239 research outputs found
Transforming structures by set interpretations
We consider a new kind of interpretation over relational structures: finite
sets interpretations. Those interpretations are defined by weak monadic
second-order (WMSO) formulas with free set variables. They transform a given
structure into a structure with a domain consisting of finite sets of elements
of the orignal structure. The definition of these interpretations directly
implies that they send structures with a decidable WMSO theory to structures
with a decidable first-order theory. In this paper, we investigate the
expressive power of such interpretations applied to infinite deterministic
trees. The results can be used in the study of automatic and tree-automatic
structures.Comment: 36 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
The continuity of monadic stream functions
Brouwerâs continuity principle states that all functions from infinite sequences of naturals to naturals are continuous, that is, for every sequence the result depends only on a finite initial segment. It is an intuitionistic axiom that is incompatible with classical mathematics. Recently MartÌĂn EscardĂł proved that it is also inconsistent in type theory. We propose a reformulation of the continuity principle that may be more faithful to the original meaning by Brouwer. It applies to monadic streams, potentially unending sequences of values produced by steps triggered by a monadic action, possibly involving side effects. We consider functions on them that are uniform, in the sense that they operate in the same way independently of the particular monad that provides the specific side effects. Formally this is done by requiring a form of naturality in the monad. Functions on monadic streams have not only a foundational importance, but have also practical applications in signal processing and reactive programming. We give algorithms to determine the modulus of continuity of monadic stream functions and to generate dialogue trees for them (trees whose nodes and branches describe the interaction of the process with the environment)
First Order Theories of Some Lattices of Open Sets
We show that the first order theory of the lattice of open sets in some
natural topological spaces is -equivalent to second order arithmetic. We
also show that for many natural computable metric spaces and computable domains
the first order theory of the lattice of effectively open sets is undecidable.
Moreover, for several important spaces (e.g., , , and the
domain ) this theory is -equivalent to first order arithmetic
Gradual Certified Programming in Coq
Expressive static typing disciplines are a powerful way to achieve
high-quality software. However, the adoption cost of such techniques should not
be under-estimated. Just like gradual typing allows for a smooth transition
from dynamically-typed to statically-typed programs, it seems desirable to
support a gradual path to certified programming. We explore gradual certified
programming in Coq, providing the possibility to postpone the proofs of
selected properties, and to check "at runtime" whether the properties actually
hold. Casts can be integrated with the implicit coercion mechanism of Coq to
support implicit cast insertion a la gradual typing. Additionally, when
extracting Coq functions to mainstream languages, our encoding of casts
supports lifting assumed properties into runtime checks. Much to our surprise,
it is not necessary to extend Coq in any way to support gradual certified
programming. A simple mix of type classes and axioms makes it possible to bring
gradual certified programming to Coq in a straightforward manner.Comment: DLS'15 final version, Proceedings of the ACM Dynamic Languages
Symposium (DLS 2015
Thin MSO with a Probabilistic Path Quantifier
This paper is about a variant of MSO on infinite trees where:
- there is a quantifier "zero probability of choosing a path pi in 2^{omega} which makes omega(pi) true";
- the monadic quantifiers range over sets with countable topological closure.
We introduce an automaton model, and show that it captures the logic
On Almost Future Temporal Logics
Dedicated to Yuri Gurevich on the occasion of his 75th birthday Abstract. Kampâs theorem established the expressive completeness of the temporal modalities Until and Since for the First-Order Monadic Logic of Order (FOMLO) over real and natural time flows. Over natural time, a single future modality (Until) is sufficient to express all future FOMLO formulas. These are formulas whose truth value at any moment is determined by what happens from that moment on. Yet this fails to extend to real time domains: here no finite basis of future modalities can express all future FOMLO formulas. Almost future formulas extend future formulas; they depend just on the very very near past, and are independent of the rest of the past. For almost future formulas finiteness is recovered over Dedekind complete time flows. In this paper we show that there is no temporal logic with finitely many modalities which is expressively complete for the almost future fragment of FOMLO over all linear flows.
Combinatorial Expressions and Lower Bounds
A new paradigm, called combinatorial expressions, for computing functions expressing properties over infinite domains is introduced. The main result is a generic technique, for showing indefinability of
certain functions by the expressions, which uses a result, namely Hales-Jewett theorem, from Ramsey theory. An application of the technique for proving inexpressibility results for logics on metafinite structures is given. Some extensions and normal forms are also presented
- âŠ