Computing languages by (bounded) local sets

1. Computing languages by (bounded) local sets Dora Giammarresi Università di Roma “Tor Vergata” Italy

2. 1. Put in a unique framework some know (disjoint) results and get a: Characterization of Chomsky’s hierarchy by local sets + alphabetic projections 2. Insert new families into Chomsky’s hierarchy by introducing new types of local sets: Bounded-Grids Context-Sensitive languages Summary of the talk

3. w with border # 0 1 0 0 1 # finite set of strings of length 2 over Γ # 0 0 1 0 0 1 Θ = … 0 1 # # allowed substrings Local (string) languages… string w over Γ= {0, 1} w= 0 1 0 0 1 Definition: String language L is local if all substrings of length 2 are in a finite set Θ. (L=L(Θ) )

4. Local set + projection = Finite automaton … to define Regular languages , Γ alphabets π: Γ alphabetic projection Theorem :L is regular   local set of strings S such that L=π(S). Proof: local alphabet Γ = edges of automaton set Θ = pairs of consecutive edges π gives labels of the edges

5. 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 0 1 1 finite set of 2x2 pictures over Γ # 1 0 0 0 1 1 Θ = 0 0 0 # … # # # # allowed subpictures picture p over Γ= {0, 1} p with border Local Picture Languages [GR’94] # # # # # # # # # p= # # # # # # # # # # # Definition: Picture language L is local if all subpictures of size 2x2 are in a finite set Θ . (L=L(Θ) )

6. # # # # # # # 0 # 2 1 2 0 # 2 0 1 0 2 # # 0 1 # 0 2 1 # # # # # # # # …to define Context-sensitive languages p= fr(p) = frontier of p Theorem [F69 – LS97]: L is context-sensitive   local set of pictures S such that L=π(fr(S)). Proof: () given an accepting run of an LBA, take all instantaneous configurations and write them in order, one above the others. This gives a local picture. () given a local set of 2x2 squares, define corresponding context-sensitive grammar rules such that derivations correspond to the local pictures.

7. finite set of 3-vertices trees over Γ # Θ = 0 0 0 0 # # # 0 # # # 0 0 1 0 0 1 1 1 # 1 1 1 # $ 0 1 # 0 1 # 1 # # allowed sub-trees … Γ= {0, 1, $} Local sets of binary trees… tree t with border Definition: Tree set L is local if all 3-vertices sub-trees belong to a finite set Θ . (L=L(Θ) )

8. t= # # # # # # # # # # # # # # $ …to define Context-free languages fr(t) = frontier of t Theorem [MW67]: L is context-free   local set of binary trees S such that L=π(fr(S)). Proof: Notice that a derivation tree of a context free grammar in Chomsky's Normal Form is actually a local binary tree (possibly after some minor modifications).

9. elementary line: ● Local sets of lines ●Local sets of binary trees elementary binary tree: y y t y y t y x z x x z x z x ●Local sets of grids elementary grid: Look at them all together…

10. 0 0 0 0 # # # # # # # # # # # # # # # # # # # # # # # # # # # # 0 0 0 0 0 0 0 0 0 0 0 0 Γ= {0,1} ● Sets of line graphs look at them all together… frontier = all non-border vertices ● Sets ofbinary trees frontier = vertices adjacent to the leaves ● Sets ofgrid graphs frontier = lowest non-border row

11. Chomsky’s hierarchy by local sets Proposition: Let L be a (string) language. Then: 1. L is regular  L is projection of the frontier of a local set of lines; 2. L is context-free  L is projection of the frontier of a local set of binary trees; 3. L is context-sensitive  L is projection of the frontier of a local set of grids;

12. # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # Local machines! Local sets as computations… Context-sensitive Context-free Regular

13. # # # # # # # # # # # # # # # # # # # • Lines left-to-right # # # # # # • Trees leaves-to-root # # $ # # # # # # # # # # # # # # # # # # # # # # # # # • Grids bottom-to-top Remark on local computations Size of local graph is measure ofTIMEof computation NOTmeasure ofSPACE! No need to keep the all graph space:

14. # # # # # # # # # # # # # # # # # # # # # # # # # $ # # # # # # # # # # # # # # # # # # # # # # # # # # # Context-sensitive New families into Chomsky’s hierarchy? Context-free Regular Define a new type of local sets …. ….. and get a new family of string languages!

15. # # # # # # # # # # # # # # # # # # # # # # # # # $ # # # # # # # # # # # # # # # # # # # # # # # # # # # BIG GAP! Context-sensitive What are the differences? Context-free Regular degree≤ 2≤ 3≤ 4 not bounded by n n+2 O(1) ≤ 4nO(n) size

16. size of local grid for w, ≤ (n+2)2 2 # # # # # # # # # # # # # # # # # # # # # # # # 1 1 0 0 0 0 0 0 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 1 0 1 0 0 Bounded Grids Local Sets Computations Definition: grids with 2-sides frontiers length of w n Exact size depends on the position where we turn the string!

17. Bounded Grids Local Sets No more space for istantaneous configurations of a run of a LBA automaton! Need to exploit geometric local properties of patterns defined in the pictures…. Use local picture languages theory techniques [GR’94]

18. # # # # # # # # # # # # # # # # # # # # # # # # # # An example = {a,b} L= anbn | n≥0 } Γ= {0,1,4} 1 1 1 1 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 π:Γ → projection 4→ a 0 → b 4 4 4 4 4

19. Another example (use same idea!) = {a,b,c} L= anbncn | n≥0 } Γ= {0,1,4,2,3,5} 1 1 1 1 0 3 3 3 3 2 1 1 1 0 0 3 3 3 2 2 1 1 0 0 0 3 3 2 2 2 1 0 0 0 0 3 2 2 2 2 0 0 0 0 0 2 2 2 2 2 π:Γ → projection 4→ a 5 → b 2→ c 4 4 4 4 4 5 5 5 5 5

20. b1 b1 b1 b1 b1 b0 b0 b0 b0 b0 Another example L= wwR | w*} palindromes = {a,b} a1 a1 a1 a1 a1 a1 a0 Γ= {a0, b0, a1, b1} a0 a1 a1 a1 a1 a1 a0 a0 a1 a1 π:Γ → projection a0, a1→ a b0, b1→ b b1 b1 a0 a0 a0 a0 a1 a0 a0 b0 a0 a0 b0 a0 a0

21. Bounded Grids Computations Theorem: If L is a projection of the 2-sides frontier of a local picture language, then L is context-sensitive. Bounded-grids context sensitive (Bgrid-CS).

22. 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 # # # # # # # # # # # # # # # Proof: Let w=a1a2....an Idea: define a LBA that “behaves” as a local machine: all the writing operations effectively build the picture! • non deterministically rewrite w=x1x2....xn ,π(xi)=ai • put w as frontier of a picture (non deterministically choose i and put xi in the BR-corner of a picture) • check that all bottom and right border subpictures are in Θ • finish to build the picture by elements of Θ

23. L=anb2nc(n+3) | n≥0 } • L=an | n≥0 }; L=a2 | n≥0 } • L=ap | p prime}; L=af | f not prime} • L=ww | w* } • L=w|w| | |w|>1} More examples n 2

24. Closure properties Theorem: Bgrid-CS languages are closed under concatenation and Kleene's star. Theorem: Bgrid-CS languages are closed under Boolean union and intersection.

25. Open problem ? ● Bgrid-CS languages = CS languages Remark 1:[ R.Book71] In 1971 R. Book defined infinite hierarchy of subfamilies of context-sensitive languages corresponding to different time bounding functions leaving open question whether hierarchy collaps. Remark 2:[ Gladkij] there are CS-languages with no linear bounded derivations

26. Open problem …recall that CS languages are closed under complement. ● Are Bgrid-CS languages closed under complement? What about deterministic versions?

27. Definition: Set of 2x2 grids Θ is deterministic when, y x3  x1,x2,x3  Γ{#} there is at most one  Θ x1 x2 Deterministic Local B-grid (machines) Open question: are deterministic B-grid CS languages equivalent to non-deterministic ones?

28. ● Bgrid-CS languages are “deterministic” ● Bgrid-CS languages = CS languages DSPACE(n)=NSPACE(n) Remark

29. Open problem (the last one!) …turning back to characterization for the Chomsky's hierarchy ……. ● Can we define a “local set” to characterize recursive languages?

30. The end

31. p p q q # p0 p1 q1 q0 # q1 q1 p0 p0 p0 p0 p1 # p0 p1 q0 p1 q0 q0 q1 # q1 p1 p0 , q0 p1 , q1 0 1 π: 0 = {0, 1} 1 Proof (by example) 0 Γ = {p0, p1, q0, q1} q p 1

32. Look at them more generally… #1,…, #k Γ border symbol Γ alphabet • embedded labeled graph over Γ {#1,…, #k } • border vertices = vertices carryng #1,…, #k • frontier = (labels of) a path of non-border vertices (string over Γ) • elementary graph = “small” graph shape • local setof graphs = (labels of) a path of non-border vertices

33. Given a typology of graphs, fix “shape” of elementary graph Local sets Definition: A set of graphs S over Γ {#} is local if there exists a finite set of elementary graphs Θ over Γ {#} such that, for all s S, every subgraph of elementary size belongs to Θ. We write S= L (Θ).