1/25 An Infinite Automaton Characterization of Double Exponential Time Gennaro Parlato University of Illinois at Urbana-Champaign Università degli Studi di Salerno Salvatore La Torre (U. Salerno) P. Madhusudan (U. Illinois U-C)

2/25 Infinite automata  Input alphabet   Fix an alphabet  States – all words *  Initial and final states are defined by two regular languages INIT, FINAL  transitions defined using rewriting: for each a   there is a transducer T a transforming words to words u u’ iff T a transforms u to u’ a

3/25 Example: Infinite automaton for { a n b n c n | n>0}  T a defines { (x n, x n+1 ) / n > 0}  {($, x)}  T b defines { (y m x n, y m+1 x n-1 ) / n > 0, m  0}  T c defines { (y m x n, y m+1 x n-1 ) / n > 0, m  0} $ x xx xxx  yxx yyx yyy yx yy y a a a b b b b b b c c c

4/25 Infinite automata using regular trasducers  Transitions defined using regular trasducers A regular trasducer reads an input word and writes an output word using finitely many states A regular trasducer has edges of the form q –a/b  q’ where a,b є  {} Eg. {(a n,(bc) n )} is a regular relation q0q0 q1q1 a/b  /c

5/25 Infinite automata & computational complexity  A remarkable theorem (Morvan-Stirling ’01) Note: no ostensible bounds of time or space are placed on the machine! Theorem Infinite automaton with regular transducers precisely define context-sensitive languages (i.e. NLINSPACE)

6/25 Infinite automata with pushdown transducers  Consider infinite automata with rewriting using pushdown automata  Pushdown transducer transform words to words using a finite-state control and a work-stack  Eg. {(a n, b n c n ) / n > 0 } can be effected (non regular relation)  Infinite automata with pushdown transducer relations define r.e. languages (undecidable)

7/25 Infinite automata with restricted pushdown transducers  Restricted pushdown transducers: Each transducer can switch between read the input tape and popping the stack only a bounded number of times Still powerful  Eg. {(a n, b n c n ) / n > 0 } can still be effected Theorem Infinite automata with restricted pushdown transducers define precisely the class 2ETIME ( in 2 2 O(n) time )  Note: Again, no ostensible space or time limits  States on a run can run for a very long time  A logical characterization of 2ETIME by restricting the power of rewriting

8/25 Known results  Rational graphs capture Context-Sensitive Languages (Morvan-Stirling, 2001)  Synchronized rational graphs are sufficient to capture CSL’s (Rispal, 2002)  Term-automatic graphs capture ETIME (Meyer, 2007)  Prefix-recognizable graphs capture Context-Free Languages (Caucal, 1996)  Survey on Infinite Automata (Thomas, 2001)

9/25 Outline of the talk  Infinite automata  2ETIME Upper Bound  2ETIME Lower Bound  Conclusions

10/25 Upper Bound  Two steps: 1.(polynomial time) reduction of membership for infinite automata with restricted pushdown transducers to emptiness for bounded-phase multi-stack pushdown automata (k-MPA) 2.2Etime solution of k-MPA emptiness Improvement of the 2 |A| 2 O(poly(k)) solution given in [ LICS’07]

11/25 Bounded-phase multi-stack pushdown automata [La Torre, P.Madhusudan, Parlato, LICS’07] finite control A phase is a sub-run where only A unique stack can be popped all stacks can be pushed onto  Finite set of states Q  An initial state q 0  Q  Final states F  Q  Actions: internal move push onto one stack pop from one stack phase-switch phase-switch RUN phase1 phase2 phase3 push2push1push2 pop1 pop2push1 pop1 push1 pop1

12/25 Simulating of the Inf. Aut. on w=a 1 a 2 … a m with an MPA  We reduce membership of BPTIAs to emptiness of k-MPAs Reduction a1a1 u0u0 u1u1 umum u2u2 a2a2 u m-1 amam c1c1 c1c1 Ta1Ta1 c2c2 Ta2Ta2 cmcm TamTam c2c2 cmcm INITFINALRES original stackS IN S OUT control states RES Guess u 0 є L(INIT) and push it into S IN For every i=1,2, …, m: 1) simulate T a i reading from S IN and writing onto S OUT 2) Empty the original stack 3) Move S OUT into S IN Accept if u m є L(FINAL) can be accomplished with O(|w|) phases

13/25 Emptiness for k-MPAs  Reduction to emptiness of tree automata  The key idea is the use of Stack trees

14/25 a e b a a’ a’ b a b’ e b’ a’ a, a’ – push/pop 1 st st b, b’ – push/pop 2 nd st e – internal Nested edges become local Linear edges lost!! Stack Trees [LICS’07] (a’,1) (a,1) (e,1) (a’, 1) (b,1) (a,1) (a’,3) 12 (b’,2) 11 (b’,2) (e,2)

15/25 Emptiness for k-MPAs  Reduction to emptiness of tree automata  The key idea is the use of stack-trees  Two main parts: the set of stack-trees is regular  linear order (Tree Aut. of size 2 O(k) ) Simulation of a k-MPA on the stack-trees  Successor (Tree Walking Aut. of size 2 O(k) )

16/25 linear order (x<y) a) x and y in diff phases c) x and y of the same phase -- easy: phase(x) < phase(y) but in diff subtrees: -- hard: b) x and y of same phase and same tree –- easy x precedes y in the prefix traversal of the tree x p y where p x =ParentRoot(x) p y = ParentRoot (y) p x > p y x < y Intuition for case c) push push pop pop … p y … p x … root(x) … x … root(y) … y

17/25 a) x and y in diff phases c) x and y of the same phase -- easy: phase(x) < phase(y) but in diff subtrees: -- hard: b) x and y of same phase and same tree –- easy x precedes y in the prefix traversal of the tree … … zyzy zxzx ≤ #phases Tree automata for the linear order xy Inductive definition Base case : a) and b) Inductive step : c) Tree automaton for c) Simulate the TA for a) or b) to check z x, z y Check if z x reach x and z y reaches y with the same phase sequence (guessed in the root)  State space 2 O(k) p x p y x p y where p x =ParentRoot(x) p y = ParentRoot (y)

18/25 Successor Motivation for case c) push push pop pop root x x y … p y … … p x a)if x and y in diff phases --easy EndPhase(x) and y=NextPhase(x) b) If x and y of same phase and same tree –- easy x is the predecessor of y in the prefix traversal of the tree c) x and y of the same phase but in diff subtrees: --hard z  ParentRoot(x); z’  Predecessor(z) while phase T (RightChild(z’)) ≠ phase T (x) z’  Predecessor(z); y = RightChild(z’);}

19/25 Tree walking automaton for Successor Procedure Successor(x) if EndPhase(x) then return NextPhase(phase T (x)); elseif (y  PrefixSucc(x) exists) then return(y); else { z  ParentRoot(x); z’  Predecessor(z) while phase T (RightChild(z’)) ≠ phase T (x) z’  Predecessor(z); return RightChild(z’);} Procedure Predecessor(x) if BeginPhase(x) then return PrevPhase(phase T (x)); elseif (y  PrefixPred(x) exists) then return(y); else { z  ParentRoot(x); z’  Successor(z) while phase T (RightChild(z’)) ≠ phase T (x) z’  Successor(z); return RightChild(z’);}  Look at Successor and Predecessor as a recursive program P  At most k (=#phases) alive calls at any time  Since k is fixed, P can be executed with finite memory O(k)  The tree walking automata simulates P in its control  Size of the tree walking automata = 2 O(k)

20/25 Outline of the talk  Infinite automata  2ETIME Upper Bound  2ETIME Lower Bound  Conclusions

21/25 Lower bound  Direct simulation of Turing machines is unfeasible: Turing machines are usually two way and have read and write moves infinite automata accept a word w in “real-time”, i.e. in |w| steps double exponentially many steps of computation should be carried out by a single bounded-phase pushdown rewriting

22/25 Lower bound (reduction)  Reduction from the membership problem for alternating machines working in 2 O(w) space to the membership problem for BPTAs a1a1 amam Guess a TM run on w a2a2 … Check all consecutive confs are correct Extract all the consecutive confs from the TM run

23/25 Checking the moves The sequence of moves: #u 1 #v 1 #u 2 #v 2... #u m #v m  Checking if u i  v i is similar to checking if c = c’  In each step transform every #u i #v i into two pairs of half size. Repeat until we get only pairs of length 1. (O(|w|) steps)  For j=1,…,|w| For i=1,…,m  Push all even symbols of u i /v i onto the stack  Write all odd symbols of u i /v i on the output tape Copy the stack content on the output tape  Check symbol equality #u 1 #v 1 #u 2 #v 2... #u m #v m #even(v m ) #even(u m )... #even(v 2 ) #even(u 2 ) #even(v 1 ) #even(u 1 ) #odd(v m ) #odd(u m )... #odd(v 2 ) #odd(u 2 ) #odd(v 1 ) #odd(u 1 ) output tape phases required log 2 |w| = |w|

24/25 Outline of the talk  Infinite automata  2ETIME Upper Bound  2ETIME Lower Bound  Conclusions

25/25 Conclusions  Characterization of 2ETIME with infinite automata  Alternate characterizations of complexity classes using rewriting theory  Rewriting is classic (Thue, Post) but never applied to complexity theory  Exact computational complexity of the emptiness problem for bounded-phase multi-stack pushdown automata

26/25 Conclusions  Can we show alternate proofs of classic results using infinite automata? Eg. NL=co-NL NLINSPACE= co-NLINSPACE …  Can we capture P or NP?  Expressive power of deterministic infinite automata