132 research outputs found
The umbilical cord of finite model theory
Model theory was born and developed as a part of mathematical logic. It has
various application domains but is not beholden to any of them. A priori, the
research area known as finite model theory would be just a part of model theory
but didn't turn out that way. There is one application domain -- relational
database management -- that finite model theory had been beholden to during a
substantial early period when databases provided the motivation and were the
main application target for finite model theory.
Arguably, finite model theory was motivated even more by complexity theory.
But the subject of this paper is how relational database theory influenced
finite model theory.
This is NOT a scholarly history of the subject with proper credits to all
participants. My original intent was to cover just the developments that I
witnessed or participated in. The need to make the story coherent forced me to
cover some additional developments.Comment: To be published in the Logic in Computer Science column of the
February 2023 issue of the Bulletin of the European Association for
Theoretical Computer Scienc
An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity
Previous work of the author [Rossmann\u2708] showed that the Homomorphism Preservation Theorem of classical model theory remains valid when its statement is restricted to finite structures. In this paper, we give a new proof of this result via a reduction to lower bounds in circuit complexity, specifically on the AC0 formula size of the colored subgraph isomorphism problem. Formally, we show the following: if a first-order sentence of quantifier-rank k is preserved under homomorphisms on finite structures, then it is equivalent on finite structures to an existential-positive sentence of quantifier-rank poly(k). Quantitatively, this improves the result of [Rossmann\u2708], where the upper bound on quantifier-rank is a non-elementary function of k
Primary Facets Of Order Polytopes
Mixture models on order relations play a central role in recent
investigations of transitivity in binary choice data. In such a model, the
vectors of choice probabilities are the convex combinations of the
characteristic vectors of all order relations of a chosen type. The five
prominent types of order relations are linear orders, weak orders, semiorders,
interval orders and partial orders. For each of them, the problem of finding a
complete, workable characterization of the vectors of probabilities is
crucial---but it is reputably inaccessible. Under a geometric reformulation,
the problem asks for a linear description of a convex polytope whose vertices
are known. As for any convex polytope, a shortest linear description comprises
one linear inequality per facet. Getting all of the facet-defining inequalities
of any of the five order polytopes seems presently out of reach. Here we search
for the facet-defining inequalities which we call primary because their
coefficients take only the values -1, 0 or 1. We provide a classification of
all primary, facet-defining inequalities of three of the five order polytopes.
Moreover, we elaborate on the intricacy of the primary facet-defining
inequalities of the linear order and the weak order polytopes
Recommended from our members
Philosophical aspects of chaos: definitions in mathematics, unpredictability, and the observational equivalence of deterministic and indeterministic descriptions
This dissertation is about some of the most important philosophical aspects of chaos research, a famous recent mathematical area of research about deterministic yet unpredictable and irregular, or even random behaviour. It consists of three parts.
First, as a basis for the dissertation, I examine notions of unpredictability in ergodic theory, and I ask what they tell us about the justification and formulation of mathematical definitions. The main account of the actual practice of justifying mathematical definitions is Lakatos's account on proof-generated definitions. By investigating notions of unpredictability in ergodic theory, I present two previously unidentified but common ways of justifying definitions. Furthermore, I criticise Lakatos's account as being limited: it does not acknowledge the interrelationships between the different kinds of justification, and it ignores the fact that various kinds of justification - not only proof-generation - are important.
Second, unpredictability is a central theme in chaos research, and it is widely claimed that chaotic systems exhibit a kind of unpredictability which is specific to chaos. However, I argue that the existing answers to the question "What is the unpredictability specific to chaos?" are wrong. I then go on to propose a novel answer, viz. the unpredictability specific to chaos is that for predicting any event all sufficiently past events are approximately probabilistically irrelevant.
Third, given that chaotic systems are strongly unpredictable, one is led to ask: are deterministic and indeterministic descriptions observationally equivalent, i.e., do they give the same predictions? I treat this question for measure-theoretic deterministic systems and stochastic processes, both of which are ubiquitous in science. I discuss and formalise the notion of observational equivalence.
By proving results in ergodic theory, I first show that for many measure-preserving deterministic descriptions there is an observationally equivalent indeterministic description, and that for all indeterministic descriptions there is an observationally equivalent deterministic description. I go on to show that strongly chaotic systems are even observationally equivalent to some of the most random stochastic processes encountered in science. For instance, strongly chaotic systems give the same predictions at every observation level as Markov processes or semi-Markov processes. All this illustrates that even kinds of deterministic and indeterministic descriptions which, intuitively, seem to give very different predictions are observationally equivalent. Finally, I criticise the claims in the previous philosophical literature on observational equivalence
- …