rst approach is to circumscribe the singular properties of triangles, that is, to compute
9
6
^
^
^
6
^
6
^
6
^
^
^
[
Circum(G O(a); S1; S2):
(3)
^
Unfortunately, this circumscription does not entail S1(a) S2(a), so that we cannot conclude that a is a generic triangle. In fact, models = (M; : : : ; OM ; S1M ; S2M ; aM ) of G O(a) with aM S1M (or S2M ) are <S1 ;S2 minimal. This is so since the theory contains de nitions of the singular predicates, and the extensions in the model of these predicates are completely :
^:
M
^
2
3
determined by the extension of M and the extensions of the basic predicates. In particular, if 0 is a model of G O(a) and 0 S1 ;S2 , then 0 = . This can be seen syntactically as well since the following is valid: Circum(G O(a); S1; S2) G O(a): This circumscription does not prevent a from belonging to S1 S2. We have seen that there are <S1 ;S2 -minimal models where a is a rightangled triangle (or an isosceles triangle) and others where this is not the case. a is xed in the circumscription (3) above. The possibility of allowing a to vary in the circumscription process (so that it can be accommodated outside the minimized Si's), i.e. considering the circumscription Circum(T ; S1; S2; a); (4) with variable a, does not help either. It can be proved that the minimization in (4) does not depend on a; the same argument preceding this remark applies in this case. Notice that in the circumscriptions above the basic predicates and are xed, and this could be the reason for not obtaining the desired conclusion. The natural question is whether we can do better by allowing and to vary in the circumscription. Since O is de ned in terms of and , we must allow this predicate to vary also. Therefore, we consider the circumscription: M
^
M
^
M
M
M
^
[
Circum(G O(a); S1; S2; ; ; O) (5) We now show that circumscription with variable predicates does not give the desired result either. We will use the predicate (x; y; z) whose intended meaning is \(x; y; z) determine a triangle with right angle at x". As we have seen, this predicate can be de ned in terms of and (formula (2)). Let D be this axiom that de nes . Instead of (5), we will consider the apparently simpler circumscription: Circum(G0 D DO O(a); ; ; ; O) (6) That is, our axioms consist of the usual axioms G0 for geometry, the axiom de ning a class of singular triangles, and the axiom DO de ning \trianglehood" (axiom (1)). By means of circumscription (6) we would like to conclude that (a; b; c) does not form a right-angled triangle, i.e. (a; b; c), (c; a; b) and (b; c; a). This conclusion is impossible: ^
^
^
^
:
:
:
4
Theorem 1 Circum(G0 D DO O(a; b; c); ; ; ; O) = (a; b; c). Proof: There is a model = (M; M ; M ; M ; OM ; aM ; bM ; cM ) of G00 := ^
^
^
6j
:
M
G0 D DO O(a; b; c) with (aM ; bM ; cM ) M . We show that this model is
