Relation Graphs:

A Structure for Representing Relationsin Contextual Logic of Relations

Silke Pollandt

Technische Universitat Darmstadt, Fachbereich MathematikSchloßgartenstr. 7, D–64289 Darmstadtpollandt@mathematik.tu-darmstadt.de

Abstract. Contextual Logic of Relations (CLR) is based on FormalConcept Analysis (FCA) and Peircean Algebraic Logic (PAL). The aimof the paper is to introduce relation graphs as an algebraic structure forrepresenting relations and operations on relations in Contextual Logic ofRelations. It is shown that each relation graph of a relation context familyuniquely represents an extent of a relation described by a first order logicformula. In reverse, each compound relation of a relation context familycan be represented by a relation graph. A graphical representation ofrelation graphs highly corresponds to a graphical system of PAL. Theformal definition of relation graphs is intended to clarify correspondencesbetween PAL and (existential semi-) concept graphs as well as existentialgraphs.

1 Introduction

Contextual Logic of Relations (CLR) can be seen as one part of ContextualLogic (especially Contextual Judgment Logic). Sowa’s theory of conceptualgraphs [So92] has been combined with Formal Concept Analysis [GW99a] in[Wi97, PrW99, Wi01] to design a mathematical Logic of Judgment in the frame-work of Contextual Logic [Wi00b]. Contextual Logic of Relations is mainly basedon Peircean Algebraic Logic (PAL) which R.W.Burch reconstructed in [Bu91],but also influenced by Universal Algebra (cf. [Jo88, Jo91, Md91a, Md91b]) aswell as relational methods in Computer Science (cf. [Ma83, BKS97, AHV95]).

Sowa has introduced a conceptual graph (cf. [So99]) as a bipartite graph thathas two kinds of nodes called concepts and conceptual relations. In the frame-work of Contextual Logic (cf. [Kl01]), an existential semiconcept graph has beenformalized as a directed multi-hypergraph where concept names are assigned tothe vertices and relation names are assigned to the edges. The main interest isdirected to conceptual dependencies. In this paper we focus our considerationson relations. To develop the Logic of Relations, it is useful to introduce relationgraphs as multigraphs where relations are assigned to the vertices (and, possibly,(semi-)conceptual instances are assigned to the edges). Simple relation graphscan be translated to special existential semiconcept graphs by exchanging therole of vertices and edges.

U. Priss, D. Corbett, and G. Angelova (Eds.): ICCS 2002, LNAI 2393, pp. 34–47, 2002.c© Springer-Verlag Berlin Heidelberg 2002

Relation Graphs: A Structure for Representing Relations 35

1684-1720Anna Magdalena Wilcken

1701-1760

Carl Philipp Emanuel1714-1788

Wilhelm Friedemann1710-1784

Johann Christian1735-1782

Johann Christoph Friedrich

Maria Barbara Bach

1732-1795

(2)

Wilhelm Friedrich Ernst1759-1845

(1)

1685-1750Johann Sebastian Bach

Fig. 1. Family tree of Johann Sebastian Bach (section)

Contextual Logic of Relations is based on Peircean Algebraic Logic (see[Bu91]). Two systems of graphical syntax for PAL have been proposed by Burch.We introduce relation graphs as an algebraic structure suitable for describing re-lations and operations on relations in Contextual Logic of Relations. A graphicalrepresentation of relation graphs coincides mostly with the first graphical sys-tem of PAL. Our contextual logic of relations has the same expressiveness asPAL. And we get the same close resemblance to Peirces existential graphs asPAL gets. Relation graphs are algebraic structures corresponding to Burch’sgraphical syntax of PAL.

On the other hand, relation graphs are closely connected to power contextfamilies. In [PoW00] and [Wi00a] a contextual logic of relations has been de-veloped as a contextual attribute logic [GW99b] on the relational contexts of apower context family. In this sense, each relation graph can be interpreted as acompound relation of a suitable relation context family.

2 Contextual Logicof Relations on Power Context Families

Contextual Logic of Relations has been developed as Contextual Attribute Logicon power context families in [PoW00] and [Wi00a] within the theory of FormalConcept Analysis (see [GW99a] for the mathematical foundations of FCA). Itsaim is to support knowledge representation and knowledge processing.

The basic structure is a data table (or “formal context”). It can representsimply objects and attributes as well as relational connections. We start with anexample.1 Figure 1 represents a small part of the Bach’s family tree, a family offamous composers and musicians (see [Me90]). The lines in such a family treeindicate two binary relations, child·of and married·to. From the respective(first) names the unary relations man and woman can be derived. These rela-tions are sufficient to determine the family relationships between each two ormore of these persons. Relations like mother·of, grandfather·of, brother·of,1 This example is similar to an example in [Po01]. Regarding a family tree, we use the

advantage that it is well-known how to read the picture, and how to derive arbitraryfamily relationships from the tree structure.

36 Silke Pollandt

K 1 IK2

child of

Johann SebastianMaria Barbara

Wilhelm FriedemannCarl Philipp Emanuel

I

Johann Christoph Friedrich

Anna Magdalena Wilcken

Johann ChristianWilhelm Friedrich Ernst

wom

anm

an

xxx

xxxxx

(Johann Sebastian, Maria Barbara)(Maria Barbara, Johann Sebastian)

(Wilhelm Friedemann, Johann Sebastian)(Carl Philipp Emanuel, Johann Sebastian)

(Johann Sebastian, Anna Magdalena W.)(Anna Magdalena W., Johann Sebastian)

xx

xxxx

mar

ried

to

Fig. 2. Relational power context family

or mother-father-child can be derived (see [Ox89], Appendix 8, for tree de-scriptions, and [Po01] for operational descriptions). The relational informationof this family tree is represented in the two data tables in Figure 2.

Contextual Logic is based on the mathematical notion of a formal context,which is defined as a triple K := (G,M, I) consisting of a set G of objects, a set Mof attributes, and a binary relation I ⊆ G×M . The relation I between G and Mcan be read as “the object g has the attribute m” for gIm (i.e., (g,m) ∈ I). Foreach attribute m ∈ M of a formal context (G,M, I), the extent is defined as theset

mI := {g ∈ G| gIm}of all objects of (G,M, I) that have this attribute. Analogously, for each setA ⊆ M of attributes, the extent is defined as the set

AI := {g ∈ G| ∀m∈A gIm} =⋂

{mI | m ∈ A}

of all objects of (G,M, I) that have all these attributes. Dually, by exchangingobjects and attributes, we get the intent of an object (set). Using this prime op-eration, relationships between formal attributes can be expressed. For example,we say an attribute m implies an attribute n if the extent of m is a subset of theextent of n (i.e., mI ⊆ nI). A formal concept of (G,M, I) is a pair (A,B) withA ⊆ G, B ⊆ M , A′ = B and B′ = A. A is called the extent and B the intentof (A,B). We write A=Ext(A,B). The set of all formal concepts of (G,M, I) isdenoted by B(G,M, I).

In order to have more expressiveness in Contextual Attribute Logic com-pound attributes of a formal context (G,M, I) have been introduced in [GW99b]by using the operational elements ¬,

∧and

∨for negation, conjunction and

disjunction. This idea has been extended in [PoW00] and [Wi00a] to relationcontexts of power context families. In [Wi00a], relation algebras of power contextfamilies have been introduced. The paper is mainly based on Peircean AlgebraicLogic. Wille has shown, that the expressiveness of the introduced language ofrelation algebras reaches the expressiveness of first order logic. This is in accor-dance to Burch’s thesis in [Bu91]: “All procedures of relational constructions

Relation Graphs: A Structure for Representing Relations 37

are formalizable in PAL.” That is a reason for us to choose the operations ofPAL for our investigations.

Semantically, Contextual Logic of Relations is based on power context fami-lies. Here we modify the definition of power context families.

Definition 1. A relation context family on a set G is a sequence −→K :=

(K0,K1, . . .) consisting of the formal contexts Kk := (Gk,Mk, Ik) with Gk ⊆ Gk

for k = 0, 1, . . . (and G0 := {∅}. The formal concepts of Kk have k-ary relationsas extents and are therefore called k-ary relational concepts.

The data tables in Figure 2 can be understood as a representation of a relationcontext family −→

K := (K1,K2) with

G := { Johann Sebastian, Maria Barbara, Anna Magdalena Wilcken, Jo-hann Christoph Friedrich, Johann Christian, Wilhelm Friedemann, CarlPhilipp Emanuel, Wilhelm Friedrich Ernst}.

For the context K1 of unary relations we choose G1 := G and M1 :={man, woman}. The context K2 of binary relations is given by G2 := G2 and M2 :={child·of, married·to}. The relation Ik (k = 1, 2) can be read from the datatables in Figure 2 in the following way: For the k-tuple (g1, . . . , gk) ∈ Gk andthe k-ary relation m ∈ Mk holds (g1, . . . , gk)Ikm if and only if the k-th datatable contains a cross in the field, where the row is denoted by (g1, . . . , gk) andthe column is denoted by m.

Now we start to describe, in analogy to [GW99b] and [PoW00], ContextualAttribute Logic of relation context families −→

K by recursively defining compoundattributes for −→

K with the operational elements ¬, ◦, �, and π. For a uniformnotation in the rest of this paper, for l > k we define (l, . . . , k) := ∅ and(gl, . . . , gk) := ∅, resp. We write N0 := N ∪ {0}.

– Each attribute m ∈ Mk (k = 0, 1, . . .) is a compound attribute and also the“constants” ⊥k, �k, and idk with the extents

(⊥k)Ik := ∅, (�k)Ik := Gk, and (idk)Ik := {(g, . . . , g) ∈ Gk| g ∈ G}.(id0 = �0 for k = 0.)

– For each attribute m ∈ Mk (k = 0, 1, 2), we define its negation ¬m to be acompound attribute with

(¬m)Ik := Gk \mIk .

Thus, (g1, . . . , gk) is in the extent of ¬m if and only if (g1, . . . , gk) is not inthe extent of m.

– For each two attributes m ∈ Mk and n ∈ Ml, and 1 ≤ i ≤ k, k+1 ≤ j ≤ k+l(i, j, k, l ∈ N), we define the i, j-concatenation m(i ◦ j)n to be the compoundattribute with

(m(i ◦ j)n)Ik+l−2 :={(g1, . . . , gi−1, gi+1, . . . , gk, g1 . . . , gj−1, gj+1, . . . , gl) ∈ Gk+l−2|∃g ∈ G : (g1, . . . , gk) ∈ mIk , (g1, . . . , gl) ∈ mIl with gi = gj = g}

38 Silke Pollandt

– For each attribute m ∈ Mk, and 1 ≤ i < j ≤ k, (i, j, k ∈ N), we define thei, j-coupled deletion m(i�j) to be the compound attribute with

(m(i�j))Ik−2 := {(g1, . . . , gi−1, gi+1, . . . , gj−1, gj+1, . . . , gk) ∈ Gk−2|∃g ∈ G : (g1, . . . , gk) ∈ mIk with gi = gj = g}

– For each attribute m ∈ Mk, and each permutation πk =(

1, . . . , ki1, . . . , ik

)

(k ∈ N), we define the permutation mπk to be the compound attribute with

(mπk)Ik := {(gi1 , . . . , gik) ∈ Gk| (g1, . . . , gk) ∈ mIk}.

(Note: For each element of the symmetric group Sk, we have one separateoperation. That is in accordance with [Bu91], and convenient for our furtherinvestigations. As shown in [Wi00a], they can be substituted by a finite set ofoperations generating the full symmetric group and working for all k ∈ N.)

– Iteration of the above compositions leads to further compound attributes,the extents of which are determined in the obvious manner.

Contextual logic of relation context families is seen in the framework ofPeircean Algebraic Logic which R. W. Burch created as “an attempt to amal-gamate various systems of logic that Peirce developed over his long career”(see [Bu91]). The relational operations occuring in the recursive definition of thecompound attributes coincide with the basic operators of Peircean AlgebraicLogic:

CLR m ∈ Mk ⊥k �k idk ¬ (i ◦ j) (i�j) πk

PAL Rk 0k Uk 1k NEG Ji,j+k2 Ji,j

1 PERMkπ

(NULLk) (UNIVk) (IDk)

In [Bu91] it is shown that it is sufficient to use ⊥k, �k, idk for k = 3, but forconvenience we admit them for k ∈ N0.

3 Simple Relation Graphs

In order to prepare the definition of relation graphs, we formally define theneeded basic structure.

Definition 2. A multigraph with ordered valences and k (k ∈ N0) pendingedges is a triple G := (V, deg, ε) where

– V is a non-empty finite set of vertices,– deg : V → N0 is a mapping, and– ε : H(G) → H(G) is a bijective, involutory mapping fulfillingε(v, i) �= (v, i) and ε(∞, i) ∈ H∗(G)

Relation Graphs: A Structure for Representing Relations 39

with the sets- val(v) := {v} × {1, . . . ,deg(v)} of valences of v,- H∗(G) :=

⋃v∈V val(v) of proper half-edges or valences,

- H∞(G) := {∞} × {1, . . . , k} of pending half-edges,- H(G) := H∗(G) ∪H∞(G) of half-edges,- E(G) := {{h1, h2} ∈ P(H(G))| ε(h1) = h2} of edges,- E∗(G) := E(G) ∩ P(H∗(G)) of proper edges,- E∞(G) := E(G) \E∗(G) of pending edges of G.

Each multigraph G := (V, deg, ε) with ordered valences and k pending edges(k ∈ N0) is uniquely determined by the triple (V,E∗, E∞).

Definition 3. A subgraph G′ with l (l ∈ N0) pending edges of a multigraphG := (V, deg, ε) with ordered valences and k (k ∈ N0) pending edges (we writeG′ <∼ G)) is a multigraph G′ := (V ′, deg′, ε′) with ordered valences and l pendingedges, where

– V ′ ⊆ V (V ′ �= ∅),– deg′ : V ′ → N0 is the restriction of deg to V ′,– E∗(G′) ⊆ E∗(G) holds.

The subgraph relation only depends on the sets of vertices and proper edgesbelonging to the graph, but not on the (ordering of) pending edges. We definethe equivalence relation

G′ ∼ G :⇐⇒ G′ <∼ G and G <∼ G′,

and write G′ <�∼ G for G′ <∼ G and G′ �∼ G.

Definition 4. A multigraph G := (V, deg, ε) with ordered valences and k (k ∈N0) pending edges is called connected if for all v, w ∈ V with v �= w thereexists a sequence ((vt, it))t=0,1,...,2s+1 with s ∈ N and (vt, it) ∈ H∗(G) for t =0, 1, . . . , 2s + 1, where v0 = v, v2s+1 = w, v2t−1 = v2t for t = 1, . . . , s, andε(v2t, i2t) = (v2t+1, i2t+1) for t = 0, . . . , s.

Each such connected multigraph is intended to describe a relation.Now, all preparations to define relation graphs are done. In the following, let−→

K := (K0,K1, . . .) be a relation context family on a set G consisting of the formalcontexts Kl := (Gl,Ml, Il) with Gl ⊆ Gl for l = 0, 1, . . . (we set G0 := {∅}).

Definition 5. A k-ary (simple) relation graph (k ∈ N0) of a relation contextfamily −→

K := (K0,K1, . . .) is a structure G := (V, deg, ε, κ), where

– G := (V, deg, ε) is a connected multigraph with ordered valences and k pend-ing edges,

– κ : V →.⋃

l=0,1,... B(Kl) with Kl := ((G0)l,.

M l, Il),.

M l:= Ml ∪ {�l,⊥l, idl}is a mapping with κ(v) ∈ B(Kdeg(v)).

40 Silke Pollandt

to married11 2

T1

b)2

child of11 2

child of1 2

c)

1 womanchild of1 22

id1

23

1d)

a)

e)

child of1 2

id1

23 1

man

2

3

2id

child of1 2

id1

23

1

13

1 woman

2 2 to married11

Fig. 3. Simple relation graphs (corresponding terms on the next page)

In the graphical representation of a relation graph each vertice v is labeled by theformal concept κ(v) ∈ B(Kdeg(v)) (attribute concepts usually are denoted by thename of its generating attribute), the half-edges are numbered. In Figure 3 somesimple relation graphs of the relation context family in Figure 2 are drawn. Thereis a correspondence between simple relation graphs and compound attributes ofa relation context family −→

K := (K0,K1, . . .). Our aim is to define a mapping ϕ

from the set of all simple relation graphs of −→K to the set⋃

l∈N0P(Gl), where ϕ

maps each k-ary simple relation graph of −→K to an element of P(Gk), i.e. to anextent of Kk. Let G := (V, deg, ε, κ) be an arbitrary k-ary simple relation graph,and G := (V, deg, ε). First, we choose a bijective mapping τ : E∗(G) ∪E∞(G) →{1, . . . , k + |E∗(G)|} with τ((∞, i), ε(∞, i)) = i, and define

ϕ(G) := {(g1, . . . , gk) ∈ Gk| ∃(gk+1, . . . , gk+|E∗(G)|) ∈ G|E∗(G)|

∀v ∈ V : (ge(v,1), . . . , ge(v,deg(v))) ∈ Ext(κ(v))}

with e(v, i) := τ({(v, i), ε(v, i)}). The mapping ϕ is τ -invariant. We say the k-arysimple relation graph G of −→K represents the k-ary relation of −→K with the extentϕ(G). For instance, for the relation graph represented by Figure 3d holds

ϕ(G) := {(g1, g2) ∈ G2| ∃(g3, g4) ∈ G2 : (g2, g3) ∈ child · ofI2

∧ (g4) ∈ womanI1 ∧ (g3, g1, g4) ∈ id3I3}

= {(g1, g2) ∈ G2| (g2, g1) ∈ child · ofI2 ∧ (g1) ∈ womanI1}.

In other words, the relation graph in Figure 3d represents the binary relationwith the extent{(Anna Magdalena W., Johann Christoph Friedrich),(Anna Magdalena W., Johann Christian),(Maria Barbara, Wilhelm Friedemann), (Maria Barbara, Carl Philipp Emanuel)}of the relation context family in Figure 2. With the common pre-knowledge aboutfamily relationships we can understand the relations represented in Figure 3 asa) married · to (binary), b) is · married (unary), c) grandchild · of (binary),d) mother · of (binary) and e) mother− father− child (ternary).

The above considerations yield the following theorem.

Relation Graphs: A Structure for Representing Relations 41

Theorem 1. Each k-ary simple relation graph of a relation context family −→K :=

(K0,K1, . . .) uniquely represents an extent of Kk described by a first order logicformula over

.⋃l∈N0

.

M l (without negation).

4 Relation Graphs with Negation

In this section we extend the definitions of the structures introduced in section3 in order to include negation.

Definition 6. A multigraph with ordered valences, k (k ∈ N0) pending edges,and nested subgraphs is a structure G := (V0, deg0, ε0, (S, <∼)) where

– G0 := (V0, deg0, ε0) is a multigraph with ordered valences and k pendingedges,

– (S, <∼) is a family of subgraphs of G0 with S := (Gt)t∈T , Gt := (Vt, degt, εt),and T := {1, . . . , n} (n ∈ N0),

– Gs<�∼ Gt implies s < t for each s, t ∈ T , and

– Vs ∩ Vt �= ∅ implies Gs<∼ Gt or Gt

<∼ Gs for each s, t ∈ T .

With s, t ∈ T0 := T ∪{0} we introduce the following tree order on S ∪{G0} withthe greatest element G0

Gs ≤ Gt :⇐⇒ Gs<�∼ Gt or (Gs ∼ Gt and s ≤ t),

and the lower neighbour relation

Gs ≺ Gt :⇐⇒ s �= t and (Gr ≥ Gs =⇒ Gr ≥ Gt) for each r ∈ T0 \ {r}.The definition of connected multigraphs can be transfered in a natural way.

Definition 7. A multigraph G := (V0, deg0, ε0, (S, <∼)) with ordered valences, k(k ∈ N0) pending edges, and nested subgraphs (with S := (Gt)t∈T and Gt :=(Vt, degt, εt)) is called connected if the multigraph G0 := (V0, deg0, ε0) with or-dered valences and k pending edges as well as Gt for all t ∈ T are connected.

Negation can be included in the definition of a relation graph in the followingway.

Definition 8. A k-ary relation graph (with negation) (k ∈ N0) of a relationcontext family −→

K is a structure G := (V0, deg0, ε0, (S, <∼), κ) where

– G := (V0, deg0, ε0, (S, <∼)) is a connected multigraph with ordered valences,k pending edges, and nested subgraphs, and

– G0 := (V0, deg0, ε0, κ) is a k-ary simple relation graph.

Each simple relation graph G := (V, deg, ε, κ) can be understood as a relationgraph (with negation) G := (V, deg, ε, (S, <∼), κ) with S = ∅.

In the graphical representation of a relation graph G with negation, eachsubgraph Gt (t ∈ T ) is indicated by a simple closed curve enclosing the edges

42 Silke Pollandt

child of1 2

child of1 2

id

1

3

2 1id

2

1

b)a)

3

id

child of1 2

1

child of1 2

1 2child of

12

3

id

221

c)

1 woman1

Fig. 4. Relation graphs with negation (corresponding terms on the next page)

and vertices belonging to the subgraph as well as each whole curve indicating asubgraph Gs ∈ S with Gs ≤ Gt. In Figure 4 some relation graphs with negationof the relation context family in Figure 2 are drawn.

We extend the mapping ϕ (see section 3) to the set of all relation graphs (withnegation). In other words, we define a mapping ϕ from the set of all relationgraphs with negation of −→

K to the set⋃

l∈N0P(Gl), where ϕ maps each k-ary

relation graph with negation of −→K to an element of P(Gk), i.e. to an extent ofKk. Let G := (V0, deg0, ε0, (S, <∼), κ) (with S := (Gt)t∈T , Gt := (Vt, degt, εt), andT := {1, . . . , n} (n ∈ N0)) be an arbitrary k-ary relation graph with negation,and G0 := (V0, deg0, ε0). To simplify the notation we use the mappings

– α : T0 → P(T ) with α : s → {t ∈ T | Gt ≺ Gs} for each s ∈ T0,– αV : {Vt| t ∈ T0} → P(V0) with αV : Vs → Vs\

(⋃t∈α(s) Vt

)for each s ∈ T0,

– α∗E : {E∗(Gt)| t ∈ T0} → P(E∗(G0)) with α∗

E : E∗(Gs) → E∗(Gs) \(⋃t∈α(s) E

∗(Gt))

for each s ∈ T0.

Then we choose a bijective mapping τ : E∗(G0)∪E∞(G0) → {1, . . . , k+|E∗(G0)|}with τ((∞, i), ε(∞, i)) = i, and define

Φs := (∃(gt)t∈τ(α∗E(E∗(Gs))) ∈ G|α∗

E(E∗(Gs))| ∀v ∈ αV (Vs) :

(ge(v,1), . . . , ge(v,deg(v))) ∈ Ext(κ(v)) ∧ ∧

t∈α(s)

(¬Φt)

(with e(v, i) := τ({(v, i), ε(v, i)})) for each s ∈ T0 (each Φs is well-defined,because the subgraphs form a finite tree), and finally,

ϕ(G) := {(g1, . . . , gk) ∈ Gk| Φ0}.

We say the k-ary relation graph with negation G of −→K represents the k-ary

relation of −→K with the extent ϕ(G). Thus we get the following theorem.

Theorem 2. Each k-ary relation graph (with negation) of a relation contextfamily −→

K := (K0,K1, . . .) uniquely represents an extent of Kk described by afirst order logic formula over

.⋃l∈N0

.

M l.

Relation Graphs: A Structure for Representing Relations 43

For the relation graph represented in Figure 4c, it holds that

ϕ(G) := {(g1, g2) ∈ G2| ∃(g3, g4, g5, g6, g7) ∈ G5 : ¬((g1, g4, g5) ∈ id3I3

∧ (g6, g2, g7) ∈ id3I3 ∧ ¬((g4, g3) ∈ child · ofI2 ∧ (g3, g6) ∈ child · ofI2) ∧

¬((g5, g7) ∈ child · ofI2))= {(g1, g2) ∈ G2| ∃(g3) ∈ G : ((g1, g3) ∈ child · ofI2 ∧ (g3, g2) ∈ child · ofI2)∨ ((g1, g2) ∈ child · ofI2)).

With the common pre-knowledge about family relationships, we can understandthe relations represented in Figure 4 as a) man (unary), b) has · two · children(unary), and c) grandchild · or · child · of (binary).

In the graphical representation of a relation graph G, it is allowed to omitthe index describing the arity of a relation, because it corresponds to the degreeof the vertex it is assigned to. For instance, we write only �, ⊥, id instead of �k,⊥k, idk for k-ary relations. An equivalence relation on a set of relation graphs(the ones with negation as well as the simple ones) of a relation context family−→K is defined by

G1ΘG2 :⇐⇒ ϕ(G1) = ϕ(G2).

In the graphical representation of relations by relation graphs the following sim-plifications are allowed in a way that each picture of a relation graph G uniquelydetermines ϕ(G), i.e. the equivalence class [G]Θ.

– It is not necessary to enumerate half-edges (v, i) ∈ H∗(G) with deg(v) = 1or with κ(v) generated by �k, ⊥k or idk (k ∈ N).

– The k-ary identity relation (for k ≥ 2) can be represented simply as a startingpoint of k half-edges, e.g. we draw Figure 5b instead of 5a.Note: It yields for k = 2 the representation in Figure 5d instead of 5c. (Fork = 1 holds id1 = �1.)

With these simplifications our graphical representation of a relation is in accor-dance with the first graphical system of Burch in his book [Bu91]. The onlydiffenrence is the numbering of the half-edges. The relations represented in Fig-ures 3e and 4b can be drawn as represented in Figures 5e and 5f.

1

3

2id id

1 2child of

1

f)

21

2

woman

child of1 2

man

3

e)a)

b)

c)

d)

child of1 2

child of1 2

1

Fig. 5. Simplified representation of relation graphs

44 Silke Pollandt

5 Iterative Construction of Relation Graphs

In the preceding chapters we have shown that each k-ary relation graph of arelation context family −→

K uniquely represents a k-ary relation with an extent ofKk. Now, our aim is to show that each compound attribute of −→K can be repre-sented by a suitable relation graph of −→K . We follow the iterative construction ofcompound attributes (see section 2).

– Each compound attribute m ∈ .

Mk= Mk

.∪ {�k,⊥k, idk} can be representedby the k-ary relation graph G := (V0, deg0, ε0, (S, <∼), κ) with V0 := {v},deg0(v) := k, ε0(v, i) := (∞, i), ε0(∞, i) := (v, i) (for i ∈ {1, . . . , k}) andκ(v) := (m′,m′′). Its graphical representation for k = 3 is drawn in Figure6a.

Let G := (V0, deg0, ε0, (S, <∼), κ) be an arbitrary k-ary relation graph represent-ing a compound attribute m ∈ Mk, and G := (V0, ˆdeg0, ε0, (S, <∼), κ) be anarbitrary l-ary relation graph representing a compound attribute m ∈ Ml of −→K .Then we get the following constructions.

– The compound attribute ¬m is represented by the k-ary relation graph G′ :=(V0, deg0, ε0, (S′, <∼), κ) with S′ := S .∪ G0 (with G0 := (V0, deg0, ε0)). Itsgraphical representation for k = 4 is drawn in Figure 6b.

– The compound attribute m(i ◦ j)n is represented by the (k + l − 2)-aryrelation graph G′ := (V0

.∪ V0, deg′0, ε

′0, (S

.∪ S, <∼), κ′) where deg′0 and κ′ are

the natural combinations of the corresponding mappings in G and G, but themapping ε′0 is changed by ε′0(ε0(∞, i)) := ε0(∞, j), ε′0(ε0(∞, j)) := ε0(∞, i),and renumbering the pending edges, moreover. Its graphical representationfor k = 4, l = 3, i = 3 and j = 1 is drawn in Figure 6c.

– The compound attribute m(i�j) is represented by the (k − 2)-ary relationgraph G′ := (V0, deg0, ε

′0, (S, <∼), κ) where the mapping ε0 is changed to ε′0 by

setting ε′0(ε0(∞, i)) := ε0(∞, j), ε′0(ε0(∞, j)) := ε0(∞, i), and renumberingthe pending edges. The graphical representation for k = 4, i = 2 and j = 3is drawn in Figure 6d.

– For each permutation πk =(

1, . . . , ki1, . . . , ik

), the compound attribute mπk is

represented by the k-ary relation graph G′ := (V0, deg0, ε′0, (S, <∼), κ) where

the mapping ε0 is changed to ε′0 by setting ε′0(ε0(∞, j)) := (∞, ij) and inreverse, ε′0(ε0(∞, ij)) := ε0(∞, j) for each j ∈ {1, . . . , k}. The graphical

representation for k = 4 and πk =(

1 2 3 44 1 3 2

)is drawn in Figure 6e.

– Iteration of the above constructions of relation graphs corresponds to theiteration of operations on compound attributes.

With these constructions we have shown the following theorem.

Theorem 3. Each compound attribute of Kk of a relation context family −→K :=

(K0,K1, . . .) can be represented by a k-ary relation graph (with negation) of −→K .

Relation Graphs: A Structure for Representing Relations 45

233

11

2m

a)

m

12

4

12

433

b)

211

2

3

e)

12

33

4

35

m n

c)

m

11

2

432

d)

m1

2

43

14

23

Fig. 6. Relation graphs representing compound attributes

The iterative construction of compound attributes (representable by relationgraphs) corresponds to the fundamental notions of PAL (representable by usingBurchs graphical syntax), cf. [Bu91]. The elements of

.⋃k∈N0

.

Mk correspond tothe primitive terms of PAL, the further compound attributes to the elements ofPAL. An array of PAL can be understood as a collection of compound attributesrepresented by a collection of relation graphs (or by one relation graph, whenwe omit the condition “connected” in the definition of a relation graph). Suchcollections of relation graphs can be neglected, because a “direct product” (cor-responding to the KPRODUCT operators, “by which any array of PAL may bereplaced by a single element of PAL” (see [Bu91], page131) can be generated bythe operations above.

6 Further Research

In the framework of this paper the aim was to introduce relation graphs asan algebraic structure for representing relations and operations on relations inContextual Logic of Relations. This formalization seems to be suitable to clarifysome open questions concerning the logic of relations as well as existential graphs.

– The (simplified) graphical representation of relation graphs highly corre-sponds to the graphical system of PAL. In [Bu91] (at the beginning of chap-ter 11) Burch states “PAL is designed specifically to accord as closely aspossible with the system of Existential Graphs that Peirce developed in thelate 1890s. . . . Of course, the exact relation each of the systems has toPeirce’s existential graphs is a matter that must be determined by ongoingscholarship.”

– Each k-ary (k ∈ N0) relation graph can be transformed into a 0-ary relationgraph exchanging each (∞, i) by (v∞, i). Let G := (V0, deg0, ε0, (S, <∼), κ) bea k-ary relation graph, and G′ := (V ′

0 , deg′0, ε

′0, (S, <∼), κ) the relation graph

obtained with V ′0 := V0

.∪ {v∞}, extending deg0 by deg′0(v∞) := k, and

changing ε0 by setting ε′0(ε0(∞, i)) := (v∞, i) and ε′0(v∞, i) := ε0(∞, i).Then G′ is a 0-ary relation graph. Extending the simplification of graphicalrepresentation of idk (in Section 4 done for k ≤ 2) to k = 1 we get the samepictures as in Section 4, but to understand as 0-ary relation graphs. Referringto [Ro73] for the beta part of Peirces existential graphs, this graphicalrepresentation of 0-ary relation graphs closely resembles existential graphs.In the case of ϕ(G′) = {∅} we get a direct correspondence, in the case ofϕ(G′) = ∅ it corresponds to the existential graph, where a cut enclosing the

46 Silke Pollandt

whole graph G′ is added. To determine this accordance more clearly is a fieldof further research.

– Relation graphs can be transformed to special concept graphs by dualiza-tion. Exchanging vertices and edges in a multigraph with pending edges andordered valences, we get a directed multi-hypergraph, the basic structureof concept graphs. This transformation promises to be useful for a betterunderstanding of correspondences between existential graphs (and relationgraphs) and concept graphs. Such investigations should especially be madefor existential semi-concept graphs (cf. [Kl01]) and concept graphs with cuts(cf. [Da01]).

– Logically, a basic question is whether two relation graphs are equivalentwith respect to Θ. Regarding the relation context families, the question iswhether the extents of compound attributes are equal or not. In analogyto [GW99b] and [PoW00] we can define extensional and global equivalenceof compound attributes. Then the question is to find effective methods fordeciding whether compound attributes are equivalent or not. This problemis not decidable in general. But the question arises, in what cases we candecide that two relation graphs are equivalent with respect to Θ. Such inves-tigations yield “inference rules” for relation graphs. (For instance, equivalentsubgraphs in S with respect to ∼ can be omitted pairwise.)Different terms in PAL may correspond to the same relation graph. Anddifferent relation graphs may represent the same (extent of a) relation. Thequestion is, in what framwork term-equivalence in PAL and Θ-equivalenceon relation graphs may be decidable.

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