514 research outputs found
Infinitary Logic and Inductive Definability Over Finite Structures
The extensions of first-order logic with a least fixed point operators (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu [AV91b] investigated the relation of these two logics in the absence of an ordering, using a mchine model of generic computation. In particular, they showed that the two languages have equivalent expressive power if and only if P = PSPACE. These languages can also be seen as fragments of an infinitary logic where each formula has a bounded number of variables, Lω∞ω (see, for instance, [KV90]). We present a treatment of the results in [AV91b] from this point of view. In particular, we show that we can write a formula of FO + LFP and P from ordered structures to classes of structures where every element is definable. We also settle a conjecture mentioned in [AV91b] by showing that FO + LFP in properly contained in the polynomial time computable fragment of Lω∞ω, raising the question of whether the latter fragment is a recursively enumerable class
On the Scope of the Universal-Algebraic Approach to Constraint Satisfaction
The universal-algebraic approach has proved a powerful tool in the study of
the complexity of CSPs. This approach has previously been applied to the study
of CSPs with finite or (infinite) omega-categorical templates, and relies on
two facts. The first is that in finite or omega-categorical structures A, a
relation is primitive positive definable if and only if it is preserved by the
polymorphisms of A. The second is that every finite or omega-categorical
structure is homomorphically equivalent to a core structure. In this paper, we
present generalizations of these facts to infinite structures that are not
necessarily omega-categorical. (This abstract has been severely curtailed by
the space constraints of arXiv -- please read the full abstract in the
article.) Finally, we present applications of our general results to the
description and analysis of the complexity of CSPs. In particular, we give
general hardness criteria based on the absence of polymorphisms that depend on
more than one argument, and we present a polymorphism-based description of
those CSPs that are first-order definable (and therefore can be solved in
polynomial time).Comment: Extended abstract appeared at 25th Symposium on Logic in Computer
Science (LICS 2010). This version will appear in the LMCS special issue
associated with LICS 201
Tameness in least fixed-point logic and McColm's conjecture
We investigate four model-theoretic tameness properties in the context of
least fixed-point logic over a family of finite structures. We find that each
of these properties depends only on the elementary (i.e., first-order) limit
theory, and we completely determine the valid entailments among them. In
contrast to the context of first-order logic on arbitrary structures, the order
property and independence property are equivalent in this setting. McColm
conjectured that least fixed-point definability collapses to first-order
definability exactly when proficiency fails. McColm's conjecture is known to be
false in general. However, we show that McColm's conjecture is true for any
family of finite structures whose limit theory is model-theoretically tame
Class forcing, the forcing theorem and Boolean completions
The forcing theorem is the most fundamental result about set forcing, stating
that the forcing relation for any set forcing is definable and that the truth
lemma holds, that is everything that holds in a generic extension is forced by
a condition in the relevant generic filter. We show that both the definability
(and, in fact, even the amenability) of the forcing relation and the truth
lemma can fail for class forcing. In addition to these negative results, we
show that the forcing theorem is equivalent to the existence of a (certain kind
of) Boolean completion, and we introduce a weak combinatorial property
(approachability by projections) that implies the forcing theorem to hold.
Finally, we show that unlike for set forcing, Boolean completions need not be
unique for class forcing
Recent Advances in Σ-definability over Continuous Data Types
The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Sigma-formulas intended to represent computability over the real numbers. In order to adequately represent computability we extend the reals by the structure of hereditarily finite sets. In this setting it is crucial to consider the real numbers without equality since the equality test is undecidable over the reals. We prove Engeler's Lemma for Sigma-definability over the reals without the equality test which relates Sigma-definability with definability in the constructive infinitary language L_{omega_1 omega}. Thus, a relation over the real numbers is Sigma-definable if and only if it is definable by a disjunction of a recursively enumerable set of quantifier free formulas. This result reveals computational aspects of Sigma-definability and also gives topological characterisation of Sigma-definable relations over the reals without the equality test. We also illustrate how computability over the real numbers can be expressed in the language of Sigma-formulas
Linear Datalog and Bounded Path Duality of Relational Structures
In this paper we systematically investigate the connections between logics
with a finite number of variables, structures of bounded pathwidth, and linear
Datalog Programs. We prove that, in the context of Constraint Satisfaction
Problems, all these concepts correspond to different mathematical embodiments
of a unique robust notion that we call bounded path duality. We also study the
computational complexity implications of the notion of bounded path duality. We
show that every constraint satisfaction problem \csp(\best) with bounded path
duality is solvable in NL and that this notion explains in a uniform way all
families of CSPs known to be in NL. Finally, we use the results developed in
the paper to identify new problems in NL
Logic in the Tractatus
I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named.
There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably infinite, then the property of being a tautology is \Pi^1_1-complete. But third, it is only granted the assumption of countability that the class of tautologies is \Sigma_1-definable in set theory.
Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects
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