Completeness of an Euler Diagrammatic System with Constant and Existential Points

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1 omleteness of n uler igrmmtic System with onstnt nd xistentil Points Ryo Tkemur Nihon University, Jn. tkemur.ryo@nihon-u.c.j strct We extend uler digrmmtic system of Minshim, Okd, & Tkemur (2012) y distinguishing constnt nd existentil oints. onstnts oints corresond to constnt symols, nd existentil oints corresond to ound vriles ssocited with the existentil quntifier of the first-order lnguge in symolic logic. We rove the comleteness theorem of our extended uler digrmmtic system. 1 Introduction Since the 1990s, resoning with uler nd Venn digrms hs een studied from mthemticl nd logicl viewoint. uler nd Venn digrms re rigorously defined s syntctic ojects, like formuls in symolic logic, nd inference systems re formlized, which re equivlent to some symolic logicl systems. Then, fundmentl logicl roerties such s soundness nd comleteness re investigted..g., [Shin, 1994, Howse et l., 2005, Mineshim et l., 2012]. See [Stleton, 2005] for survey. Mineshim, Okd, & Tkemur (2012) introduced sic uler digrmmtic system, clled GS, in which digrm consists of nmed circles nd oints. Then, the soundness nd comleteness theorems of the system re estlished. Furthermore, it is shown tht the trditionl ristotelin ctegoricl syllogisms re esily simulted in the system. In the simultion, n existentil sentence such s Some re is trnslted into n uler digrm through sentence c is nd c is for some constnt c. lthough this trnsltion works well for the simultion of the syllogisms, if we consider the negtion of whole digrm eyond the syllogisms, the ove trnsltion my cuse some rolem: while No re is the negtion of Some re, it is no longer the negtion of c is nd c is. In this er, we extend the uler digrmmtic system of [Mineshim et l., 2012] y introducing existentil oints, which corresond to ound vriles ssocited with the existentil quntifier of the first-order lnguge in symolic logic. The introduction of existentil oints, distinguished from constnt oints, mkes our uler digrmmtic system more exressive, nd enles us to trnslte syllogistic sentences more nturlly. In Section 2, we introduce the syntx, semntics, nd inference rules of our uler digrmmtic system. Then, in Section 3, y extending the roof given in [Mineshim et l., 2012], we rove the comleteness theorem of our system. 1

2 2 System of uler digrms Our uler digrm, clled UL-digrm, is defined s lne with nmed circles nd oints. ch digrm is secified y toologicl (inclusion nd exclusion) reltions mintined etween circles nd oints. Thus digrms re syntcticlly equivlent when the sme reltions hold on ech. sed on the interrettion of circles (res. oints) s susets (res. elements) of certin set-theoreticl domin, digrms re interreted in terms of reltions tht hold on them. In Section 2.1, we introduce syntx nd semntics of UL-digrms with existentil oints. In Section 2.2, we introduce our inference rules. 2.1 Syntx nd semntics of UL Our uler digrm in this er is slight extension of the most sic one of [Mineshim et l., 2012]: s well s constnt oints, which corresond to constnt symols of the first order lnguge FOL in symolic logic, we here introduce existentil oints, which corresond to ound vriles ssocited with the existentil quntifier of FOL. efinition 2.1. n UL-digrm is lne (R 2 ) with finite numer, t lest two, of nmed simle closed curves (simly clled nmed circles, nd denoted y,,,... ) nd constnt oints (denoted y,, c,... ), nd existentil oints (denoted y x, y, z,... ), where no two nmed circles, s well s oints, re comletely concurrent, nd no two nmed circles, s well s oints, hve the sme nme. onstnt oints nd existentil oints re collectively clled (nmed) oints, nd denoted y, q, 1, 2,.... Nmed circles nd oints re collectively clled (digrmmtic) ojects, nd denoted y s, t, u,.... We use rectngle to reresent the lne for digrm. igrms re denoted y,, 1, 2,.... When is digrm, we denote y t() the set of nmed oints of, y cr() the set of nmed circles of, y o() the set of ojects of. digrm consisting of only two ojects is clled miniml digrm, nd these re denoted y, β, γ,.... Our digrms re investigted in terms of the following toologicl reltions etween digrmmtic ojects. efinition 2.2. UL-reltions re the following reflexive symmetric inry reltion, nd irreflexive symmetric inry reltions nd : the interior of is inside of the interior of, the interior of is outside of the interior of, there is t lest one crossing oint etween nd, is inside of the interior of, is outside of the interior of, q is outside of q (i.e. is not locted t the oint of q). The set of UL-reltions tht hold in digrm is uniquely determined, nd we denote this set y rel(). In the descrition of rel(), we omit the trivil reflexive reltion s s for ech oject s. We consider n equivlence clss of lne digrms in terms of the UL-reltions. efinition 2.3. ny ir of UL-digrms nd re (syntcticlly) equivlent if rel() = rel(). 2

3 In wht follows, the digrms which re syntcticlly equivlent re identified, nd they re referred y single nme. See [Mineshim et l., 2012], for discussion out our slightly rough eqution of digrms. sed on the ove equivlence on digrms, we refer to ny miniml digrm, sy where s t holds, y the non-trivil UL-reltion holding on it, s s t. UL-digrms re interreted in terms of UL-reltions tht hold on them. efinition 2.4. model M is ir (U, I), where U is non-emty set (the domin of M), nd I is n interrettion function which ssigns to ech circle or constnt oint non-emty suset of U such tht I() is singleton for ny constnt oint, nd I() I() for ny constnt oints, of distinct nmes. In order to void the comlexity cused y existentil oints, we define our interrettion for set of digrms insted for single digrm. efinition 2.5. Let e set of digrms 1,..., n such tht rel( ) = rel( 1 ) rel( n ) = {R 1,..., R i, x 1 1,..., x 1 k,..., x l 1,..., x l k }, where is or, nd no existentil oints er in ech of R 1,..., R i. M = (U, I) is model of, written M =, if nd only if: for every R j rel( ) (1 j i), I(s) I(t) holds if R j is s t; nd I(s) I(t) = holds if R j is s t: for every x j (1 j l), there exists m j M such tht m j I( 1 ) nd nd m j I( k ) hold, where is if x j rel( ), nd it is if x j rel( ). From the symolic logicl viewoint, the ove interrettion corresonds to the usul set-theoreticl interrettion of the following formul: R 1 R i x 1 ((x 1 1 ) (x 1 k )) x l ((x l 1 ) (x l k )) Remrk 2.6. y efinition 2.5, the UL-reltion does not contriute to the truthcondition of UL-digrms. Informlly seking, might e understood s I() I() = or I() I(), which is true in ny model. The semntic consequence reltion, = etween UL-digrms is defined s usul in symolic logic. (See [Mineshim et l., 2012] for detiled descrition.) 2.2 Inference system GS Mineshim, Okd, & Tkemur (2012) introduced n uler digrmmtic inference system, clled Generlized igrmmtic Syllogistic inference system GS. It consists of two kinds of inference rules: eletion nd Unifiction. eletion llows us to delete digrmmtic oject from given digrm. Unifiction (rules of U1 U10, PI) llows us to unify two digrms into one digrm in which the semntic informtion is equivlent to the conjunction of the originl two digrms. Two kinds of constrint re imosed on unifiction. One is the constrint for determincy, which locks the disjunctive miguity with resect to loctions of nmed oints. The other is the constrint for consistency, which locks reresenting inconsistent informtion in single digrm. 3

4 ch unifiction rule is descried y secifying (i) remise digrms, one of which is required to e miniml; (ii) digrmmtic oertions to introduce new oject into, or to rerrnge configurtion of ojects of, one of the remise digrms. We lso give schemtic illustrtions nd concrete exmles of lictions of rules. We further secify the set of UL-reltions rel() of the unified digrm. In the following, we introduce our xiom, nd of the eleven unifiction rules, we descrie U5 nd U7. The other unifiction rules re found in endix nd [Mineshim et l., 2012]. We further introduce nother rule of Ren (Renming). In wht follows, in order to void nottionl comlexity in digrm, we exress ech nmed oint, sy, simly y its nme. efinition 2.7 (Inference rules of GS). xiom: For ny ir of circles nd, ny miniml digrm where holds is n xiom. ny UL-digrm tht consists only of oints is n xiom. For ny existentil oint x, nd ny circle, ny miniml digrm where x holds is n xiom. Unifiction: U5 Premises: miniml digrm on which holds; nd digrm such tht cr() (ut cr()). onstrint for determincy: holds for ll t(). Oertion: dd the circle to (with reservtion of ll reltions on ) so tht the following conditions re stisfied on : (1) holds; (2) X holds for ll circles X ( ) such tht X or X holds on. The set of reltions rel() of the unified digrm is secified s follows: rel() rel() { X X or X rel(), X } { X X rel()} {X X rel()} { t()} Schem of U5 F xmle of U5 U5 F U5 U7 Premises: miniml digrm on which holds; nd digrm such tht cr() (ut cr()). onstrint for determincy: holds for ll t(). 4

5 Oertion: dd the circle to so tht the following conditions re stisfied on : (1) holds; (2) X holds for ll circles X ( ) such tht X or X or X holds on. rel() rel() { X X or X or X rel(), X } {X X rel()} { t()} Schem of U7 xmle of U7 U7 U7 Ren (Renming) Premise: digrm contining oint such tht x for n existentil oint x. Oertion: Relce the oint of with the existentil oint x. The set of reltions rel([ x]) of the resulting digrm is secified s follows: (rel() \ { s {, }, s o()}) {x s s rel()} {x s s rel()} xmle of Ren Ren x [ x] Remrk 2.8 (oy). omining Ren nd PI rules, we re le to dulicte ny oint in digrm s illustrted in the following: Ren x PI x Note tht, in generl, n dditionl oint is restricted to e existentil, nd it is locted in the sme region s the originl oint. The notion of digrmmtic roof (or, d-roof for short) is defined inductively s tree structures consisting of Unifiction, eletion, nd Renming stes. (f. Fig. 1 in xmle 3.11.) 5

6 efinition 2.9. Let e set of UL-digrms. n UL-digrm is rovle from, written s, if there is d-roof of in GS from 1,..., m such tht: (1) i ; (2) U1, U2 rules (unifiction y shred oint) re not lied to ny existentil oint rovided y Ren rule; (3) the existentil oint of ny xiom 3 ers only in digrms directly follow the xiom. Remrk In our system, we ostulte the interrettion I() of circle is non-emty. ccording to this definition, lso in our inference rules, we dot ny miniml digrm such tht x holds s n xiom (3). Our definition corresonds to the existentil imort in the literture of syllogisms. Without this ostulte, two digrms 1 in tht holds, nd 2 in tht holds, re consistent, when denotes the emty set. (f. Lemm 3.3.) However, it is difficult to exress 1 nd 2 in single digrm in our frmework. Introduction of nother device such s shding, which exress the corresonding region is emty, my coe with this rolem rtly. However, the question remins whether we drw the shded circle inside or outside. Thus, in our sic uler digrmmtic system, we ssume the existentil imort. Lemm The following hold in GS: 1. If u s nd s t, then u t; 2. If u t nd s t, then u s; 3. If u s nd s t nd v t, then u v. 3 omleteness of GS In this section, we rove soundness (Theorem 3.1) nd comleteness (Theorem 3.10) of GS with resect to our forml semntics. The soundness is shown y induction on the height of given d-roof s usul. Theorem 3.1 (Soundness). Let, e UL-digrms. If is rovle from the remises ( ) in GS, then is n semntic consequence of ( = ). For the comleteness, we imose the following condition for remise digrms: efinition 3.2. set of digrms is consistent if it hs model. Without this condition, ny digrm, sy where holds, is vlid consequence of n inconsistent set of remise digrms 1 nd 2 where nd hold, resectively, lthough there is no d-roof of from 1 nd 2 in GS. This is ecuse GS does not hve rule tht corresonds to the surdity rule of usul nturl deduction: we cn infer ny digrm from ir of inconsistent digrms. It is ovious tht the soundness theorem lso holds under the ssumtion of the consistency of the remises. The following is n imortnt consequence of the consistency: Lemm 3.3 (onsistency). Let e set of digrms which is consistent. Then none of the following holds in GS for ny ojects s nd t: 1. s t nd s t. 2. There is n oject u such tht s t nd u s nd u t. 6

7 Proof. (1) y the soundness of GS, in ny model M = (U, I) of, we hve I(s) I(t), nd I(s) I(t) =. y the definition of our interrettion, we hve I(s), nd hence, we hve I(s) I(t), which contrdicts to I(s) I(t) =. (2) ssume u is nmed oint. Then, y the soundness of GS, in ny model M = (U, I) of, we hve I(s) I(t) = nd I(s) I(t), ut it is imossile. When u is circle, y the definition of our interrettion, I(u). Hence, I(u) I(s) nd I(u) I(t) imly I(s) I(t). Thus, this cse is lso imossile s the revious cse. In order to show the comleteness theorem of GS, we construct two kinds of syntctic models, clled cnonicl models, in similr wy s the construction of Lindenum lgers in the literture of lgeric semntics for vrious roositionl logics. We first define the simler one. efinition 3.4 (nonicl model M ). Let e set of miniml digrms which is consistent. cnonicl model M = (M, I ) for is defined s follows: The domin M is the set of digrmmtic ojects (nmed circles nd oints) which occur in ny miniml digrm. I is n interrettion function such tht, for ny oject t, I (t) = {s s t in GS} {t}. Oserve tht in the ove definition of I, when t is nmed oint, sy, its interrettion I () is the singleton {} since s for ny oject s y soundness. Lemm 3.5 (nonicl model M ). Let e set 1,..., n of miniml digrms which is consistent. Then M is model of. Proof. We divide into the following cses ccording to the reltion R rel( ). Since the cse R is of the form s t is ovious, we ssume R s t in wht follows. 1. When R is of the form s t, in which s is not n existentil oint, we hve s t in GS since s t rel( ). We show M = s t, i.e., I (s) I (t). Let u I (s). () When u s, we immeditely hve s I (t) y the fct s t. () Otherwise, y the definition of I (s), we hve u s. y comosing it with s t s seen in Lemm 2.11(1), we hve u t in GS, tht is, u I (t). 2. When R is of the form s t, in which s nor t is n existentil oint, we hve s t in GS. We show M = s t, i.e., I (s) I (t) =. When oth s nd t re constnt oints, the clim is trivil. Otherwise, ssume to the contrry tht some u I (s) I (t). () When u s, we hve s I (t), i.e., s t. This, together with s t, is contrdiction y Lemm 3.3(1). The sme lies to the cse u t. () Otherwise, s u t, we hve u s nd u t y the definition of I (s) nd I (t). They contrdict s t y Lemm 3.3(2). 3. When ll reltions of n existentil oint x of rel( ) re x 1,..., x i, x 1,..., x j, we show tht there exists m M such tht m I ( 1 ) I ( i ) I ( 1 ) I ( j ), where X denotes the comlement of set X. 7

8 For every 1 k i, we hve x k, since x k rel( ). Hence, y the definition of I, we hve x I ( k ). For every 1 l j, to show x I ( l ), we ssume to the contrry tht x I ( l ). Then, y the definition of I, we hve x l. On the other hnd, since x l rel( ), we hve x l. They re contrdiction y Lemm 3.3(1). Thus, we hve x I ( 1 ) I ( i ) I ( 1 ) I ( j ). s n illustrtion of the cnonicl model, let us consider the following exmle. xmle 3.6. Let e the following miniml digrms 1, 2, 3, 4 : c 1 2 Oserve tht we hve nd. In such cse, we sy tht the oint is indeterminte with resect to the circle. Let us construct the cnonicl model for the y defining: I () = {,, x, y, z,... }, where x, y, z,... denotes ll existentil oints, nd I () = {, c, x, y, z,... }. Note tht the indeterminte oint w.r.t. is not contined in the interrettion of. With this interrettion, for ny nmed oint I (), we hve. In generl, vlidity of -reltion in the model M imlies rovility of -reltion. In the ove model, however, I () does not necessrily imly ; ecuse we do not hve, while I (). Thus, in the cnonicl model M of efinition 3.4, vlidity of -reltion does not imly rovility of -reltion, nd hence the model is not enough to estlish comleteness. Let us try to modify the ove model M so tht the indeterminte oint w.r.t. is contined in the interrettion I () of : I () = {,, x, y, z,... } nd I () = {, c,, x, y, z,... }. This definition lso rovides model of, nd we hve for ny oint I (). However, in this model, I () does not necessrily imly ; ecuse we do not hve, while I (). lthough one of our two cnonicl models lone is insufficient to estlish comleteness, we cn otin our comleteness result in the following mnner: we construct cnonicl model M (efinition 3.4 ove) for vlidity of -reltion, which imlies rovility of - reltion, nd model M, (efinition 3.7 elow) for vlidity of -reltion, which imlies rovility of -reltion. We now construct nother cnonicl model. In contrst to the revious model M of efinition 3.4, we include, for fixed circle, indeterminte ojects w.r.t. into the interrettion of the circle in the following model M,. efinition 3.7 (nonicl model M, ). Let e set of miniml digrms which is consistent. Let e fixed nmed circle. cnonicl model M, = (M,, I, ) for is defined s follows: The domin M, is the sme set s M of efinition 3.4. I, is n interrettion function defined s follows: For ny oject t, when t or t holds, I, (t) = I (t) {s s nd s nd s }; otherwise, I, (t) = I (t)

9 Lemm 3.8 (nonicl model M, ). Let e set 1,..., n of miniml digrms which is consistent. Let e fixed nmed circle. Then M, is model of. Proof. We divide into the following cses ccording to the form of R rel( ). We write s t when none of s t, t s, nd s t holds. Since the cse R is of the form s t is trivil, we ssume R is not of the form. 1. When R is of the form s t, we hve s t since s t rel( ). We show I, (s) I, (t). Let u I, (s). Since the cse u s is immedite, we ssume u s. ssume s or s hold. Then, y the definition of I, (s), we hve (i) u s or (ii) u. (i) imlies, together with s t, tht u t, y Lemm 2.11(1), i.e., u I, (t). For (ii), s nd s t imly t y Lemm 2.11(1). Hence, in conjunction with u, we hve u I, (t) y the definition of I,. The other cse (s nd s) is similr. 2. When R is of the form s t, we hve s t since s t rel( ). We ssume s t nd s u t since the other cses re similr. We show tht I, (s) I, (t) =. When oth s nd t re oints, the clim is trivil. Otherwise, ssume to the contrry tht some u I, (s) I, (t). We divide into the following cses: (i) s nd t; (ii) s nd t; (iii) s nd t; (iv) s nd t. (i) contrdicts s t. For (ii), y the definitions of I, (s) nd I, (t), we hve u s nd u t, which contrdict s t. For (iii), y the definition of I, (s), we hve u s. y the definition of I, (t), we hve (iii-1) u t or (iii-2) u. (iii-1), together with u s, contrdicts s t. For (iii-2), u s, s t, nd t imly, y Lemm 2.11(3), tht u, which contrdicts u. (iv) is similr to (iii). 3. When ll reltions of n existentil oint x of rel( ) re x 1,..., x i, x 1,..., x j, we show tht there exists some m M, such tht m I, ( 1 ) I, ( i ) I, ( 1 ) I, ( j ). For every 1 k i, we hve x k since x k rel( ). Hence, we hve x I ( k ), tht imlies x I, ( k ). For every 1 l j, in order to show x I, ( l ), we ssume to the contrry tht x I, ( l ). Let l or l. Then, y the definition of I, ( l ), we hve (i) x l or (ii) x. (i) leds to contrdiction since we hve x l y the fct x l rel( ). For (ii), y the fct x l nd l, we hve x, which contrdicts to (ii). The other cse ( l nd l ) is immedite. Therefore, we hve x I, ( 1 ) I, ( i ) I, ( 1 ) I, ( j ). Using the two kinds of cnonicl models introduced so fr, we rove the following tomic comleteness, from which comleteness (Theorem 3.10) of GS is derived. Proosition 3.9 (tomic comleteness). Let 1,..., n e set of digrms which is consistent. Let β e miniml digrm. If 1,..., n = β, then 1,..., n β in GS. Proof. We first consider the cse where the remise digrms 1,..., n re restricted to miniml digrms 1,..., n. Then we extend to the generl cse. We denote y the set of given miniml digrms. ssume = β. When β is s t, we immeditely hve s t since it is n xiom. Otherwise, we divide into the following two cses ccording to the form of β. 9

10 (1) When β is of the form s t, y the ssumtion = s t, we hve, in rticulr for the cnonicl model of efinition 3.4, M = M = s t. Then, since M = y Lemm 3.5, we hve M = s t, i.e., I (s) I (t). Since s I (s) y efinition 3.4, we hve s I (t), tht is, s t in GS. (2) When β is of the form s t, oserve tht if s nd t re oth oints, then the ssertion is trivil since β is n xiom in tht cse. Otherwise, we ssume, without loss of generlity, tht t is nmed circle. y the ssumtion = s, we hve, in rticulr for the cnonicl model of efinition 3.7, M, = M, = s. Then since M, = y Lemm 3.8, we hve M, = s, i.e., I, (s) I, () =. Hence we hve s I, () nd I, (s). Then y the definition of I, () nd I, (s) of efinition 3.7, we hve s, nd s nd s for some {,, }. Therefore, we hve s in GS. Next, we extend the remises to generl digrms 1,..., n insted of miniml digrms. Let 1,..., n = β. Let rel( ) = {R 1,..., R m }, then there is some such tht rel( ) = rel( ). Hence, when M =, y the ove rgument, we hve β, i.e., there is d-roof from 1,..., m to β in GS. Since ech j is derived from some i y some lictions of eletion rule, we hve 1,..., n β. y extending the conclusion digrm β of tomic comleteness to generl (not restricted to miniml) digrm, we estlish our comleteness theorem. Proosition 3.10 (omleteness). Let 1,..., n, e UL-digrms. Let 1,..., n e consistent. If 1,..., n =, then 1,..., n in GS. Proof (sketch). We define cnonicl wy to construct d-roof of from the given remise digrms 1,..., n (see lso xmle 3.11 elow): (I) Miniml rt: y using the tomic comleteness theorem, we derive ll of: (1) ointfree miniml digrms ering in the conclusion; (2) ointed miniml digrms consisting of every oint contined in remise; (3) ointed miniml digrms otined y comining the digrms of (1) with n xiom 3 of the form x for fresh oint x nd for circle contined in remise or the conclusion. (II) Venn rt: y lying U1, U2 rules for the ove ointed miniml digrms of (I-2) nd (I-3), we construct, for every oint of the Miniml rt, Venn-like digrm, in which holds for ny ir of circles in it, nd which consists of nd ll circles of the conclusion. If it is not ossile to construct Venn-like digrm for oint ecuse of the constrint for determincy, we do not construct it. When no Venn-like digrm with oint is constructed, y lying U8 rule nd the xiom 1, we construct Venn-like digrm (without ny oint) which consists of ll circles of. (III) Modifiction rt: y using U9, U10, nd the oint-free miniml digrms otined t the Miniml rt (I-1), we modify forms nd ositions of circles of ech Venn-like digrm ove so tht they corresond to those of the conclusion. (IV) PI rt: y lying PI, we unify ll digrms otined t the Modifiction rt. 10

11 (V) Renming rt: We finlly otin y lying Ren, el, nd the coy oertion descried in Remrk 2.8. y the definition of our inference rules, oint in the conclusion comes, ossily with lictions of Ren, from the following wys: (1) it is contined in remise; (2) it is otined y lying the coy oertion of Remrk 2.8; (3) it is rovided y lying the xiom 3. Thus, t the Miniml rt, we derive ll ossile ointed miniml digrms consisting of these oints. lso t the Renming rt, we otin the conclusion y lying Ren, el, nd the coy oertion. t the Venn rt, we re le to ignore oint for which we cnnot construct Venn-like digrm. This is ecuse, for ny oint contined in the conclusion, the reltions etween the oint nd ll circles of re determined. t the Modifiction rt, we re le to ly U9, U10 rules undisturedly. This is ecuse, if digrms 1 nd 2 could not e unified ecuse of the constrint for consistency of U9, U10 rules, since 1 nd 2 re oth derived from the remises, it mens the remises re inconsistent. t the PI rt, we re le to unify ll digrms otined so fr. This is ecuse ll circles nd oints of those digrms re essentilly the sme s those of the conclusion excet for their nmes. The correctness of the ove cnonicl construction is shown in similr wy s [Mineshim et l., 2012]. Fig. 1 is n exmle of the cnonicl d-roof. xmle 3.11 (nonicl d-roof of GS). s n illustrtion of the cnonicl construction of d-roofs, let us consider the following digrms 1, 2, 3, nd : 1,, 2 We hve cnonicl d-roof of from 1, 2, 3 s in Fig. 1, where the derivtion of 15 is omitted since it is similr to tht of 13. We first derive ointed miniml digrms 1, 4, 5, 6 consisting of oint contined in remise (I-2), nd 7, 8 consisting of fresh oint comes from n xiom 3 (I-3). Next, t the Venn rt, with U1 nd U2 rules, we construct Venn-like digrms 9, 10 nd 11 ech of which consists of the ove oint (,, z) nd ll circles nd of. Then, t the Modifiction rt, with U10 rule, we modify the ove Venn-like digrms so tht ositions of circles fit to those of the conclusion. Then, t the PI rt, with PI rule, we unify 12, 13, 14, nd 15 to otin 18. Finlly, y renming the nmes of the oints nd z, nd y deleting w, we otin the conclusion. 3 x y References [Howse et l., 2005] J. Howse, G. Stleton, nd J. Tylor (2005) Sider igrms, LMS Journl of omuttion nd Mthemtics, London Mthemticl Society, Volume 8,

12 2 2 1 U2 4 9 U U7 PI 5 3 U U PI z PI x z 3 w z 18 Ren & el y U7 7 Fig. 1 nonicl d-roof z z 3 U1 z 11 z 8 U10 z 14 } 3 Miniml }. w 15 Venn Renming } Modifiction PI [Mineshim et l., 2012] K. Mineshim, M. Okd, nd R. Tkemur (2012) igrmmtic Inference System with uler ircles, Journl of Logic, Lnguge nd Informtion, 21, 3, [Shimojim, 1996]. Shimojim (1996) On the fficcy of Reresenttion, Ph. thesis, Indin University. [Shin, 1994] S.-J. Shin (1994) The Logicl Sttus of igrms, mridge University Press. [Stleton, 2005] G. Stleton (2005) survey of resoning systems sed on uler digrms, Proceedings of uler 2004, lectronic Notes in Theoreticl omuter Science, 134, 1, [Tkemur, 2013] R. Tkemur (2013) Proof theory for resoning with uler digrms: logic trnsltion nd normliztion, Studi Logic, 101, 1,

13 Unifiction rules of GS Unifiction rules re divided into three grous, Grou (I), (II), nd (III). The rules in Grou (I) nd (II) re clssified ccording to the numer nd tye of ojects shred y digrm nd miniml digrm. The rule in Grou (III) is the PI rule, where neither of two remise digrms is restricted to e miniml. (I) The cse nd shre one oject: U1 Premises: holds on, nd t(). onstrint for determincy: t() = {}. Oertion: dd the circle to (with reservtion of ll reltions on ) so tht the following conditions re stisfied on +: (1) holds; (2) X holds for ll circles X of. The set of reltions rel() is secified s rel() rel() { X X cr()}. Schem of U1 U1 xmle of U1 U1 U2 Premises: holds on, nd t(). onstrint for determincy: t() = {}. Oertion: dd the circle to so tht the following conditions re stisfied on : (1) holds; (2) X holds for ll circles X of. rel() = rel() rel() { X X cr()} Schem of U2 U2 xmle of U2 U2 U3 Premises: holds on, nd cr(). onstrint for determincy: X or X holds for ll circles X of. Oertion: dd the oint to so tht the following conditions re stisfied on : (1) holds; (2) q holds for ll oints q such tht q holds on. rel() =rel() rel() { X X rel()} { X X rel()} { q q t()} 13

14 U4 U3 U3 Premises: holds on, nd cr(). onstrint for determincy: X holds for ll circles X of. Oertion: dd the oint to so tht the following conditions re stisfied on : (1) holds; (2) q holds for ll oints q such tht q holds on. rel() = rel() rel() { X X rel()} { q q t()} U4 U4 U6 Premises: holds on, nd cr(). onstrint for determincy: holds for ll t(). Oertion: dd the circle to so tht the following conditions re stisfied on : (1) holds; (2) X holds for ll circles X( ) such tht X or X or X holds on. rel() = rel() rel() {X X or X or X rel(), X } {X X rel()} { t()} U6 U6 U8 Premises: holds on, nd cr(). onstrint for determincy: t() =. Oertion: dd the circle to so tht X holds for ll circles X of. rel() = rel() rel() { X X cr()} U8 14 U8

15 (II) The cse nd shre two circles: U9 Premises: holds on, nd holds on. onstrint for consistency: There is no oject s such tht s nd s hold on. Oertion: Modify ll circles X (including ) of such tht X holds so tht the following conditions re stisfied on +: (1) X holds; (2) X t holds with {,,, } for ll oject t of such tht t, X t rel(). rel(+)= ( rel() \ {X Y X nd Y rel()} \ {X Y X nd Y rel()} ) {X Y X nd Y rel()} {X Y X nd Y rel()} U9 U9 U10 Premises: holds on, nd holds on. onstrint for consistency: There is no oject s such tht s nd s hold on. Oertion: Modify ll circles X (including ) nd Y (including ) of such tht X nd Y, resectively hold on so tht the following conditions re stisfied on : (1) X holds; (2) X t holds with {,,, } for ll oject t of such tht t, X t rel(); (3) Y holds; (4) Y s holds with {,,, } for ll oject s of such tht s, Y s rel(). rel() = ( rel() \ {X Y X nd Y rel()} ) {X Y X nd Y rel()} F U10 U10 F (III) Neither of two remise digrms is restricted to e miniml: PI (Point Insertion) Premises: X Y rel( 1 ) iff X Y rel( 2 ) holds for ny circles X, Y with {,,, }, nd t( 2 ) = {} such tht t( 1 ). Oertion: dd the oint to 1 so tht the following conditions re stisfied on : (1) t of rel( 2 ) holds for ll ojects t; (2) q holds for ll q t( 1 ). rel( ) = rel( 1 ) rel( 2 ) { q q t( 1 )} 15

16 c 1 2 c el (eletion) Premise: contins n oject s. onstrint: is not miniml. Oertion: elete the oject s from. rel( s) = rel() \ {s t t o(), {,,, }} 18 Leonhrd uler 1990 Mineshim, Okd, & Tkemur (2012) Mineshim, Okd, & Tkemur (2012) 16