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Two-dimensional Rational A utomata : a bridge unifying 1d and 2d language theory

Two-dimensional Rational A utomata : a bridge unifying 1d and 2d language theory. Marcella Anselmo Dora Giammarresi Maria Madonia Univ . of Salerno Univ . Roma Tor Vergata Univ . of Catania ITALY. Overview. Topic : recognizability of 2d languages

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Two-dimensional Rational A utomata : a bridge unifying 1d and 2d language theory

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  1. Two-dimensional Rational Automata: a bridge unifying 1d and2dlanguagetheory Marcella Anselmo Dora Giammarresi Maria Madonia Univ. of SalernoUniv. Roma Tor Vergata Univ. of Catania ITALY

  2. Overview • Topic:recognizabilityof2d languages • Motivation: putting in a uniformsettingconcepts and resultstillnowpresentedfor 2d recognizablelanguages • Results:definitionofrationalautomata. Theyprovide a uniformsetting and allowtoobtainresults in 2d just usingtechniques and results in 1d

  3. Two-dimensionalstring (or picture)over a finite alphabet: • finite alphabet •  **picturesover • L **2d language Two-dimensional (2d) languages Problem: generalizingthe theory of recognizabilityofformal languages from 1dto 2d

  4. 2d literature Since ’60 severalattempts and differentmodels • 4NFA, OTA, Grammars, TilingAutomata, WangAutomata, Logic, Operations REC family • Mostaccreditatedgeneralization:

  5.          p = p =        • A 2d languageLislocalifthereexists a set oftiles (i. e. squarepicturesofsize22) suchthat, forany p in L, anysub-picture 22of p isin   REC family I • REC family isdefined in termsof2dlocallanguages • Itisnecessarytoidentify the boundaryofpicture • p usinga boundarysymbol

  6. REC family II • L  **isrecognizablebytiling systemifL = (L’) whereL’ G**is a locallanguage and is a mappingfrom the alphabetofL’to the alphabetofL • (, , , )iscalledtilingsystem • RECis the family oftwo-dimensionallanguagesrecognizablebytiling system

  7. Example ConsiderLsq the set ofallsquaresoverS = {a} • Lsqisnotlocal. Lsqisrecognizablebytiling system. • Lsq= (L’) whereL’is a locallanguageoverG = {0,1,2} and  issuchthat(0)=(1)=(2)=a  L’ Lsq p = (p) =

  8. Whyanothermodel? REC family hasbeendeeplystudied • Notions: unambiguity, determinism … • Results: equivalences, inclusions, closure properties, decidability properties … but … ad hoc definitions and techniques

  9. From 1d to 2d Thisnew model ofrecognitiongives: • a more naturalgeneralizationfrom 1d to 2d • auniformsettingfor allnotions, results, techniquespresentedin the 2d literature StartingfromFinite Automataforstringswe introduce RationalAutomataforpictures

  10. In thissetting • Some notions become more «natural» (e.g. different forms of determinism) • Some techniques can be exported from 1d to 2d (e.g. closure properties) • Some results can be exported from 1d to 2d (e.g. classical results on transducers)

  11. From Finite AutomatatoRationalAutomata We take inspirationfrom the geometry: 2d 2d 1d 1d Symbols Points Strings Lines Pictures Planes • Finitesetsofsymbols are usedtodefinefiniteautomatathatacceptrationalsetsofstrings • Rationalsetsofstrings are usedtodefinerationalautomatathatacceptrecognizablesetsofpictures

  12. From Finite AutomatatoRationalAutomata Finite Automaton A= (S, Q, q0, d, F) Sfinite set ofsymbols Qfiniteset of states q0initial state dfinite relation on (Q X S) X 2Q Ffinite set of finalstates SymbolString Finite Rational RationalAutomaton!!

  13. RationalAutomata (RA) A = (S, Q, q0, d, F) Sfinite set ofsymbols Qfinite set ofstates q0initial state dfinite relation on (Q X S) X 2Q Ffinite set offinalstates RationalautomatonH= (AS, SQ, S0, dT, FQ) AS= S+rationalset ofstringson S SQQ+rationalset of states S0 = q0+initialstates dTrationalrelation on (SQ X AS) X 2SQ computed by transducerT FQrationalset of finalstates SymbolString Finite Rational

  14. RA RationalAutomata (RA) ctd. H = (AS, SQ, S0, dT, FQ) dTrationalrelation on (SQ X AS) X 2SQ computed by transducerT Whatdoesitmean??? SQQ+ AS= S+ • Ifs = s1 s2 … smSQ and a = a1 a2 … am AS thenq=q1 q2 … qm dT(s , a) • ifqisoutputof the transducerT • on the string (s1,a1) (s2,a2) … (sm,am) over the alphabetQXS

  15.  S++ picture Recognitionby RA • A computation of a RA on a picturepS++, pofsize(m,n),isdoneas in a FA,just considering p as a string over the alphabet of the columns AS= S+ i.e. p = p1 p2 …pnwith pi  AS Example: string p p1 p2 p3 p4

  16. Recognitionby RA (ctd.) • The computation of a RA H on a picturep, ofsize(m,n),startsfromq0m, initial state, and readsp, as a string, column by column, from left to right. FQisrational pisrecognizedbyHif, at the end of the computation, a state qfFQisreached. L(H)= languagerecognizedbyH L(RA)= classoflanguagesrecognizedbyRA

  17. Example 1 RA recognizingLsqset ofallsquaresoverS = {a} • LetQ = {q0,0,1,2} and Hsq= ( AS, SQ, S0, dT,FQ)with • AS= a+ , SQ = q0+ 0*12*Q+ , S0 = q0+,FQ = 0*1, dTcomputedby the transducerT T L(Hsq) =Lsq

  18. Example1:computation Computation on p = T dT(q04, a4) = outputofT on (q0,a) (q0,a) (q0,a) (q0,a) = 1222 dT(1222, a4) = 0122dT(0122, a4) = 0012dT(0012, a4) = 0001FQ p L(Hsq)=Lsq

  19. RA and REC Thisexamplegives the intuitionfor the following Theorem A picturelanguageisrecognizedbya RationalAutomatoniffitistilingrecognizable • RemarkThistheoremis a 2d versionof a classical (string) theoremMedvedev ’64: • Theorem A stringlanguageisrecognizedbya Finite Automatoniffitis the projectionof a locallanguage

  20. Furthermore • In the previousexample the rationalautomatonHsqmimics a tiling system forLsq but … • in general the rationalautomatacan exploit the extra memoryof the statesof the transducersas in the followingexample.

  21. Example 2 ConsiderLfr=fc the set ofallsquaresoverS = {a,b} with the first rowequalto the first column. • Lfr=fcL(RA) • The transitionfunctionisrealizedby a transducerwithstatesr0, r1, r2, ry, dyforanyyS

  22. Similaritywithothermodels • RationalGraphs • IterationofRationalTransducers • Matz’s Automatafor L(m)

  23. Studying REC by RA • Closureproperties • Determinism:definitions and results • Decidabilityresults

  24. Closureproperties • PropositionL(RA)isclosedunder union,intersection,column-and row-concatenationandstars. • ProofTheclosureunder row-concatenationfollowsbypropertiesoftransducers. • The otherones can beprovedbyexportingFAtechniques.

  25. Determinism in REC The definitionofdeterminism in RECisstillcontroversial Now, in the RAcontext, allofthem assume a natural position in a common settingwithnon-determinism and unambiguity Differentdefinitions Differentclasses: DREC, Col-Urec, Snake-Drec The “right” one?

  26. Determinism:definition Twodifferentdefinitionsofdeterminism can begiven • The transductionis a function (i.e. dT on (SQ X AS) X SQ) DeterministicRationalAutomaton (DRA) The transductionisleft-sequential StronglyDeterministicRationalAutomaton (SDRA) Col-UREC DREC

  27. Determinism:results • Theorem • Lis in L(DRA)iffLis in Col-UREC • Lis in L(SDRA)iffLis in DREC RemarkItwasprovedCol-UREC=Snake-Drecwithad hoctechniquesLonati&Pradella2004. In the RAcontextCol-UREC=Snake-Drec followseasilyby a classicalresult on transducersElgot&Mezei1965

  28. Decidabilityresults • PropositionItisdecidablewhether a RAisdeterministic(stronglydeterministic, resp.) • ProofItfollowsveryeasilyfromdecidabilityresults on transducers.

  29. Conclusions Despite a rationalautomatonis in principle more complicated than a tiling system, ithas some major advantages: • It unifies concepts coming from different motivations • It allowsto use results of the string language theory Further steps: look for other results on transducers and finite automata to prove new properties of REC.

  30. Grazie per l’attenzione!

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