嵌套卵石自动机捕获传递闭包一阶逻辑
Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure
摘要：
在（确定性）对数空间中可识别的字符串语言的特征是双向（确定性）多头自动机，或者遵循Immerman，具有（确定性）传递闭包的一阶逻辑。在这里，我们详细阐述了这个结果，并匹配的头的数量传递闭包的arity。更确切地说，具有k元确定性传递闭包的一阶逻辑与具有k个头的确定性自动机在其输入上行走具有相同的能力，另外使用有限的嵌套卵石集。这个结果对于字符串、有序树以及具有固定自动机的图族都是有效的，该自动机可用于遍历图族中每个图的节点。此类族的其他示例包括轴网、圆环和矩形迷宫。对于非确定性自动机，逻辑仅限于传递闭包的正出现。树的k=1的特殊情况表明，具有嵌套卵石的单头确定性树行走自动机具有一元确定性传递闭包的一阶逻辑特征。这改进了我们先前的结果，将这些自动机之间的一阶和一元二阶逻辑树。
英文摘要：
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.
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