18 research outputs found
Automatic structures of bounded degree revisited
The first-order theory of a string automatic structure is known to be
decidable, but there are examples of string automatic structures with
nonelementary first-order theories. We prove that the first-order theory of a
string automatic structure of bounded degree is decidable in doubly exponential
space (for injective automatic presentations, this holds even uniformly). This
result is shown to be optimal since we also present a string automatic
structure of bounded degree whose first-order theory is hard for 2EXPSPACE. We
prove similar results also for tree automatic structures. These findings close
the gaps left open in a previous paper of the second author by improving both,
the lower and the upper bounds.Comment: 26 page
An optimal construction of Hanf sentences
We give the first elementary construction of equivalent formulas in Hanf
normal form. The triply exponential upper bound is complemented by a matching
lower bound
Observation and Distinction. Representing Information in Infinite Games
We compare two approaches for modelling imperfect information in infinite games by using finite-state automata. The first, more standard approach views information as the result of an observation process driven by a sequential Mealy machine. In contrast, the second approach features indistinguishability relations described by synchronous two-tape automata.
The indistinguishability-relation model turns out to be strictly more expressive than the one based on observations. We present a characterisation of the indistinguishability relations that admit a representation as a finite-state observation function. We show that the characterisation is decidable, and give a procedure to construct a corresponding Mealy machine whenever one exists
The First-Order Theory of Ground Tree Rewrite Graphs
We prove that the complexity of the uniform first-order theory
of ground tree rewrite graphs is in ATIME(2^{2^{poly(n)}},O(n). Providing a matching lower bound, we show that there is some
fixed ground tree rewrite graph whose first-order theory is hard
for ATIME(2^{2^{poly(n)}},poly(n)) with respect to logspace reductions. Finally, we prove that there exists a fixed ground tree rewrite graph together with a single unary predicate in form of a regular tree language such that the resulting structure has a non-elementary first-order theory
The First-Order Theory of Ground Tree Rewrite Graphs
We prove that the complexity of the uniform first-order theory of ground tree
rewrite graphs is in ATIME(2^{2^{poly(n)}},O(n)). Providing a matching lower
bound, we show that there is some fixed ground tree rewrite graph whose
first-order theory is hard for ATIME(2^{2^{poly(n)}},poly(n)) with respect to
logspace reductions. Finally, we prove that there exists a fixed ground tree
rewrite graph together with a single unary predicate in form of a regular tree
language such that the resulting structure has a non-elementary first-order
theory.Comment: accepted for Logical Methods in Computer Scienc
Ehrenfeucht-Fraïssé goes elementarily automatic for structures of bounded degree
International audienceMany relational structures are automatically presentable, i.e. elements of the domain can be seen as words over a finite alphabet and equality and other atomic relations are represented with finite automata. The first-order theories over such structures are known to be primitive recursive, which is shown by the inductive construction of an automaton representing any relation definable in the first-order logic. We propose a general method based on Ehrenfeucht-Fraïssé games to give upper bounds on the size of these automata and on the time required to build them. We apply this method for two different automatic structures which have elementary decision procedures, Presburger Arithmetic and automatic structures of bounded degree. For the latter no upper bound on the size of the automata was known. We conclude that the very general and simple automata-based algorithm works well to decide the first-order theories over these structures
The Cayley-graph of the queue monoid: logic and decidability
We investigate the decidability of logical aspects of graphs that arise as Cayley-graphs of the so-called queue monoids. These monoids model the behavior of the classical (reliable) fifo-queues. We answer a question raised by Huschenbett, Kuske, and Zetzsche and prove the decidability of the first-order theory of these graphs with the help of an - at least for the authors - new combination of the well-known method from Ferrante and Rackoff and an automata-based approach. On the other hand, we prove that the monadic second-order of the queue monoid's Cayley-graph is undecidable