Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure
String languages recognizable in (deterministic) log-space are characterized either by two-way (deterministic) multi-head automata, or following Immerman, by first-order logic with (deterministic) transitive closure. Here we elaborate this result, and match the number of heads to the arity of the tr...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Logical Methods in Computer Science e.V.
2007-04-01
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| Series: | Logical Methods in Computer Science |
| Subjects: | |
| Online Access: | https://lmcs.episciences.org/2220/pdf |
| _version_ | 1797268782113095680 |
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| author | Joost Engelfriet Hendrik Jan Hoogeboom |
| author_facet | Joost Engelfriet Hendrik Jan Hoogeboom |
| author_sort | Joost Engelfriet |
| collection | DOAJ |
| description | String languages recognizable in (deterministic) log-space are characterized
either by two-way (deterministic) multi-head automata, or following Immerman,
by first-order logic with (deterministic) transitive closure. Here we elaborate
this result, and match the number of heads to the arity of the transitive
closure. More precisely, first-order logic with k-ary deterministic transitive
closure has the same power as deterministic automata walking on their input
with k heads, additionally using a finite set of nested pebbles. This result is
valid for strings, ordered trees, and in general for families of graphs having
a fixed automaton that can be used to traverse the nodes of each of the graphs
in the family. Other examples of such families are grids, toruses, and
rectangular mazes. For nondeterministic automata, the logic is restricted to
positive occurrences of transitive closure.
The special case of k=1 for trees, shows that single-head deterministic
tree-walking automata with nested pebbles are characterized by first-order
logic with unary deterministic transitive closure. This refines our earlier
result that placed these automata between first-order and monadic second-order
logic on trees. |
| first_indexed | 2024-04-25T01:37:57Z |
| format | Article |
| id | doaj.art-a87ade0e3c9245fb82bfb7ef568d2507 |
| institution | Directory Open Access Journal |
| issn | 1860-5974 |
| language | English |
| last_indexed | 2024-04-25T01:37:57Z |
| publishDate | 2007-04-01 |
| publisher | Logical Methods in Computer Science e.V. |
| record_format | Article |
| series | Logical Methods in Computer Science |
| spelling | doaj.art-a87ade0e3c9245fb82bfb7ef568d25072024-03-08T08:46:40ZengLogical Methods in Computer Science e.V.Logical Methods in Computer Science1860-59742007-04-01Volume 3, Issue 210.2168/LMCS-3(2:3)20072220Automata with Nested Pebbles Capture First-Order Logic with Transitive ClosureJoost EngelfrietHendrik Jan HoogeboomString languages recognizable in (deterministic) log-space are characterized either by two-way (deterministic) multi-head automata, or following Immerman, by first-order logic with (deterministic) transitive closure. Here we elaborate this result, and match the number of heads to the arity of the transitive closure. More precisely, first-order logic with k-ary deterministic transitive closure has the same power as deterministic automata walking on their input with k heads, additionally using a finite set of nested pebbles. This result is valid for strings, ordered trees, and in general for families of graphs having a fixed automaton that can be used to traverse the nodes of each of the graphs in the family. Other examples of such families are grids, toruses, and rectangular mazes. For nondeterministic automata, the logic is restricted to positive occurrences of transitive closure. The special case of k=1 for trees, shows that single-head deterministic tree-walking automata with nested pebbles are characterized by first-order logic with unary deterministic transitive closure. This refines our earlier result that placed these automata between first-order and monadic second-order logic on trees.https://lmcs.episciences.org/2220/pdfcomputer science - logic in computer sciencef.1.1f.4.1f.4.3 |
| spellingShingle | Joost Engelfriet Hendrik Jan Hoogeboom Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure Logical Methods in Computer Science computer science - logic in computer science f.1.1 f.4.1 f.4.3 |
| title | Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure |
| title_full | Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure |
| title_fullStr | Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure |
| title_full_unstemmed | Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure |
| title_short | Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure |
| title_sort | automata with nested pebbles capture first order logic with transitive closure |
| topic | computer science - logic in computer science f.1.1 f.4.1 f.4.3 |
| url | https://lmcs.episciences.org/2220/pdf |
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