Pebble automaton explained

In computer science, a pebble automaton is any variant of an automaton which augments the original model with a finite number of "pebbles" that may be used to mark tape positions.

History

Pebble automata were introduced in 1986, when it was shown that in some cases, a deterministic transducer augmented with a pebble could achieve logarithmic space savings over even a nondeterministic log-space transducer (ie, compute in

log log n tape cells functions for which the nondeterministic machine required log n tape cells), with the implication that a pebble adds power to Turing machines whose functions require space between loglog n and log n. ^[1] Constructions were also shown to convert a hierarchy of increasingly powerful stack machine models into equivalent deterministic finite automata with up to 3 pebbles, showing additional pebbles further increased power.

Tree-walking automata with nested pebbles

A tree-walking automaton with nested pebbles is a tree-walking automaton with an additional finite set of fixed size containing pebbles, identified with

\{ 1 , 2 , ... , n \} . Besides ordinary actions, an automaton can put a pebble at a currently visited node, lift a pebble from the currently visited node and perform a test "is the i-th pebble present at the current node?". There is an important stack restriction on the order in which pebbles can be put or lifted - the i+1-th pebble can be put only if the pebbles from 1st to i-th are already on the tree, and the i+1-th pebble can be lifted only if pebbles from i+2-th to n-th are not on the tree. Without this restriction, the automaton has undecidable emptiness and expressive power beyond regular tree languages.

The class of languages recognized by deterministic (resp. nondeterministic) tree-walking automata with n pebbles is denoted

DPA _{n} (resp. PA _{n} ). We also define DPA = c up _{n} DPA _{n} and likewise PA = c up _{n} PA _{n} .

Properties

• there exists a language recognized by a tree-walking automaton with 1 pebble, but not by any ordinary tree walking automaton; this implies that either

TWA \subsetneq DPA or these classes are incomparable, which is an open problemPA \subsetneq REG , i.e. tree-walking automata augmented with pebbles are strictly weaker than branching automata

• it is not known whether

DPA = PA , i.e. whether tree-walking pebble automata can be determinized

• it is not known whether tree-walking pebble automata are closed under complementation
• the pebble hierarchy is strict for tree-walking automata, for every n

PA _{n} \subsetneq PA _{n+1} and DPA _{n} \subsetneq DPA _{n+1}

Automata and logic

Tree-walking pebble automata admit an interesting logical characterization.^[2] Let

FO + TC denote the set of tree properties describable in transitive closure first-order logic, and FO + pos TC the same for positive transitive closure logic, i.e. a logic where the transitive closure operator is not used under the scope of negation. Then it can be proved that PA \subseteq FO + TC and, in fact, PA = FO + pos TC - the languages recognized by tree-walking pebble automata are exactly those expressible in positive transitive closure logic.

See also

• Tree walking automata
• Branching automata
• Transitive closure logic
• Pebble game

Notes and References

1. Chang . Jik . Ibarra . Oscar . Palis . Michael . Ravikumar . B . On Pebble Automata . Theoretical Computer Science . November 1986 . 44 . 10.1016/0304-3975(86)90112-X. free .
2. Engelfriet . Joost . Hoogeboom . Hendrik Jan . Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure . Logical Methods in Computer Science . 26 April 2007 . 3 . 2 . 10.2168/LMCS-3(2:3)2007. free . cs/0703079 .