24 research outputs found
Linearly bounded infinite graphs
Linearly bounded Turing machines have been mainly studied as acceptors for
context-sensitive languages. We define a natural class of infinite automata
representing their observable computational behavior, called linearly bounded
graphs. These automata naturally accept the same languages as the linearly
bounded machines defining them. We present some of their structural properties
as well as alternative characterizations in terms of rewriting systems and
context-sensitive transductions. Finally, we compare these graphs to rational
graphs, which are another class of automata accepting the context-sensitive
languages, and prove that in the bounded-degree case, rational graphs are a
strict sub-class of linearly bounded graphs
The synchronized graphs trace the context-sensitive languages
International audienceMorvan and Stirling have proved that the context-sensitive languages are exactly the traces of graphs de ned by transducers with labelled nal states. We prove that this result is still true if we restrict to the traces of graphs de ned by synchronized transducers with labelled nal states. From their construction, we deduce that the context-sensitive languages are the languages of path labels leading from and to rational vertex sets of letter-to-letter rational graphs
Covering of ordinals
The paper focuses on the structure of fundamental sequences of ordinals
smaller than . A first result is the construction of a monadic
second-order formula identifying a given structure, whereas such a formula
cannot exist for ordinals themselves. The structures are precisely classified
in the pushdown hierarchy. Ordinals are also located in the hierarchy, and a
direct presentation is given.Comment: Accepted at FSTTCS'0
The equational theory of the natural join and inner union is decidable
The natural join and the inner union operations combine relations of a
database. Tropashko and Spight [24] realized that these two operations are the
meet and join operations in a class of lattices, known by now as the relational
lattices. They proposed then lattice theory as an algebraic approach to the
theory of databases, alternative to the relational algebra. Previous works [17,
22] proved that the quasiequational theory of these lattices-that is, the set
of definite Horn sentences valid in all the relational lattices-is undecidable,
even when the signature is restricted to the pure lattice signature. We prove
here that the equational theory of relational lattices is decidable. That, is
we provide an algorithm to decide if two lattice theoretic terms t, s are made
equal under all intepretations in some relational lattice. We achieve this goal
by showing that if an inclusion t s fails in any of these lattices, then
it fails in a relational lattice whose size is bound by a triple exponential
function of the sizes of t and s.Comment: arXiv admin note: text overlap with arXiv:1607.0298
Model Checking Synchronized Products of Infinite Transition Systems
Formal verification using the model checking paradigm has to deal with two
aspects: The system models are structured, often as products of components, and
the specification logic has to be expressive enough to allow the formalization
of reachability properties. The present paper is a study on what can be
achieved for infinite transition systems under these premises. As models we
consider products of infinite transition systems with different synchronization
constraints. We introduce finitely synchronized transition systems, i.e.
product systems which contain only finitely many (parameterized) synchronized
transitions, and show that the decidability of FO(R), first-order logic
extended by reachability predicates, of the product system can be reduced to
the decidability of FO(R) of the components. This result is optimal in the
following sense: (1) If we allow semifinite synchronization, i.e. just in one
component infinitely many transitions are synchronized, the FO(R)-theory of the
product system is in general undecidable. (2) We cannot extend the expressive
power of the logic under consideration. Already a weak extension of first-order
logic with transitive closure, where we restrict the transitive closure
operators to arity one and nesting depth two, is undecidable for an
asynchronous (and hence finitely synchronized) product, namely for the infinite
grid.Comment: 18 page
Synchronous Subsequentiality and Approximations to Undecidable Problems
We introduce the class of synchronous subsequential relations, a subclass of
the synchronous relations which embodies some properties of subsequential
relations. If we take relations of this class as forming the possible
transitions of an infinite automaton, then most decision problems (apart from
membership) still remain undecidable (as they are for synchronous and
subsequential rational relations), but on the positive side, they can be
approximated in a meaningful way we make precise in this paper. This might make
the class useful for some applications, and might serve to establish an
intermediate position in the trade-off between issues of expressivity and
(un)decidability.Comment: In Proceedings GandALF 2015, arXiv:1509.0685
Boolean Algebras from Trace Automata
We consider trace automata. Their vertices are Mazurkiewicz traces and they accept finite words. Considering the length of a trace as the length of its Foata normal form, we define the operations of level-length synchronization and of superposition of trace automata. We show that if a family F of trace automata is closed under these operations, then for any deterministic automaton H in F, the word languages accepted by the deterministic automata of F that are length-reducible to H form a Boolean algebra. We show that the family of trace suffix automata with level-regular contexts and the subfamily of vector addition systems satisfy these closure properties. In particular, this yields various Boolean algebras of word languages accepted by deterministic vector addition systems
Concurrency Makes Simple Theories Hard
A standard way of building concurrent systems is by composing several
individual processes by product operators. We show that even the simplest notion of product operators (i.e. asynchronous products) suffices to increase the complexity of model checking simple logics like Hennessy-Milner (HM) logic and its extension with the reachability operator (EF-logic) from PSPACE to nonelementary. In particular, this nonelementary jump happens for EF-logic when we consider individual processes represented by pushdown systems (indeed, even with only one control state). Using this result, we prove nonelementary lower bounds on the size of formula decompositions provided by Feferman-Vaught (de)compositional methods for HM and EF logics, which reduce theories of asynchronous products to theories of the components. Finally, we show that the same nonelementary lower bounds also hold when we consider the relativization of such compositional methods to finite systems
Cuts for circular proofs: semantics and cut-elimination
One of the authors introduced in [Santocanale, FoSSaCS, 2002] a calculus of circular proofs for studying the computability arising from the following categorical operations: finite products, finite coproducts, initial algebras, final coalgebras. The calculus presented [Santocanale, FoSSaCS, 2002] is cut-free; even if sound and complete for provability, it lacked an important property for the semantics of proofs, namely fullness w.r.t. the class of intended categorical models (called mu-bicomplete categories in [Santocanale, ITA, 2002]).
In this paper we fix this problem by adding the cut rule to the calculus and by modifying accordingly the syntactical constraint ensuring soundness of proofs. The enhanced proof system fully represents arrows of the canonical model (a free mu-bicomplete category). We also describe a cut-elimination procedure as a a model of computation arising from the above mentioned categorical operations. The procedure constructs a cut-free proof-tree with possibly infinite branches out of a finite circular proof with cuts