1,182 research outputs found
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 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.Comment: Paper for Logical Methods in Computer Science, 27 pages, 1 figur
Advanced Automata Minimization
We present an efficient algorithm to reduce the size of nondeterministic
Buchi word automata, while retaining their language. Additionally, we describe
methods to solve PSPACE-complete automata problems like universality,
equivalence and inclusion for much larger instances (1-3 orders of magnitude)
than before. This can be used to scale up applications of automata in formal
verification tools and decision procedures for logical theories. The algorithm
is based on new transition pruning techniques. These use criteria based on
combinations of backward and forward trace inclusions. Since these relations
are themselves PSPACE-complete, we describe methods to compute good
approximations of them in polynomial time. Extensive experiments show that the
average-case complexity of our algorithm scales quadratically. The size
reduction of the automata depends very much on the class of instances, but our
algorithm consistently outperforms all previous techniques by a wide margin. We
tested our algorithm on Buchi automata derived from LTL-formulae, many classes
of random automata and automata derived from mutual exclusion protocols, and
compared its performance to the well-known automata tool GOAL.Comment: 15 page
Sweep Complexity Revisited
We study the sweep complexity of DFA in one-way jumping mode answering
several questions posed earlier. This measure is the number of times in the
worst case that such machines have to return to the beginning of their input
after having skipped some of the symbols. The class of languages accepted by
these machines strictly includes the regular class and constant sweep
complexity allows exactly the acceptance of regular languages. However, we show
that there exist machines with higher than constant complexity still only
accepting regular languages and that in general the sweep complexity of an
automaton does not distinguish between accepting regular and non-regular
languages. We establish separation results for asymptotic classes defined by
this complexity measure and give a surprising exponential/logarithmic relation
between factors of certain inputs which can be verified by such machines.Comment: 12 pages, 8 figure
Revisiting the Complexity of Stability of Continuous and Hybrid Systems
We develop a framework to give upper bounds on the "practical" computational
complexity of stability problems for a wide range of nonlinear continuous and
hybrid systems. To do so, we describe stability properties of dynamical systems
using first-order formulas over the real numbers, and reduce stability problems
to the delta-decision problems of these formulas. The framework allows us to
obtain a precise characterization of the complexity of different notions of
stability for nonlinear continuous and hybrid systems. We prove that bounded
versions of the stability problems are generally decidable, and give upper
bounds on their complexity. The unbounded versions are generally undecidable,
for which we give upper bounds on their degrees of unsolvability
One Theorem to Rule Them All: A Unified Translation of LTL into {\omega}-Automata
We present a unified translation of LTL formulas into deterministic Rabin
automata, limit-deterministic B\"uchi automata, and nondeterministic B\"uchi
automata. The translations yield automata of asymptotically optimal size
(double or single exponential, respectively). All three translations are
derived from one single Master Theorem of purely logical nature. The Master
Theorem decomposes the language of a formula into a positive boolean
combination of languages that can be translated into {\omega}-automata by
elementary means. In particular, Safra's, ranking, and breakpoint constructions
used in other translations are not needed
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