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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 ${\displaystyle \log \log n}$ tape cells functions for which the nondeterministic machine required ${\displaystyle \log n}$ tape cells), with the implication that a pebble adds power to Turing machines whose functions require space between ${\displaystyle \log \log n}$ and ${\displaystyle \log n.}$ 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 ${\displaystyle \{1,2,\dots ,n\}}$. 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 ${\displaystyle DPA_{n}}$ (resp. ${\displaystyle PA_{n}}$). We also define ${\displaystyle DPA=\bigcup _{n}DPA_{n}}$ and likewise ${\displaystyle PA=\bigcup _{n}PA_{n}}$.

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 ${\displaystyle TWA\subsetneq DPA}$ or these classes are incomparable, which is an open problem
• ${\displaystyle PA\subsetneq REG}$, i.e. tree-walking automata augmented with pebbles are strictly weaker than branching automata
• it is not known whether ${\displaystyle DPA=PA}$, 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 ${\displaystyle PA_{n}\subsetneq PA_{n+1}}$ and ${\displaystyle DPA_{n}\subsetneq DPA_{n+1}}$

Automata and logic

Tree-walking pebble automata admit an interesting logical characterization. Let ${\displaystyle FO+TC}$ denote the set of tree properties describable in transitive closure first-order logic, and ${\displaystyle FO+{\text{pos}}\,TC}$ 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 ${\displaystyle PA\subseteq FO+TC}$ and, in fact, ${\displaystyle PA=FO+{\text{pos}}\,TC}$ - 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

References

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

• Trees (data structures)
• Automata (computation)