Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exact descriptional complexity of determinization of input-driven pushdown automata

Olga Martynova

Abstract

The number of states and stack symbols needed to determinize nondeterministic input-driven pushdown automata (NIDPDA) working over a fixed alphabet is determined precisely. It is proved that in the worst case exactly 2^{n^2} states are needed to determinize an n-state NIDPDA, and the proof uses witness automata with a stack alphabet Γ= {0,1} working on strings over a 4-symbol input alphabet (Only an asymptotic lower bound was known before in the case of a fixed alphabet). Also, the impact of NIDPDA determinization on the size of stack alphabet is determined precisely for the first time: it is proved that s(2^{n^2}-1) stack symbols are necessary in the worst case to determinize an n-state NIDPDA working over an input alphabet of size s+5 with s left brackets (The previous lower bound was only asymptotic in the number of states and did not depend on the number of left brackets).

Exact descriptional complexity of determinization of input-driven pushdown automata

Abstract

The number of states and stack symbols needed to determinize nondeterministic input-driven pushdown automata (NIDPDA) working over a fixed alphabet is determined precisely. It is proved that in the worst case exactly 2^{n^2} states are needed to determinize an n-state NIDPDA, and the proof uses witness automata with a stack alphabet Γ= {0,1} working on strings over a 4-symbol input alphabet (Only an asymptotic lower bound was known before in the case of a fixed alphabet). Also, the impact of NIDPDA determinization on the size of stack alphabet is determined precisely for the first time: it is proved that s(2^{n^2}-1) stack symbols are necessary in the worst case to determinize an n-state NIDPDA working over an input alphabet of size s+5 with s left brackets (The previous lower bound was only asymptotic in the number of states and did not depend on the number of left brackets).
Paper Structure (6 sections, 4 theorems, 18 equations, 4 figures)

This paper contains 6 sections, 4 theorems, 18 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Input-driven pushdown automata
  3. The exact bound on the number of states
  4. The lower bound on the number of stack symbols: one left bracket
  5. The lower bound on the number of stack symbols: several left brackets
  6. Conclusion

Key Result

Theorem 1

Let $A$ be an $n$-state NIDPDA, working over an alphabet $\Sigma = \Sigma_0 \cup \Sigma_{+1} \cup \Sigma_{-1}$. Then, there is a DIDPDA with $2^{n^2}$ states and with $|\Sigma_{+1}|(2^{n^2}-1)$ stack symbols that recognizes the language $L(A)$.

Figures (4)

  • Figure 1: The accepting computation of the automaton $A_n$ on the string $\#xw_Ry_j\#x'y_i$, where $(i,j) \in R$.
  • Figure 2: The automaton $A_n$ begins reading the string $y_i$ in the state $i$ and deterministically leaves this string in the same state $i$.
  • Figure 3: An accepting computation of the automaton $B_n$ on a string $f_{R_1,\ldots,R_m}g_{i,j,k,m}$, for $(i,j) \in R_k$. First, it is shown in the figure how the automaton moves through the substring ${<}w_{R_k}{<}$ of $f_{R_1,\ldots,R_m} = {<}w_{R_1}{<}w_{R_2}\ldots{<}w_{R_m}{<}$, pushing onto the stack $\overrightarrow{i}\widehat{j}$. Then $B_n$ enters the second part $g_{i,j,k,m}$ of the string, that begins in the figure with a symbol '${\#}$' at the top, and it is illustrated how $B_n$ gets through a substring $\#y_0{\gg}y_j\#y_1{\gg}y_i$ of $g_{i,j,k,m}$ popping symbols $\widehat{j}$ and $\overrightarrow{i}$ out of the stack.
  • Figure 4: An accepting computation of the automaton $B_{n,s}$ on a string $f_{R_1,\ldots,R_m,\ell_1,\ldots,\ell_{m+1}}h_{k,x,m}$, in a case of $\ell_k[x] = 1$.

Theorems & Definitions (23)

  • Definition 1: Mehlhorn Mehlhorn, von Braunmühl and Verbeek vonBraunmuehl_Verbeek, Alur and Madhusudan AlurMadhusudan
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Claim 1
  • Claim 2
  • Claim 3
  • Claim 4
  • Theorem 3
  • ...and 13 more