1,265 research outputs found
Decidability of the Clark's Completion Semantics for Monadic Programs and Queries
There are many different semantics for general logic programs (i.e. programs
that use negation in the bodies of clauses). Most of these semantics are Turing
complete (in a sense that can be made precise), implying that they are
undecidable. To obtain decidability one needs to put additional restrictions on
programs and queries. In logic programming it is natural to put restrictions on
the underlying first-order language. In this note we show the decidability of
the Clark's completion semantics for monadic general programs and queries.
To appear in Theory and Practice of Logic Programming (TPLP
Temporalized logics and automata for time granularity
Suitable extensions of the monadic second-order theory of k successors have
been proposed in the literature to capture the notion of time granularity. In
this paper, we provide the monadic second-order theories of downward unbounded
layered structures, which are infinitely refinable structures consisting of a
coarsest domain and an infinite number of finer and finer domains, and of
upward unbounded layered structures, which consist of a finest domain and an
infinite number of coarser and coarser domains, with expressively complete and
elementarily decidable temporal logic counterparts.
We obtain such a result in two steps. First, we define a new class of
combined automata, called temporalized automata, which can be proved to be the
automata-theoretic counterpart of temporalized logics, and show that relevant
properties, such as closure under Boolean operations, decidability, and
expressive equivalence with respect to temporal logics, transfer from component
automata to temporalized ones. Then, we exploit the correspondence between
temporalized logics and automata to reduce the task of finding the temporal
logic counterparts of the given theories of time granularity to the easier one
of finding temporalized automata counterparts of them.Comment: Journal: Theory and Practice of Logic Programming Journal Acronym:
TPLP Category: Paper for Special Issue (Verification and Computational Logic)
Submitted: 18 March 2002, revised: 14 Januari 2003, accepted: 5 September
200
On Decidability Properties of Local Sentences
Local (first order) sentences, introduced by Ressayre, enjoy very nice
decidability properties, following from some stretching theorems stating some
remarkable links between the finite and the infinite model theory of these
sentences. We prove here several additional results on local sentences. The
first one is a new decidability result in the case of local sentences whose
function symbols are at most unary: one can decide, for every regular cardinal
k whether a local sentence phi has a model of order type k. Secondly we show
that this result can not be extended to the general case. Assuming the
consistency of an inaccessible cardinal we prove that the set of local
sentences having a model of order type omega_2 is not determined by the
axiomatic system ZFC + GCH, where GCH is the generalized continuum hypothesi
An Application of the Feferman-Vaught Theorem to Automata and Logics for<br> Words over an Infinite Alphabet
We show that a special case of the Feferman-Vaught composition theorem gives
rise to a natural notion of automata for finite words over an infinite
alphabet, with good closure and decidability properties, as well as several
logical characterizations. We also consider a slight extension of the
Feferman-Vaught formalism which allows to express more relations between
component values (such as equality), and prove related decidability results.
From this result we get new classes of decidable logics for words over an
infinite alphabet.Comment: 24 page
Expansions of MSO by cardinality relations
We study expansions of the Weak Monadic Second Order theory of (N,<) by
cardinality relations, which are predicates R(X1,...,Xn) whose truth value
depends only on the cardinality of the sets X1, ...,Xn. We first provide a
(definable) criterion for definability of a cardinality relation in (N,<), and
use it to prove that for every cardinality relation R which is not definable in
(N,<), there exists a unary cardinality relation which is definable in (N,<,R)
and not in (N,<). These results resemble Muchnik and Michaux-Villemaire
theorems for Presburger Arithmetic. We prove then that + and x are definable in
(N,<,R) for every cardinality relation R which is not definable in (N,<). This
implies undecidability of the WMSO theory of (N,<,R). We also consider the
related satisfiability problem for the class of finite orderings, namely the
question whether an MSO sentence in the language {<,R} admits a finite model M
where < is interpreted as a linear ordering, and R as the restriction of some
(fixed) cardinality relation to the domain of M. We prove that this problem is
undecidable for every cardinality relation R which is not definable in (N,<).Comment: to appear in LMC
Spectra of Monadic Second-Order Formulas with One Unary Function
We establish the eventual periodicity of the spectrum of any monadic
second-order formula where:
(i) all relation symbols, except equality, are unary, and
(ii) there is only one function symbol and that symbol is unary
Monadic Second-Order Logic with Arbitrary Monadic Predicates
We study Monadic Second-Order Logic (MSO) over finite words, extended with
(non-uniform arbitrary) monadic predicates. We show that it defines a class of
languages that has algebraic, automata-theoretic and machine-independent
characterizations. We consider the regularity question: given a language in
this class, when is it regular? To answer this, we show a substitution property
and the existence of a syntactical predicate.
We give three applications. The first two are to give very simple proofs that
the Straubing Conjecture holds for all fragments of MSO with monadic
predicates, and that the Crane Beach Conjecture holds for MSO with monadic
predicates. The third is to show that it is decidable whether a language
defined by an MSO formula with morphic predicates is regular.Comment: Conference version: MFCS'14, Mathematical Foundations of Computer
Science Journal version: ToCL'17, Transactions on Computational Logi
Undecidable First-Order Theories of Affine Geometries
Tarski initiated a logic-based approach to formal geometry that studies
first-order structures with a ternary betweenness relation (\beta) and a
quaternary equidistance relation (\equiv). Tarski established, inter alia, that
the first-order (FO) theory of (R^2,\beta,\equiv) is decidable. Aiello and van
Benthem (2002) conjectured that the FO-theory of expansions of (R^2,\beta) with
unary predicates is decidable. We refute this conjecture by showing that for
all n>1, the FO-theory of monadic expansions of (R^2,\beta) is \Pi^1_1-hard and
therefore not even arithmetical. We also define a natural and comprehensive
class C of geometric structures (T,\beta), where T is a subset of R^2, and show
that for each structure (T,\beta) in C, the FO-theory of the class of monadic
expansions of (T,\beta) is undecidable. We then consider classes of expansions
of structures (T,\beta) with restricted unary predicates, for example finite
predicates, and establish a variety of related undecidability results. In
addition to decidability questions, we briefly study the expressivity of
universal MSO and weak universal MSO over expansions of (R^n,\beta). While the
logics are incomparable in general, over expansions of (R^n,\beta), formulae of
weak universal MSO translate into equivalent formulae of universal MSO.
This is an extended version of a publication in the proceedings of the 21st
EACSL Annual Conferences on Computer Science Logic (CSL 2012).Comment: 21 pages, 3 figure
Infinite and Bi-infinite Words with Decidable Monadic Theories
We study word structures of the form where is either
or , is the natural linear ordering on and
is a predicate on . In particular we show:
(a) The set of recursive -words with decidable monadic second order
theories is -complete.
(b) Known characterisations of the -words with decidable monadic
second order theories are transfered to the corresponding question for
bi-infinite words.
(c) We show that such "tame" predicates exist in every Turing degree.
(d) We determine, for , the number of predicates
such that and
are indistinguishable.
Through these results we demonstrate similarities and differences between
logical properties of infinite and bi-infinite words
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