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Pebble automaton

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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

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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 tape cells functions for which the nondeterministic machine required tape cells), with the implication that a pebble adds power to Turing machines whose functions require space between and [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

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A tree-walking automaton with nested pebbles is a tree-walking automaton with an additional finite set of fixed size containing pebbles, identified with . 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 (resp. ). We also define and likewise .

Properties

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  • 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 or these classes are incomparable, which is an open problem
  • , i.e. tree-walking automata augmented with pebbles are strictly weaker than branching automata
  • it is not known whether , 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 and

Automata and logic

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Tree-walking pebble automata admit an interesting logical characterization.[2] Let denote the set of tree properties describable in transitive closure first-order logic, and 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 and, in fact, - the languages recognized by tree-walking pebble automata are exactly those expressible in positive transitive closure logic.

See also

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References

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  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.