Category Theory Approach to Fusion of Wavelet-Based Features

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1 Catgory Thory Approach to Fusion of Wavlt-Basd Faturs Scott A. DLoach Air Forc Institut of Tchnology Dpartmnt of Elctrical and Computr Enginring Wright-Pattrson AFB, Ohio Miczyslaw M. Kokar Northastrn Univrsity Dpartmnt of Elctrical and Computr Enginring Boston, Massachustts 02115k Abstract This papr discusss th application of catgory thory as a unifying concpt for formally dvlopd information fusion systms. Catgory thory is a mathmatically sound rprsntation tchniqu usd to captur th commonaltis and rlationships btwn objcts. This fatur maks catgory thory a vry lgant languag for dscribing information fusion systms and th information fusion procss itslf. Aftr an initial ovrviw of catgory thory, th papr invstigats th application of catgory thory to a wavlt basd multisnsor targt rcognition systm, th Automatic Multisnsor Fatur-basd Rcognition Systm (AMFRS), which was originally dvlopd using formal mthods. 1. Introduction Th goal of information fusion is to combin multipl pics of data in a way so that w can infr mor information than what is containd in th individual pics of data alon. This rquirs that w b abl to dtrmin how th individual pics of data ar rlatd. It would also b nic if w could dscrib this rlationship btwn data in a formal way so that w can automatically rason ovr th information fusion procss without th us of unrliabl and brittl huristics. In this papr w prsnt catgory thory as a unifying concpt for formally dfining information fusion systms. Th goal of catgory thory is to dfin th rlationships btwn objcts within a catgory of rlatd objcts. Catgory thory also provids oprators that allow us to rason ovr ths rlationships. Th first sction of th papr is a tutorial on algbraic spcifications and catgory thory. Nxt w dscrib a formally dfind fusion systm, th Automatic Multisnsor Fatur-basd Rcognition Systm (AMFRS), and dscrib how w could incorporat catgory thory constructs to provid a provably corrct tchniqu for implmnt th systm. 2. Thoris and Spcifications Th notation gnrally usd to captur th formal dfinitions of systms is a formal spcification. Thr ar two typs of formal spcifications commonly usd to dscrib th bhavior of softwar: oprational and dfinitional. An oprational spcification is a rcip for an implmntation that satisfis th rquirmnts whil a dfinitional spcification dscribs bhavior by listing th proprtis that an implmntation must posss. Dfinitional spcifications hav svral advantags ovr oprational spcifications. 1. Dfinitional spcifications ar gnrally shortr and clarr than oprational spcifications. 2. Dfinitional spcifications ar asir to modulariz and combin. 3. Dfinitional spcifications ar asir to rason about, which is th ky rason thy ar usd in automatd systms. It is gnrally rcognizd that crating corrct, undrstandabl formal spcifications is difficult, if not impossibl, without th us of som structuring tchniqu or mthodology. Algbraic thoris provid th advantags of dfinitional spcifications along with th dsird structuring tchniqus. Algbraic thoris ar dfind in trms of collctions of valus calld sorts, oprations dfind ovr th sorts, and axioms dfining th smantics of th sorts and oprations. Th structuring of algbraic thoris is providd by catgory thory oprations and provids an lgant way in which to combin smallr algbraic thoris into largr, mor complx thoris. Catgoris ar an abstract mathmatical construct consisting of catgory objcts and catgory arrows. In gnral, catgory objcts ar th objcts in th catgory of intrst whil catgory arrows dfin a mapping from th intrnal structur of on catgory Pag 1 of 9

2 objct to anothr. In our rsarch, th catgory objcts of intrst ar algbraic spcifications and th catgory arrows ar spcification morphisms. In this catgory, Spc, spcification morphisms map th sorts and oprations of on algbraic spcification into th sorts and oprations of a scond algbraic spcification such that th axioms in th first spcification bcom provabl thorms in th scond spcification. Thus, in ssnc, a spcification morphism dfins an mbdding of on spcification into a scond spcification Algbraic Spcification In this sction, w dfin th important aspcts of algbraic spcifications and how to combin thm using catgory thory oprations to crat nw, mor complx spcifications. As dscribd abov, catgory thory is an abstract mathmatical thory usd to dscrib th xtrnal structur of various mathmatical systms. Bfor showing its us in rlation to algbraic spcifications, w giv a formal dfinition [8]. Catgory. A catgory C is comprisd of a collction of things calld C-objcts; a collction of things calld C-arrows; oprations assigning to ach C-arrow f a C-objct dom f (th domain of f) and a C-objct cod f (th codomain of f). If a = dom f and b = cod f this is displayd as f f: a b or a b an opration,, calld composition, assigning to ach pair g, f of C-arrows with dom g = cod f, a C- arrow g f: dom f cod g, th composit of f and g such that th Associativ Law holds: Givn th configuration f g h a b c d of C-objcts and C-arrows, thn h (g f) = (h g) f. an assignmnt to ach C-objct, b, a C-arrow, id b: b b, calld th idntity arrow on b, such that th Idntity Law holds: For any C-arrows f: a b and g: b c id b f = f and g id b = g Th Catgory of Signaturs In algbraic spcifications, th structur of a spcification is dfind in trms of an abstract collction of valus, calld sorts and oprations ovr thos sorts. This structur is calld a signatur [9]. A signatur dscribs th structur that an implmntation must hav to satisfy th associatd spcification; howvr, a signatur dos not spcify th smantics of th spcification. Th smantics ar addd latr via axiomatic dfinitions. Signatur. A signatur Σ = S, Ω, consists of a st S of sorts and a st Ω of opration symbols dfind ovr S. Associatd with ach opration symbol is a squnc of sorts calld its rank. For xampl, f:s 1,s 2,...,s n s indicats that f is th nam of an n-ary function, taking argumnts of sorts s 1, s 2,, s n and producing a rsult of sort s. A nullary opration symbol, c: s, is calld a constant of sort s. An xampl of a signatur is shown in Figur 1. In th signatur RING thr is on sort, ANY, and fiv oprations dfind on th sort. signatur Ring is sorts ANY oprations plus : ANY ANY ANY tims : ANY ANY ANY inv : ANY ANY zro : ANY on : ANY nd Figur 1. Ring Signatur In our rsarch, signaturs dfin th rquird structur for formally dscribing wavlt-basd modls. Signaturs provid th ability to dfin th intrnal structur of a spcification; howvr, thy do not provid a mthod to rason about rlationships btwn spcifications. To crat a thory of information fusion using algbraic spcifications, oprations to dfin rlations btwn spcifications must b availabl. Thr must b a wll-dfind thory about how to rason about th xtrnal structur of ths spcifications (i.., how thy rlat to on anothr). As might b xpctd, signaturs (as th C-objcts ) with th corrct C-arrows form a catgory that is of grat intrst in our rsarch. For signaturs, th C- arrows ar calld signatur morphisms [9]. Signaturs and thir associatd signatur morphisms form th catgory, Sign. Signatur Morphism. Givn two signaturs Σ = S, Ω and Σ ' = S ', Ω ', a signatur morphism σ : Σ Σ ' is a pair of functions σ S : S S', σ Ω : Ω Ω ', mapping sorts to sorts and oprations to oprations such that th sort map is compatibl with th ranks of th oprations, i.., for all opration symbols f:s 1,s 2,...,s n s in Ω, th opration symbol σ Ω (f):σ S (s 1 ), σ S (s 2 ),...,σ S (s n ) σ S (s) is in Ω'. Th composition of two signatur morphisms, obtaind by Pag 2 of 9

3 composing th functions comprising th signatur morphisms, is also a signatur morphism. Th idntity signatur morphism on a signatur maps ach sort and ach opration onto itslf. Signaturs and signatur morphisms form a catgory, Sign, whr th signaturs ar th C-objcts and th signatur morphisms ar th C- arrows. Givn th signaturs RING from Figur 1 and RINGINT from Figur 2, a signatur morphism σ : RING RINGINT, is shown in Figur 3. As rquird by th dfinition of a signatur morphism, σ consists of two functions, σ S and σ Ω as shown. σ S maps th sort ANY to Intgr whil σ Ω maps ach opration to an opration with a compatibl rank. Spc RingInt is sorts Intgr oprations + : Intgr Intgr Intgr : Intgr Intgr Intgr - : Intgr Intgr 0 : Intgr 1 : Intgr nd Figur 2. Intgr Ring Signatur σ S = {ANY Intgr} σ Ω = {plus +, tims, inv -, zro 0, on 1} Figur 3. Signatur Morphism Signatur morphisms map sorts and oprations from on signatur into anothr and allow th rstriction of sorts as wll as th rstriction of th domain and rang of oprations. Howvr, to build up mor complx signaturs, introduction of nw sorts and oprations into a signatur is rquird. This is accomplishd via a signatur xtnsion [1]. Extnsion. A signatur Σ 2 = S 2, Ω 2 xtnds a signatur Σ 1 = S 1, Ω 1 if S 1 S 2 and Ω 1 Ω 2. Signatur xtnsions allow th dfinition of ntirly nw signaturs and th growth of complx signaturs from xisting signaturs Th Catgory of Spcifications To modl smantics, signaturs ar xtndd with axioms that dfin th intndd smantics of th signatur oprations. A signatur with associatd axioms is calld a spcification [9]. Spcification. A spcification SP is a pair Σ, Φ consisting of a signatur Σ = S, Ω and a collction Φ of Σ-sntncs (axioms). Although a spcification includs smantics, it dos not implmnt a program nor dos it dfin an implmntation. A spcification only dfins th smantics rquird of a valid implmntation. In fact, for most spcifications, thr ar a numbr of implmntations that satisfy th spcification. Implmntations that satisfy all axioms of a spcification ar calld modls of th spcification [9]. To formally dfin a modl, w first dfin a Σ- algbra [9]. Σ-algbra or Σ-modl. Givn a signatur Σ = S, Ω, a Σ- algbra A = A S, F A consists of two familis: a collction of sts, calld th carrirs of th algbra, A S = {A s s S}; and a collction of total functions, F A = {f A f Ω} such that if th rank of f is s 1,s 2,..., s n s, thn f A is a function from A s1 A s2 A sn to A s. (Th symbol indicats th Cartsian product of sts hr.) Modl. A modl of a spcification SP = Σ, Φ is a Σ- algbra, M, such that M satisfis ach Σ-sntnc (axiom) in Φ. Th collction of all such modls M is dnotd by Mod[SP]. Th sub-catgory of Mod(Σ) inducd by Mod[SP] is also dnotd by Mod[SP]. An xampl of a spcification is shown in Figur 4. This spcification is th original RING signatur of Figur 1 nhancd with th axioms that dfin th smantics of th oprations. Valid modls of this spcification includ th st of all intgrs, Z, with addition and multiplication as wll as th st of intgrs modulo 2, Z 2 = {0, 1}, with th invrs opration (-) dfind to b th idntity opration. As signaturs hav signatur morphisms, spcifications also hav spcification morphisms. Spcification morphisms ar signatur morphisms that nsur that th axioms in th sourc spcification ar thorms (ar provabl from th axioms) in th targt spcification. Showing that th axioms of th sourc spcification ar thorms in th targt spcification is a proof obligation that must b shown for ach spcification morphism. Spcifications and spcification morphisms nabl th cration and modification of spcifications that corrspond to valid signaturs within th catgory Sign. Howvr, bfor w can formally dfin a spcification morphism, w must first dfin a rduct [9]. Pag 3 of 9

4 spc Ring is sorts ANY oprations As dfind in Figur 2 axioms a,b,c ANY a plus (b plus c) = (a plus b) plus c a plus b = b plus a a plus zro = a a plus(inv a) = zro a tims (b tims c) = (a tims b) tims c a tims on = a on tims a = a a tims (b plus c) = (a tims b) plus (a tims c) (a plus b) tims c = (a tims c) plus (b tims c) nd Figur 4. Ring Spcification Rduct. Givn a signatur morphism σ:σ Σ ' and a Σ '- algbra A', th σ-rduct of A', dnotd A' σ, is th Σ- algbra A = A S, F A dfind as follows (with Σ = S, Ω ): A S = A σ(s) ' for s S, and f A = (σ(f)) A', for f Ω A rduct dfins a nw Σ-algbra (or Σ-modl) from an xisting Σ'-algbra. It accomplishs this by slcting a st or functions for ach sort or opration in Σ basd on th signatur morphism from Σ to Σ '. Thus if w hav a signatur, Σ ', and a Σ '-modl, w can crat a Σ-modl for a scond signatur, Σ, by dfining a signatur morphism btwn thm and taking th rduct basd on that signatur morphism. A rduct is now usd to xtnd th concpt of a signatur morphism to form a spcification morphism [9]. Spcification Morphism. A spcification morphism from a spcification SP = Σ, Φ to a spcification SP' = Σ ', Φ' is a signatur morphism σ: Σ Σ ' such that for vry modl M Mod[SP'], M σ Mod[SP]. Th spcification morphism is also dnotd by th sam symbol, σ: Σ Σ '. W now turn to th dfinition of thoris and thory prsntations. Basically a thory is th st of all thorms that logically follow from a givn st of axioms [8]. A thory prsntation is a spcification whos axioms ar sufficint to prov all th thorms in a dsird thory but nothing mor. Put succinctly, a thory prsntation is a finit rprsntation of a possibly infinit thory. To formally dfin a thory and thory prsntation w must first dfin logical consqunc and closur [8]. Logical Consqunc. Givn a signatur Σ, a Σ-sntnc ϕ is said to b a logical consqunc of th Σ-sntncs ϕ 1,...,ϕ n, writtn ϕ 1,...,ϕ n = ϕ, if ach Σ-algbra that satisfis th sntncs ϕ 1,...,ϕ n also satisfis ϕ. Closur, Closd. Givn a signatur Σ, th closur, closur(φ), of a st of Σ-sntncs Φ is th st of all Σ- sntncs which ar th logical consqunc of Φ, i.., closur(φ) = {ϕ Φ = ϕ}. A st of Σ-sntncs Φ is said to b closd if and only if Φ = closur(φ). Thory, prsntation. A thory s a pair Σ, closur(φ) consisting of a signatur Σ and a closd st of Σ-sntncs, closur(φ). A spcification Σ, Φ is said to b a prsntation for a thory Σ, closur(φ). A modl of a thory is dfind just as for spcifications; th collction of all modls of a thory s dnotd Mod[T]. Thory morphisms ar dfind analogous to spcification morphisms. Spcification morphisms complt th basic tool st rquird for dfining and rfining spcifications. This tool st can now b xtndd to allow th combination, or composition, of xisting spcifications to crat nw spcifications. This is whr catgory thory is xtrmly usful in information fusion. Oftn two spcifications that wr originally xtnsions from th sam ancstor nd to b combind. Thrfor, th dsird combind spcification consists of th uniqu parts of two spcifications and som shard part that is common to both spcifications (th part dfind in th shard ancstor spcification). This combining opration is calld a colimit [8]. Th colimit opration crats a nw spcification from a st of xisting spcifications. This nw spcification has all th sorts and oprations of th original st of spcifications without duplicating th shard sorts and oprators. To formally dfin a colimit, w must first dfin a con (or cocon) [8]. Con. Givn a diagram D in a catgory C and a C-objct c, a con from th bas D to th vrtx c is a collction of C- arrows {f i : d i c d i D}, on for ach objct d i in th diagram D, such that for any arrow g: d i d j in D, th diagram shown in Figur 5 commuts i.., g f j = f i. Colimit. A colimit for a diagram D in a catgory C is a C- objct c along with a con {f i : d i c d i D} from D to c such that for any othr con {f i ': d i c' d i D} from D to a vrtx c', thr is a uniqu C-arrow f: c c' such that for vry objct d i in D, th diagram shown in Figur 6 commuts (i.., f f i = f i '). Pag 4 of 9

5 d i f i g c Figur 5. Con Diagram f i d i f c c f j f i Figur 6. Colimit Diagram Concptually, th colimit of a st of spcifications is th shard union of thos spcifications basd on th morphisms btwn th spcifications. Ths morphisms dfin quivalnc classs of sorts and oprations. For xampl, if a morphism for spcification A to spcification B maps sort α to sort β, thn α and β ar in th sam quivalnc class and thus is a singl sort in th colimit spcification of A, B, and th morphism btwn thm. Thrfor, th colimit opration crats a nw spcification, th colimit spcification, and a con morphism from ach spcification to th colimit spcification. Ths con morphisms satisfy th condition that th translation of any sort or opration along any of th morphisms in th diagram lading to th colimit spcification is quivalnt. An xampl of th colimit opration is shown in Figur 7 and Figur 8. Givn th BIN-REL, REFLEXIVE, and TRANSITIVE spcifications in Figur 7, th colimit spcification would b th PRE- ORDER spcification as shown in th diagram in Figur 8. Notic that th sorts E, X, and T blong to th sam quivalnc class in PRE-ORDER. Likwis, th oprations, =, and < also form an quivalnc class in PRE-ORDER. Thus PRE-ORDER dfins a spcification with on sort, {E, X, T} and on opration, {, =, <}, which is both transitiv and rflxiv. Th spcification BIN-REL dfins th shard parts of th colimit but adds nothing to th final spcification. d j spc Bin-Rl is sorts E oprations : E, E Boolan nd spc Rflxiv is sorts X oprations = : X, X Boolan axioms x X x = x nd spc Transitiv is sorts T oprations < : T, T Boolan axioms x, y, z T (x < y y < z) x < z nd spc Pr-Ordr is sorts {E, X, T} oprations {, =, <} : {E, X, T}, {E, X, T} Boolan axioms x, y, z {E, X, T} x {, =, <} x (x {, =, <} y y {, =, <} z) x {, =, <} z nd Figur 7. Spcification Colimit Exampl Rflxiv {E X, =} c Bin-Rl c Pr-Ordr {E T, <} c Transitiv Figur 8. Exampl Colimit Diagram A catgory in which th colimit of all possibl C- objcts and C-arrows xists is calld cocomplt. As shown by Gogun and Burstall [2, 3], th catgory Sign and Spc ar both cocomplt; thrfor, th colimit opration may b usd frly within th catgory Spc to dfin th construction of complx thoris from a group of simplr thoris. Using morphisms, xtnsions, and colimits as a basic tool st, thr ar a numbr of ways that spcifications can b constructd [9, 4]: Pag 5 of 9

6 1. Build a spcification from a signatur and a st of axioms; 2. Form th union of a collction of spcifications; 3. Translat a spcification via a signatur morphism; 4. Hid som dtails of a spcification whil prsrving its modls; 5. Constrain th modls of a spcification to b minimal; 6. Paramtriz a spcification; and 7. Implmnt a spcification using faturs providd by othrs. Many of ths mthods ar usful in spcifying and implmnting information fusion systms. For instanc, if w can dfin th shard part of two typs of data, w can formally combin thm using a colimit Functors Th prvious sctions dfind th basic catgoris and construction tchniqus usd to build larg-scal softwar spcifications. In this sction, w xtnd ths concpts furthr to dfin modls of spcifications and how thy ar rlatd to th construction tchniqus usd to crat thir spcifications. Bfor dscribing this rlationship, w dfin th concpt of a functor that maps C-objcts and C-arrows from on catgory to anothr in such a way that th idntity and composition proprtis ar prsrvd [7]. Functor. Givn two catgoris A and B, a functor F: A B is a pair of functions, an objct function and a mapping function. Th objct function assigns to ach objct X of catgory A an objct F(X) of B; th mapping function assigns to ach arrow f: X Y of catgory A an arrow F(f) : F(X) F(Y) of catgory B. Ths functions satisfy th two rquirmnts: F(1 X ) = 1 F(X) for ach idntity 1 x of A F(g f) = F(g) F(f) for ach composit g f dfind in A Basically a functor is a morphism of catgoris. Actually, w hav alrady prsntd two functors: th rduct functor that maps modls of on spcification (in th catgory Mod[X 1 ]) into modls of a scond spcification (in th catgory Mod[X 2 ]) and th modls functor that maps spcifications in th catgory Spc to thir catgory of modls, Mod[X], in Cat, th catgory of all sufficintly small catgoris. 3. Automatic Multisnsor Fatur-basd Rcognition Systm To show applicability of th catgory thortic notions dscribd abov to information fusion systms, w will discuss a cas study of Automatic Multisnsor Fatur-basd Rcognition Systm (AMFRS) [6], which was originally dvlopd using a modl-basd approach. In this cas study, w transform th AMFRS framwork into an quivalnt systm using a catgory thortic approach. First w will discuss th original systm and thn show its quivalnt structur using algbraic spcifications and catgory thory Modl-Thory Basd Framwork In th original modl-basd dvlopmnt approach, wavlt-basd modls wr dvlopd for intgration into th AMFRS to hlp rcogniz targts. AMFRS uss a modl-basd framwork to dscrib how to combin information containd in th wavlts for us in th systm. Within this framwork, modls wr dvlopd to hlp rcogniz targts basd on wavlt cofficints that could b intrprtd as maningful faturs of th targt. In this framwork, modls wr dvlopd basd on a languag and its associatd thory that dscribd th smantics of th languag. To combin languags and thoris, thr oprators ar usd: rduction, xpansion, and union. In gnral, th rduction oprator rmovs symbols from a languag along with all th sntncs in which it xists in its associatd thory. Expansion is th opposit. Expansion allows us to add symbols and nw sntncs about thos symbols to th languag. Finally, th union oprator combins th symbols and sntncs from two diffrnt languag/thory pairs into a singl languag and a singl thory. Using ths oprators, Korona cratd a framwork for combining languags and thoris about two diffrnt typs of snsor data into a singl fusd languag and thory. This framwork is shown in Figur 9. In Figur 9, w show only th languag composition procss. Th thory fusion procss is idntical. In this xampl, w assum thr ar two snsors whos data is dscribd by two languags L r and L i. Ths languags ar xtndd to th languags L r and L i by adding symbols dnoting oprations on a subst of th wavlt cofficints usd to dscrib th snsor data. Ths substs of cofficints rprsnt thos cofficints that will b Pag 6 of 9

7 part of th final fusd languag. Th cofficints ar slctd by th dsignr basd on knowldg of th wavlt cofficints and thir rlationship to faturs in targts of intrst. L r E L r R L r r U L ri L ri L f E R L i R L i r E L i E - xpansion R - rduction U - union Figur 9. Modl-Thory Basd Framwork Aftr th ncssary symbols hav bn addd to th languags, L r and L i ar rducd by rmoving all th symbols not rlatd to th cofficints slctd for us in th final fusd languag. Th nw rducd languags, L r r and L i r, ar thn combind into a singl languag, L ri, by th union opration. This languag contains all th symbols rprsnting th cofficints and oprations on thm rquird to construct th final fusd languag. Th last two stps in th procss crat our final fusd languag, L f. First, L ri is xtndd to L ri by adding symbols dnoting oprations that combin th cofficints from L r r and L i r. Thn, w crat L f by rmoving th symbols dnoting thos oprations that do not work on th fusd st of cofficints An Equivalnt Catgory Thortic Framwork Bfor convrting th AMFRS modl-basd framwork into a catgory thortic framwork, a fw obsrvations ar ncssary. First, th languag and thory combination usd in AMFRS is basically quivalnt to an algbraic spcification. An algbraic spcification dfins a st of sorts, oprations ovr thos sorts, and axioms that dfin th smantics of th oprations. Constants, rlations and functions dfind via languag symbols ar dfind as oprations in an algbraic spcification. Sntncs of a thory translat to axioms in an algbraic spcification. Algbraic sorts dfin a collction of valus usd in th oprations. Th modl-basd xpansion, rduction, and union oprators also hav countrparts in catgory thory. Th basic oprator in catgory thory is th morphism. In th catgory of Spc, which includs all possibl algbraic spcifications, ths morphisms ar spcification morphisms that dfin how on spcification is mbddd in a scond spcification. That is, it dfins a mapping from th sorts and oprations of th first spcification into th sorts and oprations of th scond spcification in such a way as to nsur th axioms of th first spcification ar thorms of th scond spcification (i.., th axioms hold in th scond spcification undr th dfind mapping of sorts and oprations). Thus a spcification morphism can b usd to dfin an xpansion as wll as a rduction (thy ar basically invrss of ach othr). If w hav an xpansion of spcification A into spcification B, in ffct w hav a morphism from A to B. Likwis, a rduction of spcification A to spcification B, indicats morphism from B to A. Th languag union oprator can also b modld asily using th catgory thory colimit opration. Th colimit opration combins two (or mor) spcifications, automatically crating a morphism btwn th original spcifications and th rsulting colimit spcification. If two spcifications bing combind using a colimit opration shar common parts (.g., thy both us intgrs), ths parts can b spcifid as common by dfining morphisms from th common, or shard, spcification to th individual spcifications. This shard spcification, along with th associatd morphisms, ar includd in th colimit opration. Th rsult of this is that th shard parts of th two spcifications ar not duplicatd. Th convrsion of th modl-basd framwork into a catgory thortic framwork is shown in Figur 10. In this framwork, th languags and thir associatd thoris ar convrtd to algbraic spcifications (or thory prsntations) and rductions and xtnsions ar convrtd to morphisms. Not that a rduction from A to B rsults in a morphism from B to A. Th union opration is convrtd to a colimit opration. Th S spcification dnots any shard part of r spcifications and i. In this cas it might includ domain information about wavlts, targts, tc. Figur 11 rprsnts a simplification of th catgory thortic stting shown in Figur 10. Basically, th morphisms σ 3, σ 4, and σ 8 from Figur 10 hav bn combind into morphism σ 15 of Figur 11. This is possibl sinc all th sorts, oprations, and axioms rmovd by σ 3 and σ 4 can b carrid along without changing th smantics. As w s whn w gt to th modl cration phas, carrying along ths xtra sorts, oprations, and axioms will bcom an advantag. Pag 7 of 9

8 σ 1 S σ 2 S σ 3 σ 4 r r σ 12 σ 13 σ 5 σ 6 σ 7 d σ 14 T f -as- T f Figur 10. Catgorical Framwork S σ 10 σ 11 σ 8 σ 12 σ 13 σ 14 σ 15 T f Figur 11. Simplifid Catgorical Stting Figur 12 is an vn furthr simplification of th catgory thortic stting of Figur 10. In Figur 12, th morphisms σ 1, σ 2 and σ 7 from Figur 10 hav bn combind into morphism σ 14. In this framwork, w combin th two basic spcifications togthr via th colimit opration bfor w insrt any knowldg about which wavlt cofficints corrspond to which intrprtabl faturs. Sinc all th oprations usd to xpand th basic spcifications hav a wll dfind intrprtation in th xpandd spcifications (cf. [6]), th morphism σ 14 bcoms a dfinitional xtnsion and th subdiagram containd in th dottd box bcoms an intrprtation. An intrprtation basically says that w can build a modl of T f from a modl of. This is a powrful construct in catgory thortic softwar dvlopmnt tools such as Spcwar [5]. T f σ 15 Intrprtation Figur 12. Thory Intrprtation Finally Figur 13 dscribs how w crat modls in our catgory thortic framwork. In Figur 13, rprsnts th modl functor, which taks spcifications from th catgory Spc and maps thm to a valid catgory of modls, dnotd [Spc], in th catgory Cat (th catgory of all sufficintly small catgoris). Th nic part about th catgory thortic framwork w hav com up with is that ach morphism, σ: Α Β, inducs a rduct functor, σ, that automatically maps modls of B to modls of A. Thrfor if w crat a valid modl for B, w automatically gt a valid modl for A! Following th flows of rduct functors in Figur 13, w now s that if w can crat a valid modl of T f -as- (M ri as pointd at by th larg arrow in Figur 13) w can automatically crat th valid, consistnt modls M r, M i, M ri, and M f for,,, and T f rspctivly. 4. Implications Thr ar many positiv implications of putting th AMFRS dsign into a catgory thortic stting. First, thr is no information loss in translating languags and thoris into algbraic spcifications. In fact, w gain modling ability by adding th notion of a sort. By using sorts, w can prcisly dfin opration signaturs. Also, th notions of morphisms, dfinitional xtnsions, colimits, and intrprtations giv us a wid varity of tools with wll-dfind manings. W can prov whn morphisms and dfinitional xtnsions xist as wll as construct th rsulting colimit spcification basd on a st of spcifications and morphisms. All in all, catgory thory provids us a much gratr capability to prov rlationships btwn spcifications. Finally, th catgorical stting allows us to construct, Pag 8 of 9

9 in a provably corrct mannr, consistnt sts of modls rquird by th AMFRS systm. All w hav to do is construct on spcific modl and th modls rquird by AMFRS can b gnratd automatically. Th bottom lin is, you los nothing and gain a lot by using algbraic spcifications and catgory thory in th dvlopmnt of formal information fusion systms such as AMFRS. M r σ3 S σ 12 σ 13 d σ 14 M ri σ4 M i 6. Korona, Z. Modl-Thory Basd Fatur Slction for Multisnsor Rcognition. Ph.D. Thsis, Northastrn Univrsity, MacLan, Saundrs and Birkhoff. Algbra. Nw York, NY: Chlsa Publishing Company, Srinivas, Yllamraju V. Catgory Thory Dfinitions and Exampls. Tchnical Rport, Dpartmnt of Information and Computr Scinc, Univrsity of California, Irvin, Fbruary TR Srinivas, Yllamraju V. Algbraic Spcification: Syntax, Smantics, Structur. Tchnical Rport, Dpartmnt of Information and Computr Scinc, Univrsity of California, Irvin, Jun TR T f -as- σ 15 σ5 Mri T f σ6 M f Figur 13. Modl Cration using Thory Intrprtation 5. Rfrncs 1. Grkn, Mark J. Spcification and Dsign Thoris for Softwar Architcturs. PhD dissrtation, Graduat School of Enginring, Air Forc Institut of Tchnology (AU), Gogun, J. A. and R. M. Burstall. Som Fundamntal Algbraic Tools for th Smantics of Computation Part I: Comma Catgoris, Colimits, Signaturs and Thoris, Thortical Computr Scinc, 31: (1984). 3. Gogun, J. A. and R. M. Burstall. Som Fundamntal Algbraic Tools for th Smantics of Computation Part II: Signd and Abstract Thoris, Thortical Computr Scinc, 31: (1984). 4. Gogun, J. A. Rusing and Intrconncting Softwar Componnts, IEEE Computr, (Fbruary 1986). 5. Jullig, Richard and Yllamraju V. Srinivas. Diagrams for Softwar Synthsis. Procdings of th Knowldg Basd Softwar Enginring Confrnc. IEEE Pag 9 of 9