An Infinite Automaton Characterization of Double Exponential Time

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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)

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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 Ta
  transforming words to words
• u u iff Ta transforms u to u

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Example Infinite automaton for
anbncn ngt0

• Ta defines (xn, xn1) / n gt 0 ? (, x)
• Tb defines (ym xn, ym1 xn-1) / n gt 0, m ?
  0
• Tc defines (ym xn, ym1 xn-1) / n gt 0, m ?
  0

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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 ? ???
• Eg. (an,(bc)n) is a regular relation

a/b
q0
q1
? /c
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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)
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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. (an, bn cn) / n gt 0 can be effected
• (non regular relation)
• Infinite automata with pushdown transducer
  relations define r.e. languages (undecidable)

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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. (an, bn cn) / n gt 0 can still be effected

Theorem Infinite automata with restricted
pushdown transducers define precisely the
class 2ETIME ( in 22O(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

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Known results

• Rational graphs capture Context-Sensitive
  Languages (Morvan-Stirling, 2001)
• Synchronized rational graphs are sufficient to
  capture CSLs (Rispal, 2002)
• Term-automatic graphs capture ETIME (Meyer, 2007)
• Prefix-recognizable graphs capture Context-Free
  Languages (Caucal, 1996)
• Survey on Infinite Automata (Thomas, 2001)

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Outline of the talk

• Infinite automata
• 2ETIME Upper Bound
• 2ETIME Lower Bound
• Conclusions

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Upper Bound

• Two steps
• (polynomial time) reduction of
• membership for infinite automata with restricted
  pushdown transducers
• to
• emptiness for bounded-phase multi-stack
  pushdown automata (k-MPA)
• 2Etime solution of k-MPA emptiness
• Improvement of the 2A 2O(poly(k)) solution
  given in LICS07

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Bounded-phase multi-stack pushdown automataLa
Torre, P.Madhusudan, Parlato, LICS07

• Finite set of states Q
• An initial state q0?Q
• Final states F ? Q
• Actions
• internal move
• push onto one stack
• pop from one stack

finite control
phase-switch
phase-switch RUN phase1
phase2
phase3
pop1
pop1
pop1
push2
push1
push2
pop2
push1
push1

• A phase is a sub-run where only
• A unique stack can be popped
• all stacks can be pushed onto

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Reduction

• We reduce membership of BPTIAs to emptiness of
  k-MPAs

am
a1
a2
u0
u1
um
u2
um-1


c1

c2
cm
Ta1
Ta2
Tam
Simulating of the Inf. Aut. on wa1a2 am with
an MPA
Guess u0 ? L(INIT) and push it into SIN For every
i1,2, , m 1) simulate Tai reading from
SIN and writing onto SOUT
2) Empty the original stack 3) Move SOUT
into SIN Accept if um ? L(FINAL)


control states

c1
c2
cm
INIT
FINAL
RES
RES
RES
SOUT
original stack
SIN
can be accomplished with O(w) phases
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Emptiness for k-MPAs

• Reduction to emptiness of tree automata
• The key idea is the use of Stack trees

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Stack Trees LICS07

• a e b a a a b a b e b
  a
• a, a push/pop 1st st
• b, b push/pop 2nd st
• e internal
• Nested edges become local
• Linear edges lost!!

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10
4
1 2 3 5 6 7
9 11 12
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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 2O(k))
• Simulation of a k-MPA on the stack-trees
• Successor (Tree Walking
  Aut. of size 2O(k))

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linear order (xlty)

• a) x and y in diff phases c) x and y
  of the same phase
• -- easy phase(x) lt 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

xlty ? px gt py where px ParentRoot(x)

py ParentRoot (y)
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Tree automata for the linear order
a) x and y in diff phases c) x and y
of the same phase -- easy phase(x) lt
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
xlty ? px gt py where px ParentRoot(x)

py ParentRoot (y)

• Tree automaton for c)
• Simulate the TA for a) or b) to check zx, zy
• Check if zx reach x and zy reaches y with the
  same phase sequence (guessed in the root)
• State space 2O(k)

Inductive definition Base case a) and
b) Inductive step c)
px py
x
y
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Successor
c) x and y of the same phase but in diff
subtrees --hard z ?
ParentRoot(x) z ?
Predecessor(z) while
phaseT(RightChild(z))
? phaseT(x) z ?
Predecessor(z) y RightChild(z)

• if x and y in diff phases
• --easy
• EndPhase(x) and
• yNextPhase(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

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Tree walking automaton for Successor

• 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 2O(k)

Procedure Successor(x) if EndPhase(x) then
return NextPhase(phaseT(x)) elseif (y ?
PrefixSucc(x) exists) then
return(y) else z ? ParentRoot(x)
z ? Predecessor(z) while
phaseT(RightChild(z))
? phaseT(x) z ?
Predecessor(z) return RightChild(z)
Procedure Predecessor(x) if BeginPhase(x) then
return PrevPhase(phaseT(x)) elseif (y ?
PrefixPred(x) exists) then
return(y) else z ? ParentRoot(x)
z ? Successor(z) while
phaseT(RightChild(z))
? phaseT(x) z ?
Successor(z) return RightChild(z)
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Outline of the talk

• Infinite automata
• 2ETIME Upper Bound
• 2ETIME Lower Bound
• Conclusions

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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

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Lower bound (reduction)

• Reduction from
• the membership problem for alternating machines
  working in 2O(w) space
• to
• the membership problem for BPTAs

Extract all the consecutive confs from the TM
run
Guess a TM run on w
a1
am
a2

Check all consecutive confs are correct
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Checking the moves

• The sequence of moves u1v1 u2v2 ...
  umvm
• Checking if ui?vi is similar to checking if c
  c
• In each step transform every uivi into two
  pairs of half size. Repeat until we get only
  pairs of length 1. (O(w) steps)

• For j1,,w
• For i1,,m
• Push all even symbols of ui/vi onto the stack
• Write all odd symbols of ui/vi on the output tape
• Copy the stack content on the output tape
• Check symbol equality

u1v1 u2v2 ... umvm
even(vm) even(um) ... even(v2)
even(u2) even(v1) even(u1)
odd(vm) odd(um) ... odd(v2) odd(u2) odd(v1)
odd(u1)
phases required log 2w w
output tape
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Outline of the talk

• Infinite automata
• 2ETIME Upper Bound
• 2ETIME Lower Bound
• Conclusions

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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

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Conclusions

• Can we show alternate proofs of classic results
  using infinite automata?
• Eg. NLco-NL
• NLINSPACE co-NLINSPACE
• Can we capture P or NP?
• Expressive power of deterministic infinite
  automata