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A quadratic lower bound for 2DFAs against one-way liveness

Kehinde Adeogun, Christos Kapoutsis

Abstract

We show that every two-way deterministic finite automaton (2DFA) that solves one-way liveness on height h has Omega(h^2) states. This implies a quadratic lower bound for converting one-way nondeterministic finite automata to 2DFAs, which asymptotically matches Chrobak's well-known lower bound for this conversion on unary languages. In contrast to Chrobak's simple proof, which relies on a 2DFA's inability to differentiate between any two sufficiently distant locations in a unary input, our argument works on alphabets of arbitrary size and is structured around a main lemma that is general enough to potentially be reused elsewhere.

A quadratic lower bound for 2DFAs against one-way liveness

Abstract

We show that every two-way deterministic finite automaton (2DFA) that solves one-way liveness on height h has Omega(h^2) states. This implies a quadratic lower bound for converting one-way nondeterministic finite automata to 2DFAs, which asymptotically matches Chrobak's well-known lower bound for this conversion on unary languages. In contrast to Chrobak's simple proof, which relies on a 2DFA's inability to differentiate between any two sufficiently distant locations in a unary input, our argument works on alphabets of arbitrary size and is structured around a main lemma that is general enough to potentially be reused elsewhere.
Paper Structure (14 sections, 17 theorems, 16 equations, 4 figures)

This paper contains 14 sections, 17 theorems, 16 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preparation
  3. Two-way finite automata
  4. One-way liveness
  5. Properties
  6. A lower-bound lemma
  7. Generic strings
  8. Exit sizes
  9. The lemma
  10. The lower bound
  11. The connectivities
  12. Some basic facts
  13. The desired facts
  14. Conclusion

Key Result

Lemma 1

Let $C,C'\in\mathbb{B}^{h\times h}$. If $C\neq C'$ then $P(C),P(C')$ are separated by $\text{\scshape\mdseries\rmfamily owl}_h$.

Figures (4)

  • Figure 1: Three symbols in $\varSigma_5$ ( L); their string ( M); and its connectivity ( R).
  • Figure 2: (a) A computation on $u\vartheta v$ with $m+2=7$ crucial points; and the corresponding $m+1=6$ computation infixes $c_1,\dots,c_6$, odd (bold) and even (dashed). (b) The computation $c'$ on $u\vartheta(x\vartheta)^tv$ (here $t=3$), created by joining the concatenations $c_i'$ of the $c_i$ and $d_i$. (z) How $\alpha:=\alpha_{\vartheta,x\vartheta}$ (solid lines) permutes $A:=Q_\textup{lr}(\vartheta)$; and $\alpha_{\vartheta,(x\vartheta)^t}$ (solid & dotted) permutes $A$, too, sometimes as identity (here $t=2)$.
  • Figure 3: Some of the connectivities $C_0,C_1,\dots,C_N$ for $h=5$. Here, $U=10$ and $N=15$. In each $C_t$, we have underlined every 'fresh' $1$, i.e., every $1$ that is not also present in $C_{t-1}$ ($t\neq0$).
  • Figure 4: Some of the auxiliary matrices for building $C_0,C_1,\dots,C_N$ for $h=5$.

Theorems & Definitions (47)

  • Definition 1
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Lemma 1
  • proof
  • Lemma 2: ka07jalc
  • Definition 2: ka07jalc
  • Lemma 3: ka07jalc
  • ...and 37 more