Semantic Integration in the IFF

Semantic Integration in the IFF Robert E. Kent Ontologos rekent@ontologos.org Abstract The IEEE P160 0.1 Stand ard Upp er Ontology (SUO) project aims to specify an upper ontol ogy that will p rovide a structure and a set of gene ral conc epts upon which do- main ontologies could be constructed. The Informa tion Flow Framework (IFF), which is being developed under the auspices of the SUO Working Group, represents the structural aspect of the S UO. The I FF is ba sed on ca tegory theory. Semanti c integration of object-level ontologies in the IFF is represented with its fusion construct ion * . The IFF mainta ins ontologies using powerful composition pri- mitives, which includes the fusion co nstruction. 1. The Information Flow Fram ework The IEEE P16 00.1 Standard Upp er Ontology (SUO) 1 proj ect aims to specify an upper ontolo gy that wi ll provide a structure and a set of general concepts upon which ob- ject-level domain ontologies could be constructed. These object-level domain ontologies will utilize the SUO for “applications such as data interopera bility, information search and retrieval, automated inferencing, and natural language processing”. A central purpose of the SUO proj ect is interoperability. The Information Flow Framework (IFF) 2 is being de- veloped to represent the structural aspect of the SUO. It aims to provide semantic interopera bility among various object-level ontologies. The IFF supports this interopera- bility by its architecture and its use of a part icular branch of mathematics known as category theory (Mac Lane, 1971). A major reason that the IFF uses the architecture and formalism s that it does is to support modular ontology development. Modularity facilitates the development, test- ing, maintenance, and use of ontologies. The categorical approach of the IFF provide s a principled framew ork for modular design via a structural metatheory of object-level ontologies. Such a metatheory is a method for repre senting the structural relationships between ontologies. The IFF provides m echanisms for the principled foun- dation o f a metalevel ontological framework – a framew ork for sharing ontologies, manipul ating ontologies as objects, relating ontologies through morphisms, partitioning ontol- * Throughout this paper, we use the intuitive terminology of mathematical context, passage/construction, pa ir of invertible passages an d fusion for the mathematical concepts of category , functor, adju ncti on and colimit, respectively . ogies, composing ontologies via fusi ons, noting dependen- cies between ontologies, declari ng the use of other ontolo- gies 3 , etc. The IFF takes a building blocks approach to- w ards the development of obj ect-level ontological struc- ture. This is a rather elaborat e categorical approa ch, which uses insights and ideas from the theory of distributed logic know n as information flow (Barwise and Seligm an, 199 7) and the theory of formal concept analysis (Ganter and Wille, 1999 ). The IFF represents metalogic, and as such operates at the structural level of ontologies. In the IFF, there is a precise boundary between the metalevel and the object level. The m odular architecture of the IFF consists of metale- vels, namespaces and meta-ontologies. Ther e are three me- talevels: top, upper and lower. This partition, which cor- responds to the set-theoretic distinction between small (sets), large (classes) and generic collections, is permanent. Each m etalevel services the level belo w by pro viding a language that is used to declare and axiomatize that level. The top metalevel services the upper metalevel, the upper metalevel services the lower met alevel, and the lower meta- level services the object-level. Within each metalevel, the terminology is partitioned into namespaces † . The num ber of namespaces and the content may vary over time: new nam espaces may be created or old namespaces m ay be de- precated, and new terminology and axiomatization within any particular namespace m ay change. In addition, within each level, various namespaces are collected together into meaningf ul composites called meta-ontologies. At any par- ticular metalevel, these m eta-ontologies cover all the na- mespaces at that level, but they may overlap. The num ber of meta-ontologies and the content of any meta-ontology may vary over time: new m eta-ontologies may be created or old meta-ontologies m ay be deprecated, and new na- mespaces w ithin any particular m eta-ontology m ay change (new versions). The top IFF metalevel provides an interface between the simple IFF-KIF language and the other IFF terminolo- gy. By analogy, the sim ple IFF-KIF language is like a ma- chine language and the top IFF metalevel is like an assem- bly language. There is only one namespace and one meta- ontology in the top m etalevel: the To p Core (meta) Ontol- ogy. This meta-ontology represents generic collections. In a sense, it boo tstraps the rest of the IF F into existence. The single namespace, the m eta-ontology and the top metalevel can be identified with each other. The upper and lower IFF metalevels represent the structural aspect of the SUO. By analogy, the structural aspect of t he SUO is like a high lev- † The I FF terminology is disa mbiguated via the disjoin t uni on of local namespace terminology . A fully qua lified term in the IFF is of the form “  $  ”, where the namespace prefix label “  ” is a “.” se parated sequence of alphabetic strings that uniquely represents an I FF namespace, and the local unqua lified term “  ” is a uniqu e lowercase alphanu meric-dash string within that n amespace. For example: the term “ th.col.psh$coe qualizer-diagra m ” represents the coe qualizer diagram underly ing a pus hout diagram of theories within the theory pus hout namespace in th e low er IF F me talevel. el programming language such as Lisp, Java, ML, etc. There are three permanent meta-ontologies in the upper metalevel: the Upper Core (meta) Ontology represents the large collections called classes; the Category Theory (me- ta) Ontology represents category theory; and the Upper Classification (meta) Ontology repre sents information flow and formal concep t analysis. There will eventually be many meta-ontologies situated in the lower IFF m etalevel ‡ . Cur- rently there are only four: the Lower Core (meta) Ontology represents the small collections called sets; the Lower Classification (meta) Ontology is a sm all and more specia- lized version of i ts upper c ounterpart; the Algebraic T heory (meta) Ontology repre sents equational logic; and the On- tology (meta) Ontology represents first order logic and model theory. All versions of these meta-ontologies are listed as links in the SUO IFF site map 4 . The IFF, which is situated at the metalevel, represents form. The ontologies, which are situated at the o bje ct level, represent content § . By analogy, the content aspect of the SUO is like the various software appli cations, such as word processors, browsers, spreadsheet software, databases, etc. The distinction between content and form is basic in the general gramm ar of natural languages, in logic and in on- tology. In all of these realms, but especially in logic and ontology, the IFF offers a coherent principled approach to form. Such form is realized in the structuring, mapping and integration of ontologies. The IFF offers axiomatization and techniques for the hierarchical structuring of obje ct- level ontologies via the lattice of theories, the m apping between ontologies via sy ntax directed translation, and the semantic integration of ontologies via mediating or refer- ence ontologies. T o p araphrase Jo hn Sowa 5 , developing the tools and methodologies for extending, refining, and shar- ing obje ct-level ontologies is more important than d evelop- ing the content for those ontologies. ‡ A module in the IFF lower metalevel should represe nt a well-resear ched area. In addition to the IFF -OO, which represents first order logic and model theory , other non-core lower metaleve l modules are also being considered: a module for the “sof t computat ion” of both rough sets and fuzzy logic; a module for theories of semiotics; a module for game- theoretic semantics; etc. § Many current object-level ontologies contain generic axiomatizations for notions such as binary relations, partial orders, etc. I n the IFF , these are not needed, since such axiom at izations are includ ed in the Low er Core (meta) Ontology, etc. W hen compliant w ith the IFF , object-level ontologies can concentrate on their core axiomatics. 2. Basic Concepts of the IFF-OO The metalevel axiomatic framework f or ob ject-level on- tologies r epresented in first o rder logic and model theory is concentrated in the lower metalevel IFF Ontology (meta) Ontology (IFF-OO). The IFF-OO is a generic framework for the representation and manipulation of obj ect-level on- tologies. The architecture of the IFF-OO (Figure 1) con- sists of four central m athematical contexts * interconnected by five pairs of inv ertible passages * . Each of the four con- texts repre sents a basic concept axiomatized in the IFF- OO. These four concepts are language, theory, model and logic. The context of first order logic langu ages 6 sits at the base of the IFF-OO – everything depends upon it. The three other contexts – models, theories and logics – are situated above the language context. Models provide the interpretive semantics for obje ct-level ontolo gies, theories provide the formal or axiomatic sem antics, and logics pro- vide the combined sem antics. Any theory is based on a language, and the context of theories is connected to the context of languages by the base passage. An object-level ontology is populated wh en it has instance data. Unpopu- lated object-level ontolo gies are represented by IFF theo- ries, wh ereas pop ulated obje ct-level ontologies are represented by IFF logics. This paper deals only with for- mal, axiomatic sem antics for obj ect-level ontologies. Inter- pretive sem antics w ill be combined wi th this in futu re w ork. The concept of an IFF language is many- sorted – the definition follows (Enderton, 1972 ), generalizing the stan- dard notion of a single-sorted language. The IFF terminol- ogy is somewhat different from Enderton – it uses the tw o polarities of entities versus relations and instances versus types: an IFF entity type corre sponds to a sort, an IFF rela- tion type corre sponds to a predi cate, and an IFF function type corre sponds to a function symbol. In this paper, we ignore function types for simplicity – these are adequately handled in the IFF Algebraic Theo ry (meta) Ontology. Note that an IFF language deals only with type informa- tion. Constants are regarded as nullary function types. Lan- guages are comparable via langu age morphisms , and theo- ries are comparable via theory m orphisms . Any language L determines a lattice of theories fiber ( L ) ** , a base passage fiber †† . Any language morphism f : L   L  determines a function expr ( f ) : expr ( L  )  expr ( L  ) by induction, and from this a lattice morphism of the ories fiber ( f ) =  inv ( f )  dir   : fiber ( L  )  fiber ( L  ), * * The lat tice of theo ries fiber ( L ) for a language L is the complete lattice of all t heories with base language L usi ng enta ilment order between theo- ries: T   T  means that T  is more specialized than T  in the sense that T  is contained in the closure of T  ; or equivalently, that any theorem of T  is entailed by the axioms of T  . †† A fib er of a passage P : C  B for fixed object b  B is analogous to the inverse image of b along P , thus for ming the sub -context fi- ber P ( b )  C of all C -objects th at map to b and all C -morphisms that ma p to the identity at b . Language Logic Model Theory Figure 1: IFF-OO Architecture the fiber invertible passages of direct/inverse imag e opera- tors – the ( existential ) direct image operator dir  ( f ) = (  expr ( f )) op : fiber ( L  )  fiber ( L  ) ‡‡ and the inverse image operator inv ( f ) = ( expr ( f )  ) op : fiber ( L  )  fiber ( L  ). The mapping of unpopulated obje ct-level ontologies is represented by IFF language/theory morphisms. In particu- lar, the IFF repre sents ontology m apping as the movemen t of theories back and forth between lattices of theories by using the above lattice morphism of theories over a lan- guage m orphism. A recent vote by the SUO Working Group approved a proposal by Jo hn Sowa to develop a library of modules structured in a hierarchy. This library of modules w ill in- clude modules derived from other obj ect-level ontologies. The hierarchical structure framing such a library of mod- ules is a lattice of theories. Sowa has offered a step-wise approach for building a library of modules 7 . However, the processing involved here can be applied to any system of ontologies, and each step of Sowa’s proce ss of “building the hierarchy” is represented in the IFF. To do this we represent a module as an IFF theory. A library of modules, regarded as a generalization-specialization hierarchy, is conceptually situated within the context o f a lattice of theo- ries ** and its correlat ed structure known as the truth con- cept lattice §§ . In the IFF, an unpopulated monolithic obje ct- level ontology is represented as an IFF theory, the same as a module. The IFF regards a library of modules to be an unpopulated modularized obj ect-level ontology. This is represented in the IFF as a diagram of theo ries *** . In other terminology, an IFF diagram of theories repre sents a sy s- tem of object-level ontologies. Diagrams of theories are comparable via theory diagram morphisms 6 . Any diagram of theories T indexed by a shape graph G has a base dia- gram of languages L = base ( T ) of the same shape, where the language (language morphism) at any indexing node (edge) of graph G is the underlying base language (lan- guage morphism) of the theory (theory morphism) at that node (edge). Generalizing the fiber over a language, any language diagram L : G  |L anguage| determines a lat- tice of theory diagrams fiber ( L ) 6 . Generalizing the fiber adjoint pair over a language morphism, any language dia- ‡‡ In the fol lowing, we abbreviate this as dir ( f ) = dir  ( f ). §§ Intuitively, the truth conc ept latt ice is the latt ice of closed theories. The lattice order is reverse subset inclusion. The truth concept lattice is the concept lattice for the truth classificati on , the fund amental example 4.6 introduced in (Ba rwise and Seligman, 1997). ** * A diagram of theo ries T : G  |Theory| consists of two collections, theories and theory morphisms, indexed by a shape graph G : each G - node n indexes a theory T n and each G -edge e : m  n ind exes a theory morphism T e : T m  T n . The size of a diagram corresponds to the cardi- nality of the node a nd ed ge sets of its s hape graph. Although these c an be infinite, in most prac tical situations they are finite – there are empty diagrams, single theory diagrams, diagrams with only two theories and one theory morphism, etc. gram morphism φ determines a lattice morphism of theo ry diagrams fiber (φ) 6 . 3. Fusion of a System of Ontologies The IFF can utilize the fusion construction * in various mathema tical contexts. Since this pape r only discusses the formal, axiomatic semantics of integration, here we limit ourselves to the fusion construction for la nguages and theo- ries. The fusion of t heories is d efined in t erms of the fusion of languages (Table 1). Table 1: The Fusion Co nstruction ††† 1. I nformally , ident ify the th eories to be used in the construction. 2. Formall y, create a diagram of theories T of shape (indexing) graph G that indica tes this selection. This diagram of theories is tran- sient, since it will be used only for this computation. Other dia- grams could be used for other fusion constructions. 3. Form the fusion theory T · =  T of this diagram of theories, with theory fusion c ocone  : T  T · . a. Compute the base diagram of languages L = base ( T ) with the same shape. In more detail, L = base ( T ) =  L n    L e : L m  L n  =  base ( T n )    base ( T e ) : base ( T m )  base ( T n )  . b. Form the fusion language Ŀ =  L of this diagram, with lan- guage fusion cocone  : L  Ŀ . In more detail,  =  n : L n  Ŀ  , sat isfying the conditions  m = L e ·  n for G - edge e : m  n . c. Move (the individual theories  T n  in) the diagram of theo- ries T from the lattice of theory diagrams fiber ( L ) along the language morphisms in the fusion cocone  : L  Ŀ to the lattice of theories fiber ( Ŀ ) using the direct image function, getting the homogeneous diagram of theor ies dir (  )( T ) with the same shape G , where each theory dir (  )( T ) n = dir (  n )( T n ) has the same base language Ŀ (the meaning of homog ene- ous). d. Compute the meet (union) of the diagram dir (  )( T ) w ith in the lattice fiber ( Ŀ ) getting the fusion theory T · =  T = meet ( Ŀ )( dir (  )( T )). e. The language fusion cocone is the base of the theory fusion cocone  = base (  ) : base ( T )  base ( T · ). As m entioned before , any diagram of theories T has a base diagram of languages L = base ( T ) of the same shape. It is i mportant to not e that the indexed theories within T do not necessarily have the same base language. To semanti- cally compare these theories and to conceptually situate them wi thin a lattice of theories, w e move them to the lat- tice of theories over the fusion language Ŀ =  L , w ith this movemen t guided along the language morphism s in the fusion cocone  : L  Ŀ . The latter is a node ( G )-indexed collection of language morphisms, whose source is the lan- guage diagram L and whose target is the language Ŀ . For any diagram of theories T in fiber ( L ), the direct image fiber operator dir (  ) moves T along the fusion coco ne to ††† The two operations of (1) forming sums of theories and (2) specifying endorelati ons and then computing their quotien ts , offer an alternate method for the fusion construction of diagrams of theories: coequali zers of theories can be constructed a s qu otients of endorelations; and pusho uts of theories can be constructed in terms of sums of components and then quotients of endorelations. dir (  )( T ), a homogeneous diagram of s hape G in the lattice of theories over Ŀ . Homogeneous means that all the in- dexed theories in dir (  )( T ) have the same base language Ŀ , and hence can be semantically compared via the theory entailment order . The fusion of the diagram of theories T resolves into  T = meet ( Ŀ )( dir (  )( T )) – the fiber direct image dir (  ) along the base diagram fusion cocone, fol- lowed by the meet meet ( Ŀ ) in the lattice of theories over Ŀ , the base diagram fusion language. Two new ideas have emerged recently in the discussion of the SUO Working Group: the idea of a polycosmos and the id ea of mapping closure. Bo th of these idea s are impor- tant in the theory of semantic integration. However, it was not possible to succinctly express these ideas without the use of theory fusions. ○ The idea of a polycosmos ‡‡‡ was first expressed 8 by Patrick Cassidy: a polycosmos is a n unpop ulated mod- ular object-level “ontology that has a provision for al- ternative possible worlds, and includes some alterna- tive logically contradictory theories as applying to al- ternative possible worlds”. The math ematical formula- tion of polycosmic 9 w as immediately given by the au- thor in terms of the fusion of a diagram of theories. A diagram of theories T is mono cosmic when the fusion theory  T is consistent. A diagram of theories T is pointwise consistent when each indexed theory in dir (  )( T ) is consistent. A monocosmic diagram of theories is pointwise consistent by default. A diagram of theories T is polycosmic when it is pointwi se consis- tent, but not monocosmic; that is, when there are (at least) tw o consistent b ut mutually inconsistent theories in dir (  )( T ). In the IFF §§§ , there are some extreme po- lycosmic diagrams of theories, w here any two theories are either equivalent or mutually inconsistent. Each of the theories in these diagrams lies at the lowest level in the lattice of theories, strictly above the botto m incon- sistent theory containing all expressions. ○ The idea of map ping closure was first expressed 10 by the author. Any mapping of ontologies involves this notion of m apping closure. For any morphism of lan- guages f : L   L  , the mapping closure of f applied to any source theory T   fiber ( L  ) is the closure asso- ciated w ith the fiber adj oint pair : clo ( f )( T  ) = inv ( f )( dir ( f )( T  )). Since languag e m orphisms and en- dorelations are in a sense equivalent **** , the idea of mapping closure is also induced by a l anguage endore- ‡‡‡ According to the dictionary, a cosmos is an orderly harmonious sys - tematic universe. §§§ Since I FF model s have a set of tuples (= relation instances) as one component, they are more refined than traditional model-theor etic struc- tures and are better able to represe nt the intui tive notion of cont ext – some I FF models even have only one tuple. ** * * Any language morphism has a kernel (equivalence) endorelation based on the source language , w here two source typ es are equivale nt when they are mapped to the same target type. Conversely, any language endorelation generates an epimorphic language morphism onto the quo- tient language of the endorelation. lation. An endorelation based on a language L defines by induction an equivalence rel ation on variables, enti- ty types, relation types and expressions. One expres- sion is equivalent to another expression when the con- stituent terms †††† in each are equivalent. Any expres- sion that is equivalent to a theorem of a theory T  fiber ( L ) is included in the mapping closure ‡‡‡ ‡ . Any morphism of languages f : L   L  determines a lattice morphism of the ories  dir  ( f )  inv ( f )  : fiber ( L  )  fiber ( L  ) w ith the ( universal ) direct image opera tor dir  ( f ) =  expr ( f ) op : fiber ( L  )  fiber ( L  ) and the inverse image operator. In summ ary, for any morphism of languages f : L   L  there are two linked pairs of invertible monotonic functions: dir  ( f ) ⊣ inv ( f ) ⊣ dir  , with dir  ( f ) and inv ( f ) preserving join s (intersections), and inv ( f ) and dir ( f ) = dir  preserving meets (unions). Two questions arise. ( 1) W hat is the significance of the mapping closure? (2) Which quantificational direct image operator should be used for moving theories? In the IFF view , mapping theories along a langua ge mor- phism requires a commitme nt to mapping closure. In other words, if one is willing to use a language mor- phism to map a theory, then one is committing oneself to the mapping closure of t hat theory; that is, o ne is e s- sentially asserting all of the additional axioms in the difference between the theory and its mapping closure. The existential direct image operator is seen to be im- portant by its use in the fusion construction. Howeve r, wh at about the universal direct ima ge oper ator? The fact is that the two operators are identical on the map- ping closure of a theory. Hence, if we comm it our- selves to the mapping closure of a theory, it does not matter which direct image operator w e use, since they are bo th equal in this case. 4. Maintenance of a System of Ontologies This section discusses how the notions of modularity and centralization are represented in the IFF. As the author has discussed 11 and demonstrated 12 , each step of Sow a’s process of “building the hierarchy” 7 is represented in the IFF. All steps take pl ace in the co ntext of theories. Howev- er, in the general maintenance of a diagram of theories, these processing steps can be used in any fashion deemed necessary . The follow ing are various operations that are †††† By terms, we mean the variables and the entity, relation and funct ion types used in the language L . Constants are nullary function symbols. ‡‡‡‡ The IF F notion of language endorelation is a theory of relative syn- onymy – synonymy relative to the base language, and hence relative to the conceptu al structures of whatever community ow ns and manages the corresponding ontology . Such a theory of relative sy nonymy may be related to any linguisti c/ph ilosophical discussion of synonymy , such as (Quine, 1951). possible in the IFF in order to practically maintain a dia- gram of theories. ○ Consistency checking: Any theory in a homogeneous diagram of theories may be inconsistent (equivalent to the bott om of the lattice of theories). A basic and non- trivial opera tion is to check for the consistency of the indexed theories in a diagram. Of course, any theory that comes with its own special model is already con- sistent. ○ Sum th eory: This is a procedure for distinguishing the various terms used in a discrete diagram of theories. Every theory in such a diagram has a unique theory in- dex, and all terms in the standard theory sum are dis- tinguished by ‘labeling” with the index of their theory of origin. This is the process of forming the sum in the context of theories and the underlyin g context of lan- guages. ○ Endorelation and Quotient theory: The q uotient of a theory is based upon an endorelation over that theory §§§§ . The identification of pairs of terms †††† cor- responds to the mathematical proce ss of forming the quotient of the sets of terms in a theory via a suitable endorelation . This is the pro cess of forming the quo- tient in the context of theories and the underlying con- text of type languages. ○ Subtheory: Often it is helpful in maintaining a dia- gram of theories to extract smaller (and hence more generic) subtheories from larger more specific ones. This makes the diagram of theories more flexible to use. In particular, when fusing theories, one may need to only use some smaller more generic parts. Each ex- tracted theory is more general than its theory of origin, and thus high er in the lattice hierar chy. ○ Alignment: For alignment in particular and integration in general, we follow the definitions of the ontolo gy w orking group of the NCITS T2 Committee on Infor- mation Interchange and Interpretation as reco rded by Sowa 3 . Ontological alignment consists of the sharing of comm on termin ology and semantics through a me- diating or reference ontology (Kent, 200 0). The intent of alignment is that mapped t ypes ar e equivalent. Such equivalence can be automatically computed via the FCA-Merge pro cess (Stumm e and Mädche, 2001) ***** . To formalize this, we represent an equivalence pair of §§§§ This is a systematic procedure for specifying the pairs of terms to be semantically identi fied. One can assume that the terms in the sum of a (discrete) diagram of theories are coordinated with one another in the foll owing sense. In a theory sum, (1) an y two terms from independently developed component theories should not be identi fied; howeve r, (2) two identical terms from different component theories should be identified if these theories originated by subsetting from a third more specialized theory. ** * ** In fact, although we recognize that it can serendipitously dis cover new rel ationships, we view FCA -Merg e as predominately an automatic process for ontology alignment. I t is important to note that FCA-Merg e requires interpretative or combined sem antics, since it crucially depends upon instan ce data and classifications. Hence, th is ap proach to alignment uses logics, not just theories. types as a single type in a mediating or reference theory, wi th tw o mappings from this new type back to the participant theory types. Thus, alignment is represented as a span or ‘Λ’ -sha ped diagram of three theories and two theory m orphisms. The m ediating or reference ontology in the middle repre sents both the equivalenced ty pes and the axiomatization needed for the d esired degree of compatibility with the particip ant ontologies, whether part ial or complete. Since the theoretical alignm ent links preserve this axiomatiza- tion, compatibility w ill be enforced ††††† . ○ Sum diagram: Given two diagrams o f theories T  and T  of shapes G  and G  , respectively, the sum diagram of theories T =  T   T   has the sum shape G   G  w ith obj ect function obj ( T ) that maps nodes in node ( G   G  ) = node ( G  )  node ( G  ) according to component: obj ( T )( n  ) = obj ( T  )( n  ) and obj ( T )( n  ) = obj ( T  )( n  ); similarly for edge s. ○ Removal: Any theory in a diagram migh t be m ark for deletion for various reasons – the theory m ay have been proven inconsistent, or the theory may no longer be of interest to the community federation m aintaining the sy stem of ontologies. ○ Fusion (or Unificati on): It may be desirab le at any time to create a customi zed theory. One exam ple of such a customized theory is a “great big hierar chy with modules copied in, frozen into place, and relabeled to avoid inconsistencies” as describ ed 13 by Jo hn Sowa. This is built as the fusion construction of a sub- diagram of theories (Table 1). The fusion T •, the de- sired theory to be constructed, is just another theory. The other theories in the diagram being m aintained have been left in place undisturbed. Forming the meet is a special case of t he fusion construction for a homo- geneous sub-diagram of theories. ○ Theory Creation: Often a small theory of specialized axioms is needed. This may occur when defining a custom ized theory as the fusion of a diagram w ith the small theory as one indexed component. 5. Future Prospects Full semantic integration involves the notion of infor- mation flow (Barwise and Seligm an, 199 7). Special cases of this have appeared in the pape rs (Kent, 2000 and 2003), (Kalfoglou and Schorlemm er, 2002) and (Schorlemm er and Kalfoglou, 2003). In particular, the papers by the au- thor argue that the semantic integration of ontolo gies is the two-step proce ss of alignment and unification. Ontological †††† † I n general, alignment acts through communit y ontology port(al)s. Before two ontol ogies can be aligned, it may be necessary to introduce new subtypes or supertypes of terms in either ontolog y in order to provide suitable targets for alignment. In addition, when any p artici pant ontology has some distinct instan ce data , alignment may quotient that participant. Hence, alignment is represe nted by a ‘W ’-shaped diagram, with the orig- inal parti cipating ontolog ies at the two upper outer vertices, the mediat- ing or reference ontolog y at the upper center vertex, and the partic ipan t port(al) ontolog ies at the two low er vertices. alignm ent consists of the sharing of comm on terminology and semantics through a mediating or reference ontology. Ontological unification, concentrated in a virtual ontology of comm unity connections, is fusi on of the alignment dia- gram o f participant community ontologies – the quotient of the sum of the particip ant port(al)s modulo the ontolo gical alignm ent structure. The current pape r contributes to this “inf ormation flow approa ch to semantic integration” by describing how the IFF represents formal semantic integra- tion through its general fusion construction and situates formal semantic integration i n the on-going maintenance of a system of ontologies. Howev er, true inf ormation flow, and hence combined semantic integration, both formal and interpretive, occurs at the level of logics. The corre ct for- mulation of this requir es the notion of free logics and the notion of fusions of logics. The current version of the IFF- OO has axiomatizations for free logics and for fusions of theories. However, fusions of logics cannot be constructed. The pro blem is that the current version of the IFF-OO fol- lows too closely Enderton’s notion of a sorted language. In particular, IFF languages using reference functions (sort functions in Enderton’s termin ology) cause pro blems when trying to construct the copr oduct of models or logics. This has been remedied in the new version of the IFF-OO to be posted soon. See the discussion of the “IFF Work in Progress” 14 for more on this. In sum mary, w e argue that the principled framew ork of the IFF realizes the informa tion flow approach to semantic integration, and we hope that this theoretical approach and its implementation ‡‡‡ ‡ ‡ will contribute to realizatio n of the “gold standard for semantic integration” (Uschold and Gruninger, 2002). 6. References Barwise, Jon and Jerry Seligma n. Information Flow: The Logic of Distributed Systems. Cambridge T racts in Theoretical Com- puter Science 44. Cambridge University Press. 199 7. Enderton, Herbert B. A Mathematical Introductio n to Logic. New York: Academic Press. 1972. Ganter, Bernhard and Ru dolf Wille. Formal concept analysis: mathema tical foundations. Heidelb erg: Springer. 199 9. Kalfoglou, Yannis and Marco Scho rlemmer. Information-Flow- based Ontology Mapping. On the Move to Meani ngful Inter- net Systems 2002: CoopIS, DOA, and ODBASE, Lecture Notes in Computer Science 2519, 113 2–1151. Sp ringer. 20 02. Kent, Robert E. The Information Flow Foun dation for Concep- tual Knowledge Organization. In: Dynamism and Stability in Knowledge Organization. Proceedings of the Sixth In terna- tional ISKO Conference. Advances in Knowledge Organiza- tion 7, 111 –1 17. Ergon Verlag, Würzburg. 20 00. Kent, Robert E. The IFF Fo undation for Ontolo gical Knowledge Organization. In: Knowledge Organization and Classification ‡‡‡ ‡‡ The I FF takes the high ro ad to implementation. There is work in progress on an I FF representation of the Meta Object Facility (MOF) and the Model Driven Architecture (MDA) of the Object Management Group (OMG). In the other direction, there are on-going explorations to demon- strate how the MOF can be used for a high level specification of the I FF . in Internation al Information Retrieval. Cataloging and Classi- fication Quarterly. The Haworth Press Inc., Binghamton, New York. 2003. Mac Lane, Saunders. Categories for the Working Mathematician, New York/Heidelberg/Berlin: Springer-Verlag. 1971. (New edition 1998). Quine, Willard V.O. Two Dogmas of Empiricism. Philosophical Review. 1951. Schorlemmer, Marco and Yannis Kalfoglou. Using Information- Flow Theory to Enable Semantic Interoperability. Sisè Congrés Català ďI ntelligència Artific ial. P alma de Mallorca, 2003. Sowa, John F. Knowledge Representation : Logical, Philosop hical and Computational Fo undati ons. Brookes/Coles. 2000 . Stumme, Ge rd and Alexander Mädche. FCA-Merge: Bottom-Up Merging of Ontologies. In: B. Nebel (Ed.): P roc. 17 th Intl . Conf. on Artificial Intelligence (IJCAI ‘01). Seattle, WA, USA, 2001, 2 25–23 0. Long version: On tology Merging for Federated Ontologies for the Semantic Web. In: E. Fr anconi, K. Barker, D. Calvanese (Eds.): Proc. Intl. Workshop o n Foundation s of Models for Information Integration (FMII’01) , Viterbo, It aly, Sept. 16–18, 2 001. LNAI, Springer 2002. Uschold, Michael and Michael Gruninger. Creating Semantically Integrated Communities on the World Wide Web. Invited pa- per. International Workshop o n the Semantic Web. 2002 . 1 “I EEE P1600.1 St andard Upper Ontology (SUO) Wo rking Group”. http://suo.ieee.org/ . 2 “The SUO I nformation Flo w Fr amework (SUO IFF )”. http://suo.ieee.org/I FF / . 3 Sowa, John F . B uilding, Sharing, an d Merging Ontologies. Unpub- lished paper. (2001) htt p://www .jfsowa.com/ontology /ontoshar.htm . 4 “The SUO I F S ite Map”. htt p://suo.ieee.org/I FF /site-map.html . 5 Sowa, John F . “Reply: Ontolog y Structure & Content”. (13 January 2001) htt p://grouper.ieee.org /groups/suo/email/msg02765.html . 6 Kent, Robert E. “I FF L attice of Theorie s (LOT ) Glossary ”. http://suo.ieee.org/I FF /work-in-progre ss/LOT/g lossary.pdf . 7 Sowa, John F . “Bui lding the hierarchy”. (16 May 200 3) http://suo.ieee.org/email/msg09453.html . 8 Cassidy, Patrick. “Reply: “SUO Ball ot with 2 Questions – ‘mono- lithic’?”. (10 Jun 20 03) http://grouper.ieee.org/gr oups/suo/email/msg09701.h tml . 9 Kent, Robert E. “Mathematical Definitions for Monocosmic and Poly cosmic”. (14 Jun 2003) http://grouper.ieee.org/gr oups/suo/email/msg09768.h tml . 10 Kent, Robert E. “Mapping Closure”. (5 Jul 2003 ) http://grouper.ieee.org/gr oups/suo/email/msg10356.h tml . 11 Kent, Robert E. “Building the Hierarchy by L anguage and Module Processing”. (2 Jun 2003) http://grouper.ieee.org/gr oups/suo/email/msg09555.h tml . 12 Kent, Robert E. “L anguage and Module Processing” (31 May 2003) htt p://suo.ieee.org/I FF /work-in-progres s/LOT/language -module- processing.html . 13 Sowa, John F. “Reply : Charter vs. Consensus”. (27 Jun 2 003) http://grouper.ieee.org/gr oups/suo/email/msg10139.h tml . 14 Kent, Robert E. “I FF Work in Progress”. (4 April 2003) http://suo.ieee.org/I FF /work-in-progre ss/ .