Equivalence of Deterministic One-Counter Automata is NL-complete

Equivalence of Deterministic One-Counter Automata is NL-complete Stanislav B¨ ohm (Ostrava) Stefan G¨ oller (Bremen) Petr Janˇcar (Ostrava)

Highlights 2013, Paris September 19th

Deterministic One-counter automaton (DOCA) Input: (

a (

(

a )

)

[ [

a a (

Stack: (Counter)

)

A doca language

Decision problem

Input: Question:

Language equivalence PDA (CFL)

Undecidable

DPDA (DFCL)

Decidable [Sénizergues 1997] [Valiant, Paterson 1975]

DOCA

(REG)

NFA DFA

PSPACE complete NL complete

Language equivalence PDA (CFL)

Undecidable

DPDA (DFCL)

Decidable [Sénizergues 1997] [Valiant, Paterson 1975]

DOCA

(REG)

NFA DFA

[BGJ, STOC 2013]

PSPACE complete NL complete

NL Complete

Con guration pairs in 3D space

2D projection

Nonequivalence witness paths and equivalence levels

Nonequivalence witness paths and equivalence levels

Con guration q(413) (of a doca) Input:

Input: a

Counter:

Con guration Mod(q(413)) (of the extended doca)

...

a

Con guration q(413) (of a doca)

Con guration Mod(q(413)) (of the extended doca) Input:

Input:

a

a

Counter:

...

reset Input: a

Counter:

Transition system of extended doca

counter

Linearity of "independence levels"

steps

Quadruples (b, l, r, o) of eq-levels

First step of the proof

at least 2 minimums -1 each step

Form: small values only few values

Second step of the proof

slope

o set

Thank you for your attention!