MODELLING MEASUREMENTS AS TIMED INFORMATION PROCESSES IN SIMPLEX DOMAINS Graçaliz P. Dimuro 1, Antônio Carlos R. Costa 12 and Vladik Kreinovich 3

Transcription

1 h IEKO C7 Inernaonal Symposum June July 4 San-Peersburg Russa ODELLING EASUREENS AS IED INFORAION PROCESSES IN SIPLEX DOAINS Graçalz P Dmuro Anôno Carlos R Cosa and Vlad Krenoch Escola de Informáca Unersdade Caólca de Peloas 96-4 Peloas Brazl PPGC Unersdade Federal do Ro Grande do Sul Poro Alegre Brazl Deparmen of Compuer Scence Unersy of exas a El Paso El Paso X USA Absrac hs paper presens a doman-heorec model for measuremens and measurng nsrumens by mang explc n smplex-doman srucures wo mporan aspecs of measuremen processes: he noon of sandard represenaon relaon esablshed beween he (physcal alues ha are beng measured and he meanngs of he readngs (semanc alues of he measurng nsrumens used o measure hem and he me underlyng eery measuremen process n a way ha s possble o race he hsory of eery measurng process We also presen he modellng of measuremens performed by combned measurng nsrumens synchronzed n me Fnally he doman-heorec modellng of a sample measurng process s presened o llusrae he approach Keywords: qualae domans smplex domans measuremen nsrumens fuson processes INRODUCION Sco and Srachey [] nroduced Doman heory as a mahemacal framewor for he semancs of programmng languages he man dea s ha programmng language semancs can be formally specfed n erms of obecs of domans conceed as parally ordered ses of obecs wh ceran properes such ha he order (called nformaon order models he noons of approxmaon beween obecs and he obecs hemseles model paral resuls of compuaon seps hs means ha (parally compued obecs can be compared by he qualy of nformaon hey carry wh respec o some oally compued obec (a maxmal obec n he doman called oal obec In hs sense f an obec x approxmaes anoher obec y hen y conans a leas he same nformaon carred by x he leas elemen or boom (requred for eery doman models he absence of nformaon represenng he begnnng of any compuaonal sep; n conras oal obecs represen he fnal resul of a fne compuaon or a lm when hey are produced by nfne compuaons he man subec addressed by Doman heory s precsely he modellng of compuaons performed oer obecs ha can only be produced by nfne processes herefore became he deal srucure for modellng compuaons oer he real numbers [] and real nerals [] Snce hen doman-le srucures hae been used n seeral applcaons mos of hem n he conex of compuaon and mahemacs [4] hose feaures of Doman heory suppor he man nuon behnd our wor he possbly of deelopng a doman-heorecal model of uncerany n measurng nsrumens and processes In [5] we nroduced he smplex and smplcal complex domans qualae domans whose obecs are respecely smplexes and smplcal complexes ha are coheren n a ceran sense allowng a doman-heorecal modellng of measuremen processes based on hose domans We showed ha qualae domans of coheren smplexes can be used o model he seps of (possbly nfne measurng processes In [6] we used he same approach o model a smple compee sensor daa fuson [7]-[8] where seeral subsequen percepons obaned by a sngle percepon module are fused In boh papers me s no consdered explcly n he doman-heorec model In hs paper we exend he nal wor [5]-[6] done on smplex domans where he noons of qualae domans and smplexes were frsly combned We nroduce a general model for a measurng nsrumen (measurng dece sensor ec as a smplex doman whose obecs are consruced from readng eens labelled wh emporal nformaon and nerpreed oer a semanc doman allowng for he defnon of a emporal order beween hem n addon o he nformaon order A sandard represenaon relaon beween sources of physcal alues and semanc domans s defned so ha he noon of a calbraed measurng nsrumen for each se of physcal alues can be defned showng consrucely (sep-by-sep how nformaon eens abou a physcal alue are gahered and nerpreed n he semanc doman An exponenal consrucor s nroduced for modellng he hsores of measurng processes he new feaures nroduced n hs paper can suppor he defnon of some med smplex-doman See [8] for sandard defnons and resuls concernng Doman heory [] for qualae domans and [] for smplexes he elemens of he semanc doman model he se of possble fnal measuremens ha shall be aen no consderaon by he user For nsance n agens (or robos hey model he nernal represenaons (nernal daa srucures used for mang decsons or comng o conclusons [7]-[8]

2 consrucors ha allow he modellng of composed measuremen nsrumens ha are able o synchronze and o produce fused measuremens wh less measuremen uncerany he paper s organzed as follows Secon presens basc defnons and summarzes preous wor Secon nroduces he noon of measurng nsrumen and he calbraon procedure Readng processes are dscussed n Secon 4 allowng emporal nformaon he exponenal consrucor s nroduced n Secon 5 Secon 6 dscusses measurng processes performed by combned nsrumens A sample of a measurng process s presened n Secon 7 and Secon 8 comes wh he Concluson SIPLEX DOAINS A qualae doman = ( D D s a collecon of ses D ordered under he ncluson relaon such ha: ( D s non-empy: he empy se D s he boom of D; ( D s closed under dreced unons: S D S dreced * S D ; ( D s downward closed: ab ( a D b a b D he nformaon order s he ncluson relaon he nformaon uny s called oen and he se of oens s gen { } by α { α} D = D = * D A se x D s sad o be coheren f x D A smplex s a counable se A smplex p σ of ω dmenson p s a fne se of p+ elemens σ represens a smplex of nfne dmenson and σ s he empy non-dmensonal smplex 4 A smplcal complex s a counable se K of smplexes such ha f s n K so are all s faces (subses he dmenson of K s he larges of he dmensons of s he smplexes Le B be a counable se of oens hang an assocaed nerpreaon n a semanc doman gen by a complee lace = ( F where s he leas elemen F s he greaes elemen and F : B s sad o be a oen nerpreaon func- B F on f and only f ( he noon of coherence n B s defned n erms of a oen nerpreaon funcon ha s a smplex of σ = Bs sad o be coheren f oens { } and only f { σ} F where means he supremum of he ndcaed se he collecon of (paral or oal coheren smplexes nduced on CohSmp B B by he nerpreaon s denoed by he nerpreaon of a coheren smplex σ CohSmp B s gen by he smplex nerprea- A dreced se s a parally ordered se where each wo elemens always hae a supremum 4 he noon of smplcal complex [] was exended n [5] o consder boh denumerable and non-dmensonal smplexes on funcon : CohSmp( B ( σ = { σ} heorem [5] ( CohSmp( B defned by s a qualae doman called he smplex doman nduced on B by an nerpreaon funcon EASURING INSRUENS AND CALIBRAION FUNCIONS he se of physcal alues s conceed as cpo 5 = wh elemens represened by alues layng n a semanc doman I s possble o defned many dfferen relaons beween and bu one of hem shall be consder as a sandard for calbraon procedures o defne such sandard relaon denoed by consder he ses: { s} { s s} { } { } = = ( s X = s X s ( Y = s Y s ( Defnon A sandard represenaon relaon for n s a relaon such ha: Eery well-defned physcal alue has a well-defned represenaon n ha s: ( s s s F s (4 sasfes he src-le properes: ( ( s s s = ; = (5 sasfes he order-preserng properes: ( s s s s s s ( (6 s s ; s s s s (7 sasfes he connuous-le properes: ( F s ( X X s X dreced X ; (8 X ( Y Y (9 Y Gen a cpo of physcal alues and a semanc doman a measurng nsrumen for n denoed by ( s any smplex doman = ( CohSmp( ( where he counable se s called he oen nernal scale wh respece nerpreaon funcon : A coheren smplex 5 A cpo (or a complee paral order s a parally ordered se hang a leas elemen where each dreced subse has a supremum X

3 σ ( s called coheren smplex of readngs or coheren readng for shor he se of all (paral or oal coheren readngs s denoed by CohSmp he coheren readngs are nerpreed n he semanc doman by he smplex ner- : CohSmp If σ s a preaon funcon possble coheren readng n ( hen s nerpreaon ( σ s called he measuremen gen by σ An mporan as s o now weaher or no a measurng nsrumen s calbraed so ha a measuremen can be relable assocaed o a physcal alue In hs sense a measurng nsrumen s consdered calbraed f s capable o nerpre s coheren readngs accordng o a sandard represenaon relaon Defnon : A measurng nsrumen s calbraed for he sandard f and only f he followng condons hold: ( ( σ σ ( ( σ σ ( ( A calbraed measurng nsrumen s denoed by Consderng a sandard and gen a = ( measurng nsrumen ( CohSmp a calbraon procedure for s any operaor : ha s able o adus he [ ] [ ] scale nerpreaon scale such ha ( ( σ ( ( σ σ ( σ ( where ( D { } σ = σ s he calbraed smplex nerpreaon funcon Any measurng nsrumen s lable o be calbraed f and only one succeeds n fndng such a calbraon procedure 4 READING PROCESSES he wo fundamenal noons of Doman heory are hose of paral obecs and he approxmaon relaon Paral obecs represen paral nformaon abou a subec and he approxmaon relaon orders such obecs accordng o he degree of compleeness of her nformaon conen Gen an approxmaon order beween he paral coheren readngs n a measurng nsrumen a dreced se of such obecs can be undersood as a readng process where each readng n he dreced se can be seen as resulng from a parcular sep n he progresse process of accumulang nformaon abou he physcal alue Defnon Le = { } be a (possbly nfne sequence of dscree nsans of me A readng process of physcal alue wh me duraon performed by a measurng nsrumen s a funcon : ( { } wh ( = = where s he oen nernal scale and s he nal me of he process We use he noaon o say ha ( = s called he readng een ha occurs a he me and s he null nformaon een ha occurs a he begnnng of eery readng process a he nal me he se of readngs eens ha occur a any me s denoed by he readng process nerpreaon funcon : nduced by he nerpreaon funcon : of he measurng nsrumen on a s gen by readng process = I follows ha a smplex of readngs σ s coheren wh respec o a readng process nerpreaon { } f and only f ( σ s a coheren readng wh respec o hen s reasonable o use he sandard represenaon relaon of he measurng nsrumen o oban a subse of paral coheren readngs ha are relean for he readng process abou a gen physcal alue : Defnon 4 A non-empy coheren readng σ for σ s sad o be a -readng f σ { ( σ } = he empy smplex σ s called a null-readng hey are denoed by σ he subse of readng eens ha are relean for he readng process of s gen by { σ ( σ } = he se of - readngs ogeher wh he null-readng s denoed by CohSmp( readng CohSmp( he me duraon of any paral - σ s hen gen by f σ = σ { n } Γ max f s fne σ = σ σ σ undefned oherwse (4 heorem = ( CohSmp( s a qualae doman he Doman of -readngs nduced on he calbraed measurng nsrumen by a readng process abou a physcal alue A -readng smplex n he doman s a oally emporally ordered se represenng he order of occurrence of s readng eens n he readng process Defne an equalence relaon ~ on as ' ' '' '' ' = '' For each n he scale

4 of he calbraed measurng nsrumen he se { ( } Denoe by = { X } X n = s an equalence class he collecon of non-empy equalence classes Obsere ha snce { } for any ( he duraon Γ X = max X For each Γ = of a X s de- fned as ( { } such ha X s possble o defne s maxmal occurrence me Γ : by ( max{ } Γ = (5 Consder an nerpreaon funcon defned on as : X = I follows ha heorem X = ( CohSmp( so ha s a qualae doman of coheren smplexes somorphc o a { } sub-doman N = CohSmp( N N of he N calbraed measurng nsrumen N { } X N called he -measurng nsrumen nduced by a readng process on he measurng nsrumen { } N N represens he par of a measurng nsrumen ha shall be noled n a measurng process of a ceran physcal alue From heorem s possble o assocae o each oen N an nformaon abou he me of s laes occurrence usng (5 5 RACING EASURING PROCESSES he exponenal consrucor s used o model he measurng processes n me generang he se of all possble hsores of all measurng processes I s essenal eg o model sensor daa fuson [6][7][8] n order o allow he synchronzaon of successe and/or parallel sensng processes performed by seeral sensors For a calbraed measurng nsrumen consder a fne K CohSmp ( fn s A smplcal complex K s sad o be coheren f and only f ' K ' F he se of coher- { } { } en complexes s denoed by CohComp( horem 4 = ( CohComp (! s a qualae doman called he exponenal of CohComp shows he hsory of he he exponenal doman! of coheren smplcal complexes represens he measurng processes of physcal alues Any coheren smplcal complex of ( approxmaon of s maxmal elemens n he oal obecs n! are coheren complexes represenng complee hsores of measuremens; paral obecs ndcaes measuremens parally performed 6 COBINING INSRUENS In hs secon we brefly show some examples of how o oban a modellng for measurng processes performed by combned measurng nsrumens (le for example n sensor daa fuson based on some specal consrucors of domans of coheren smplexes ha were defned on he bass of he sandard coherence spaces consrucors [] 6 Compee easuremens Compee measuremen processes perform he fuson of parally redundan nformaon abou he same aspec of he subec obanng he nformaon usng dfferen nsrumens n order o smulae a more accurae measurng nsrumen (by reducng he uncerany manly sysemac errors lmed resoluon ec presen n he nddual measurng nsrumen are sad o be compable sandard represenaon relaons wh respec o a sandard relaon f and only f and s s ( s s ( s s (6 wo calbraed measurng nsrumens and are sad o be compable f and only f her sandard represenaon relaons are compable wh respec o some sandard Consder hen he dson unon of he oen scales and gen by ( { } ( { } = and an assocaed nerpreaon funcon of ndexed oens : defned by ( ( n = f n = ; f n = A compee smplex ( Φ s coheren f ( ( n ( n (7 { Φ} Φ = { n n Φ } F and s nerpreaon s heorem 4 he compee combnaon of calbraed measurng nsrumens and s gen by & = CohSmp a doman ( ( ( whch s also calbraed for he sandard 6 Complemenary easuremens Complemenary measurng processes perform he fuson of ndependen nformaon abou separae

5 aspecs of a subec obanng he nformaon usng dfferen measurng nsrumens n order o smulae a larger mul-faceed measurng nsrumen Consder wo calbraed measurng nsrumens and and he caresan produc of her oen scales wh and an assocaed nerpreaon func- on : defned by ( ( ( he semanc doman A subse ( = where s Ψ s a coheren complemenary smplex f { ( ( ( Ψ} ( F F and s nerpreaon s gen by { } ( ( Ψ = Ψ (9 heorem5 he complemenary fuson of he measurng nsrumens and s he doman = ( CohSmp( ( n each componen calbraed for he sandards 6 Cooperae easuremens Cooperae measurng processes perform he fuson of ndependen nformaon abou separae aspecs of a subec obanng he nformaon usng dfferen measurng nsrumens n order o esmae a he nerpreaon leel a sngle measuremen ha deraes from he orgnal ones Because of lac of space we om here furher deals of s formulaon See Secon 7 for a conexualzed example 7 A SAPLE EASURING PROCESS Consder he case of a box of whch we wan o measure he hegh H he wdh W and he area A of s fronal face 6 o oban a sandard for calbrang measurng nsrumens for H and W consder a semanc doman of real nerals gen by = x ; x x x 5 ordered under { [ ] } { } he reerse ncluson relaon (meanng ha he leas dameer he bes nformaon wh = [ 5] (meanng no nformaon and F = (meanng conradcory nformaon he sandard represenaon relaons beween and he hegh H and he wdh W of he box are gen by s HW HW s Assume ha we hae hree calbraed nsrumens ha we wan o combne compeely ( & ( & o measure he hegh H complemenarly o measure also he wdh W We also shall combne hem cooperaely o oban he area A a he nerpreaon leel Each nsrumen can 6 H and W lay n a fla cpo of real numbers r 5 produce readngs (oens ha are nerpreed n he semanc doman he nerpreaon funcons of he nsrumens are shown n able I able II shows one possble readng process for each nsrumen operang durng he me nsans { } = he maxmal smplexes of coheren readngs and he respece bes semanc alues ha such processes produce a me ABLE I Inerpreaon funcons for oens measurng nsrumens 4 [ ;5 ] [ ;6 ] [ 4;7 ] [ ] [ 5;55 ] [ 5;4 ] [ 8;4 ] [ ] [ 5;5 ] [ 8;8 ] [ ; ] [ ] ABLE II Readng processes for nsrumens bes resuls a me axmal Readngs { } of 5;8 5;65 4;4 and Bes Semanc Value [ 4;5 ] { } 4 { } 4 [ 8;4 ] [ 4; ] he maxmal coheren smplexes of readngs obaned a me are regsered by each exponenal doman! n he maxmal smplcal complexes ha represen he complee hsores of he readng processes endng a me n each nsrumen For example for he nsrumen n he exponenal (also shown n able II represens he doman! we fnd ha he nal fragmens of he hsory of measurng H are gen by he smplcal complexes shown n able III he maxmal coheren complex among hem has me duraon I represens he complee hsory of he measurng process and he nerpreaon of s maxmal smplex { } bes nformaon gahered abou he hegh H when usng only he measurng nsrumen he maxmal coheren compee smplexes produced by he fuson process performed by he combnaon & a each me nsan and he respece nerpreaons n he semanc doman are shown n able IV In able V we presen he maxmal coheren ensor smplexes produced by he fuson process performed by he complemenary combnaon & a each me nsan and he respece nerpreaons n he semanc doman

6 me ABLE III Inal fragmens of he measurng process of he nsrumen Smplcal Complex { } { { σ } { { } { } { } σ { { } { } { } { } σ { } { } { }} me ABLE IV Compee fuson process n he doman & for measurng H σ axmal Compee Fuson Smplex Inerpreaon H [ ( = [ ;5 ] 5;4 ] { } { } { } [ 8;4 ] [ 4;4 ] ABLE V Complemeary fuson frocess n he doman & axmal Complemenary Fuson Smplex ( Inerpreaon HW ( = ( [ ;5 ] ;5 [ ] σ σ ( [ 5;4; ] [ ] { ( ( } { ( ( ( ( } { ( ( ( ( ( ( } 4 4 ( [ 8;4; ] [ ] ( [ 4;44; ] [ ] able VI shows he cooperae measuremen & process performed n he doman obanng he fronal area A usng a he nerpreaon leel he neral operaon [] f : defned by f ( HW = H W ha esmaes he area A ABLE VI Cooperae fuson process n he doman & where A f [ ;5 ] [ 75;88 ] [ 48;88 ] [ 56;88 ] f 8 CONCLUSION hs paper nroduced a doman-heorec framewor o model measurng processes We defne he noon of calbraon of measurng nsrumens and deeloped he necessary formal machnery o allow he modellng of measurng processes performed by combned nsrumens Such nsrumens can operae n parallel and synchronzed n me o aend dfferen nds of requremens: o oban beer resuls by reducng he uncerany; o reduce sysemac measuremen errors; o smulae a mul-faceed nsrumen ec A sample (compee complemenary and cooperae fuson process was modelled o help fgurng ou n a concree way how he elemens of he arous domans loo le n he consrucons of he formal model REFERENCES [] DS Sco C Srachey owards a mahemacal semancs of compuer languages Oxford Unersy Compung Lab 97 [] H Escardó "PCF exended wh real numbers" heorecal Compuer Scence 6( [] GP Dmuro ACR Cosa D Claudo "A coherence space of raonal nerals for a consrucon of IR" Relable Compung 6( 9-78 [4] A Edala "Domans for compuaon n mahemacs physcs and exac real arhmec" Bullen of Symbolc Logc ( [5] GP Dmuro ACR Cosa "oward a doman-heorec modellng of measurng processes" In: J Abe and JIS Flho (eds Logc Arfcal Inellgence and Robocs Froners n Arfcal Inellgence and Applcaons 7 IOS Press Amserdam [6] GP Dmuro ACR Cosa "oward a doman-heorec modellng of sensor daa fuson" In: GL orres e al (eds Adances n Inellgen Sysems and Robocs Froners n Arfcal Inellgence and Applcaons IOS Press Amserdam - [7] HF Durran-Whye Inegraon coordnaon conrol of mulsensor robo sysems Kluwer Boson 988 [8] JV Dam Enronmen modellng for moble robos: neural learnng for sensor fuson Amserdam 998 [9] V Solenberg-Hansen L Lnsröm E Grffor ahemacal heory of domans Cambrdge Unersy Press Cambrdge 994 [] J-Y Grard "he sysem F of arable ypes - 5 years laer" heorecal Compu Scence [] S Lefschez Inroducon o opology Prnceon Unersy Press 949 [] AS roelsra "Lecures on Lnear Logc" CLSI Lecures Noes 9 99 [] RE oore ehods and Applcaons of Ineral Analyss SIA Publ Phladelpha 979 Auhors: GP Dmuro and ACR Cosa are parally suppored by CINFO/CNPq and FAPERGS hs wor was done durng her say a UEP n July V Krenoch s suppored n par by NASA under cooperae agreemen NCC5-9 he Ar Force Offce of Scenfc Research gran F NSF grans EAR-968 EAR- 567 and by he ARL gran DA-5--C-46 e-mals: {lzrocha}@alasucpelchebr lad@csuepbr