A Representation Framework for Continuous Dynamic Systems

1. ID.ti'OducttoD

JU -.m- r"- · ..,. focua CD two llpiCII. Pint. -"'-' ~ ,_.,., of ot}«:U w «<Mda II a plllplrty at maay objlctl. ~ M 11qlll* ud ..- ud tlllir -am.. Slcalld. _,._, ~ of pro,..., .a .. _,_Ciani, ..-- ad localkD, "'*' bodl -=­ - ...,. ad ccatbualty at acdaDI C&1llbla ·· tnlllitima

Scbmoa. [7) ... .... that pnYiolla .-mtatka thaorW do llllt dlaibe bow -a.- pQIIcal objlctl - ~ at putt. Scbmolal dlaibtl IUCil puta .. CXIIIIildJI& at "pam&1ea" that cu be -.Ibid popertllt u.ld for Infer­ rille popertllt at tha wbale fftllll til.- at tha putt.

We P1111D1 1D foaad a .-mtatka thaor7 for IUCil ~ CD llwlt'l lll­ -ble theor7 [.1, 4l r-ble theor7 ... eat theory by lntiOduc:ill& -- -~~~ee ""- putll - -~~~ee (I.e. at tha - ~. Wlllb ~- at .-). Aa -aM II -~ Iff It baa proper -~ putt. 0... cu dlaibt objlctl wiG ccatill- w dllcmt putt. .a .. (Uqliid) watar with 1M* at altilla II& AJIIItblr -pll II fiDe.pillld 1U1V that II --um. ,.,.. ... llqllld. ... ,... fllrwlq thlouP · - ad --um. ... coi­ J..ama at ~ "::r:t-,... dilt::ullia& tha pala ... bodl - - .. Jailed, .. an1a .. now pl'Olllftl& r-~~~e theor7 with tha ·­ ....... fi thll ,.,. provide · fOWidatka 1D eztcd eat theor7. ad 1D ICCOWlt for ~ dlfflcu.J.t to dlaibe ill - at 'puWII', WI!.- anauJ- an limply tUa 1D be -bla

We appq thaee _,a~ to tha .-mtatka f.-.-k DRBAM ('l):yumlc ~ ad WadaiiD(). DRJ!AM .... pll)'llca1 objlctl ....... ~ IDe -~ dllcnte or miDd -bleL After utatldiDJ -ble thaory with liOtioal at cca1:illuoua ltnaml and me..-, we allow bow auch mathemati­ cal _,c. cu be U&lllfernd Into DREAM. DREAM .._ mecbontmo from ·betnct data type opeclflcatka and object«<entecl 1aDpap, tak.illJ objectl 1D be c:arrlln at illformatka (putt. etateo) toptiYr with tha fadllti· to IIWiipu.lata IIICil illformatka (time, ectioDI, illfenncee), thUI comblnlna the tmnlnoloJic:al ad -aca.aJ. 1AIYala [l~ ~ an '-porally ezteDded pattenuo at litUI­ tkllll &lid eventa dlaibln& d)'IW!Ik: clwiJeL

2. The DREAM Representation Framework

2.1 AJeaa

DIU!AW .-mta objectl u araa. edoptiDa an object«ientecl viewpoint wban cbaqel an bloup.t about by qenta carryiDJ out operadou. Aa operatka may clwlp tha 11a1 etata at an qat, ad communicate illformatka ltlmulatine other qenta to aecute operatloal. Tbll illformatka flow ~· c:aUIHffect relatloublpo. &lid C#IUIIl ptll""""' If It proceed& aJcac epeclfk: rout& State. camprile tha illformatka that, Ulptber with operatka epeclfic:a­ IIID, determiDe an qenn behavior. The behavior ol an apnt cbaqe1 becaUII Ita etata cbaDJeL Operatioaa quantize all potaDtlal activity at an apnt.

Pftllll objlct-orieatecl camputatlcD, w. allo odopt tha DOdcD ol ~ or typal ol qenta with ldeatk:al opeclflcatloaa ol operatkllll &lid etate opaca Apnta at tha - type may differ DOt ODly ill etata, but mUit differ ill piOpltCiM that Nllllla I&DCIIaqed M loq u they a1at, -bllllllna an apnt u an IMM4MIIL A llilllplol &lid effk:Mat ~ pro~ II a UlllqUI 111o111e. Type-wide -..t popertllt &lid IDdlvlduatloll _,c. an opeclfled ill a CHARACI"1!1l claua.

For -pie, qenta at type 'valve' CPic· 2-t) have a COIIIWlt maximal open1nJ ..., ad receive IDdlvlduallty by nama Sta111 an ds:rlbed by biDdiDp at etata vuiablee ccutraiDed by proportlee ill a WHEIIE-da\118. For 1D3' COIIIWlt or vuiabla :r, lCD IDdkateo that x may Olliy - velUM ill a climeDiion D defm.t a-where.

Opaatkae an opeclfled ill tarma of tha followiJia c1a .... (we IDdlcate cia.._ by c:apitallad byword&): A PRECOND!tka Ia a propalitka on etateo det"lllillJ wbell tha operatka Ia applk:abla. The PRECONDitka IIIOdl DOt be true after tha operatka baa beJu.n OUCiltka. A "l"!alllltka INVariant' ds:rlbtl cbaqe1 wbell tha opaatka II be1DJ c:arrled out, opec:ifyiDJ ccadltloaa that per* u loq M the opaatka Ia be1DJ aecutecl. A Tl!llMIDatka COND!tka Ia a propalitka about etateo that, wbea it become~ true, Immediately tamiiDateo ex­ OCiltioD. A POSI'ccadltka ds:rlbtl .--.!tine etateo u far u they an 1111t lm­ pliecl by tha other claiiiM. DURatloll provldee illformatloll about bow much tinM tha opaatka epeDda Oil OUCiltizlc, eopeclally Wilen thll depeDda Oil dJ1WIIi<:ally cb.a11c111J CODditloal. The P ARTS-d&Uie eotab~ a parc ... bpart hle!uchy. Aacla may be dJ1WIIi<:ally CREATEd, u ill a c~ reo<:t1on. illformatioll. II communicated via typed 1/Q-por~a.

To ds:rlbe tha operatkae at 'valve', w. aeod Ollly tbiM of tbMe claiiiM. A valve cu oaJy 'opa' wbea 118 prior etata II ·~·. The flm tftlllitloll ill­ vuiaat tabe tha c11acreta, or qualitative part at the temporal derivative (we writa a for dlcW &lid - It to -1 to -n that the open~nc 'llfldcL State -.!at (3) lmpllle that tha valve II ill etata 'opeoed' ""'-&tlly after the operatka 'opa' baa bepa, even before It bu tamiiDatecl. Opaatka 'flow' II thanby lll&bl.ed II tha In port 'Ill' II DOt · (I.e. empty). AA apnla will Ollly ODCUia opaatlaao wi1ea they an beillc told, the valve mUit tall ltalf to let tha liquid floW, &lid it doM ., ill the CAUSI!S-<Ia~~~e. 'flow' lmpllee that a '*"- at iDcomllla llquid corr.pclllda to an ln._ at outaolnl llquld, ·­ JIIIIIlna that liquid 'f1owl' tbrou&h tha valve. A phylical COilltra.lnt -.11y terminatinc tha 'opa'~tka II 'Cipllliq' ~ !Ia maximal value. The epeciflcatka doM allo allow an e:rtamal event 'te' ("tennnnatine event") to ter­ millata tha operatiOD, wban 'ta' II explicitly commWiicatecl to tha 'valve'.....,t wbell be1DJ inltructecl to 'open'.

2.2 m.a.aa o1 Claaqc ,_ lltutioaa aa4 Bnaa

We ~t ....,_ ill a ..,- u ......,, that daiiOta etata tftlllitiODI broupt about by -tine opaatiODI over tlmo illtarvala. Sltutioaa an a dual DOtiOil .-cribiq tha ,....._. at etalll over tlnM illtarvala. Both nly Oil a temporal Joalc.

We briefly Introduce til.- coocepa ol the d)'IW!Ik: temporal Jo&k: DTn. [6) that an ·'-lueatly a.dad. D'IU. -bedl the tlnM illtatval Jo&k: m. Into a d)'IW!Ik: Jo&k: that OOIIliiiCia tlmo illtarvall with litUitioDa and eveniL m. Ia · "temporal framework" utelldlnc an arbitrary flm«der Jo&k: l;o(T,P,R) ol - T, formulae P, &lid ru1ee R Inductively with illtatval tarma n and Inter­ val formu.lu PI, wban

(1) Clntv, :S) Ia an illfiDlta, partially ordend eet of tiJM illtarvala.

(2) llltvCTI llnlilrvala - tha llilllpiMt intatval tl:fiDI}.

(3) v IJ en. Ia {IJ}, (iJ}, be&(l), endCi) e n.

(4) VIJETI. 1Jll$ll-j, llj, 11,1 E Pl.

n.. ota-t below =-· the 'valve'....,.t vl, wlua ill- &lid out-por111 an thG1 ~ 10 Jlil* p} IUid Dl.

1;j II the IICI-dal compcoldoll <..tarlll with bepll of I. endl with eDd of j), &lid (IJ). {lJ} an pua11e1 compcoldoal of time illterftla (lor {l.j}, I &lid j mlllt overlap. for (IJ). I &lid j mlllt 110t be $-<XJmpuable). tJea(l) &lid Olld(l) deiiOte besiA &lid eDd of L Uj Iff I - j [ll llj iff I &lid j an CODCIIJTIIIlt ( $-IDc:cmpuable), u.J iff I &lid j an limulw.eoua (overlappilla). TIL ulomaW. illtervala 10 be opaa. $ II a chaln-<lOIIIplata panlal order 10 that a time illltant, IIICh M IJe&(l), II the .. premiUD of all illtervala It meetiD& L 'f II the lattlce­ thooratlc joill IDduced by $ with illltante .. 111111-«<eaaeelta.

To ~t temporal atru<:tuJW, eopoclally for cycUc, ncune~~t activitiel, TIL baa ... bt.terY.J en- eztalllloD 10 the abcm framework. AD illterval typa rr .s-. m illfllllte .. pply of illtervala all related 10 the - activity or property. llltervala ill rr differ ill duradoll &lid bepa/llldo. ~ 10 all parlodl over wblcll m operadoll may be carried out, or - llate property panlota. for .ampl.a, all time perioda of COIIlllllltill& 10 WDrlt .ell w.tday 1110111111& will be ~ted with the - illterval typa IT-commllte, altholl&h all time illtervala of thla typa an cllff~n~~t (cllffereat da:JI> tlmea of day vary, duratlcM clepead CD tnfflo, ell:.) A -por.J 1tr110t11n II 1llta m illterval term, blat with - illterval typaa illatead of illtervala. for .ample, T-worltday · IT~-bnaltfut;IT -commllta;IT-wort;IT-commllta;IT._t

11 the temporal ltriiCtlln of worltda:JI> &lid rr -workday · that of work w..u. AD illterval - d.crlbUI& a pu1kular workday II obtaiMd by a4111UIIIIU ~ of illtervala for illterval type~. COIIItnilled by the Nqw-t that It prod.._ OD1y ldmlllible illterval -. (with propar Oldar-relatiollablpo U1111D1 llltervala). 111 our .ampl.a, &II illterval .,betltlltill& IT11eop mUIC - the illterval .. betltllted for rr -bnaltfut.

A ll-doa [,~] II a p!OpOiidoll ·:Jilla that lt&te a .,.mota throuabollt time ill­ terval L AD nat 4loslW> II a propclidoll '*'rlbUI& that oparadoll op II carried out throuabollt time illterval I &lid c11.aJ1811 otate a IIllO otata ~. Wt caD­ alder time DOt 10 aile aprlorl, blat u balD& -bllabed by ph,.al cbaJI&e. AD tve~~t 41osW> ..,. be taUil 10 th/IM time illterval I 10 bepll wbell op IIWtl IDCUtilla &lid 10 eDd wbell op tmllillatea TwD ...-lve tve~~te 41osW> &lid ~'II"> daliDate the lltlladoll [a' J1 when UJ' (j bepa wbell I eDdo, &lid &llda wbelll'bepa).

'The !Dhenmt relatiolllhlp benne~~ lltlladolla, tvellll, IUid time illtervala Ia a:­ plcl.ted by a:teDdiDa the TIL-<XIIDbillaton for illtervala IUid illterval typaa 10 lltllatlcM &lid tvellte: IJo.lol ; ~ta,.> ; liot~l d.cribea thtt after llt&ta "0 pmilll throll&h lAter­ val Jo. eDCiltllli CJFo dllriD& It clwl.a- "0 10 ·t· &lid 't thell .,.mota lllltll tlldCI,). Slldl lltlladollleftllwcructuna ..,. provide a camplata _,, of the hlollory of a ,_. a.-u..a CC111C111m1C:J &lid limultalldty by marbd braJiolllD& &lid joillilla &!1:8, llawever, yleldl IUtory INtW!crb rather th&ll ·

~~- of litllatlcM &lid tvellta.

2.3 rr-

A ~ apedflel a tamporally mnded pattem of activity performed by a .,._ of ....,ca. Euclltllla a ~ aeura· a hlll:ory llltWorlt of litllatlcM

&lid tve~~te thtt ~t f10111 ~DC~&till& operadolla at the t1mea &lid ill the cmter opeclfled by the ~ 'The key COIICipla 10 '*'ribe ~ an: (1) Ca-.1. nla A calllal n&1a relatea c1uaaa of IIN&tiolll IUid evata. AI m evat alwaya tan~~JDa· ill a lltllatiOD, the DOdoll of evente calllilla lltlladoal II alnady lllllena.t ill tvellta. Tbe other two typaa of ..... tloll ahlbit dlfflftllt nwhenlernc

• A ca ... doll n&1a odtllatioll-pattam> -> 4vellt-pattam> c1e110tea 6~ ~ when jlllt the ~ of a IIN&doll .......... tvellt.
• e A ca.,.doll n&1a 4\'ellt-pattem> -> 4\'ellt-pattam> .s-. ~ ~ when ~DCUtill& m operadoll .. ...,. thtt of aDOtber OM.

(l) Sl-tloJI/nat - &lid ,.._ AltbouaJl chaiDID& calllal ru1a1 a-n· hlll:ory llltWorb, calllal rulal do DOt cS.crlbe propart1oo of a:tlllded activity panome a:pllcl~. DREAM CCIIIMCte ~.--JI'IIUriU with C&llla1 lilllta 10 ~t lltllatiCG/evat atru<:tuJW, or ,__, ~ docu­ lll&llt propertilo. OCIIIItftillte &lid COD~ ill the -aoa. of litllaticM &lid evata. Oftall, a ~ cu. be oolla~ IIllO a 1i11aJa -t or lltlladoll -­ marlzbl& the overall effect at a hlaJler laval of abltncdoll, al1owill& 10 bllild abatncdoll bleruclllla ill ~ .-:rlpdolla.

3. El:ulembl-= Mathematica for Clmtinuous Objects

3.1 ... Ea.able T1leor7

llulc: -at of -ble theory an ez.,._ llt&lld for -bill)

1. Part-wt.m nladoa !;;;. z!;;y deiiOtea 'Z II part of y'. s;; II tnll· litlve, &II~ &lid reflalw. Then II aacdy OM -blot, the -ttJ' -ble .. thtt II part of r1V1 -1111. Nt/UIJDt&. c deiiOtea the 'lloll._pty part' reladoll. A propar _. -pt)' part of ... -ble Ia a1IO ca11ad a ..... put.
2. MRp &lid llftl'laJ facdou u, n. For a collectiCG C of e­ -· the aerp u(C) lo the .. -ble Lt. lf'mY _.. part of u(C) baa at ._ OM part thtt II a1IO a part of 101M member of C. Lt. u(C) II the milllma1 -ble thtt baa all membera of C u lte part&. 'The llftl'laJ r(C) of C II the ._ -bla CODtallliJI& all COIIllllCil putt of all memben of C.

For ... -ble X. &lid a part z!;;X. r lo the -Jiledoa of J: w.r.t. X. dtfilled u the milllma1 -bla with :rU r - X.

For &II -ble X. (C(X).u 11 ,C,.:x;J II a Bool.a alpin. when COO II the &It of all putt ill X.

1. A~ _._ &lid tile ..W.wtom nlada i· A _. ~ -'* II a~ iff It baa 110 other putt th&ll · &lid It· all. For m atomic object :11, then II a lllliqllt _.. part of :11, ca11ad the .mat. of L Tbe relatloDibip betwMil m atomic -ble &lid lte lllllcle II ca11ad the Wol .. wholt reladoll i :

VJC.(atom(z)-.3 1 y.yf.J:~ IUid

V .11.(~ atom(z)-.~3 y.y f.%~

'The ~wboW reladoll admite m a:tlllllollallty ulom atlpulatllla thtt z E y Iff all putt of z an putt of y &lid, if z Ia atomic, the lllllcle of z II the lllllcle of - part of y.

• 4 .... ~llole nlau.. E. z E y iff z II the llllicle of - part of y. lfiDaa the llllicle of m -'* z II a member of :11, relatllla _ .. 10 - theory.
• C..da- -IIlia. A IICG._pty -bla z II -da- If lf'mY l"'lliM part of z baa a propar l"'lliM put. l"- a CCD­ tilliiOIIII -ble C&lliiOt bave &ll:J milllma1 pull, &lid CODtilliiOIIII .. -IIIIa bave 110 memban.
• Dt.nte -IIlia. AD lllllllllbla II ~ Iff It II eqllal 10 the -ae of lte atomic parta. Oarly, the -pt)' -ble · II dlacnta.

O..till_,. -blee CODtaill 110 atomic pull, &lid dlocma -blee ~ Ollly of atomic putt Of ...,. ). Ellaemblee thtt an neither CODtillllOUI aor dlocma an ca11ad lll1zt4 -bl-. Every milled ~~~~~mbla ..,. be cleccmJDICI IIllO a dilcma &lid a CODtillllOUI put.

3.2 C..tbt.- .......

Olalider IIIIIJ!III two CODtillllOUI -IIIIa w of water W &lid v of vodt.a V 10 form m -ble d · w u v of dllllted boola D. By our cltfillltlcll, .ell part of d CODta!Da putt w', v'!;d of water &lid VDdlta. Bllt ill nallty, then lo 110 part of 'pun' water or vodt.a ill cl, when a Jlllft part of water II OM thtt baa 110 part of VDdlta. AI thll Ia 110t para~~tlld by the abcm -.. oparatlcll, W1l captun thll CllliiMIIpt ill a apeclallzatloa:

Dafta!doa 1: [O..tillllOUI Me!Je]

'The -tbt.- ...... of two CODtlAIIOIIII _.,blee lCX &lid y:Y II the ._ -ble z with (0) v ll!;;.r.3 W!;;J:,liCJ.M.- ...

We deDote tile oet of all enoembleo oblalnecl by contlniiOUI!y merz!na -blel from X and Y u XI!IY, i.e. 7!Z. with Z-XI!IY.

Nat·. We omit tile proof thlt z In (·) Ia Wllque (I.e. Ill Ia well.-<leflnecl). . AJJ. Important appllcatloll of contlnuoua mefJt1 Ia -blllhlna a 'cam.' con­ malnt for a dlemk:al reaction by havln& liquid or poeoua mallerlala mmp con­ tlniiOUI!y to ~

3.3 State.

How can we delcribe tile temperature cllatribution In a lake In oummer? Thla il tile phenomenon thlt, eopecially for deep lalteo, tile bnttnm rep.. Ia much cooler than feliona clooer to tile IUrface, and there Ia UIUally a ifldual U.C... of temperatu""' from bottom to IUrface, un1 .. dilturbed by underwater - Ukewile, rwo contlnuoualy merpl mallerlala may not ltUt react1na unl.- tbelr tempera~ exceed& 1011111 minimal value. In IUch cueo, we would want to ucribe a 'temperature'-property to enoembleo thlt may chanp over time.

DefiDitiOJl l: [Stateo of enoembles]

A lfatl of an enoemble il a property thlt may chanp ila value over time. We uoe two opeclfic notlona of lt&tec

• A propooitlonal atata il a propcoltion about an enoemble.
• A finHr<ler atata il a functloll from atata Tariablea to valueo In a atata .,..,..

Natatlorl. For an enoemble e, lt&teCe) denoteo ita lt&te. If we delcribe flm-order lt&teo, we wrile :u - v for tile fact thlt e Ia In a lt&le whare ltate-variable x ......,. the value v. For a type X of enoemblea, State(X) Ia tile ltale opeoe of ita membm.

App!Jina a notlllll of ltateo to enoemblea require& a deflnltloo. of how ltateo of porta relate to the whole enoemble.

DefiDitiOJl 3: [State combination function]

For a type X of contlnuoua enoemblea, a atata combbaatlo11. f1111.ctioll ""x mapo tile otale of tile porta of portltiona of any X-enaemble e to tile otale of e:

""x Ia a atate combbaatlo11. fllllctioll. for enoemble type X Iff

for any portitionln& "' of an enoemble e:X,

1. otateCe) - ""x(~te(ei). e I}), and

(l) for any X-c111111ble e',

1tateCe Ill e') - ""x(~te(e). lt&teCe')}).

{""x comblna tile otateo of all portltlona to tile lt&te of tile whole, and eJ:Chan&ina any portltion with anothlr port havln& tile ame lt&te will 11.ot chall.p tile ltate of tile whole (lllbetitut1Tlt7 propert7 of acate.). Clearly, tile combined otate of an enoemble Ia lndepeadent of tile cholen portitiooin&)}.

E%41flpl4. For enoemblea w:W of water, let ltate(w) be a functloa that 1111ip number& berween 0 and 100 to tile ltate variable 'temp'. Then, for any portltionln& p( w) - {w 1 , · · . ,w n} of w Into water enoemblea, a uoeful lt&te combination function would clelel'mlne the averap temperature of w from thooe of tile poctap It oonailla of:

whare M - liWI( w ). ""x would, however, 11.ot be a ltale comtmatloo. functloll If it took. jult tile maximum element of ita UJW1111Dt oet.

3.4 sueama

lmpaoina otructure "" contlnuoua enoemblel require& maltln& porta clllliJI&ullhable without rellnqullhlne continuity. Aa:ribln& otateo to porta -plloblo jult that. Olveo a way of cllltlnauilhlna porta, cae can imiUt order &1110111 them. For otud:Jin& COIIIleplll of flow, we need a -m of ltnaDII · opeclla1IM portltl­ tloaa. A pard.tloa of an -ble e Ia a let p 10 that each ~t of it lo .an -ble of .-J'Uia with

(1) v 1J e pJ1 <x.i> - ·· and

(l) U(p) - e.

The elementa of p are the blocb of tile portltlon.

DefiDitioa 4: A otream pard.tlo11. of an -ble e Ia a portltloll of e toptber with a linear order of ita blocb. Two otreem portltlona (p 1 ,<1) and (Pl·"l) of an enoemble e are competlbla iff, for i-1,2.

{Ul(J:1) Ia the let of all Pl-blockl c:ontalnin& porta of "1 and Ul(y1) il the let of all Pl-blockl contalnin& porta of y1}

The .- prodact p - p 1 *t'l of two compotible otreem portlticl>l (p 1 ,< 1 ) and <1'2·"2) of an -ble e Ia tile coa.- otreem portltlon of e that renn. botb, u lll111trated In Fi&· H. If a otreem portltloo. (p 1 ,< 1 ) of an -ble e refU. a otreem portltloo. (Pl·"l) of e. then both are compotible and P1 *1'2 - pt. Stna.m portltlona "chop" enoemblea Into pl-. Thla appeon to defy Intuitive not1ou of liquid atreama. wh&re any ouch "chopppna" 1a artificially impooecl, yet atlll part of the concept of a meam. By relativlzlna otreem portltlona, we ob­ tain c:ontlnuoua ltrealD&:

F1pre 3-1: Stna.m Product DefiDitloa 5: A contlnuoua enoemble · il a -dll.- .- Iff each otreem portltloo. of e hall a proper reflna*lt. A let F(e) of otreem portlticl>l of e 10 that each otreem portltloll In F(e) hall a proper reflna*lt in F(e) II a -dll.-.-ntm.-t ~ for e.

Every contlnuoua otreem refinement 1J1te111 provldeo an Infinite .. pply of ubitraril;J refined - portltloaa. Thla il the faclllty we ll.eed for 1D8rJ1n1 COD­ tlnuoua..._

The .._... · - e1 0 'l of two contlnuoua ltnaDII e1 and 'l II tha ~ lll­ -ble · a.t. for all COiltlnuoua meam reflna*lt ~ F(e1) and p('l). thare

lo a contlnuoua otreem reflnemeat 1J1te111 F(e) with

v p 1 e F(e 1 ~ e p('l).3 p e I'( e).

(1) v .x e p. 3 x1 e P1.X2 e Pl· "1 ~" " "l~x. {each J>'bloclt contalna porta that are p 1 - and Pl-blocb}

(l) ""1J1 e P1-"2J2 e Pl·

~1<1Y1 A "2"2Y2J -

31JEP. J:<J A "1-"2~" A Y1J2!;;y.

{blocU In p ,_..,. the orclen on pUla from p 1 and Pl}

Eumplea

1. Metal .U07 bar. b II a cylindrical tn. bar, made of copper and z1ne, whare cae and lo pure copper, the othlr pure zinc, and b Ia a "contlnuoua" 01/ZA..uoy In bennm. All obvioul approacll Ia delcriblna b with a functioo feu alvlna tha differential CU-<>oncentratiOil at c111tance r from ita left e11.c1, whare the re1t lo zinc:

dfcuCr)ldr - -. with fcu(o).(J, rcu<ro>-um.

We ~t b · a contlnuoua otream, whare every contlnuoua otreem reflne­ meot .,._ F(b) baa the property VpEP(b).VIJEP. :I<J - r.x<r.y

VIJ E p. J:<J - 1foC:II,z < 110&.y

1 MuJbal water .,.. aloolaol -- Cculcler ID8rJinl two contlnuoua ltnaDII w of water and a of alcobol to form a miJ:ture m - w®e. We ucribe any part of a liquid a otale thet contalna valueo for temperatura and volume; each part of Dl hall the adclltloo.al ltate variable "w ( waler c:ontenta>. 'Jbea, the 111181Je of contlnuoua water and alcobol ltnaDII ,.,._. a cleflnltiOil for cleterallnlna the ltatel of m-porta from thole of tile omerpc1 w- and a­ parta.

4. Representation of Continuous Objects

4.1 Eum.ple: Llqalda

To ~be llquldl. TA - a limple view by ·Jina that an object il c:on­ llclered a liquid iff ita -peratura ranaeo berweell. a freezlna and a boilln& temperatura that clepeDd 011 ~""' ooly, ita volume clependl "" ita tempera­ ture. and unl.- it !a In a "'table" ltale, it f1owa until it become~ lt&ble:

A UquJdO.liquld lo chancteriJ:ed by Iu.r clependencleo of volume and tempera­ ture. and f~ tempera~ and ~re. Ita otatel c:onalat of volume, ~ and temperature, oonatralned u Indicated. It hall UquldG-porta that can be portltloaed Into UquldG-porta with an obvioul otate-<omblnatloll function. It il ooly otable wben all of ita porta are otable.

:aolf Ia a lllf-ref- of any aaent of type UquldO. For brevity, POSI'COD­ cllticl>l conleln - --!op:al notatloG: (1) A (l) - IW>da for the formula obtained by ANDina the lndk:ated formul.u, and {u/:aolf} c1enoteo a .. botitutloo. In tha fnllowlna formula. By ita P ARTS-clauoe, a Uquld().qent Ia compooed of UquJdO.«pnta -*tina of Uquld().apnuo, etc. with tile ltate compclition func­ tloo. cleflned In the vJiiERE..eiaUIL J.. contlnuoua ap~~.ta can be portltioll.ed In lnflnlt.ty many waya. the otate compclitloo. functiOil mut be inclependent of any particular partitlooln&. Thla Ia obvloualy accompl.labed In tile WJIERE.<:lauoe, whare PutitiOil(:aolf) clenoteo the Infinite let of all portltlona of a 0011-· Uquld().qmc

X e Putitlon(u) : ..... 31~N.X - {:~jl e I},

and for any permutatloo. 1 of I. u - ~1Ptl'Y-

A ~· of liquid il otable, and thereby 11.ot flowlna. If all ita porta are lt&ble.

The operatlona are opeclfled In · nm-«c~er Janauaae that lncluclea part-a£ and eqllallty relaticl>l ~tina the CGrrelpOIIdl_na relaticl>l of enoemble theory: V x.:r.LiquldO. .J:Qr : ..... 3z:Liqulcl0. mmpW - y A x-y: ..... x!;;yA y!;;x A ,.rtloular wato,_.,.nt Ia HtolnM ~ lnotentlotlnt 11 pot urtMuMI $-profhH ototo l,.otlfloro, '"' ·-lfyiOf'" loltlol ototo: NlEIIT · · Wotor(.._I-4:STAT! YOI-uft , ~--· ., .... ,., ·· lt0111

.us-

We mu.aa· dll ~·tloD of tbe .a.m. ccaoepta dnelopell In -. 3 In - of JIIIICiallzadiD of UquldO:

AGEHT-1"11'[ Ll~ul,.....rtltl'"(l) · CIWUICTIII CICINST I :N .... t PMTI 1 1 :L1'1Uidl -lEI" I ....... I,IIEII

Aa;J plftitloD ol 1 II ~ by I· pe.rta aDil u iDda eot I of D&aual lliUilbln tbet c..-w I· biGcU. -..fltl E I) ...... for 0 0 0 ~10u> 0 0 0 · uoiD& 1M feet tbat -au(y) · -..:JW.

a- J!Ud.,._ ol u ..-t l:UqlddO en limllu 10 plftidall...-· for 1 witll 1M lldditloD tbat 1M 1iDiu order OD iDda - 1111...- a IIMor order OD I· pula.

Ccada- - an 8piCifW by UUiferr!D& our defiDitloD IA110 proper lllli!AiikJB- 10 l!piCif)' ·~· ""' ... -...ntiOD.

~,._,. I -w lib 110 t1w1k N. S. SridllanA for helpflll -· Oil lA -u.r clnft ol dill paper.

• [1) Alia, JJI. Malntellllq bowJedae abollt -ponl !Jltervall. ec.o-lcrllbv of tlw ACM. l6(U}:a3l-a.43, 1983.
• [l) llncbmaD. R.J. v ~. Oilben, w. t.n.que. AA -tlal 1l1brill -m, ~ KDowledae ""' oymbol 1ne1 ..,. COWl. of JatYPI'ON. iJl Proc. Slrll IJCAI, paa- 532-539. UCAl. Lat A11aeJo1, CA, Au...-. 1915.
• [3) Bwlt, B.C. M- - 11M ..-..,,_.. -"<:~. Cambrldae UIIIVWiity Prw, Cambftd&e, I!DpoDd. 19116.
• [4) Bwlt, B.C. Tile formal ,._.tieD of (q..--) OODtiJIIIOUI ccaoepta. Ill J.L Hobbl. ILC. Moan (editor}, ForMIIl 1"'-fu of tlw c-­ WorW. cbapcer l. paa- 37-1'0. Abla PubL Co.. Norwood, NJ. &9116.

[S) Ha,_, P. J.

Naiw ~ 1: OlliOIOI:J for Uquido.

Ill J.L Hobbl, ILC. Moan (editor), Ftlnlllll 1"'-fu of tlw c---· 'WorW, cbapcer 3, paa- 71-101. Abla PubL Co.. Norwood. NJ. 1985.

• [6) llaWef1, P. A d,..amlc lop: for -ponl -m, about dytWIIIo oy-. "' ,..,.... · 1917.

[7) Scbmobt, J.

~for robol&

Ill Proc. $111 N.rt C..f. on Ardflt:W liii.W,._, paa- 4HO. AAAI. pbU..Wphle, PA. AIIJUit, 19116.