Descriptive Complexity

Transcription

1 Descriptive Complexity and Nested Words 1/46 Descriptive Complexity and Nested Words Neil Immerman immerman I decided to start with a survey of Descriptive Complexity, and end with a survey of Nested Words. Please ask questions during my talk because two-way communication is more fun and allows much more understanding!

2 Descriptive Complexity Descriptive Complexity and Nested Words 1/46

3 Descriptive Complexity and Nested Words 1/46 Descriptive Complexity Input q 1 q 2 q n Computation Output a 1 a 2 a i a n k

4 Descriptive Complexity and Nested Words 1/46 Descriptive Complexity Input q 1 q 2 q n Computation Output a 1 a 2 a i a n k

5 Descriptive Complexity and Nested Words 1/46 Descriptive Complexity Input q 1 q 2 q n Computation Output a

6 Descriptive Complexity and Nested Words 1/46 Descriptive Complexity Input q 1 q 2 q n Computation Output a How much computation is needed to check if input has property S?

7 Descriptive Complexity and Nested Words 1/46 Descriptive Complexity Input q 1 q 2 q n Computation Output a How much computation is needed to check if input has property S? How rich a language do we need to express property S?

8 Descriptive Complexity and Nested Words 2/46 Encode Input as Finite Logical Structure Graph G = ({v 1,...,v n },E,s,t) s t Binary A w = ({p 1,...,p 8 },S) String S = {p 2,p 5,p 7,p 8 } w = Vocabularies: τ g = (E 2,s,t), τ s = (S 1 )

9 Descriptive Complexity and Nested Words 3/46 First-Order Logic input symbols: from τ variables: x,y,z,... boolean connectives:,, quantifiers:, numeric symbols: =,, +,, 0, max α x y(e(x,y)) L(τ g ) β x y(x y S(x)) L(τ s ) β S(0) L(τ s )

10 Descriptive Complexity and Nested Words 4/46 Second-Order Logic = first-order logic + quantifiable relational variables Φ 3 color R 1 Y 1 B 1 xy ((R(x) Y (x) B(x)) (E(x,y) ( (R(x) R(y)) (Y (x) Y (y)) (B(x) B(y))))) s d t b a c f g e

11 Descriptive Complexity and Nested Words 4/46 Second-Order Logic Fagin s Theorem: NP = SO Φ 3 color R 1 Y 1 B 1 xy ((R(x) Y (x) B(x)) (E(x,y) ( (R(x) R(y)) (Y (x) Y (y)) (B(x) B(y))))) s d t b a c f g e

12 Descriptive Complexity and Nested Words 5/46 Addition is First-Order Q + : STRUC[τ AB ] STRUC[τ s ] A a 1 a 2... a n 1 a n B + b 1 b 2... b n 1 b n S s 1 s 2... s n 1 s n ( C(i) ( j > i) A(j) B(j) ) ( k.j > k > i)(a(k) B(k)) Q + (i) A(i) B(i) C(i)

13 Descriptive Complexity and Nested Words 6/46 Parallel Machines CRAM[t(n)] = CRCW-PRAM-TIME[t(n)]-HARD[n O(1) ] P r P 1 Global P 2 Memory P 3 P 4 synchronous, concurrent read and write n O(1) processors and memory P 5

14 Descriptive Complexity and Nested Words 6/46 Parallel Machines CRAM[t(n)] = CRCW-PRAM-TIME[t(n)]-HARD[n O(1) ] P r P 1 Global P 2 Memory P 3 P 4 Quantifiers are parallel: If array A[x] : x = 1,...,r in memory, x(a(x)) write(1); proc p i : if (A[i] = 0) then P 5

15 Descriptive Complexity and Nested Words 6/46 Parallel Machines CRAM[t(n)] = CRCW-PRAM-TIME[t(n)]-HARD[n O(1) ] P r P Global Memory P 2 P P 4 Quantifiers are parallel: If array A[x] : x = 1,...,r in memory, x(a(x)) write(1); proc p i : if (A[i] = 0) then P 5

16 Descriptive Complexity and Nested Words 6/46 Parallel Machines CRAM[t(n)] = CRCW-PRAM-TIME[t(n)]-HARD[n O(1) ] P r P Global Memory P 2 P P 4 Quantifiers are parallel: If array A[x] : x = 1,...,r in memory, x(a(x)) write(1); proc p i : if (A[i] = 0) then P 5

17 Descriptive Complexity and Nested Words 7/46 Inductive Definitions E (x,y) x = y E(x,y) z(e (x,z) E (z,y)) s t

18 Descriptive Complexity and Nested Words 7/46 Inductive Definitions E (x,y) x = y E(x,y) z(e (x,z) E (z,y)) ϕ tc (R,x,y) x = y E(x,y) z(r(x,z) R(z,y)) A REACH A = (LFPϕ tc )(s,t) s t

19 Descriptive Complexity and Nested Words 7/46 Inductive Definitions E (x,y) x = y E(x,y) z(e (x,z) E (z,y)) ϕ tc (R,x,y) x = y E(x,y) z(r(x,z) R(z,y)) A REACH A = (LFPϕ tc )(s,t) Thus, REACH IND[log n]. s t

20 Descriptive Complexity and Nested Words 7/46 Inductive Definitions E (x,y) x = y E(x,y) z(e (x,z) E (z,y)) ϕ tc (R,x,y) x = y E(x,y) z(r(x,z) R(z,y)) A REACH A = (LFPϕ tc )(s,t) Thus, REACH IND[log n]. s t Next, we ll show that REACH FO[log n].

21 ϕ tc (R,x,y) x = y E(x,y) ( z)(r(x,z) R(z,y)) Descriptive Complexity and Nested Words 8/46 1. Dummy universal quantification for base case: ϕ tc (R,x,y) ( z.m 1 )( z)(r(x,z) R(z,y)) M 1 (x = y E(x,y))

22 ϕ tc (R,x,y) x = y E(x,y) ( z)(r(x,z) R(z,y)) Descriptive Complexity and Nested Words 8/46 1. Dummy universal quantification for base case: ϕ tc (R,x,y) ( z.m 1 )( z)(r(x,z) R(z,y)) M 1 (x = y E(x,y)) 2. Using, replace two occurrences of R with one: ϕ tc (R,x,y) ( z.m 1 )( z)( uv.m 2 )R(u,v) M 2 (u = x v = z) (u = z v = y)

23 ϕ tc (R,x,y) x = y E(x,y) ( z)(r(x,z) R(z,y)) Descriptive Complexity and Nested Words 8/46 1. Dummy universal quantification for base case: ϕ tc (R,x,y) ( z.m 1 )( z)(r(x,z) R(z,y)) M 1 (x = y E(x,y)) 2. Using, replace two occurrences of R with one: ϕ tc (R,x,y) ( z.m 1 )( z)( uv.m 2 )R(u,v) M 2 (u = x v = z) (u = z v = y) 3. Requantify x and y. M 3 (x = u y = v) ϕ tc (R,x,y) ( z.m 1 )( z)( uv.m 2 )( xy.m 3 )R(x,y)

24 Descriptive Complexity and Nested Words 9/46 Define the quantifier block: QB tc ( z.m 1 )( z)( uv.m 2 )( xy.m 3 ) Operator ϕ tc is equivalent to QB tc : ϕ tc (R,x,y) [QB tc ]R(x,y) Thus, for any r, ϕ r tc ( ) [QB tc] r (false) Thus, for any structure A STRUC[τ g ], A REACH A = (LFPϕ tc )(s,t) A = ([QB tc ] 1+log A false)(s,t)

25 Descriptive Complexity and Nested Words 10/46 CRAM[t(n)] = concurrent parallel random access machine; polynomial hardware, parallel time O(t(n)) IND[t(n)] = first-order, depth t(n) inductive definitions FO[t(n)] = t(n) repetitions of a block of restricted quantifiers: QB = [(Q 1 x 1.M 1 ) (Q k x k.m k )]; M i quantifier-free ϕ n = [QB][QB] [QB] M }{{} 0 t(n)

26 Descriptive Complexity and Nested Words 11/46 parallel time = inductive depth = quantifier-block iteration Thm: For all constructible, polynomially bounded t(n), CRAM[t(n)] = IND[t(n)] = FO[t(n)] Thm: For all t(n), even beyond polynomial, CRAM[t(n)] = FO[t(n)]

27 Descriptive Complexity and Nested Words 12/46 Thm: For v = 1, 2,..., DSPACE[n v ] = VAR[v + 1] Number of variables corresponds to amount of hardware. Since variables range over a universe of size n, a constant number of variables can specify a polynomial number of gates: A bounded number of variables corresponds to polynomially much hardware.

28 co r.e. complete co r.e. Arithmetic Hierarchy r.e. r.e. complete Recursive Primitive Recursive EXPTIME co NP complete PSPACE Polynomial Time Hierarchy co NP NP NP co NP NP complete P "truly feasible" NC log(cfl) NSPACE[log n] DSPACE[log n] Regular FO Descriptive Complexity and Nested Words 13/46 AC 0

29 co r.e. complete Arithmetic Hierarchy co r.e. r.e. complete Recursive r.e. Primitive Recursive co NP complete EXPTIME PSPACE Polynomial Time Hierarchy co NP NP NP co NP NP complete P "truly feasible" NC FO NC log(cfl) NSPACE[log n] DSPACE[log n] Regular Logarithmic Time Hierarchy 2 SAC Descriptive Complexity and Nested Words 14/46 1 NC AC 0 1 ThC 0

30 co r.e. complete Arithmetic Hierarchy co r.e. r.e. complete Recursive r.e. Primitive Recursive EXPTIME PSPACE co NP complete Polynomial Time Hierarchy co NP NP NP co NP SO E NP complete P FO(LFP) "truly feasible" NC FO FO(TC) NC log(cfl) NSPACE[log n] DSPACE[log n] Regular Logarithmic Time Hierarchy 2 SAC Descriptive Complexity and Nested Words 15/46 1 NC AC 0 1 ThC 0

31 co r.e. complete Arithmetic Hierarchy co r.e. r.e. complete Recursive r.e. Primitive Recursive EXPTIME O(1) n FO[2 ] PSPACE FO(PFP) co NP complete Polynomial Time Hierarchy co NP NP NP co NP SO E NP complete O(1) FO[n ] P FO(LFP) "truly feasible" O(1) FO[(log n) ] NC FO(TC) FO(DTC) FO(M) FO NC log(cfl) NSPACE[log n] DSPACE[log n] Regular Logarithmic Time Hierarchy 2 SAC Descriptive Complexity and Nested Words 16/46 1 NC AC 0 1 ThC 0

32 co r.e. complete Arithmetic Hierarchy co r.e. r.e. complete Recursive r.e. Primitive Recursive EXPTIME O(1) n FO[2 ] PSPACE FO(PFP) co NP complete SO A co NP Polynomial Time Hierarchy SO NP co NP NP SO E NP complete O(1) FO[n ] "truly feasible" P FO(LFP) SO Horn O(1) FO[(log n) ] NC FO(TC) FO(DTC) FO(M) FO NC log(cfl) NSPACE[log n] DSPACE[log n] Regular Logarithmic Time Hierarchy 2 SAC Descriptive Complexity and Nested Words 17/46 1 SO Krom NC AC 0 1 ThC 0

33 co r.e. complete Arithmetic Hierarchy co r.e. r.e. complete Recursive r.e. Primitive Recursive O(1) n FO[2 ] co NP complete SO O(1) n SO[2 ] A O(1) SO[n ] co NP EXPTIME PSPACE SO FO(PFP) Polynomial Time Hierarchy NP co NP SO(LFP) NP SO SO(TC) E NP complete O(1) FO[n ] "truly feasible" P FO(LFP) SO Horn O(1) FO[(log n) ] NC FO(TC) FO(DTC) FO(M) FO NC log(cfl) NSPACE[log n] DSPACE[log n] Regular Logarithmic Time Hierarchy 2 SAC Descriptive Complexity and Nested Words 18/46 1 SO Krom NC AC 0 1 ThC 0

34 co r.e. complete FO A (N) co r.e. Arithmetic Hierarchy FO(N) Recursive r.e. FO E r.e. complete (N) Primitive Recursive n O(1) O(1) FO[2 ] SO[n ] FO(PFP) SO(TC) co NP complete SO n O(1) SO[2 ] A co NP Polynomial Time Hierarchy NP EXPTIME PSPACE SO co NP SO(LFP) NP SO E NP complete O(1) FO[n ] "truly feasible" P FO(LFP) SO Horn O(1) FO[(log n) ] NC FO(TC) NC 2 log(cfl) NSPACE[log n] FO(DTC) DSPACE[log n] Regular FO(M) FO Logarithmic Time Hierarchy SAC Descriptive Complexity and Nested Words 19/46 1 SO Krom NC AC 1 ThC 0 0

35 Descriptive Complexity and Nested Words 20/46 We Need an Ordering on the Universe G = (V,E); V = {0, 1,...,n 1}; 0 < 1 < < n 1 An unordered graph makes sense mathematically, but you can t store such an object in a computer as far as I know. If you remove the ordering then the first-order descriptive characterizations fail: EVEN requires Ω(n) variables without ordering. Thus, EVEN FO(wo )[2 no(1) ]; (FO[2 no(1) ] = PSPACE) Fagin (SO = NP) didn t run into this problem because in SO you can guess an ordering (unless we are dealing with SO (monadic)).

36 Descriptive Complexity and Nested Words 21/46 Ehrenfeucht-Fraïssé Games Delilah: hide any differences between the two structures Two-person combinatorial game for characterizing what is expressible in a given quantifier depth. Fundamental Thm: D has a winning strategy on the m-move, k-pebble game on A, B iff A and B agree on all formulas using k variables and quantifier depth m. A k m B A k m B Samson: show a difference

37 Descriptive Complexity and Nested Words 22/46 dist(x,y) 2 k FO 3 k FO k 1 move 0 x y k 2 2 k 1 2 k k 2 2 k 2 k + 1 x y

38 Descriptive Complexity and Nested Words 22/46 dist(x,y) 2 k FO 3 k FO k 1 move 1 z(dist(x,z) 2 k 1 dist(z,y) 2 k 1 ) x z y k 2 2 k 1 2 k k 2 2 k 2 k + 1 x y

39 Descriptive Complexity and Nested Words 22/46 dist(x,y) 2 k FO 3 k FO k 1 move 1 z(dist(x,z) 2 k 1 dist(z,y) 2 k 1 ) x z y k 2 2 k 1 2 k k 2 2 k 2 k + 1 x z y

40 Descriptive Complexity and Nested Words 22/46 dist(x,y) 2 k FO 3 k FO k 1 move 2 x(dist(z,x) 2 k 2 dist(x,y) 2 k 2 ) z x y k 2 2 k 1 2 k k 2 2 k 2 k + 1 z y

41 Descriptive Complexity and Nested Words 22/46 dist(x,y) 2 k FO 3 k FO k 1 move 2 x(dist(z,x) 2 k 2 dist(x,y) 2 k 2 ) z x y k 2 2 k 1 2 k k 2 2 k 2 k + 1 z x y

42 Descriptive Complexity and Nested Words 22/46 dist(x,y) 2 k FO 3 k FO k 1 move k-1: Delilah wins dist(x,y) 2 1 x y k 2 2 k 1 2 k 2 k + 1 x y

43 Descriptive Complexity and Nested Words 22/46 dist(x,y) 2 k FO 3 k FO k 1 move k: Samson wins E(x,z) E(z,y) x z y k 2 2 k 1 2 k 2 k + 1 x y

44 Descriptive Complexity and Nested Words 23/46 These games have been used to prove many beautiful lower bounds without the ordering, and thus separate most descriptive classes without ordering. Among others: McColm and Grädel: DTC, TC, LFP Grohe: arity hierarchies for TC and LFP Etessami: lower bounds with counting and one-way local orderings Libkin: lower bounds with counting via locality arguments Grohe and Schwentick: locality of order-independent queries

45 Descriptive Complexity and Nested Words 24/46 One Historical Thread Fagin: REACH SO (monadic) and thus since REACH SO (monadic), SO (monadic) is not closed under complementation. de Rougemont: remains true with successor Schwentick: remains true with ordering! Doesn t give us a complexity lower bound, cf., Lynch: NTIME[n k ] SO (+)(arity k)

46 Descriptive Complexity and Nested Words 25/46 Ehrenfeucht-Fraïssé games: lower bounds on depth. With ordering, depth 2 + log n suffices to express any graph property for graphs on n vertices. Let G = (V,E) # i (x) exists a path of length i from 0 to x depth(log n) ϕ G i,j E i,j E x,y(# i (x) # j (y) E(x,y)) x,y(# i (x) # j (y) E(x,y)) For S an arbitrary set of graphs on n vertices: ϕ S G S ϕ G ; depth(ϕ S ) = 2 + log n

47 Descriptive Complexity and Nested Words 26/46 Newish Size Lower Bound Game Adler & I defined these games and proved a pair of optimal succinctness result for temporal logics. Karchmer and Wigderson used similar communication complexity games to prove lower bound on monotone circuits for REACH. Idea: unlike Ehrenfeucht-Fraïssé games, we play on a pair of sets of structures, A,B. We determine the size of the smallest sentence true of all of A and none of B. More recent lower bounds by Grohe and Schweikardt

48 Descriptive Complexity and Nested Words 26/46 Newish Size Lower Bound Game Adler & I defined these games and proved a pair of optimal succinctness result for temporal logics. Karchmer and Wigderson used similar communication complexity games to prove lower bound on monotone circuits for REACH. Idea: unlike Ehrenfeucht-Fraïssé games, we play on a pair of sets of structures, A,B. We determine the size of the smallest sentence true of all of A and none of B. More recent lower bounds by Grohe and Schweikardt Not surprisingly, these games are much harder to play than Ehrenfeucht-Fraïssé games...

49 Descriptive Complexity and Nested Words 27/46 Size versus Number of Variables on Line Graphs dist(x,y) 1 x = y E(x,y) dist(x,y) 2k z(dist(x,z) k dist(z,y) k) dist(x,y) n FO 3 log n ; SIZE(n)

50 Descriptive Complexity and Nested Words 27/46 Size versus Number of Variables on Line Graphs dist(x,y) 1 x = y E(x,y) dist(x,y) 2k z(dist(x,z) k dist(z,y) k) dist(x,y) n FO 3 log n ; SIZE(n) dist u (x,y) 1 x = y E(x,y) E(y,x) dist u (x,y) 2k z w((w = x w = y) dist u (z,w) k) dist u (x,y) n FO 4 log n; SIZE(O(log n))

51 Descriptive Complexity and Nested Words 27/46 Size versus Number of Variables on Line Graphs dist(x,y) 1 x = y E(x,y) dist(x,y) 2k z(dist(x,z) k dist(z,y) k) dist(x,y) n FO 3 log n ; SIZE(n) dist u (x,y) 1 x = y E(x,y) E(y,x) dist u (x,y) 2k z w((w = x w = y) dist u (z,w) k) dist u (x,y) n FO 4 log n; SIZE(O(log n)) Grohe-Schweikardt: FO 2 and FO 3 are polynomially SIZE-related Exponential Size gap between FO 3 and FO 4 Gap between FO k and FO k+1 is open for k > 3

52 Descriptive Complexity and Nested Words 28/46 FO 2 on Words: [Philipp Weis & I] FO on words reduces to FO 3 on words FO 2 on words was well studied, except alternation hierarchy was open

53 Descriptive Complexity and Nested Words 28/46 FO 2 on Words: [Philipp Weis & I] FO on words reduces to FO 3 on words FO 2 on words was well studied, except alternation hierarchy was open b a c d b b a c d Rankers

54 Descriptive Complexity and Nested Words 28/46 FO 2 on Words: [Philipp Weis & I] FO on words reduces to FO 3 on words FO 2 on words was well studied, except alternation hierarchy was open Rankers b a c d b b a c d a 1 ranker

55 Descriptive Complexity and Nested Words 28/46 FO 2 on Words: [Philipp Weis & I] FO on words reduces to FO 3 on words FO 2 on words was well studied, except alternation hierarchy was open Rankers b a c d b b a c d a b 2 ranker

56 Descriptive Complexity and Nested Words 28/46 FO 2 on Words: [Philipp Weis & I] FO on words reduces to FO 3 on words FO 2 on words was well studied, except alternation hierarchy was open Rankers b a c d b b a c d a b c 3 ranker

57 Descriptive Complexity and Nested Words 28/46 FO 2 on Words: [Philipp Weis & I] FO on words reduces to FO 3 on words FO 2 on words was well studied, except alternation hierarchy was open Rankers b a c d b b a c d a b c d 4 ranker

58 Descriptive Complexity and Nested Words 28/46 FO 2 on Words: [Philipp Weis & I] FO on words reduces to FO 3 on words FO 2 on words was well studied, except alternation hierarchy was open Rankers b a c d b b a c d a b c 3 ranker Thm: FO 2 m which m-rankers exist, and, relative order with smaller rankers.

59 Descriptive Complexity and Nested Words 28/46 FO 2 on Words: [Philipp Weis & I] FO on words reduces to FO 3 on words FO 2 on words was well studied, except alternation hierarchy was open Rankers b a c d b b a c d a b c 2, 3 ranker Thm: FO 2 m which m-rankers exist, and, relative order with smaller rankers. Thm: FO 2 k,m which k,m-rankers exist, and, relative order with smaller rankers.

60 Descriptive Complexity and Nested Words 28/46 FO 2 on Words: [Philipp Weis & I] FO on words reduces to FO 3 on words FO 2 on words was well studied, except alternation hierarchy was open Rankers b a c d b b a c d a b c 2, 3 ranker Thm: FO 2 m which m-rankers exist, and, relative order with smaller rankers. Thm: FO 2 k,m which k,m-rankers exist, and, relative order with smaller rankers. Thm: FO 2 has a strict alternation hierarchy.

61 Descriptive Complexity and Nested Words 29/46 Ultimately, want to understand the VAR-SIZE trade-off in many settings. Currently, Philipp Weis and I would be happy to settle what Grohe and Schweikardt left open, then to move on to words, and other simple graphs, 2 and 3 variables...

62 Descriptive Complexity and Nested Words 29/46 Ultimately, want to understand the VAR-SIZE trade-off in many settings. Currently, Philipp Weis and I would be happy to settle what Grohe and Schweikardt left open, then to move on to words, and other simple graphs, 2 and 3 variables... Would one day like to understand the following: PSPACE = FO[2 no(1) ] = SO[n O(1) ] NC 1 FO[log n/ log log n] NC 1 L NL sac 1 AC 1 = FO[log n]

63 co r.e. complete FO A (N) co r.e. Arithmetic Hierarchy FO(N) Recursive r.e. FO E r.e. complete (N) Primitive Recursive n O(1) O(1) FO[2 ] SO[n ] FO(PFP) SO(TC) co NP complete SO n O(1) SO[2 ] A co NP Polynomial Time Hierarchy NP EXPTIME PSPACE SO co NP SO(LFP) NP SO E NP complete O(1) FO[n ] "truly feasible" P FO(LFP) SO Horn O(1) FO[(log n) ] NC FO(TC) NC 2 log(cfl) NSPACE[log n] FO(DTC) DSPACE[log n] Regular FO(M) FO Logarithmic Time Hierarchy SAC Descriptive Complexity and Nested Words 30/46 1 SO Krom NC AC 1 ThC 0 0

64 Descriptive Complexity and Nested Words 31/46 Nested Words & Descriptive Complexity Neil Immerman immerman I ll survey work by Alur, Madhusudan, Kumar, Viswanathan, Arenas, Barceló, Etessami, and Libkin [LICS 2007].

65 Descriptive Complexity and Nested Words 32/46 Nested Words

66 Descriptive Complexity and Nested Words 32/46 Nested Words words on some alphabet, Σ well-nested edges Three kinds of nodes: call, return, and internal

67 Descriptive Complexity and Nested Words 32/46 Nested Words words on some alphabet, Σ well-nested edges Three kinds of nodes: call, return, and internal Applications: calls and returns: model checking recursive programs parse trees: linguistics XML

68 Descriptive Complexity and Nested Words 32/46 Nested Words words on some alphabet, Σ well-nested edges Three kinds of nodes: call, return, and internal Applications: calls and returns: model checking recursive programs parse trees: linguistics XML CFL s for the price of regular languages

69 Descriptive Complexity and Nested Words 33/46 Nested Word Automata: NWA An NWA, (Q, Σ,δ,s,F), is just like a finite automaton except, at return nodes, the new state depends on both previous states q q q q q q q q q q q q For example, q 4 = δ r (q 1,q 3, 0)

70 Descriptive Complexity and Nested Words 34/46 Example of a Regular Nested Language The balanced parentheses languages of nested words, in which the nesting edges go from each open parenthesis to its mate, are regular. [ ( a ) ( a ) ] [ ] a

71 Descriptive Complexity and Nested Words 35/46 Regular Nested Languages Closed under,,, concatenation, complementation. For every n-state nondeterministic NWA, there is an equivalent deterministic NWA with at most 2 n2 states. EMPTY-NWA, minimal deterministic NWA, union intersection, language containment for deterministic NWA are all polynomial time. Language containment given nondeterministic NWA s is exptime complete. Compare this with CFL s where containment is undecidable.

72 (u,i) v = insert v after letter i in u. Descriptive Complexity and Nested Words 36/46

73 Descriptive Complexity and Nested Words 36/46 (u,i) v = insert v after letter i in u. h e r e g o e s n e w

74 Descriptive Complexity and Nested Words 36/46 (u,i) v = insert v after letter i in u. h e r e g o e s n e w = h e r e n e w g o e s

75 Descriptive Complexity and Nested Words 37/46 Myhill Nerode Theorem for Nested Words x L y iff u,i ( (u,i) x L (u,i) y L )

76 Descriptive Complexity and Nested Words 37/46 Myhill Nerode Theorem for Nested Words x L y iff u,i ( (u,i) x L (u,i) y L ) Thm: [Alur, Kumar, Madhusudan, and Viswanathan] The nested word language, L, is regular iff L has finitely many equivalence classes.

77 Descriptive Complexity and Nested Words 38/46 Notation µ(i, j): i is a call node, and j is its matching return r(i) = j c(j) = i a i m j z

78 Descriptive Complexity and Nested Words 38/46 Notation µ(i, j): i is a call node, and j is its matching return r(i) = j c(j) = i a i m j z C(x) is the innermost call that x is within, e.g., C(i) = C(j) = a; C(m) = i

79 Descriptive Complexity and Nested Words 38/46 Notation µ(i, j): i is a call node, and j is its matching return r(i) = j c(j) = i a i m j z C(x) is the innermost call that x is within, e.g., C(i) = C(j) = a; C(m) = i R(x) is the innermost return that x is within, e.g., R(i) = R(j) = z; R(m) = j

80 Descriptive Complexity and Nested Words 39/46 Monadic Second-Order Logic, (MSO µ ) ϕ := Q a (x) x X x y µ(x,y) ϕ ϕ ϕ x(ϕ) X(ϕ)

81 Descriptive Complexity and Nested Words 39/46 Monadic Second-Order Logic, (MSO µ ) ϕ := Q a (x) x X x y µ(x,y) ϕ ϕ ϕ x(ϕ) X(ϕ) Thm: [Alur and Madhusudan] MSO µ = Regular Nested Languages

82 Descriptive Complexity and Nested Words 39/46 Monadic Second-Order Logic, (MSO µ ) ϕ := Q a (x) x X x y µ(x,y) ϕ ϕ ϕ x(ϕ) X(ϕ) Thm: [Alur and Madhusudan] MSO µ = Regular Nested Languages Proof: Similar to proof that MSO = Regular.

83 Descriptive Complexity and Nested Words 39/46 Monadic Second-Order Logic, (MSO µ ) ϕ := Q a (x) x X x y µ(x,y) ϕ ϕ ϕ x(ϕ) X(ϕ) Thm: [Alur and Madhusudan] MSO µ = Regular Nested Languages Proof: Similar to proof that MSO = Regular. MSO Regular: new case is µ(c, r). An NWA can check that the call related to r was c.

84 Descriptive Complexity and Nested Words 39/46 Monadic Second-Order Logic, (MSO µ ) ϕ := Q a (x) x X x y µ(x,y) ϕ ϕ ϕ x(ϕ) X(ϕ) Thm: [Alur and Madhusudan] MSO µ = Regular Nested Languages Proof: Similar to proof that MSO = Regular. MSO Regular: new case is µ(c, r). An NWA can check that the call related to r was c. Regular MSO: as usual, in MSO we guess the accepting run of the automaton. New case: we use µ to check that the return transitions are correct.

85 Descriptive Complexity and Nested Words 40/46 Next Operators: Temporal Logic for Nested Words (w,i) = ϕ (w,i + 1) = ϕ ( ) (w,i) = µ ϕ j µ(i,j) (w,j) = ϕ Past Operators: (w,j) = ϕ (w,j 1) = ϕ ( ) (w,j) = µ ϕ i µ(i,j) (w,i) = ϕ Since and Until Operators for several kinds of paths: (w,i) = αuβ path i = i 0 < i 1 < < i k (w,i k ) = β j < k(w,i j ) = α (w,i) = αsβ path i = i 0 > i 1 > > i k (w,i k ) = β j < k(w,i j ) = α

86 Descriptive Complexity and Nested Words 41/46 4 Kinds of Paths The sequence, i = i 0 < i 1 < < i k = j is a linear path if i p+1 = i p

87 Descriptive Complexity and Nested Words 41/46 4 Kinds of Paths The sequence, i = i 0 < i 1 < < i k = j is a linear path if i p+1 = i p + 1 a call path if i p = C(i p+1 ) 1 5 6

88 Descriptive Complexity and Nested Words 41/46 4 Kinds of Paths The sequence, i = i 0 < i 1 < < i k = j is a linear path if i p+1 = i p + 1 a call path if i p = C(i p+1 ) { r(i p ) if i p is a call an abstract path if i p+1 = i p + 1 otherwise

89 Descriptive Complexity and Nested Words 41/46 4 Kinds of Paths The sequence, i = i 0 < i 1 < < i k = j is a linear path if i p+1 = i p + 1 a call path if i p = C(i p+1 ) { r(i p ) if i p is a call an abstract path if i p+1 = i p + 1 otherwise a summary path if { i p+1 = r(i p ) i p + 1 if i p is a call and r(i p ) j otherwise

90 Descriptive Complexity and Nested Words 42/46 4 Resulting Kinds of Until and Since Operators linear paths: U, S call paths: U c, S c abstract paths: U a, S a summary paths: U σ, S σ (w,i) = αuβ path i = i 0 < i 1 < < i k (w,i k ) = β j < k(w,i j ) = α (w,i) = αsβ path i = i 0 > i 1 > > i k (w,i k ) = β j < k(w,i j ) = α

91 Descriptive Complexity and Nested Words 43/46 Nested Word Temporal Logic: NWTL ϕ := a ϕ ϕ ϕ ϕ µ ϕ ϕ µ ϕ ϕu σ ϕ ϕs σ ϕ

92 Descriptive Complexity and Nested Words 43/46 Nested Word Temporal Logic: NWTL ϕ := a ϕ ϕ ϕ ϕ µ ϕ ϕ µ ϕ ϕu σ ϕ ϕs σ ϕ Thm: [LICS 07] NWTL is exactly as expressive as FO over finite, and over infinite, nested words.

93 Descriptive Complexity and Nested Words 43/46 Nested Word Temporal Logic: NWTL ϕ := a ϕ ϕ ϕ ϕ µ ϕ ϕ µ ϕ ϕu σ ϕ ϕs σ ϕ Thm: [LICS 07] NWTL is exactly as expressive as FO over finite, and over infinite, nested words. Proof: NWTL FO: translate each ϕ NWTL to τ(ϕ) FO 3 s.t., (w,i) = ϕ (w,i/x) = τ(ϕ).

94 Descriptive Complexity and Nested Words 43/46 Nested Word Temporal Logic: NWTL ϕ := a ϕ ϕ ϕ ϕ µ ϕ ϕ µ ϕ ϕu σ ϕ ϕs σ ϕ Thm: [LICS 07] NWTL is exactly as expressive as FO over finite, and over infinite, nested words. Proof: NWTL FO: translate each ϕ NWTL to τ(ϕ) FO 3 s.t., (w,i) = ϕ (w,i/x) = τ(ϕ). Cor: Every formula in FO on nested words is equivalent to a formula in FO 3, i.e., using at most three variables, x,y,z.

95 Descriptive Complexity and Nested Words 43/46 Nested Word Temporal Logic: NWTL ϕ := a ϕ ϕ ϕ ϕ µ ϕ ϕ µ ϕ ϕu σ ϕ ϕs σ ϕ Thm: [LICS 07] NWTL is exactly as expressive as FO over finite, and over infinite, nested words. Proof: NWTL FO: translate each ϕ NWTL to τ(ϕ) FO 3 s.t., (w,i) = ϕ (w,i/x) = τ(ϕ). FO NWTL: using FO-completeness of a temporal logic on trees [Marx]. Cor: Every formula in FO on nested words is equivalent to a formula in FO 3, i.e., using at most three variables, x,y,z.

96 Descriptive Complexity and Nested Words 44/46 Model Checking Thm: Satisfiability of NWTL is EXPTIME complete.

97 Descriptive Complexity and Nested Words 44/46 Model Checking Thm: Satisfiability of NWTL is EXPTIME complete. Proof: [Sketch] lower bound: Previously known from Caret Paper [AEM].

98 Descriptive Complexity and Nested Words 44/46 Model Checking Thm: Satisfiability of NWTL is EXPTIME complete. Proof: [Sketch] lower bound: Previously known from Caret Paper [AEM]. upper bound: Transform formula ϕ to a nested-word automaton of size 2 O( ϕ ) and test emptiness. Cor: Model checking NWTL specifications wrt boolean programs or nested-word automata is EXPTIME complete.

99 Descriptive Complexity and Nested Words 45/46 Work To Be Done on Nested Words Fill out more of the theory of nested words. Use for model checking recursive programs. Use for developing better query languages and algorithms for XML. Use in Linguistics.

100 co r.e. complete FO A (N) co r.e. Arithmetic Hierarchy FO(N) Recursive r.e. FO E r.e. complete (N) Primitive Recursive n O(1) O(1) FO[2 ] SO[n ] FO(PFP) SO(TC) co NP complete SO n O(1) SO[2 ] A co NP Polynomial Time Hierarchy NP EXPTIME PSPACE SO co NP SO(LFP) NP SO E NP complete O(1) FO[n ] "truly feasible" P FO(LFP) SO Horn O(1) FO[(log n) ] NC FO(TC) NC 2 log(cfl) NSPACE[log n] FO(DTC) DSPACE[log n] Regular FO(M) FO Logarithmic Time Hierarchy SAC Descriptive Complexity and Nested Words 46/46 1 SO Krom NC AC 1 ThC 0 0