An Automata-Theoretical Characterization of Context-Free Trace Languages

1. An Automata-TheoreticalCharacterization ofContext-FreeTraceLanguages Benedek Nagy Friedrich Otto Department of Computer Science FachbereichElektrotechnik/Informatik Faculty of InformaticsUniversitatKassel University of Debrecen, Hungary Kassel, Germany SOFSEM 2011. Novy Smokovec, Slovakia

2. Outline • Introduction, concepts, definitions • Stateless R-automata withconstantwindowsize – withwindowsize 1 • CD-systems(of statelessdet. R-aut. withwindowsize1) withexternalPushDown • The newmodel – (an alternativeview) • Context-freetracelanguages • The title, i.e., the main result • Concludingremarks

3. Stateless Restarting (R-)automata • Restarting automata – linguisticmotivationPrinciple of analysisbyreduction • (,¢,$,k,), where  is a finite alphabet, ¢, $   markersleft, right border of the workspacek ≥ 1 is the size ofthe read/write window,  is the transition relationthereare 3 types of transitions • Deterministicmachine – at most 1 transition

4. Transitions thereare 3 types of transitions: • move-right steps (MVR), which shift the windowone step to the right, • combined rewrite/restart steps, which delete one or moresymbols from the content u of the window, thereby shortening the tape, andplace the window over the left end of the tape, and • accept steps (Accept), whichcause the automaton to halt and accept.

5. Efficiency of Restart-autom. is acceptedby a stateless deterministic R(2)-automatonforeach n. Thesemachinesaremonotone, therefore With k=9: non CF language

6. Det R-automata with k=1 • The alphabetcan be partitioned:

7. Acceptedlanguage • Withoutrewriting/restartingstep (tailcomputation) • Allowingrewriting/restartingsteps (cycles):

8. CD systems • CooperatingDistributedsystem • More than 1 device, workoneaftertheotherseveralmodesareknown • Forstatelessdeterministic R-automata weuse =1 mode (thesimplestcombination) • Weneed: initialcomponent(s) and definethesuccessor(s) foreachcomponent • In PDCD systems an externalpushdownhelpstochoosethenextcomponent

9. PushDown CD-Systems of StatelessDeterm. R(1)-Automata • with • I is a finiteset of indices • is a finite input alphabet • components statelessdet. R(1)-automata • Possiblesucessors • Initialindicesare:

10. PushDown CD-Systems of StatelessDeterm. R(1)-Automata • with • is a finitepusdownalphabet • Successorrelation :

11. Configurations of a PDCD-stl-dR(1) system • A configuration ofMis (i, ¢w$, α), where i ∈ I is the index of the activecomponent Mi,¢w$ (w ∈ Σ∗) is a restarting config. of Mi, andα ∈ ⊥· Γ∗ is the content of the PD storewith the first symbol of α at the bottom and the last symbol of α at the top. • For w ∈ Σ∗, an initial configuration of M on input w has the form (i0, cw$,⊥)for i0∈ I0, and • an accepting configuration has the form (i, Accept,⊥).

12. Work of PDCD-stl-dR(1) systems • The single-step computation relation ⇒ is defined by the following three rules, where i ∈ I, w ∈ Σ∗, α ∈ ⊥· Γ ∗,A ∈ Γ, arethe subsets of Σ: (MVR and delete)rewriting/restarting

13. Work of PDCD-stl-dR(1) systems • The single-step computation relation ⇒ is defined by the following three rules, where i ∈ I, w ∈ Σ∗, α ∈ ⊥· Γ ∗,A ∈ Γ, arethe subsets of Σ: (MVR and delete)rewrite/restartforemptystack

14. Work of PDCD-stl-dR(1) systems • The single-step computation relation ⇒ is defined by the following three rules, where i ∈ I, w ∈ Σ∗, α ∈ ⊥· Γ ∗,A ∈ Γ, arethe subsets of Σ: (MVR and delete)accept

15. Acceptedlanguage • By⇒∗ we denote the computation relation ofM, which is simply the reflexiveand transitive closure of the relation ⇒. The language L(M) accepted by M consists of all words for which M has an accepting computation, that is,

16. OC-CD-R(1) • A PD-CD-R(1)-systemis called a one-counterCD-system of statelessdeterministic R(1)-automata(OC-CD-R(1)-systemforshort), if |Γ| = 1, that is, if there is only a single pushdown symbol in additiontothebottom marker ⊥.

17. The newmodel – otherpoint of view • Letthetape is dividedto| | slices • At a ‘state’ (i.e. component) themachinecanseethefirstdesiredletter, ifonly ‘translucent’ lettersarebefore • Wehave a PDA (or a one-countermachine) thatusesuch input tape (withtranclucentletters)

18. Example L={ } 4 components: 1. delete an ‘a’ and extendthestack 1.,2.2. delete a ‘b’ (c translucent)  3.3. delete a ‘c’ (b transl.), removestack 2,0.0. accepttheempty: on $, (no translucent) Initial {0,1}

19. Example L={ } Initial {0,1}1. delete an ‘a’ and extendthestack 1.,2.

20. Example L={ } Initial 11. delete an ‘a’ and extendthestack 1.,2.

21. Example L={ } 1. delete an ‘a’ and extendthestack 1.

22. Example L={ } 1. delete an ‘a’ and extendthestack 1.

23. Example L={ } 1. delete an ‘a’ and extendthestack 1.

24. Example L={ } 1. delete an ‘a’ and extendthestack 1.

25. Example L={ } 1. delete an ‘a’ and extendthestack 2.

26. Example L={ } 2. delete a ‘b’ (c translucent)  3.

27. Example L={ } 2. delete a ‘b’ (c translucent)  3.

28. Example L={ } 2. delete a ‘b’ (c translucent)  3.

29. Example L={ } 3. delete a ‘c’ (b transl.), removestack 2,0.

30. Example L={ } 3. delete a ‘c’ (b transl.), removestack 2,0.

31. Example L={ } 3. delete a ‘c’ (b transl.), removestack 2.

32. Example L={ } 2. delete a ‘b’ (c translucent)  3.

33. Example L={ } 2. delete a ‘b’ (c translucent)  3.

34. Example L={ } 2. delete a ‘b’ (c translucent)  3.

35. Example L={ } 3. delete a ‘c’ (b transl.), removestack 2,0.

36. Example L={ } 3. delete a ‘c’ (b transl.), removestack 2.

37. Example L={ } 2. delete a ‘b’ (c translucent)  3.

38. Example L={ } 2. delete a ‘b’ (c translucent)  3.

39. Example L={ } 3. delete a ‘c’ (b transl.), removestack 0,2.

40. Example L={ } 3. delete a ‘c’ (b transl.), removestack 0,2.

41. Example L={ } 3. delete a ‘c’ (b transl.), removestack 0

42. Example L={ } 0. accepttheempty: on $, (no translucent)aaaccbcbb has been ACCEPTED. L is an OC-CD-R(1) language.

43. PD-CD and OC-CD languages • Withonecounterthemachine is more powerfulthanwithoutanyadditionaldatastructure (example L) • Bysimulation of a PDA (OC), everycontext-free/one-counterlanguage is inourfamilies (no need of translucency)

44. PD-CD and OC-CD languages • The proofgoesbysimulation of PDA basedonquadraticGreibach normal formgrammar (eachrule has at most 2 nonterminalsonitsright-handside)

45. Normalfrom (NF) machine • 1. canacceptonlyemptystring. • 2. Foreveryothercomponents candeleteexactly 1 letter, no acceptanceonanyletters (noron $) There is an equivalentmachineforeachin NFOnecanconstructequivalent NF machine..

46. Semi-linearity contains a letterequivalentcontext-freesublanguage. (alsofor OC-CD and OC)Idea: wordsacceptedin a waythatthefirstletter is being deletedineverystepform a languagethat is acceptedby a PDA (OC). is notinourclass.

47. Tracelanguages A dependency relation D is a binary relation on an alphabet that is reflexiveand symmetric. Then is the corresponding independencerelation. Itis irreflexiveand symmetric. It induces a bin.relation that is definedas the smallest congruence relationcontaining the For , the congruenceclass of w mod D is denoted by [w]D. These congruence classes are called traces,and the factor monoidis a trace monoid.

48. Alph(w) is theset of lettersoccurin w. • Extensiontowords: if and onlyif

49. Rationaltracemonoids • A subset S of a trace monoid M(D) isrational if it can be obtained from singleton sets by a finitenumber ofunions,products, and star operations. • It follows that S M(D) is rationalif and only if there exists a regular language L over such that. • By RAT(M(D)) we denote the set of rational subsets of M(D).

50. OC tracelanguages • A language L ⊆ Σ∗ isa one-counter trace language, if there exist - a dependency relation D on Σand - a one-counter language R ⊆ Σ∗ such that • denotestheset of one-counter trace languages obtained from (Σ,D).