multitape nfa: weak synchronization of the input heads · multitape nfa: weak synchronization of...
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MotivationPrevious Work
Main ResultsAn Open Problem
Multitape NFA: Weak Synchronizationof the Input Heads
Ömer Egecioglu1, Oscar H. Ibarra1 and Nicholas Tran2
1Department of Computer ScienceUniversity of California at Santa Barbara
{omer,ibarra}@cs.ucsb.edu
2Department of Mathematics & Computer ScienceSanta Clara [email protected]
SOFSEM 2012
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
MotivationPrevious Work
Main ResultsAn Open Problem
Outline
1 Motivation
2 Previous Work
3 Main Results
4 An Open Problem
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
MotivationPrevious Work
Main ResultsAn Open Problem
Motivation & FA Models
String synchronization:
Vulnerability analysis (SQL Injection Attacks)
Verification
Reachability (Reachability Analysis of String Systems)
Finite Automata Models of String Systems:
Single-track DFA to represent individual strings [Xu et al.],[Shannon et al.]
Multi-track DFA to represent groups of strings [Yu et al.]
Multi-tape and multi-head FA/PDA to represent groups ofstrings [Ibarra et al.]
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
MotivationPrevious Work
Main ResultsAn Open Problem
NFA, Stack and Counter Machines
NFA: Nondeterministic, one-way FA
PDA: Nondeterministic, one-way FA with a stack
CM: PDA where stack alphabet is unary
Reversal-bounded CM: the stack height function has abounded number of local maxima/minima
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
MotivationPrevious Work
Main ResultsAn Open Problem
Multitrack vs. Multitape Automata
Multitape automata are more expressive but haveTuring-complete decision problems.
Best of both worlds: expressiveness of multitape butdecidability of multitrack automata.
Naturally leads to synchronization/synchronizability:
Does a multitape automaton "behave" like a multitrackautomaton?
Can a multitape automaton be converted to an equivalentmultitrack automaton?
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
MotivationPrevious Work
Main ResultsAn Open Problem
Basic Model
2-tape NFA, input (a1a2a3a4a5a6a7$,b1b2b3b4b5$).
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
MotivationPrevious Work
Main ResultsAn Open Problem
Weak/Strong Synchronization & Synchronizability
Given an n-tape NFA M,M is strongly k-synchronized if no two heads, neither ofwhich is on $, are more than k cells apart for anycomputation (accepting or not).
M is weakly k-synchronized if for each accepted input tuplethere exists an accepting computation in which no twoheads, neither of which is on $, are more than k cells apart.
M is strongly/weakly synchronized if it is strongly/weaklyk-synchronized for some k .
M is strongly/weakly k-synchronizable if there is astrongly/weakly k-synchronized M ′ such thatL(M) = L(M ′).
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
MotivationPrevious Work
Main ResultsAn Open Problem
Strong Synchronization 6= Weak Synchronization
L = {(am$,bn$) | m,n > 0}.
M: 2-tape NFA. On input (am$,bn$), executes one of:1 M reads am$ on tape 1 until head 1 reaches $, and then
reads bn$ on tape 2 until head 2 reaches $, then accepts.2 M reads the symbols on the two tapes simultaneously until
one head reaches $. Then the other head scans theremaining symbols on its tape and accepts.
M is not strongly synchronized, but it is weakly synchronized(weakly 0-synchronized).
Strongly synchronized → weakly synchronized, but notconversely.
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
MotivationPrevious Work
Main ResultsAn Open Problem
A Strong Synchronization Result [DCFS11]
TheoremIt is decidable to determine, given an n-tape NFA M, whether itis strongly k-synchronized for some k. If this is the case, thesmallest such k can be found.
Recall:M is strongly k-synchronized if no two heads, neither of whichis on $, are more than k cells apart for any computation(accepting or not).
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
MotivationPrevious Work
Main ResultsAn Open Problem
Sample Strong Synchronization Results [DCFS11]
One-way 2-tape 2-counter DCM (for a fixed k):undecidable
One-way n-tape PDA + reversal-bounded counters (forsome k): decidable
Two-way 2-tape DFA (for some k): undecidable
Two-way n-tape NFA (for a fixed k): decidable
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
MotivationPrevious Work
Main ResultsAn Open Problem
Definitions: Bounded and Unary Inputs
DefinitionLet a1,a2, . . . ,ak be arbitrary symbols.
A string x is
unary if x ∈ a∗
1;
bounded if x ∈ a∗
1a∗
2 · · · a∗
k .
A tuple of strings (x1, . . . , xn) is
unary if each xi is unary;
bounded if each xi is bounded;
all-but-one bounded (ABO) if all xi except one is bounded.
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
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Definitions: Semilinear sets
Definition
Linear set in Nk : Q = {v0 + t1v1 + · · · + tnvn | t1, . . . , tn ∈ N}.
v0 constant vector, v1, . . . , vn period vectors.
Semilinear set: Finite union of linear sets.
Properties:
Every finite subset of Nk is semilinear
Semilinear sets are closed under (finite) union,complementation and intersection
Disjointness, containment, and equivalence problems forsemilinear sets are decidable
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
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Definitions: Parikh mapping
Definition
Σ = {a1, . . . ,ak}, w ∈ Σ∗.|w |ai : the number of occurrences of ai in w .
Parikh image of a word: P(w) = (|w |a1 , . . . , |w |ak ) ∈ Nk
Parikh image of a language L: P(L) = {P(w) | w ∈ L} ⊆ Nk .
Properties:
The Parikh image of a language L accepted by a PDA is aneffectively computable semilinear set [Parikh].
The Parikh image of a language L accepted by a PDA with1-reversal counters is an effectively computable semilinearset [Ibarra].
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
MotivationPrevious Work
Main ResultsAn Open Problem
Weak Synchronization & Synchronizability Results
2-tape 2-ambiguous NFA is k-synchronized for a given k?
2-tape 2-ambiguous NFA has an equivalent 0-synchronized version?
unde
cida
ble
decidable
n-tape unambiguous NFA + reversal-bounded counters isk-synchronized for some k?
n-tape ABO NFA is k-synchronized for a given k?
2-tape unary NFA is k-synchronized for some k?
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
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Main ResultsAn Open Problem
Helpful Fact: 0-synchronization and Regularity
Definition
Let (x1, . . . , xn) be a tuple of strings. Define 〈x1, . . . , xn〉 to bean n-track string where the symbols of xi ’s are left-justified andthe shorter strings are right-filled with blanks (λ) to make alltracks the same length.
For a language L of n-tuples, 〈L〉 = {〈x〉 : x ∈ L}.
Lemma
Let L be a set of n-tuples. L is accepted by a weakly0-synchronized n-tape NFA iff 〈L〉 is regular.
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
MotivationPrevious Work
Main ResultsAn Open Problem
Proof Sketch of an Undecidability Result
TheoremIt is undecidable to determine whether a 2-tape NFA M is0-synchronized.
ProofBy reduction from the Post Correspondence Problem: given aPCP instance I = (u1, . . . ,un); (v1, . . . , vn) we construct a2-tape NFA M to accept
L = {(xc i , yd j) : i , j > 0, x 6= y}∪{(xc i , xd j) : i , j > 0, j = 2i , x ∈ solution(I)}
such that M is weakly 0-synchronized iff I has no solutions.
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
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Proof Sketch (cont.): A 2-tape NFA for L
L = {(xc i , yd j) : i , j > 0, x 6= y}∪{(xc i , xd j) : i , j > 0, j = 2i , x ∈ solution(I)}
Define a 2-tape NFA M as follows: on input w = (xc i , yd j), Mnondeterministically does one of the following:
M verifies that x 6= y by moving the two heads in0-synchrony from left to right until both heads reach theirendmarkers $ and noting that x and y differ at some point;
M guesses a sequence of indices i1, . . . , ik , and verifies(one index at a time) that x = ui1 · · · uik and y = vi1 · · · vikand that j = 2i .
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
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Proof Sketch (cont.)
L = {(xc i , yd j) : i , j > 0, x 6= y}∪{(xc i , xd j) : i , j > 0, j = 2i , x ∈ solution(I)}
If I has no solutions, the second part is empty and hence 〈L〉 isregular.
If I has a solution, then L contains a string w = (xc i , xd2i) forsome x and i . If 〈L〉 is regular, pump w to obtain〈xc i+k , xd2i+k〉 /∈ 〈L〉 for some k , a contradiction.
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
MotivationPrevious Work
Main ResultsAn Open Problem
A General Undecidability Result
Theorem
The following problems are undecidable, given a 2-tape (andhence multitape) NFA M:
1 Is M weakly k-synchronized for a given k?2 Is M weakly k-synchronized for some k?3 Is there a 2-tape (multitape) NFA M ′ that is weakly
0-synchronized (or weakly k-synchronized for a given k, orweakly k-synchronized for some k) such thatL(M ′) = L(M)?
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
MotivationPrevious Work
Main ResultsAn Open Problem
An Undecidability Result for 2-ambiguous NFA
Theorem
The following problems are undecidable, given a 2-ambiguous2-tape (and hence multitape) NFA M:
1 Is M weakly k-synchronized for a given k?2 Is M weakly k-synchronized for some k?3 Is there a 2-tape (multitape) NFA M ′ that is weakly
0-synchronized (or weakly k-synchronized for a given k, orweakly k-synchronized for some k) such thatL(M ′) = L(M)?
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
MotivationPrevious Work
Main ResultsAn Open Problem
Some Decidability Results: ABO and Unary Cases
Theorem
It is decidable to determine, given a 3-tape ABO NFA M and anonnegative integer k, whether M is weakly k-synchronized.
Unary case: input is of the form (an$,bm$).
Theorem
Suppose M is a unary 2-tape NFA. Then it is decidable whetheror not M is weakly synchronized.
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
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Unary Case
Transitions of M → letters from Σ = {a,b, $} → NFA M ′.Computation paths of M → words over Σ. Any acceptingcomputation path w for input (an$,bm$) has P(w) = (n,m).
There are two types of accepting computations:1 w = x$y$ with x ∈ {a,b}∗ and y ∈ {b}∗,2 w = x$y$ with x ∈ {a,b}∗ and y ∈ {a}∗.
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
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Unary Case (cond.)
Whether M is weakly k-synchronized is equivalent to showingthat for any given word w ∈ L(M ′), there is a stronglyk-synchronized word u such that either
ubj ∈ L(M ′) and P(w) = P(u) + (0, j), oruai ∈ L(M ′) and P(w) = P(u) + (i ,0).
Segueway to regular languages.Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
MotivationPrevious Work
Main ResultsAn Open Problem
Synchronization and Regular Languages
Definition
w ∈ Σ∗ is strongly k-synchronized if for any factorization x = uv
−k ≤ |u|a − |u|b ≤ k .
A language L over Σ is:
Strongly k-synchronized if all of its words are stronglyk-synchronized.
Strongly synchronized if it is strongly k-synchronized for somek .
Weakly k-synchronized if for every w ∈ L, there is a w ′ ∈ Lsuch that P(w ′) = P(w) and w ′ is strongly k-synchronized.
Weakly synchronized if it is weakly k-synchronized for some k .
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
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Balanced Cycles and Periods
Balanced cycles of length 4:
For a linear set
{(a0,b0) + k1(a1,b1) + · · ·+ ks(as,bs) | k1, k2, . . . , ks ≥ 0},
that appears in P(L), a period (ai ,bi), i > 0 is balanced ifai = bi .
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
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Synchronization and Regular Languages
Theorem
For a regular language L over Σ = {a,b}, the following areequivalent:
1 L is strongly synchronized2 L is weakly synchronized3 The minimum state DFA for L has no unbalanced simple
cycles4 The Parikh image P(L) has no unbalanced periods
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
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In Closing
An easy to state open problem on weak synchronization ofmultitape NFA.
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
MotivationPrevious Work
Main ResultsAn Open Problem
An Open Decidability Problem
ProblemIs it decidable to determine, given a 2-tape NFA whose tapesare over bounded languages, whether it is weaklyk-synchronized for some k?
Note:
We only know this (in the positive) for the bounded unarycase.
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata
MotivationPrevious Work
Main ResultsAn Open Problem
Sample Strong Synchronization Results [DCFS11]
One-way 2-tape 2-counter DCM (for a fixed k):undecidableOne-way n-tape PDA + reversal-bounded counters (forsome k): decidable
One-way 2-head DFA (for some k): undecidableOne-way n-head PDA + reversal-bounded counters (for afixed k): decidable
Two-way 2-tape DFA (for some k): undecidableTwo-way n-tape NFA (for a fixed k): decidable
Two-way 2-head DCM (for a fixed k): undecidableTwo-way n-head NFA (for a fixed k): decidableTwo-way 2-head reversal-bounded DCM (for a fixed k):decidable
Egecioglu, Ibarra & Tran Weakly Synchronized Multitape Automata