Conceptual Graph Aboutness Theo Huibers1 and Iadh Ounis2 and Jean-Pierre Chevallet2 1
Dept. of Comp. Science, Utrecht University, PO Box 80089, Utrecht, The Netherlands, Tel: (31) 30 2 53 21 80, Fax: (31) 30 2 51 37 91, email:
[email protected] 2 CLIPS-IMAG, BP 53, 38041 Grenoble Cedex, France, Tel: (33) 76 63 56 87, Fax: (33) 76 44 66 75, email:(ounis,chevallet)@imag.fr Abstract. In this paper we present an information retrieval model based on conceptual graphs named E LEN [1]. In E LEN a conceptual graph is a representative of the information inherent in the document and query. Graph operators can be used to determine whether a graph is a generalisation of another graph, in which case the information carried by the graph is said to be about the information carried by the generalisation of the graph. In this article we want to compare this conceptual graph aboutness with aboutness derivations of other information retrieval models. The aboutness derivations are first mapped to a situated logic framework. Within this framework, the axioms that drive the retrieval process can be filtered out. The retrieval mechanism can then be compared with other approaches according to which axioms they are governed by. Using this method we can show that E LEN is indeed closely related to other classical aboutness derivations, and that in some aspects it has even more potential from an information retrieval perspective.
1 Introduction Information retrieval (IR) begins with a user having an information need (N ) that she wishes to fulfil. The information need is materialised in the form of a query , denoted q, which is given to an automatic IR system using query languages. Answers are in the form of documents that suit the user’s need according to the IR system. The content of each document d is represented by a specific set of descriptors (denoted by (d)), taken from a descriptor language. The set (d) is obtained by a process termed indexing . IR is driven by a process called matching, the request is compared with the documentdescriptions. If the matching operation deems a document as being sufficiently similar to the query, then the document is assumed likely to be relevant and returned to the user. Relevance is then a connection between a document and an information need. In recent years several attempts to define a logic for IR have been carried out [1, 2, 3]. The earliest approaches were directed at the use of classical logics, like Boolean logic [4]. The basis of a logical model for IR originates in the work of Cooper who introduced an objective notion of relevance termed logical relevance [5]. Cooper provides a formal definition of relevance in terms of logical consequences. If the query is a logical consequence of the document, then this document is deemed to be relevant. In order to forestall confusion, we will call the logical relevance relation aboutness. An aboutness derivation is rigid and strict: either a document is logically relevant with respect to a query, or it is not. Therefore aboutness has no degrees and documents are not classified with respect to the plausibility of their aboutness to the query. Van Rijsbergen [2] pointed out this problem when he proposed the use of a non-classical logic for IR.
However, van Rijsbergen leaves open the problem of the choice of an appropriate logic to represent documents, queries, system’s knowledge and the inference process that can deal with the involved uncertainty. As a consequence, the key problem of the research paradigm is the selection of an adequate logic for this task. The choice depends mainly on the ability of the logic to model an uncertain retrieval process. Other criteria may also be considered, like the expressiveness and the computation complexity, especially when multimedia documents are involved. Some authors suggest that expressive knowledge representation formalisms can be used for modelling complex information and logical processes. For example knowledge-based formalisms which are based on conceptual information [6, 7]. Recently, Chevallet [1] argued that the conceptual graphs formalism, as presented by Sowa [8], is suitable for modelling relevance along the guidelines of the logical model. He proposed an IR-system, named E LEN, which is based on the conceptual graphs approach. The reason for this choice is threefold: (1) the formalism has a great expressive power that can represent all components of an IR system; (2) it has a powerful formal semantics (first-order logic) and (3) it supports efficient implementation. Moreover, recent extensions of this theory [9] allow to deal with uncertain knowledge which is a very important notion for designing IR models and their uncertain-matching functions. The idea seems to be attractive, but is it more convenient than the well-known classical IR models? In previous research Huibers & Bruza presented a neutral framework for studying IR mechanisms based on situation theory [10]. Within this framework, axioms have been outlined that reflect assumptions made by IR models. The framework allows the comparison of retrieval models at an abstract level, according to which axioms they are governed by. The theme of this paper is the study of the aboutness derivations of E LEN. The information carried by a graph A is said to be about the information carried by another graph B , if graph A is a specialisation of graph B . E LEN can be mapped to a neutral framework, then the axioms which this system is governed by will be described. This article will highlight the derivation-steps of the E LEN model as to how it derives whether a graph (as an information carrier) is about another graph. These derivations seem to be logically valid (as they can be mapped to first-order logic) but are they informationally valid? The advantages of this study are threefold. Firstly we can show where the IR potential of the model arises. Secondly it will allow us to compare E LEN with other retrieval approaches according to which axioms they are governed by, here we focus on the Boolean and the coordinate level matching models. Lastly, it will allow us to make suggestions for the improvement of the model. The rest of this paper is structured as follows. In Sect. 2 we give a brief introduction to E LEN and its basic approach. In Sect. 3 we specify how this approach can be mapped onto the situated framework. In Sect. 4 two IR mechanisms, will be theoretically compared with E LEN. Finally we discuss the axiomatisation of E LEN and give several possible improvements to this system.
2 The Elen-system Several IR models are already designed to deal with multimedia and complex information through an appropriate knowledge representation formalism. Good examples of such formalisms, experienced within the RIME [7] and MIRTL [6] projects, are based
on a formalism derived from the notion of Conceptual Dependency and on Terminological Logics, respectively. They attempt, with varying degrees of success, to build operational models for IR. It is in that direction that Chevallet in [1] proposed the use of the conceptual graphs formalism to instantiate the logical model suggested in [2]. The latter is only a formal framework for designing retrieval systems involving knowledge and deduction mechanisms. Chevallet motivated his choice by the fact that this formalism can represent all components of an IR system: documents and queries can be represented using conceptual graphs, the general domain knowledge that can be reduced to ontologies is represented using concept type and conceptual relation type lattices, and the matching function is implemented using an extension of the projection operator. This idea led to the system E LEN (g´eniE logicieL & recherchE d’informatioNs). For sake of simplicity, E LEN adopts the additional constraint that every document and every query is characterised by one single simple graph [8]. Only dyadic relations are used in the system. E LEN considers graphs furthermore to be the information carriers for documents and queries. The matching process takes place based on the decision whether a graph A is derivable from another graph B , using one or more construction operators. In fact the partial order, , defined on the canonical graphs, is of prime importance in this choice. Indeed, a document d indexed by a conceptual graph (d) is about a query represented by a conceptual graph q , if (d) is a specialisation of q , i.e. the graph (d) expresses something more detailed than q . Phrased differently, the information contained in graph q is also contained in graph (d). This means that (d) q but also that ( (d)) (q) according to the formula operator introduced by Sowa. Going back to IR considerations, one can say that the formula related to the graph q is a logical consequence of the formula related to the graph (d). In the case of conceptual graphs the partial order plays the role of the deduction connective in the logical model. Definition 2.1 (Conceptual Graph Aboutness) Let D be a set of documents, with d 2 D. Furthermore, let G be a set of single conceptual graphs, with q and (d) 2 G . Then, d about q if and only if (d) q. Such a retrieval mechanism is implemented through the projection operator, which is used as a basis for defining a matching function between documents and query representations. Indeed, the conceptual graphs theory tells that if a graph u is a specialisation of a graph v , then there must exist a projection of v in u [8]. Mugnier proves the converse of this basic theorem and shows that if there is a projection of v in u, then u v [11]. This shows that the projection operator may be viewed as the basic retrieval operator: retrieving documents that imply query q comes down to retrieving documents that contain a projection of q . Example 1. For instance, a user wants to retrieve all documents dealing with ‘a Unix command that searches for an object in a structure’. The system has to retrieve all the documents d represented by a specialisation of the query q . The manual of grep constitute a relevant document for the user, its description is a specialisation of q (see figure below).
A Conceptual Graph for the Query: "A Unix command that searches an object in a structure"
q
UNIX-MAN
Agt
ActsOn
SEARCH
STRUCT
IncludeIn
Obj
OBJECT
A Conceptual Graph for the document: "Grep search on a file for a string or a regular expression"
d
UNIX-MAN:grep
Agt
ActsOn
SEARCH
IncludeIn
Obj
STRING
FILE
Chr
EXPRESSION
Fig. 1. The projection operation
Here, the subgraph of (d) which contains darkened nodes corresponds to the projection of q . Note that in this projection, concepts “FILE” and “EXPRESSION” are restrictions of the concepts “STRUCTURE” and “OBJECT” of q , respectively.
3 Situated Information Retrieval Paradigm In this section we propose a situated framework as introduced in [10]. This framework is based on concepts emerging from the field of Situation Theory. So called infons represent information items. Sets of infons form situations which are used to model the information born by objects such as documents. An arbitrary IR model can be mapped onto the framework. Special transformation functions are defined for this purpose depending on the model at hand. An important aspect is the aboutness relation which is, in the framework, a relation between situations. This aboutness can be derived by a so called derivation system, which consists of a set of postulates. This set of postulates is presented as a series of axioms and rules which establish properties of the aboutness relation between situations. This offers the possibility to compare IR systems according to which axioms and rules they satisfy. In Situation Theory [12], a cognitive agent’s world divides up into a collection of situations. In documents various situations are present. For example, the situations present
in ‘Romeo and Juliet’ by Shakespeare consist of love-situations, disappointmentsituations, etc. Situations like these are termed real situations, which are ‘parts of the world’, picked out by some individuate scheme. We are all (as human beings) able to individuate some items of information of such a situation. Real situations are hard to formalise. Therefore Situation Theory distinguishes one other sort of situations, called abstract situations, which are mathematical constructs and consist of a set of information items. One can see the descriptor set of a document as an abstract situation, which is an approximate representation of the real situations presented in the document. Some specific notion of an ‘item of information’ to work with is needed. Following Devlin [12], we introduce the notion of an infon: Definition 3.1 (Infon) An infon is a structure hhR,a1 ,...,an; iii that represents the information that the relation R holds (if i = 1) or does not hold (if i = 0) between the objects a1 ,...,an . In IR the objects in this definition are referring to atomic information items, such as keywords, noun phrases, or boolean formulas. The set of relationships can for instance consist of semantic relations [7], conceptual relations [1], or informational relations [13]. Consider a document containing the information “The Sioux defeated General Custer’s cavalry at Little Big Horn in 1876.”. The information carried by this document is modelled by the infon hhdefeat,Sioux,General Custer’s Cavalry ; 1ii. We say that the document-situation supports this infon. The relation defeat denotes a specific relationship between the terms Sioux and General Custer’s Cavalry. Such relationships are used to provide contextual information for driving IR. An example of an abstract situation is given by: fhhdefeat,Sioux,General Custer’s Cavalry ; 1ii, hhposition,Little Big Horn,
iig
1876; 1
The concept of parameters in infons used within Situation Theory 3 is useful for an IR meta-theory, for instance, if we want to make clear that “Sioux” are I NDIANS or that the relation defeat is between I NDIANS and C AVALRY . We use a; _ b_ , etc. to denote these parameters. The situation fhhdefeat,a_ ,General Custer’s Cavalry ; 1ii, hhnamed,Sioux,a_ ; 1iig represents the fact that the object a_ that defeat “General Custer’s Cavalry” is named “Sioux” (because both infons are labelled with the same parameter a_ ). In the situated IR framework we have the notion of information inference. Inference takes place if additional information can be inferred from one infon, information that is implicit in the information item (nested information Dretske [14]). Commonly in the terminology of IR, this inference is based on the notion of containment. Following Barwise & Etchemendy [15] we present the information inference (denoted by !) as follows: Definition 3.2 Given two infons '; , the formula '! holds if and only if infon can be derived from the infon ' within the framework of the underlying IR model. Infons constitute the lowest level of information granularity. At a higher level of granularity we find the situations, or in IR terminology, the documents or queries. Two situations can be composed in various ways to form a new situation. 3
For detailed information see [12], page 50.
Situation union takes all the infons of the first situation together with those of the second one. The union of the situations S and T , written as S [T , represents all the infons of both sets. If situation S [ T supports an infon, situation S or situation T supports this infon. We can use this operator to model the fact that we combine unrelated situations. A situation fusion operator (denoted with ) can be seen as creating new situations from existing ones by putting them in relation with each other. By way of illustration, consider two situations, one in which “Custer’s Cavalry was defeated” and another situation in which “the Sioux defeat someone”. It is possible to create new information using these situations by stating that “the Sioux defeat Custer’s Cavalry”. This is based on the assumption that there is a certain relation, viz. that Custer’s Cavalry is the someone in the second situation, between the two situations. There are different ways to define this kind of situation fusion. One possible definition of situation fusion can be proposed by the composition of particular infons. For example, take the sets S = fhhdefeat,p_ ,q_; 1ii, hhnamed,Sioux,p_; 1iig and T = fhhdefeat,r_ ,s_; 1ii, hhnamed,Custer’s Cavalry,s_; 1iig. To make clear that “the Sioux defeat Custer’s Cavalry” we have to state that in the union of S and T the parameters p_ and r_ (respectively q_ and s_ ) are identical. The result is the set S T which is fhhdefeat,r_,s_; 1ii, hhnamed,Sioux,r_; 1ii, hhnamed,Custer’s Cavalry,s_; 1iig. This kind of parameter exchange can be defined as follows: _ y_ ) represents the replacement of the parameter x_ in set Definition 3.3 The notation S (x; S by the parameter y_ . The properties of the parameters exchange are defined as follow:
S (w;_ x_ )(y;_ z_ ) =def (S (w;_ x_ ) )(y;_ z_ )
So, the representation of the previous fusion example can be given as: S T =def _ r_ )(q; _ s_ ) (S [ T )(p; with p; _ q_ and r; _ s_ are parameters used in S and T respectively. Various fusion operators can be proposed, and therefore, we define the x -operator that is using variant x of combining. The next step is the translation of the conceptual graph representation, as used in E LEN into abstract situations. A graph carries information, and as such it can be seen as a situation. What are the infons of this situation? In E LEN a simple conceptual graph is constructed out of concepts, referents, and dyadic relations. All parts have a specific role in the description of information. For instance, the concepts describe the types of objects. Following this idea, we propose to translate each item of a graph (concept, referent, relation) into a specific infon. We have to be careful to conserve the information that, for instance, a referent infon representing “Sioux” is connected with a particular type infon representing I NDIAN, in order to distinguish between the indians “Sioux” and a person named “Sioux”. Given two conceptual graphs g and h 2 G , and let S be the set of all abstract situations, the translation function trans : G ! S is defined as follows: – If R is a dyadic relation between two subgraphs g and h then trans (gRh ) = (trans (g ) rel fhhR; p_ ; q_ ; 1iig rel trans (h )) with p_ and q_ two unique parameters. An infon of the form hhR,p_ ,q_ ; 1iiis called a relation infon. The ith parameter of the infon is the identifier of the concept linked to the ith arc of R. The situation fusion operator rel is explained later on.
– For each individual concept u with a referent r of graph G the function trans (u : r ) has as result trans (u ) ref fhhRef; r;p_ ; 1iig with p_ as a unique parameter and r the referent of concept u. An infon of the form hhRef,r,p_ ; 1iiis called a referent infon. The situation fusion operator ref is explained later on. – For each generic concept u of graph G the function trans (u ) has as result fhhType,T YPE - OF-U,p_ ; 1iig with p_ as a unique parameter and T YPE - OF-U the concept-type of u. An infon of the form hhType,T YPE - OF -U,p_ ; 1iiis called a concept infon. The translation uses two situation fusion operators, namely, rel and ref . The former relates two concepts, the latter connects a referent to a concept. Take the first operator, assume we have two situations S , T and we want to connect them by a situation R representing the information about a relation. The translation function was defined as (S rel R rel T ). We have to make clear that the parameters used in R are connected to a concept in S and to a concept in T . Therefore we define the situation fusion as follows: (S rel fhhR; p_ ; q_ ; 1iig rel T ) =def ((S [ fhhR; p_ ; q_ ; 1ii)g)(s_ ;p_ ) ) [ T )(_t;q_ ) with s_ and _t parameters used in a concept infon of S and T , respectively. The situation fusion for a referent infon ( ref ) is defined as follows: fhhRef; r; p_ ; 1iig ref S =def (fhhRef; r; p_ ; 1iig [ S )(p_ ;q_ ) with q_ a parameter used in a concept infon of S . For example, trans ([I NDIANS : 0 Sioux0 ]!(loc)![P LACE]) results in the situation fhhRef,Sioux,s_; 1ii, hhType,I NDIANS,s_; 1ii, hhloc,s_,q_ ; 1ii, hhType,PLACE,q_ ; 1iig. If we have two infons that are sharing the same parameter then we call this corresponding infons. For instance, the infons hhType,I NDIANS,p_ ; 1ii and hhRef,Sioux,p_ ; 1ii are corresponding. Definition 3.4 (Graph Situation) In the case of E LEN, a set S of infons is called a graph situation if and only if it satisfies for its elements: 1. For each parameter used in a relation infon there exists a corresponding concept infon. 2. For each concept infon there exists at most one corresponding referent infon. 3. If there is more than one concept infon in the set then for each concept infon there exists a corresponding relation infon 4. For each referent infon there exists a corresponding concept infon. 5. Each relation infon has exactly two parameters. 6. Each pair of relation infons ri and rj has a parameter in common, or there exist a list of pairs (ri ; rk ) : : : (rl ; rj ) such that rn 2 S and each pair has a parameter in common. The information containment holds between two concept infons '! if the concept corresponding to is defined as a generalisation of the concept corresponding to ' in the concept taxonomy. For example, if G ROUP I NDIANS is defined in the taxonomy, then hhType,G ROUP,p_ ; 1ii!hhType,I NDIAN,p_ ; 1ii holds for any parameter p_ . In order to create a platform for a discussion about aboutness derivations, we have to make an explicit assumption of aboutness, viz. that aboutness can be derived with some sort of logic. We represent the aboutness-relation with the symbol 2;. Informally, S 2; T means that situation S is about situation T , and S 2 6; T that S is
not about T . We suggest that aboutness can be viewed as a more or less logical derivation. For instance, in this way given the fact that S [ T is about S , we can derive that situation fhhdefeat,Sioux,Cavalry; 1ii, hhfighting,Sioux,Cavalry; 1iig is about the situation fhhdefeat,Sioux,Cavalry; 1iig. The aboutness derivation of a situation should depend on its meaning, not on its form. Therefore we need a set equivalence relation which is defined as follows: given two situations S; T the equivalence relation is defined as: S T =def (' 2 (S [ T )(p_1;q_1 ):::(p_n ;q_n ) ) , (' 2 T ) and (' 2 (T [ S )(r_1 ;s_1 ):::(r_n;s_n ) ) , (' 2 S ). For example, S = fhhRef,Sioux,p_ ; 1iig and T = fhhRef,Sioux,q_ ; 1iig are equivalent because (' 2 (S [ T )(p;_ q_) ) , (' 2 T ) and (' 2 (T [ S )(q;_ p_) ) , (' 2 S ). In our framework we formalise aboutness by a set of axioms and rules. First we define the derivation system’s language and the derivation system: Definition 3.5 The language L is defined as a triplet (I ,S ,Ab) such that each part is defined as follows: The set I of infons is taken from an infon language4. The set of (abstract) situations S is defined as follows: 1. if ' 2 I then f'g 2 S 2. if S; T 2 S then S [ T , S x T 2 S (for any definition of the fusion x) 3. only that which can be generated by clauses 1 and 2 in a finite number of steps is an element of S The set of aboutness formula Ab is defined as follows: 1. if '; 2 I then '! ,'6! 2 Ab 2. if S; T 2 S then S T; S 2; T; S 26; T 2 Ab 3. only that which can be generated by clauses 1 and 2 in a finite number of steps is an element of Ab Definition 3.6 Given a language L as defined in 3.5, a derivation system A is a pair of the form (Ax; Rule), with Ax a set of axioms in Ab and Rule a set of rules of the form R(T1 ; : : : ; Tk ; Tk+1 ). Here, T1 ; : : : ; Tk are the premises of the rule and Tk+1 is the conclusion, which are elements from the set Ab.
A The representation of the following rules is as usual in logical systems, i.e. B means that if A is valid in an IR model, then B is also valid. Let us inspect now some postulates which are valid in E LEN. The first axiom is reflexivity, i.e. aboutness as defined in E LEN is reflexive, which means that every situation is about itself. In E LEN every graph is projected in itself. Reflexivity seems to be an inherent property of aboutness in many IR models. Reflexivity
S 2;S
The first rule is the Left Monotonic Union (LMU). The term “monotonicity” stems from the fact that aboutness is preserved under informational union. An example of the 4
Representing the descriptor language, which could be based on keywords, conceptual graphs, etc.
left monotonic union is the following. Given that the set fhhdefeat,Sioux,Cavalry; 1iig is about fhhdefeat,Sioux,Cavalry; 1iig (for instance by using reflexivity), a new situation is formed by informationally uniting the first set with fhhbattle,Sioux,Cavalry; 1iig. LMU allows us to conclude that this new united set is also about fhhdefeat,Sioux,Cavalry; 1iig. Left Monotonic Union
S 2;T S[U 2;T
So far we proposed postulates without taking into account the properties of the infons. However the symbol ! is defined as information derivation and may be useful for the aboutness derivation. The following rule Containment
! f g 2;f g
is valid in E LEN. A graph with only one concept is said to be about a generalisation of the concept, if the generalisation is defined in the taxonomy. The next two rules are Left Set Equivalence and Right Set Equivalence, which are jointly referred to as Set Equivalence (SE). It expresses the requirement that set equivalence sets have exactly the same aboutness derivations. Set Equivalence
S 2;T T U S 2;U
S 2;T SU U 2;T
With this rule we can for instance derive that, given S [ T 2; S it is allowed to deduce that T [ S 2; S . In the conceptual graph model this implies that it does not matter how the graph is constructed: two identical graphs have the same aboutness derivations. The Cut rule, which is valid in E LEN, is common in logical systems. Using it we are able to reduce the left side of an aboutness formula. Cut
S[T 2;U S 2;T S 2;U Next we define the set of postulates which completely describes the E LEN aboutness derivation. Definition 3.7 The derivation system E corresponding to the aboutness derivation of E LEN is defined as the set of postulates fReflexivity, Left Monotonic Union, Containment, Set Equivalence, Cutg. Of course there are more postulates that are valid in E LEN, like for example the rule of transitivity, Transitivity
S 2;T T 2;U S 2;U
It states that if S 2; T and T 2; U are concluded, then it is allowed to draw the conclusion that S 2; U . Transitivity is a derived rule in any derivation system containing LMU, SE and Cut. The question arises whether the proposed set of postulates is powerful enough. Theorem 1 The derivation system E LEN is complete. That is, for all g , h 2 G (with G a set of conceptual graphs) it holds that if g h then trans (g ) 2; trans (h ).
Proof 1 Sketch: assume g h, this means that g is constructed out of h using the four graph operators. For each constructor operator we have to inspect whether this is governed by our derivation system. For a detailed proof see appendix A.1. Theorem 2 The derivation system E LEN is sound. That is, for all g , h set of graphs) it holds that if trans (g ) 2; trans (h ) then g h.
2 G (with G a
Proof 2 Sketch: we have to proof whether each postulate is governed by one ore more construction operators. For a detailed proof see appendix A.2.
4 Comparison of Information Retrieval Models In this section we briefly introduce two other models, which are already mapped into our framework [10]. We begin by introducing the so-called strict coordinate retrieval model. In strict coordinate retrieval, aboutness is interpreted as: d is about q if and only if the keywords of q are a subset of the keywords of (d). Definition 4.1 (Strict Coordinate Aboutness) Let D be a set of documents, with d 2 be a vocabulary, with q and (d) T . Then, d about q if and
D. Furthermore, let T only if q (d)
The translation of a document d in a situation Sd is defined as follows: trans (d ) =
fhhI,t;1ii j t 2 (d )g. The relation I in the infon signifies an unspecified unary relation
reflecting the fact that the indexing process has deprived us of all knowledge of relations that the keyword was a part of. The translation of a query to a situation Sq proceeds in a similar way. The underlying derivation system of the strict coordinate retrieval is defined as follows.
Definition 4.2 (Strict Coordinate Situation Aboutness) The derivation system C corresponding to the aboutness derivation of the strict coordinate retrieval is defined as the set of postulates fReflexivity,Set Equivalence, Left Monotonic Union,Cutg. The last model that we introduce is the Boolean IR model. In Boolean retrieval the characterisation consists of a set of keywords which originate from a vocabulary T . The request is specified as a formula. These formulas are constructed from the vocabulary T using the logical connectives _,^, and :. For simplicity, we only consider postulates involving : and ^. A formula may contain negation, expressing for example that the user wants documents that are not described by a particular keyword. The Boolean retrieval inference mechanism is based on the classical derivation in addition with the Closed World Assumption rule (CWA), introduced by Reiter [16].
Definition 4.3 (Reiter 1978) The closure of a theory D, denoted by CWA(D ), is the theory D [f:t : D 6` t and t 2 T g. The set of all theorems derivable from D by CWA is identified with the set of all formulas classically derivable from CWA(D ). Definition 4.4 (Boolean Inference Aboutness) Let D be a set of documents, with d 2 D. Furthermore, let T be a vocabulary with (d) T . Let q be a logical formula constructed from the vocabulary T using the logical connectives ^ and :. Then, d about q if and only if CWA( (d )) ` q The next step is to define Boolean retrieval in terms of our framework. Translating the Closed World Assumption proceeds as follows. The document characterisation is extended with the negation of all the terms which are not contained in the characterisation. The definition of the trans-function trans 1 for the document descriptors and trans 2 for the query are as follows:
trans 1 (d ) = fhhI,t; 1ii j t 2 (d) and t 2 T g [ fhhI,t; 0ii j t 62 (d) and t 2 T g trans 2 ( 1 ^ : : : ^ n ) = ftrans 2 ( 1 )g [ : : : ftrans 2 ( n )g trans 2 (t ) = hhI,t; 1ii with t a propositional constant trans 2 (:t ) = hhI,t; 0ii with t a propositional constant Definition 4.5 (Boolean Situation Aboutness) The derivation system B corresponding to the aboutness derivation of the Boolean retrieval is defined as the set of postulates fReflexivity,Set Equivalence, Left Monotonic Union,Cutg. Comparing the derivation systems E , C , and B , we see that the postulates present in the derivation systems C and B are a subset of the postulates supported by E . The postulate of E that is not governed by the two others is Containment. This postulate derives aboutness based on the knowledge in the taxonomy. This is a powerful feature of E LEN. For instance, in the case of a user searching for the concept A NIMALS, she could of course be satisfied with a document indexed with the concept B IRD. More important is the possibility of E LEN to propose a preference on postulates. This preference can be obtained by applying a method as described in [17]. Roughly speaking, this method describes how document classes can be constructed out of a derivation system. For instance, the reflexivity axiom is much more likely to conserve the aboutness relation than the postulate LMU. In contrast with the systems C and B , we can propose a set of postulates inherent in the system that are to be preferred over Left Monotonic Union, on the basis of the semantical information present in the situation. A postulate that may be preferred over Left Monotonic Union is one using a type of fusion, for instance a Left Monotonic Related Fusion. Another point is that we can propose, on a meta-level, new rules for E LEN which are useful for IR. One of the goals of E LEN is to create a precision-oriented system in order to provide the user with highly precise “relevant” information rather than an overdose of “non-relevant” information. The aboutness decisions of E LEN are therefore strict, in the sense that there are only a few operators. As a result, often very few documents are
considered to be relevant in E LEN. In order to deliver some more documents, which still are very likely to be relevant, we can propose new rules. These rules can easily be added to the system. They are no longer based on the projection operators, but defined in terms of the general framework. Another extension of E LEN could be to allow the aboutness deduction between a graph and its restricted form, as given by the Right Monotonic Fusion rule. Right Monotonic Fusion
S 2;T S 2;U ref T
For instance, up till now it was not allowed to conclude that the graph representing “The Indians Sioux defeat a Cavalry” is about the graph representing “The Indians Sioux defeat a Cavalry named General Custer’s Cavalry” because the right graph is a specialisation of the left one (rather than the opposite). With the new Right Monotonic Fusion rule we are allowed to add referents on the left side in order to determine aboutness. Adopting this new plausible rule can be viewed as allowing the user to mislabel references. A user uses a referent in the query as an example, but maybe she is looking for more general information.
5 Conclusions The theme of this article is the theoretical study of the conceptual graph based IR model E LEN. This investigation proceeds in terms of a framework based on a theory of information. Within this framework, postulates have been outlined that reflect assumptions made by the E LEN-model. The postulates of the model have been examined. Compared with a Boolean IR model and a strict coordinate retrieval model, conceptual graphs seem to be a suitable candidate for the determination of information aboutness as it is done in IR. The possibility of adjusting aboutness inside the derivation mechanism is lacking in the other two models. The expressive power of the axiomatisation lies in the fact that we can propose new aboutness decisions rules based on the E LEN-model at a conceptual level. Another advantage is the possibility it offers to investigate the effects of ordering axioms and postulates, as it is done in [17]. This ordering could possibly provide us with a ranking of aboutness, based on a qualitative measuring rather than a quantitative one. Above all we can prove the usefulness of the addition of new rules in terms of theoretical proofs rather than with experimental results. Acknowledgements This paper was written while the first author was visiting the Mod´elisation et Recherche d’Information Multim´edia (MRIM) team in Grenoble, France. He would like to thank the people of the MRIM for their kind hospitality. His visit was partially supported by the Esprit Network of Excellence No.6606 (IDOMENEUS Research Exchange Program), which is gratefully acknowledged. The second and third author are partially funded by the European Community under the ESPRIT Basic Research schema, FERMI No. 8134. We are very grateful to Franc¸ois Paradis, Bernd van Linder, Nathalie Denos, and Peter Bruza for reading an earlier version of this paper. We also appreciate the constructive comments made by the anonymous referees.
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A Completeness and Soundness of the Derivation System A.1 Completeness of the Derivation System Theorem 3 The derivation system E LEN is complete. That is, for all g , h 2 G (with G a set of conceptual graphs) it holds that if g h then trans (g ) 2; trans (h ). Proof 3 Assume g h, this means that g is constructed out of h using the four graph operators. So, for each graph operator there should be a representative dedu ction possibility in the derivation system. – If g is a copy of h then trans (g ) is a set containing the same infons as trans (h ) but this set is labelled with another set of parameters. In this case, according to the set equivalence relation which was introduced in section 3, we have two equivalent sets. Starting with the reflexivity axiom, we can infer that trans (g ) 2; trans (h ) when they correspond to the same set of infons. – If g is obtained from h by the restriction operator then two cases are possible: we can restrict a graph if we replace a concept type by a sub-type, or when a referent is replaced by an included set. For the first case, if we change a sub-type-infon by a type-infon in trans (g ), then we assume, for example, the existence of an infon b = hhType; B;p_ ; 1ii which can derive an infon a = hhType; A;p_ ; 1ii in trans (h ) with b ! a. We build a situation trans (g )0 from situation trans (g ) by removing item “b” from trans (g ) (trans (g ) is equal by construction to trans (g )0 [fb g). Using reflexivity we got the axiom trans (g )0 [ fb g 2; trans (g )0 [ fb g. The containment-rule allows us, given the fact that hhType; B;p_ ; 1ii ! hhType; A;p_ ; 1ii, to replace the former infon with the latter in the right situation of the aboutness. Finally after using the set-equivalence rule, we obtain trans (g ) 2; trans (h ). – It is not necessary to present the simplification rule in this proof, as it is inherent of the set-theoretical properties, and therefore governed by the set-equivalence rule. – Two graphs that share a common concept can be external joined to form a new graph having this common concept. So, if g is constructed from h by a join operator then trans (g ) trans (h ) x trans (z ), where z is an arbitrary canonical graph having at least one common concept with h, and x is a situation fusion operator defined as follows: S x T =def (S [ T )(a_1 ;:::;a_n ):::(z_1 ;:::;z_n ) . With reflexivity trans (h ) 2; trans (h ), the x situation fusion operator, through the left monotonic union-rule, allows to fuse the left situation with the arbitrary situation trans (z ) corresponding to the graph z . Finally, with the set-equivalence rule we obtain trans (g ). A.2 Soundness of the Derivation System Theorem 4 The derivation system E LEN is sound. That is, for all g , h set of graphs) it holds that if trans (g ) 2; trans (h ) then g h.
2 G (with G a
Proof 4 If trans (g ) 2; trans (h ), we have to prove that for every deduction of the derivation system the graph corresponding to the right situation of the aboutness is still projected in the graph corresponding to the left situation of the aboutness. This means that we have to prove this property for each axiom and rule of the E derivation system.
– Using the reflexivity axiom then S 2 ; S . If we prove that for every situation, there is only one unique graph, then it follows that the graph h is projected in graph g (by means that a graph is projected in itself). So, the proof left, if trans (h ) = trans (g ) then graph h is identical to graph g. As we handle only graph situations, the sets contains only three kinds of infons, namely, referent-infons, relation-infons, and concept-infons, with the conditions as given in 3.4. The structure of the graph is conveyed by the infons, the edge of the relation as how it connect concepts are represented in order of the occurrence of the parameters in the relation-info n. Every concept-infon can be directly translated into one concept of the graph, similar for the referent-infons, which can be translated to referents of the concepts, according to the translation function. Therefore, every graph situation corresponds to an unique graph. – Given the conclusion that situation trans (g ) is about situation trans (h ) using left set equivalence implies that there exists a situation trans (z ), such that trans (z ) 2; trans (h ) and trans (z ) trans (g ). According to the equivalence relation definition, the situations trans (z ) and trans (g ) exhibit the same meaning and contain the same infons with just some parameters exchange. Hence, we can use the same arguments we used for the reflexivity axiom to say that the corresponding graphs z and g are identical. Going back to our aboutness decisions, there is a projection of h in z and a graph g identical to z so we have z h and then obviously g h. For right set equivalence, the proof is similar to one of the left set equivalence. – If situation trans (g ) is about trans (h ), using the cut-rule then this implies the presence of two aboutness decisions: trans (g ) 2; trans (z ) and trans (x ) 2; trans (h ) such that x g [ z . The corresponding graphs g , z , x and h are such that there is a projection of z in g and a projection of h in x, so we have g z and x h. As we handle only graph situations, the conceptual graph x consists in the graph g containing the graph z with eventually some restrictions due to the projection of z on g . Hence the information conveyed by the graph z already exists in the graph g in a more specific form. The conceptual graph x is reduced in fact into the graph g (we can say that g and x are identical as they convey the same meaning). As x h, and x is reduced to g , we have g h. – In case situation trans (g ) is about trans (h ), by the left monotonic union rule, as adding infons does not violate the projection rules, it only makes more specialization on the initial graph. – If situation trans (g ) is about trans (h ), by using the containment rule then there exists a concept-infon i in trans (g ) which is a specialisation of a concept-infon j in trans (h ). So, concept i in conceptual graph g is a restriction of concept j in graph h. Therefore graph g is a specialisation of graph h.
