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Unifying 1D and 2D Language Theory through Rational Automata

This study proposes a framework connecting one-dimensional (1D) and two-dimensional (2D) language theory using Rational Automata. By providing a uniform setting for understanding 2D languages, it seeks to generalize concepts known from 1D to 2D, such as recognizability and closure properties. The paper introduces Rational Automata as an effective model for recognizing 2D languages and outlines significant results regarding their applications, including examples and theorems that illustrate the interaction between 1D and 2D formal languages.

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Unifying 1D and 2D Language Theory through Rational Automata

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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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