11 research outputs found
Continuous Truth II: Reflections
Abstract. In the late 1960s, Dana Scott first showed how the Stone-Tarski topological interpretation of Heytingâs calculus could be extended to model intuitionistic analysis; in particular Brouwerâs continuity prin-ciple. In the early â80s we and others outlined a general treatment of non-constructive objects, using sheaf modelsâconstructions from topos theoryâto model not only Brouwerâs non-classical conclusions, but also his creation of ânew mathematical entitiesâ. These categorical models are intimately related to, but more general than Scottâs topological model. The primary goal of this paper is to consider the question of iterated extensions. Can we derive new insights by repeating the second act? In Continuous Truth I, presented at Logic Colloquium â82 in Florence, we showed that general principles of continuity, local choice and local com-pactness hold in the gros topos of sheaves over the category of separable locales equipped with the open cover topology. We touched on the question of iteration. Here we develop a more gen-eral analysis of iterated categorical extensions, that leads to a reflection schema for statements of predicative analysis. We also take the opportunity to revisit some aspects of both Continuous Truth I and Formal Spaces (Fourman & Grayson 1982), and correct two long-standing errors therein
Validity and Entailment in Modal and Propositional Dependence Logics
The computational properties of modal and propositional dependence logics have been extensively studied over the past few years, starting from a result by Sevenster showing NEXPTIME-completeness of the satisfiability problem for modal dependence logic. Thus far, however, the validity and entailment properties of these logics have remained uncharacterised to a great extent. This paper establishes a complete classification of the complexity of validity and entailment in modal and propositional dependence logics. In particular, we address the question of the complexity of validity in modal dependence logic. By showing that it is NEXPTIME-complete we refute an earlier conjecture proposing a higher complexity for the problem
Complexity thresholds in inclusion logic
Inclusion logic differs from many other logics of dependence and independence in that it can only describe polynomial-time properties. In this article we examine more closely connections between syntactic fragments of inclusion logic and different complexity classes. Our focus is on two computational problems: maximal subteam membership and the model checking problem for a fixed inclusion logic formula. We show that very simple quantifier-free formulae with one or two inclusion atoms generate instances of these problems that are complete for (non-deterministic) logarithmic space and polynomial time. We also present a safety game for the maximal subteam membership problem and use it to investigate this problem over teams in which one variable is a key. Furthermore, we relate our findings to consistent query answering over inclusion dependencies, and present a fragment of inclusion logic that captures non-deterministic logarithmic space in ordered models. (C) 2021 The Author(s). Published by Elsevier Inc.Peer reviewe
On the Complexity of Team Logic and its Two-Variable Fragment
We study the logic FO(~), the extension of first-order logic with team
semantics by unrestricted Boolean negation. It was recently shown
axiomatizable, but otherwise has not yet received much attention in questions
of computational complexity.
In this paper, we consider its two-variable fragment FO2(~) and prove that
its satisfiability problem is decidable, and in fact complete for the recently
introduced non-elementary class TOWER(poly). Moreover, we classify the
complexity of model checking of FO(~) with respect to the number of variables
and the quantifier rank, and prove a dichotomy between PSPACE- and
ATIME-ALT(exp, poly)-completeness.
To achieve the lower bounds, we propose a translation from modal team logic
MTL to FO2(~) that extends the well-known standard translation from modal logic
ML to FO2. For the upper bounds, we translate to a fragment of second-order
logic
Complexity Thresholds in Inclusion Logic
Logics with team semantics provide alternative means for logical
characterization of complexity classes. Both dependence and independence logic
are known to capture non-deterministic polynomial time, and the frontiers of
tractability in these logics are relatively well understood. Inclusion logic is
similar to these team-based logical formalisms with the exception that it
corresponds to deterministic polynomial time in ordered models. In this article
we examine connections between syntactical fragments of inclusion logic and
different complexity classes in terms of two computational problems: maximal
subteam membership and the model checking problem for a fixed inclusion logic
formula. We show that very simple quantifier-free formulae with one or two
inclusion atoms generate instances of these problems that are complete for
(non-deterministic) logarithmic space and polynomial time. Furthermore, we
present a fragment of inclusion logic that captures non-deterministic
logarithmic space in ordered models