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A lower bound on the state complexity of transforming two-way nondeterministic finite automata to unambiguous finite automata

Semyon Petrov, Alexander Okhotin

TL;DR

The paper establishes a tight-seeming lower bound on the state blow-up when transforming a two-way nondeterministic finite automaton with $n$ states into a one-way unambiguous automaton, proving a bound of $\Omega\big( n^{2n+2} / e^{2n} \big)$ states via a rank analysis of a matrix associated with a universal witness language $L_n$. The method constructs prefix/suffix table encodings and a corresponding matrix $M^{(n)}$, then applies Schmidt’s theorem and a sequence of rank-preserving reductions to show that the number of states required by any $UFA$ recognizing $L_n$ is at least the number of ordered prefix tables on $[n]$, which equals $\sum_{k=1}^{n} (k-1)! k! \genfrac\{\

Abstract

This paper establishes a lower bound on the number of states necessary in the worst case to simulate an $n$-state two-way nondeterministic finite automaton (2NFA) by a one-way unambiguous finite automaton (UFA). It is proved that for every $n$, there is a language recognized by an $n$-state 2NFA that requires a UFA with at least $\sum_{k=1}^{n} (k - 1)! \cdot k! \cdot \mathrm{stirling2}(n, k) \cdot \mathrm{stirling2}(n+1, k)$ = $Ω\big( n^{2n+2} / e^{2n} \big)$ states, where $\mathrm{stirling2}(n, k)$ denotes Stirling's numbers of the second kind. This result is proved by estimating the rank of a certain matrix, which is constructed for the universal language for $n$-state 2NFAs, and describes every possible behaviour of these automata during their computation.

A lower bound on the state complexity of transforming two-way nondeterministic finite automata to unambiguous finite automata

TL;DR

The paper establishes a tight-seeming lower bound on the state blow-up when transforming a two-way nondeterministic finite automaton with states into a one-way unambiguous automaton, proving a bound of states via a rank analysis of a matrix associated with a universal witness language . The method constructs prefix/suffix table encodings and a corresponding matrix , then applies Schmidt’s theorem and a sequence of rank-preserving reductions to show that the number of states required by any recognizing is at least the number of ordered prefix tables on , which equals $\sum_{k=1}^{n} (k-1)! k! \genfrac\{\

Abstract

This paper establishes a lower bound on the number of states necessary in the worst case to simulate an -state two-way nondeterministic finite automaton (2NFA) by a one-way unambiguous finite automaton (UFA). It is proved that for every , there is a language recognized by an -state 2NFA that requires a UFA with at least = states, where denotes Stirling's numbers of the second kind. This result is proved by estimating the rank of a certain matrix, which is constructed for the universal language for -state 2NFAs, and describes every possible behaviour of these automata during their computation.

Paper Structure

This paper contains 11 sections, 31 theorems, 6 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions
  3. Lower bound
  4. The witness language and its 2NFA
  5. Rank-preserving matrix reduction
  6. Ordered prefix tables
  7. Linear independence of rows of the matrix
  8. Rank calculation
  9. Asymptotics of the lower bound
  10. Optimality of the lower bound
  11. Conclusion

Key Result

Lemma 1

The automaton $\mathcal{A}_n$ accepts the string $(\mathtt{l}, s(f)) (\mathtt{l}, f) (\mathtt{r}, g)$ if and only if the graph $G(f) \cup H(g)$ has a path that starts in $(\mathtt{l}, s(f))$ and ends in $(\mathtt{r}, q)$ for some $q \in A(g)$.

Figures (5)

  • Figure 1: A prefix table constructed from an automaton on a prefix $x$
  • Figure 2: A suffix table constructed from an automaton on a suffix $y$.
  • Figure 4: The matrix $K^{(2)}$, which is actually equal to $M^{(2)}$ (unordered prefix and suffix tables exist only for $n \geqslant 3$).
  • Figure 5: Prefix and suffix layers of a prefix table $f$.
  • Figure 6: a) A prefix table that breaks through all layers between $i$ and $j - 1$ of $f_{\circ}$, inclusive. b) A prefix table that drops down from layer $i$ of $f_{\circ}$.

Theorems & Definitions (89)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • proof
  • Theorem A: Schmidt Schmidt, see also Leung Leung
  • proof : Sketch of a proof
  • Definition 5
  • Lemma 2
  • ...and 79 more