Conservativity for a hierarchy of Euler and Venn reasoning systems

Transcription

1 onservativity for a hierarchy of uler and Venn reasoning systems Koji Mineshima, Mitsuhiro Okada, and Ryo Takemura Department of Philosophy, Keio University, Mita, Minato-ku, Tokyo , Japan. {minesima,mitsu,takemura}@aelard.flet.keio.ac.jp stract This paper introduces a hierarchy of uler and Venn diagrammatic reasoning systems in terms of their expressive powers in topological-relation-ased formalization. t the ottom of the hierarchy is the uler diagrammatic system introduced in Mineshima-Okada-Sato-Takemura [13, 12], which is expressive enough to characterize syllogistic reasoning in terms of unification and deletion rules. t the top of the hierarchy is a Venn diagrammatic system such as Swooda-llwein s uler/venn diagrammatic system [23]. In order to understand the hierarchy uniformly, we introduce an algeraic structure, which also provides another description of our unification rule of uler diagrams. We prove that each system S of the hierarchy is conservative over any lower system S with respect to validity in the sense that S is an extension of S, and the semantic consequence relations of S and S are equivalent for diagrams of S. Furthermore, we prove that a region-ased Venn diagrammatic system is conservative over our topological-relation-ased uler diagrammatic system with respect to provaility. 1 Introduction uler diagrams were introduced y uler (1768) [1] to represent logical relations among the terms of a syllogism y topological relations among circles. Given two uler diagrams which represent the premises of a syllogism, the syllogistic inference can e naturally replaced y the task of manipulating the diagrams, in particular of unifying the diagrams and extracting information from them. For example, the wellknown syllogism named arara, i.e., ll are and ll are ; therefore ll are, can e represented diagrammatically as in Fig. 1. nother well-known diagrammatic representation system for syllogistic reasoning is Venn diagrams. In Venn diagrams a novel syntactic device, namely shading, to represent emptiness plays a central role in place of the topological relations of uler diagrams. ecause of their expressive power and their uniformity in formalizing the manipulation of comining diagrams simply as the superposition of shadings, Venn diagrams have een very well studied. f. Venn-I, II systems of Shin [19], Spider 37

2 D 1 D 2 D v 1 Dv 2 Fig.1 arara with uler diagrams v Fig.2 arara with Venn diagrams diagrams SD1 and SD2 of [9], [14], etc. For a recent survey, see [20]. However, the development of systems of Venn diagrams is otained at the cost of clarity of uler diagrams. s Venn [25] himself already pointed out, when more than three circles are involved, Venn diagrams fail in their main purpose of affording intuitive and sensile illustration. (For some discussions on visual disadvantages of Venn diagrams, see [8, 5]. See also [18] for our cognitive psychological experiments comparing linguistic, uler diagrammatic, and Venn diagrammatic representations.) Recently, uler diagrams with shading were introduced to make up for the shortcoming of Venn diagrams:.g., uler/venn diagrams of [23, 24]; Spider diagrams SD2 of [14] and SD3 of [10]. However, their astract syntax and semantics are still defined in terms of regions, where shaded regions of Venn diagrams are considered as missing regions. That is, the idea of the region-ased uler diagrams is essentially along the same line as Venn diagrams. We may point out the following complications of region-ased formalization of diagrams: 1. In region-ased diagrams, logical relations among circles are represented not simply y topological relations, ut y the use of shading or missing regions, which makes the translations of categorical sentences uncomfortaly complex. For example, ll are is expressed y a region-ased diagram through a translation to the statement There is nothing which is ut not as seen in D v 1 of Fig The inference rule of unification, which plays a central role in uler diagrammatic reasoning, is defined y way of the superposition of Venn diagrams. For example, when we unify two region-ased uler diagrams as in D 1 and D 2 of Fig. 1, they are first transformed into Venn diagrams D v 1 and D v 2 of Fig. 2, respectively; then, y superposing the shaded regions of D v 1 and D v 2, and y deleting the circle, the Venn diagram v is otained, which is transformed into the region-ased uler diagram. In this way, processes of deriving conclusions are 38

3 often made complex, and hence less intuitive, in the region-ased framework. In contrast to the studies in the tradition of region-ased diagrams, we proposed a novel approach in [13, 12] to formalize uler diagrams in terms of topological relations. Our system has the following features and advantages: 1. Our diagrammatic syntax and semantics are defined in terms of topological relations, inclusion and exclusion relations, etween two diagrammatic ojects. This formalization makes the translations of categorical sentences natural and intuitive. Furthermore, our formalization makes it possile to represent a diagram y a simple ordered (or graph-theoretical) structure. 2. Our unification of two diagrams is formalized directly in terms of topological relations without making a detour to Venn diagrams. Thus, it can directly capture the inference process as illustrated in Fig. 1. We formalize the unification in the style of Gentzen s natural deduction, a well-known formalization of logical reasoning in symolic logic, which is intended to e as close as possile to actual reasoning (Gentzen [3]). This makes it possile to compare our uler diagrammatic inference system directly with natural deduction system. Through such comparison, we can apply well-developed proof-theoretical approaches to diagrammatic reasoning. See [13] for such proof-theoretical analyses. From a perspective of proof-theory, the contrast etween the standpoints of the region-ased framework and the topological-relation-ased framework can e understood as follows: t the level of representation, the contrast is analogous to the one etween disjunctive (dually, conjunctive) normal formulas and implicational formulas; at the level of reasoning, the contrast is analogous to the one etween resolution calculus style proofs and natural deduction style proofs. From a perspective of cognitive psychology, our system is designed not just as an alternative of usual linguistic/symolic representations; we make the est use of advantages of diagrammatic representations so that inherent definiteness or specificity of diagrams can e exploited in actual reasoning. See [18] for our experimental result, which shows that our uler diagrams are more effective than Venn diagrams or linguistic representations in syllogism solving tasks. We roughly review our topological-relation-ased uler diagrammatic representation system UL in Section 2. (We also review our inference system GDS in ppendix.) lthough UL is weaker in its expressive power than usual Venn diagrammatic systems (e.g. Shin s Venn-II [19], which is equivalent to the monadic first order logic in its expressive power), UL is expressive enough to characterize asic logical reasoning such as syllogistic reasoning, see [12]. Our UL-diagrams can e astractly seen as algeraic (or graph-theoretical) structure, where inclusion relations etween diagrammatic ojects are reflexive transitive ordering relations, and exclusion relations are irreflexive symmetric relations. ased on this oservation, in Section 3, we introduce UL-structure, which provides another description and a verification of our unification rule of ppendix. In Section 4, ased on the UL-structure, we introduce a hierarchy of uler and Venn diagrammatic reasoning systems as seen in Fig. 3. The most elementary system UL considers only circles and points as diagrammatic ojects; UL is extended y considering intersection regions Y, union regions 39

4 UL with Venn UL with,, UL with, UL UL with Fig.3 Hierarchy of uler and Venn systems Y, and complement regions as diagrammatic ojects, respectively, as well as linking of points; Venn diagrams can e put at the top of the hierarchy of these extended systems. The algeraic structure thus otained for Venn diagrams is essentially the directed acyclic graph of Swooda-llwein [23]. We prove that each system S of the hierarchy is conservative over any lower system S with respect to validity in the sense that S is an extension of S, and the semantic consequence relations of S and S are equivalent for diagrams of S. Moreover, we prove that a region-ased Venn diagrammatic system is conservative over our topological-relation-ased uler diagrammatic system with respect to provaility. We also give a procedure to transform a topological-relation-ased UL-diagram through an UL-structure to a semantically equivalent region-ased Venn diagram. 2 diagrammatic representation system (UL) for uler circles and its set-theoretical semantics In this section, we review our diagrammatic representation system UL of [13, 12]. 2.1 Diagrammatic syntax of UL The following definition of diagrams is slightly different from that of [13, 12] in that (1) we regard inclusion relation as reflexive in this paper; (2) we exclude from ULdiagrams only the empty diagram, on which no topological relation holds. Definition 2.1 n UL-diagram is a plane (R 2 ) with a finite numer, at least one, of named simple closed curves (denoted y,,,... ) and named points (denoted y a,, c,... ), where each named simple closed curve or named point has a unique and distinct name. UL-diagrams are denoted y D,, D 1, D 2,.... n UL-diagram consisting of at most two ojects is called a minimal diagram. Minimal diagrams are denoted y α, β, γ,.... In what follows, a named simple closed curve is called a named circle. Moreover, named circles and named points are collectively called ojects, and denoted y s, t, u,.... We use a rectangle to represent a plane for an UL-diagram. 1 1 Several uler diagrammatic representation systems impose some additional conditions for well-formed diagrams..g., at most two circles meet at a single point, no tangential meetings or concurrency etc. f. e.g., [22]. However, for simplicity of the definition, those are all considered to e well-formed in UL. 40

5 We define the following inary topological relations etween diagrammatic ojects 2 : Definition 2.2 UL-relations are the following inary relations etween diagrammatic ojects: the interior of is inside of the interior of, a the interior of is outside of the interior of, there is an intersection etween the interior of and the interior of, is inside of the interior of, is outside of the interior of, a is outside of (i.e. a is not located at the point of ). We call -relation crossing relation. UL-relations and are symmetric, while is not. In this paper, we consider -relation as reflexive y allowing s s for each oject s. Proposition 2.3 Let D e an UL-diagram. For any distinct ojects s and t of D, exactly one of the UL-relations s t, t s, s t, s t holds. Oserve that, y Proposition 2.3, for a given UL-diagram D, the set of ULrelations holding on D is uniquely determined. We denote the set y rel(d). We also denote y pt(d) the set of named points of D, y cr(d) the set of named circles of D, and y o(d) the set of ojects of D. lthough in this section, o(d) = pt(d) cr(d), in Section 4, diagrammatic ojects are extended, in addition to named circles and points, y introducing intersection, union, and complement regions respectively. The following properties, as well as Proposition 2.3, characterize UL-diagrams. Lemma 2.4 Let D e an UL-diagram. Then for any ojects (named circles or points) s, t, u o(d), we have the following: 1. (Transitivity) If s t, t u rel(d), then s u rel(d). 2. ( -downward closedness) If s t, u s rel(d), then u t rel(d). 3. (Point determinacy) For any point x of D, exactly one of x s and x s is in rel(d). 4. (Point minimality) For any point x ( s) of D, s x rel(d). quivalence etween UL-diagrams is defined as follows. (See [13] for a more detailed explanation.) Definition 2.5 When any two ojects of the same name appear in different diagrams (planes), we identify them up to isomorphism. ny UL-diagrams D and such that o(d) = o() are syntactically equivalent when rel(d) = rel(). 2 We follow Gergonne [4] for our notations on topological relations and. 41

6 xample 2.6 (quivalence of diagrams) For example, diagrams D 1, D 2, D 3, and D 4 of Fig. 4 are equivalent since rel(d 1 ) = rel(d 2 ) = rel(d 3 ) = rel(d 4 ) = {,,, a, a, a }. In the description of a set of relations, we usually omit the reflexive relation s s for each oject s. a a a a a a D 1 D 2 D 3 D 4 D 5 D 6 Fig.4 quivalence of UL-diagrams. On the other hand, D 1 and D 5 (resp. D 1 and D 6 ) are not equivalent since different UL-relations hold on them: holds on D 5 in place of of D 1 (resp. and hold on D 6 in place of and of D 1 ). f. xample 4.5 and 4.7 of Section 4, where D 1, D 2, D 3, and D 4 are distinguished. Our equation of diagrams may e explained in terms of a kind of continuous transformation (deformation) of named circles, which does not change any of the ULrelations in a diagram. (See [13] for an explanation.) In what follows, the diagrams which are syntactically equivalent are identified, and they are referred y a single name. Remark 2.7 (xpressive power of UL) Our equation of diagrams in the asic system UL may seem to e counterintuitive since seemingly distinctive diagrams D 1, D 2, D 3, D 4 of xample 2.6 are identified. 3 However, this slightly rough equation makes the description of unification of diagrams much simpler; see ppendix. Furthermore, it is shown that UL is expressive enough to characterize asic logical reasoning such as syllogistic reasoning; see [12]. In Section 4, we consider some extensions of UL, where D 1, D 2, D 3, and D 4 are distinguished y regarding intersection and union regions respectively as diagrammatic ojects. See, in particular, xamples 4.5 and 4.7. Note that, y introducing new diagrammatic ojects in a representation system, UL-relations for these new ojects are augmented, so that the system ecomes more expressive. t the level of diagrammatic syntax, this means that more fine-grained distinctions etween diagrams are made possile. 2.2 Set-theoretical semantics of UL Our semantics is distinct from usual ones, e.g., [6, 8, 24, 10] in that diagrams are interpreted in terms of inary relations. In order to interpret the UL-relations and uniformly as the suset relation and the disjointness relation, respectively, we regard each point of UL as a special circle which does not contain, nor cross, any other ojects. 3 This is also pointed out in Fish-Flower [2] as an drawack of the relation-ased approach. 42

7 Definition 2.8 model M is a pair (U, I), where U is a non-empty set (the domain of M), and I is an interpretation function which assigns to each named circle or point a non-empty suset of U such that I(a) is a singleton for any named point a, and I(a) I() for any points a, of distinct names. Note that we assign a non-empty set to each named circle. This condition is essential for our completeness. See the paragraph on the constraint for consistency in ppendix and footnote 7 there. Definition 2.9 Let D e an UL-diagram. M = (U, I) is a model of D, written as M = D, if the following truth-conditions (1) and (2) hold: For all ojects s, t of D, (1) I(s) I(t) if s t holds on D, and (2) I(s) I(t) = if s t holds on D. Note that when s is a named point a, for some e U, I(a) = {e}, and the aove I(a) I(t) of (1) is equivalent to e I(t). Similarly, I(a) I(t) = of (2) is equivalent to e I(t). Remark 2.10 (Semantic interpretation of -relation) y Definition 2.9, the ULrelation does not contriute to the truth-condition of UL-diagrams. Informally speaking, s t may e understood as I(s) I(t) = or I(s) I(t), which is true in any model. f. also Remark 2.7. Definition 2.11 n UL-diagram is a semantically valid consequence of ULdiagrams D 1,..., D n, written as D 1,..., D n =, when the following holds: For any model M, if M = D 1 and... and M = D n, then M =. See ppendix and [13] for our Generalized Diagrammatic Syllogistic inference system GDS, whose completeness holds with respect to the semantics of this section. 3 UL-structure In this section, we introduce an algeraic structure called UL-structure for ULdiagrams. Definition 3.1 n UL-structure (D, p(d),, ) is a partially ordered structure, where D is a set whose cardinality #D 1, and p(d) D: 1. is a reflexive transitive ordering relation on D. 2. is an irreflexive symmetric relation on D. 3. ( -downward closedness) For any s, t, u D, s t and t u imply s u. 4. (Point determinacy) For any s D and x p(d), x s or x s. 5. (Point minimality) For any s D and x p(d) such that s x, not(s x). 43

8 f. Lemma 2.4. Oserve that the aove properties (i), (ii), and (iii) imply that, for any distinct pair of elements of D, at most one of the relations and holds (cf. Proposition 2.3); ecause if oth of them hold, say s t and s t, the property (iii) implies s s, which contradicts the irreflexivity of -relation. 4 s seen in Section 2.1, given an UL-diagram D, the set rel(d) of relations holding on it is uniquely determined y Proposition 2.3. rel(d) can e regarded as an ULstructure. Proposition 3.2 Let D e an UL-diagram. The set of UL-relations rel(d) gives rise to an UL-structure (o(d), pt(d),, ). For example, rel(d 1 ), rel(d 5 ) and rel(d 6 ) of Fig. 4 in xample 2.6 are expressed graphically as follows: Here the ordering relations are expressed y -edges. a rel(d 1 ) a rel(d 5 ) a rel(d 6 ) Oserve that there is no edge for -relation. Now we descrie the unification rule of Definition.1 of ppendix in terms of a graph-theoretical representation of UL-diagrams, which may assist with the understanding and motivation of our unification rule. Proposition 3.3 Let D e an UL-diagram, and α e a minimal diagram. The set of UL-relations rel(d + α), which is otained y unifying D and α, gives rise to an UL-structure. Proof. In order to descrie graphically the unification of UL-diagrams D and α, we focus on the shared oject of D and α, say, and express the UL-structure of rel(d) as follows: -edge denotes -relation -edge denotes -relation Y/y W No edge for -relation Z/z denotes one of,,, rel(d) The variales, Y, Z, W (resp. y, z) are representative circles (resp. points) which are possily related to. When it makes no difference whether a possily related oject is circle or point, we denote the oject as Y/y (instead of simply writing s). ach dotted line etween ojects expresses that there may e one of the relations,,, etween the ojects. Note that there is no edge for each -relation, as seen etween and W. We omit the trivial transitive edge Z to avoid notational complexity. In the following description of each unification rule for D and α, we give a graphical 4 Note that, y the properties (i) (iii), an UL-structure (D, p(d),, ) is an event structure of Nielsen- Plotkin-Winskel [15]. 44

9 representation of the UL-structures of rel(d) in the left-hand graph, and rel(d +α) in the right-hand graph. We egin with U3-rule since U1 and U2 rules are rather untypical cases: U3 Under the constraint of U3-rule, there is no circle Z such that Z holds, and no circle W such that W holds, which is expressed y in the graph of rel(d). ccording to U3-rule of Definition.1, rel(d + ( )) is represented y the graph on the right. Y/y W Z/z Y/y z rel(d) rel(d + ( )) It is easily seen that rel(d + ( )) is an UL-structure: I.e., the augmented edges do not violate the properties of UL-structure. Note also that, without the constraint, i.e., if there is a circle Z or W as aove, in order to preserve Point determinacy, we should fix a relation etween and Z (resp. W ) to or. However, neither of them is sound with respect to our formal semantics of UL. U4 Under the constraint of U4-rule, there is no circle such that holds, no circle Y such that Y holds, and no circle W such that W holds, which is expressed y in the graph of rel(d). ccording to U4-rule of Definition.1, rel(d + ( )) is represented y the right hand graph elow. Y/y W Z/z y Z/z rel(d) rel(d + ( )) It is easily seen that rel(d + ( )) is an UL-structure: I.e., the augmented edges do not violate the properties of UL-structure. Without the constraint, i.e., if there is a circle, Y or W as aove, in order to preserve Point determinacy, we should fix a relation etween and (resp. Y, W ) to or in rel(d + ( )). However, none of them is sound with respect to our semantics of UL. U5 Under the constraint of U5-rule, there is no point z such that z holds. ccording to U5-rule of Definition.1, rel(d + ( )) is represented y the right hand graph elow. 45

10 Y/y Z/z rel(d) W Y/y W Z rel(d + ( )) Without the constraint, i.e., if there is a point z as aove, in order to preserve Point determinacy, we should fix a relation etween z and to or. However, none of them is sound with respect to our semantics of UL. U6 Under the constraint of U6-rule, there is no point y such that y holds. ccording to U6-rule of Definition.1, rel(d + ( )) is represented y the right hand graph elow. Y/y Z/z W Y Z/z W rel(d) rel(d + ( )) Without the constraint, i.e., if there is a point y as aove, in order to preserve Point determinacy, we should fix a relation etween y and to or. However, none of them is sound with respect to our semantics of UL. U7 Under the constraint of U7-rule, there is no point y such that y holds. ccording to U7-rule of Definition.1, rel(d + ( )) is represented y the right hand graph elow. Y/y Z/z W Y Z/z W rel(d) rel(d + ( )) Without the constraint, i.e., if there is a point y as aove, in order to preserve Point determinacy, we should fix a relation etween y and to or. However, none of them is sound with respect to our semantics of UL. U8 Under the constraint of U8-rule, there is no point in D. ccording to U8-rule of Definition.1, rel(d + ( )) is represented y the right hand graph elow. Y/y Z/z W Y Z W rel(d) rel(d + ( )) 46

11 Without the constraint, i.e., if there is a point y or z as aove, in order to preserve Point determinacy, we should fix a relation etween y (resp. z) and to or. However, none of them is sound with respect to our semantics of UL. U1 Under the constraint of U1-rule, there is no point y in D other than. ccording to U1-rule, rel(d + ( )) is represented y the right hand graph elow. Y/y Y rel(d) rel(d + ( )) Without the constraint, i.e., if there is a point y as aove, in order to preserve Point determinacy, we should fix a relation etween y and to or. However, none of them is sound with respect to our semantics of UL. U2 Under the constraint of U2-rule, there is no point y in D other than. ccording to U2-rule, rel(d + ( )) is represented y the right hand graph elow. Y/y Y rel(d) rel(d + ( )) Without the constraint, i.e., if there is a point y as aove, in order to preserve Point determinacy, we should fix a relation etween y and to or. However, none of them is sound with respect to our semantics of UL. In U9, U10 rules of Definition.1, the unified diagrams D and α share two circles, which makes the graphical description of rel(d) complicated. In order to avoid notational complexity, we omit irrelevant ojects and edges, which are retained after the application of U9 and U10 rule, respectively. U9 Under the constraint of U9-rule, there is no oject s such that s and s hold on D, i.e., in the following description of rel(d), the dotted line etween Y/y and should not e, and the dotted line etween Z/z and should not e. ccording to U9-rule of Definition.1, rel(d + ( )) is represented y the right hand graph elow. Z/z Y/y Y/y Z/z rel(d) rel(d + ( )) 47

12 Oserve that, after the unification, some of the dotted lines of rel(d) are fixed to or in rel(d + ( )) according to Definition.1. We need to check that rel(d + ( )) is an UL-structure; for example, if the dotted line etween and in rel(d) is (or ), after the application of U9-rule, there are two incompatile edges (resp. ) and etween and, which violates the irreflexivity of the -relation of UL-structure. It is shown that, ecause of our constraint for U9-rule, the dotted line etween and is (i.e., no edge) or. Oserve that, if we have in rel(d), y the downward closedness of rel(d), we have Z/z in rel(d), which contradicts the constraint. If we have in rel(d), y the transitivity of rel(d), we have in rel(d), which contradicts the presupposition of U9-rule, i.e., there is no edge etween and in rel(d). Thus the dotted line etween and should e (i.e., no edge) or, either of which is compatile with the edge in rel(d + ( )). Similarly, it is shown that the other dotted lines of rel(d) are compatile with the edges of rel(d + ( )). Then it is easily checked that rel(d + ( )) satisfies Definition 3.1, and hence it is an UL-structure. U10 Under the constraint of U10-rule, there is no oject s such that s and s hold on D, i.e., in the following graph of rel(d), the dotted line etween Z /z and (and also etween Z/z and ) should not e. ccording to U10-rule, rel(d + ( )) is represented y y the right hand graph elow. Z/z rel(d) Z /z Z/z Z /z rel(d + ( )) We show that there are no incompatile edges in rel(d + ( )). For the dotted line etween Z/z and, it is not y the constraint for U10-rule. Furthermore, assume to the contrary that we have Z/z in rel(d). Then, y the transitivity of rel(d), we have in rel(d), which contradicts the presupposition of U10-rule, i.e., there is no edge etween and. Hence the dotted line etween Z/z and should e (i.e., no edge) or, either of which is compatile with the edge Z/z in rel(d + ( )). Similarly, it is shown that the other two dotted lines of rel(d) are compatile with the edges of rel(d + ( )). Then it is easily checked that rel(d + ( )) satisfies Definition 3.1, and hence it is an UL-structure. For a given UL-structure (D, p(d),, ), it can e shown that there is an ULdiagram D such that rel(d) is equivalent to (D, p(d),, ). 4 hierarchy of UL-diagrams and Venn diagrams The representation system UL is extended y introducing new diagrammatic ojects, intersection, union, and complement regions, respectively. xtended systems are strat- 48

13 ified in terms of their expressive powers. In what follows, we do not mention named points explicitly, since any named point of UL can e regarded as a special circle, which does not contain, nor cross, any other ojects. If we allow a point (as a special circle) to cross other circles, it amounts to adopting linking etween points, although it is slightly restricted compared with usual linking as in Shin [19], among others. 5 We first extend UL y considering intersection regions as diagrammatic ojects. Regions of an UL-diagram are defined recursively as usual, which are closed under intersection, union, and complement, provided that each is non-empty in a diagram. See, e.g., [10]. Definition 4.1 non-empty region r of an UL-diagram D is an intersection region when, for some { 1,..., n } cr(d), r = 1 i n in( i), where in( i ) is the interior of circle i. n UL-diagrams with intersections D is an UL-diagram where each intersection region r = 1 i n in( i) has the name 1 i n i, which is sometimes denoted y n for short. (In particular when n = 1, 1 = 1.) Note that, in an UL-diagram with intersections, a region may have two names: For example, when holds on D, circle has another name,. We define an algeraic structure for UL-diagrams with intersections. Definition 4.2 n UL-structure with greatest lower ounds (gls) (D,,, ) is an UL-structure, where for any suset { 1,..., n } D such that 1 j,k n ( j k holds on D), there is the greatest lower ound 1 i n i. lthough we regard named points as special named circles, the operation is not applied to them. Lemma 4.3 Let D e an UL-diagram with intersections. The set of relations rel(d) gives rise to an UL-structure with gls. Lemma 4.4 (UL UL+ ) Let (D,, ) e an UL-structure. It is extended, y introducing gls, to an UL-structure with gls (D,,, ). Proof. The domain D is defined as follows: D := D { 1 i n i 1 j,k n ( j k holds on D)} and relations on D are preserved on D and they are extended for any 1 i n i D as follows: Let, Y D. n n n iff i for all 1 i n n iff i for some 1 i n Y iff Y D 5 We exclude a crossing relation c d etween distinct named points, since it amounts to c = d or c d (cf. Remark 2.10) ut we always assume c d in our framework. 49

14 It is immediate that thus constructed (D,,, ) is an UL-structure, which satisfies Definition 3.1, and n is the gl of Definition 4.2. See also xample When D is an UL-diagram, we denote D an UL-diagram with intersections whose algeraic structure is constructed from the UL-structure rel(d) y Lemma 4.4. We say that the diagram D is otained from D. y introducing intersection regions as diagrammatic ojects, UL with intersections are more expressive than the asic UL of Section 2.1. Let us see the following example. xample 4.5 (UL-diagrams with intersections) The three diagrams D 1, D 2, and D 3 of Fig. 4 in xample 2.6, which are identified in the original UL, are distinguished when they are regarded as UL-diagrams with intersections. The difference among the three diagrams is more clearly seen y drawing their UL-structures with gls. (Here, for reasons of simplicity, we omit the point a and areviate -relation y stipulating that Y holds when Y rel(d ).) rel(d 1 ) = rel(d 2 ) rel(d 3 ) In a similar way as intersections, y considering union regions as diagrammatic ojects we have another extension of UL. Definition 4.6 n UL-diagrams with unions D is an UL-diagram where each union region r = 1 i n in( i) has the name 1 i n i, provided that it is connected. UL-structures with least upper ounds (lus) for UL-diagrams with unions are defined in a similar way as UL-structures with gls. UL with unions is also more expressive than UL. xample 4.7 (UL-diagrams with unions) D 1 and D 4 of Fig. 4 in xample 2.6 are distinguished when they are regarded as UL-diagrams with unions. The ULstructures with lus for these two diagrams are represented y the following different structures. rel(d1 ) = rel(d 4 ) 50

15 Definition 4.8 n UL-diagram with intersections and unions D is an ULdiagram with intersections where union regions also have names. Note that we only consider intersection (resp. union) regions of circles, and we exclude other regions such as ( ) ( D). UL-structure with gls and lus are defined y comining UL-structure with gls and UL-structure with lus. y considering the complement region of each circle as a diagrammatic oject, we further introduce UL-diagrams with intersections, unions, and complements. Definition 4.9 n UL-diagram with intersections, unions, and complements D is an UL-diagram with intersections and unions, where each complement of a circle, i.e., the exterior of, has the name. UL-structures for UL-diagrams with,, are defined as follows. Definition 4.10 n UL-structure with gls, lus, and complements (D,,,,, ) is an UL-structure with gls and lus (D,,,, ) where, for each D which is not of the form j nor j (j 2), the complement of is defined in D. lthough we regard named points as special named circles, the operations,, and are not applied to points. Lemma 4.11 Let D e an UL-diagram with,,. The set of relations rel(d) gives rise to an UL-structure with gls, lus, and complements. Lemma 4.12 (UL+ + UL+ + + ) Let (D,,,, ) e an ULstructure with gls and lus. It is extended, y introducing complements, to an ULstructure with gls, lus, and complements (D c,,,,, ). Proof. The domain D c is defined y adding complement for each D which is not of the form j nor j (j 2), and y extending gls (of the form ( j ) ( i )) and lus (of the form ( j ) ( i )) in a similar way as Lemma 4.4. and relations on D are preserved on D c and they are extended as follows: For any, D c not of the form j nor j (j 2), iff in D and iff in D For any, Y D c of the form ( j ) ( i ) (resp. ( j ) ( i )), and relations are extended to e closed under and in a similar way as Lemma 4.4. See also xample uler/venn diagrams of Swooda-llwein [23] are otained y adding shading of minimal regions and linking of points to UL-diagrams with,,. 6 6 There are some differences etween our system and Swooda-llwein s system: (i) we allow one circle to cross with another circle any numer of times; (ii) we allow union regions as diagrammatic ojects, which does not increase expressive power as compared to Swooda-llwein s system; (iii) we do not allow a circle to e completely shaded given our definition of semantics, where each circle denotes a non-empty set. 51

16 UL-structures for uler/venn diagrams, which we call Venn-structures, are the directed acyclic graphs DGs of Swooda-llwein [23]. Lemma 4.13 (UL+ + + Venn) Let (D c,,,,, ) e an ULstructure with gls, lus, and complements. It is extended to a Venn-structure D v of Swooda-llwein [23] y introducing shading and linking. When D is an UL-diagram, we denote y D v (resp. D, D, D, D c ) an uler/venn diagram (resp. UL-diagram with intersections, unions, intersections and unions, intersections and unions and complements) whose algeraic structure is constructed from the UL-structure rel(d) y Lemma 4.4, 4.12, and We say that the diagram D v (resp. D, D, D, D c ) is otained from D. Various extensions of UL introduced so far can e summarized y the following UL-hierarchy: Venn UL with,, UL with, UL with UL UL with Fig.5 UL-hierarchy Note that the semantics of UL of Section 2.2 is essentially the same as the semantics of Venn diagrams (e.g. [10, 19]), where the interpretation function I of circles is naturally extended to interpret regions: I( i ) = I( i ), I( i ) = I( i ), and I() = U \ I(). Note that the denotations of intersections, unions, and complements are not assumed to e non-empty, while those of circles and points are non-empty. Thus when D is a diagram otained from an UL-diagram D for {,,, c, v}, D and D are semantically equivalent since any relation of D is preserved in D y constructions given in Lemmas 4.4, 4.12, and 4.13: Lemma 4.14 Let D e an UL-diagram. For each {,,, c, v}, let D e a diagram otained from D. For any model M, M = D if and only if M = D. ased on Lemma 4.14, it is shown that each system of UL-hierarchy is conservative over any lower system with respect to validity. We denote y D a sequence of diagrams D 1,..., D n. Proposition 4.15 (Semantic conservativity) Let S and S e any systems of the ULhierarchy such that S is an extension of S. Let D, e diagrams of S, and D, e diagrams of S otained from D, for {,,, c, v}. Then D = iff D =. 52

17 In parallel to the extensions of representation system UL, we can otain extended inference systems of GDS of ppendix. It can e shown that each extended system is a conservative extension of the most elementary GDS with respect to provaility. In particular, for uler/venn diagrammatic inference system of Swooda-llwein [24], we have the following conservativity theorem: Theorem 4.16 (onservativity) Let D and e UL-diagrams such that D has a model. If v is provale from D v in uler/venn diagrammatic system, then is provale from D in GDS. Proof. Let D v v in uler/venn diagrammatic inference system. y soundness (cf. [24]) we have, for any model M, M = D v M = v. ssume M = D. Then we have M = D v y Lemma Thus we have M = v, that is, M =. Hence, y the completeness (Theorem.2) of GDS, we have D in GDS. The constructions of extensions of UL-structures given in Lemma 4.4, 4.12, and 4.13 provide a procedure to transform an UL-diagram to a Venn diagram. Let us see the following example: xample 4.17 Let D e an UL-diagram such that rel(d) = {,, }. y transforming the UL-structure rel(d) through an UL-structure with gls rel(d), we otain a Venn-structure rel(d) v. In rel(d) and rel(d) v elow, we omit symol and write for. In rel(d) v, we further omit lus and -relation, and represent arrows y lines in a hierarchical structure. y extracting minimal unshaded regions (,,,, ) from rel(d) v, we otain a Venn diagram D v, which is semantically equivalent to the original UL-diagram D. UL-diagram D UL-diagram with D = Venn diagram D v rel(d) rel(d) = = = = = = = = = rel(d) v In this paper, we introduced a hierarchy of uler and Venn diagrammatic reasoning systems in terms of their expressive powers in our topological-relation-ased formalization. ecause of the space limitation in this paper, we discuss our extensions of UL mainly at the level of representation and semantics. This is why our conservativity results for these systems (Proposition 4.15) are kept at the level of semantics. We leave 53

18 our explicit formalization of diagrammatic inference systems for UL-diagrams with intersections, with unions, with complements, respectively, as future work. References [1] L. uler, Lettres à une Princesse d llemagne sur Divers Sujets de Physique et de Philosophie, Saint-Pétersourg: De l cadémie des Sciences, (nglish Translation y H. Hunter, 1997, Letters of uler to a German Princess on Different Sujects in Physics and Philosophy, Thoemmes Press, 1997.) [2]. Fish and J. Flower, stractions of uler Diagrams, lectronic Notes in Theoretical omputer Science, 134, , [3] G. Gentzen, Investigations into logical deduction. In M.. Szao, ed., The collected Papers of Gerhard Gentzen, (Originally pulished as Unter suchungen uer das logische Scliessen, Mathematische Zetischrift 39 (1935): , ) [4] J. D. Gergonne, ssai de dialectique rationelle, nnales de mathématiques pures et appliqées, 7, , [5] J. Gil, J. Howse,. Tulchinsky, Positive Semantics of Projections in Venn-uler Diagrams, Journal of Visual Languages and omputing, 13(2), , [6]. Hammer, Logic and Visual Information, SLI Pulications, [7]. Hammer and N. Danner, Towards a model theory of diagrams, in G. llwein and J. arwise, eds., Working Papers on Diagrams and Logic, loomington: Indiana University, [8]. Hammer and S.-J. Shin, uler s visual logic, History and Philosophy of Logic, 19, 1-29, [9] J. Howse, F. Molina, J. Taylor, SD2: Sound and omplete Diagrammatic Reasoning System, 2000 I International Symposium on Visual Languages, , [10] J. Howse, G. Stapleton, and J. Taylor, Spider Diagrams, LMS Journal of omputation and Mathematics, Volume 8, , London Mathematical Society, [11]. Meyer, Diagrammatic evaluation of visual mathematical notations, in Diagrammatic Representation and Reasoning, P. Olivier, M. nderson, and. Meyer (ds.), Springer, , [12] K. Mineshima, M. Okada, Y. Sato, and R. Takemura, Diagrammatic Reasoning System with uler ircles: Theory and xperiment Design, Diagrams 08, LNI, Vol. 5223, Springer, , [13] K. Mineshima, M. Okada, and R. Takemura, Diagrammatic Reasoning System with uler ircles, 2009, to e sumitted. [14] F. Molina, Reasoning with extended Venn-Peirce diagrammatic systems, Ph. D Thesis, University of righton,

19 [15] M. Nielsen, G. Plotkin, and G. Winskel, Petri Nets, vent Structures and Domains, Part I, Theoretical omputer Science, Vol. 13, No. 1, pages , [16]. S. Peirce, ollected Papers IV, Harvard University Press, 1897/1933. [17] D. Prawitz, Natural Deduction, lmqvist & Wiksell, 1965 (Dover, 2006). [18] Y. Sato, K. Mineshima, R. Takemura, and M. Okada, How can uler diagrams improve syllogistic reasoning?, poster presented at ogsci 2009, msterdam, at the VU University msterdam, July 29th to ugust 1st, [19] S.-J. Shin, The Logical Status of Diagrams, amridge University Press, [20] G. Stapleton, survey of reasoning systems ased on uler diagrams, uler 2004, lectronic Notes in Theoretical omputer Science, 134(1), , [21] G. Stapleton, J. Masthoff, J. Flower,. Fish, and J. Southern, utomated Theorem Proving in uler Diagram Systems, Journal of utomated Reasoning, volume 39, issue 4, , [22] G. Stapleton, P. Rodgers, J. Howse, J. Taylor, Properties of uler diagrams, - SST 7, 2-16, [23] N. Swooda and G. llwein, Using DG transformations to verify uler/venn homogeneous and uler/venn FOL heterogeneous rules of inference, Software and System Modeling, 3(2), , [24] N. Swooda and G. llwein, Heterogeneous reasoning with uler/venn diagrams containing named constants and FOL, uler Diagrams 2004, NTS, Volume 134, lsevier, , [25] J. Venn, Symolic Logic, Macmillan, Diagrammatic inference system GDS In this section, we review Generalized Diagrammatic Syllogistic inference system GDS of [13, 12], which consists of two inference rules: unification and deletion. In order to motivate our definition of unification, let us consider the following question: Given the following diagrams D 1, D 2 and D 3 of Fig. 6, what diagrammatic information on, and c can e otained? (In what follows, in order to avoid notational complexity in a diagram, we express each named point, say c, simply y its name c.) Fig. 6 represents a way of solving the question. In Fig. 6, at the first step, two diagrams D 1 and D 2 are unified to otain D 1 + D 2, where point c in D 1 and D 2 are identified, and is added to D 1 so that c is inside of and overlaps with without any implication of a relationship etween and. We formalize such cases, where two given diagrams share one oject, y U1 U8 rules of group (I) of Definition.1. t the second step, D 1 + D 2 is comined with another diagram D 3 to otain (D 1 + D 2 ) + D 3. Note that the diagrams D 1 + D 2 and D 3 share two circles and : holds on D 1 + D 2 and holds on D 3. Since the semantic information of on D 3 is more accurate than that of on D 1 + D 2, according to our semantics of UL (recall that means just true in 55

20 c c c D 1 c D 2 D 3 D 4 D 5 Fig.7 Indeterminacy D 1 + D 2 a c (D 1 + D 2) + D 3 D 6 D 7 Fig.6 Unification D 8 D 9 Fig.8 Inconsistency our semantics), one keeps the relation in the unified diagram (D 1 + D 2 ) + D 3. We formalize such cases, where two given diagrams share two ojects, y U9 U10 rules of group (II) of Definition.1. Oserve that the unified diagram (D 1 + D 2 ) + D 3 of Fig. 6 represents the information of these diagrams D 1, D 2, and D 3, that is, their conjunction. We impose two kinds of constraints on unification. One is the constraint for determinacy, which locks the disjunctive amiguity with respect to locations of named points. For example, two diagrams D 4 and D 5 in Fig. 7 are not permitted to e unified into one diagram since the location of the point c is not determined (it can e inside or outside ). The other is the constraint for consistency, which locks representing inconsistent information in a single diagram. For example, the diagrams D 6 and D 7 (resp. D 8 and D 9 ) in Fig. 8 are not permitted to e unified since they contradict each other. Recall that each circle is interpreted y non-empty set in our semantics of Definition 2.8, and hence D 8 and D 9 are also incompatile. 7 We formalize our unification 8 of two diagrams y restricting one of them to e a minimal diagram, except for one rule called the Point Insertion-rule. Our completeness (Theorem.2) ensures that any diagrams D 1,..., D n may e unified, under the constraints for determinacy and consistency, into one diagram whose semantic information is equivalent to the conjunction of that of D 1,..., D n. We give a formal description of inference rules in terms of UL-relations: Given a diagram D and a minimal diagram α, the set of relations rel(d + α) for the unified diagram D + α is defined. It is easily checked that the set rel(d + α) satisfies the properties of Lemma 2.4 according to our constraints for determinacy and consistency, and hence locations of points are determined in a unified diagram. (See also Section 3, where we give a 7 In place of our syntactic constraint, it is possile to allow unification of inconsistent diagrams such as D 6 and D 7 (resp. D 8 and D 9 ) y extending GDS with an inference rule corresponding to the asurdity rule of Gentzen s natural deduction system: We can infer any diagram from a pair of inconsistent diagrams. (For natural deduction systems, see, for example, [3, 17].) Such a rule is introduced in, for example, [10] for spider diagrams; [7] for Venn diagrams; [23, 24] for uler/venn diagrams. However, such a rule requires linguistic symol, say, or some aritrary convention to represent inconsistency, and hence we prefer our syntactic constraint in our framework of a diagrammatic inference system. 8 The following definition of inference rules of GDS is slightly different from that of [13, 12] since we regard -relation as reflexive relation in this paper. 56

21 graph-theoretical representation of unification.) For a etter understanding of our unification rule, we also give a schematic diagrammatic representation and a concrete example of each rule. In the schematic representation of diagrams, to indicate the occurrence of some ojects in a context on a diagram, we write the indicated ojects explicitly and indicate the context y dots as in the diagram to the right elow. 9 For example, when we need to indicate only and c on the left hand diagram, we could write it as shown on the right. D F c c Definition.1 xiom, unification, and deletion of GDS are defined as follows. xiom: 1: For any circles and, any minimal diagram where holds is an axiom. 2: ny UL-diagram which consists only of points is an axiom. Unification: We denote y D + α the unified diagram of D with a minimal diagram α. D + α is defined when D and α share one or two ojects. We distinguish the following two cases: (I) When D and α share one oject, they may e unified to D + α y rules U1 U8 according to the shared oject and the relation holding on α. ach rule of (I) has a constraint for determinacy. (II) When D and α share two circles, if the relation which holds on α also holds on D, D + α is D itself; otherwise, they may e unified to D + α y rules U9 or U10 according to the relation holding on α. ach rule of (II) has a constraint for consistency. Moreover, there is another unification rule called the Point Insertion-rule (III). (I) The case D and α share one oject: U1: If holds on α and pt(d) = {}, then D and α may e unified to a diagram D + α such that the set rel(d + α) of relations holding on it is the following: U1 is applied as follows: rel(d) { } { cr(d)} D U1 α D 1 U1 D 2 D + α D 1 + D 2 9 Note that the dots notation is used only for areviation of a given diagram. For a formal treatment of such ackgrounds in a diagram, see, for example, Meyer [11]. 57

22 U2: If holds on α and pt(d) = {}, then D and α may e unified to a diagram D + α such that the set rel(d + α) of relations holding on it is the following: U2 is applied as follows: rel(d) { } { cr(d)} D U2 α D 1 U2 D 2 D + α D 1 + D 2 U3: If holds on α and cr(d), and if or holds for all circle in D, then D and α may e unified to a diagram D + α such that the set of relations rel(d + α) is the following: rel(d) { rel(d)} { rel(d)} { x x pt(d)} U3 is applied as follows: D U3 α D 1 U3 D 2 D + α D 1 + D 2 U4: If holds on α and cr(d), and if holds for all circle in D, then D and α may e unified to a diagram D + α such that the set of relations rel(d + α) is the following: rel(d) { rel(d)} { x x pt(d)} U4 is applied as follows: D U4 α D 1 U4 D 2 D + α D 1 + D 2 58

23 U5: If holds on α and cr(d), and if x holds for all x pt(d), then D and α may e unified to a diagram D + α such that the set of relations rel(d + α) is the following: rel(d) { rel(d)} U5 is applied as follows: { or rel(d)} { rel(d)} {x x pt(d)} F D U5 α D 1 F U5 D 2 D + α D 1 + D 2 U6: If holds on α and cr(d), and if x holds for all x pt(d), then D and α may e unified to a diagram D + α such that the set of relations rel(d + α) is the following: rel(d) { rel(d)} {x x pt(d)} U6 is applied as follows: { or or rel(d)} D U6 α D 1 U6 D 2 D + α D 1 + D 2 U7: If holds on α and cr(d), and if x holds for all x pt(d), then D and α may e unified to a diagram D + α such that the set of relations rel(d + α) is the following: rel(d) { rel(d)} {x x pt(d)} U7 is applied as follows: { or or rel(d)} a D U7 α D 1 U7 D 2 a D + α D 1 + D 2 59

24 U8: If holds on α and cr(d), and if pt(d) =, then D and α may e unified to a diagram D + α such that the set of relations rel(d + α) is the following: rel(d) { cr(d)} U8 is applied as follows: D U8 D + α α D 1 U8 D 2 D 1 + D 2 (II) When D and α share two circles, they may e unified to D + α y the following U9 and U10 rules. U9: If holds on α and holds on D, and if there is no oject s such that s and s hold on D, then D and α may e unified to a diagram D + α such that the set of relations rel(d + α) is the following: ( rel(d) \ {Y Y and rel(d)} \ { Y and Y rel(d)} ) {Y Y and rel(d)} { Y and Y rel(d)} U9 is applied as follows: D U9 α D 1 U9 D 2 D + α D 1 + D 2 U10: If holds on α and holds on D, and if there is no oject s such that s and s hold on D, then D and α may e unified to a diagram D + α such that the set of relations rel(d + α) is the following: ( rel(d)\{ Y and Y rel(d)} ) { Y and Y rel(d)} U10 is applied as follows: F D U10 α D 1 U10 D 2 F D + α D 1 + D 2 60

25 (III) Point Insertion: If, for any circles, Y and for any {,,, }, Y rel(d 1 ) iff Y rel(d 2 ) holds, and if pt(d 2 ) is a singleton {} such that pt(d 1 ), then D 1 and D 2 may e unified to a diagram D 1 + D 2 such that the set of relations rel(d 1 + D 2 ) is the following: rel(d 1 ) rel(d 2 ) { x x pt(d 1 )} Point Insertion is applied as follows: a c D 1 D 2 a c D 1 + D 2 Deletion: When t is an oject of D, t may e deleted from D to otain a diagram D t under the constraint that D t has at least one ojects. The notion of diagrammatic proofs (or, d-proofs) is defined inductively as tree structures consisting of unification and deletion steps. The provaility relation etween UL-diagrams is defined as usual. We denote y D a sequence of diagrams D 1,..., D n. Theorem.2 (Soundness and completeness of GDS [13]) Let D, e ULdiagrams, and let D have a model. is a semantically valid consequence of D (D = ), if, and only if, there is a d-proof of from D (D ) in GDS. 61