Random Models and Solvable Skolem Classes

Philosoph

Random Models and the Maslov Class

Philosoph

Why There is no General Solution to the Problem of Software Verification

How can we be certain that software is reliable? Is there any method that can verify the correctness of software for all cases of interest? Computer scientists and software engineers have informally assumed that there is no fully general solution to the verification problem. In this paper, we survey approaches to the problem of software verification and offer a new proof for why there can be no general solution.The National Security Agency through the Science of Security initiative contract #H98230-18-D-0009

Filtration Games and Potentially Projective Modules

The notion of a \textbf{$\boldsymbol{\mathcal{C}}$-filtered} object, where $\mathcal{C}$ is some (typically small) collection of objects in a Grothendieck category, has become ubiquitous since the solution of the Flat Cover Conjecture around the year 2000. We introduce the \textbf{$\boldsymbol{\mathcal{C}}$-Filtration Game of length $\boldsymbol{\omega_1}$} on a module, paying particular attention to the case where $\mathcal{C}$ is the collection of all countably presented, projective modules. We prove that Martin's Maximum implies the determinacy of many $\mathcal{C}$-Filtration Games of length $\omega_1$, which in turn imply the determinacy of certain Ehrenfeucht-Fra\"iss\'{e} games of length $\omega_1$; this allows a significant strengthening of a theorem of Mekler-Shelah-Vaananen \cite{MR1191613}. Also, Martin's Maximum implies that if $R$ is a countable hereditary ring, the class of \textbf{$\boldsymbol{\sigma}$-closed potentially projective modules}---i.e., those modules that are projective in some $\sigma$-closed forcing extension of the universe---is closed under $<\aleph_2$-directed limits. We also give an example of a (ZFC-definable) class of abelian groups that, under the ordinary subgroup relation, constitutes an Abstract Elementary Class (AEC) with L\"owenheim-Skolem number $\aleph_1$ in some models in set theory, but fails to be an AEC in other models of set theory

Lower Complexity Bounds for Lifted Inference

One of the big challenges in the development of probabilistic relational (or probabilistic logical) modeling and learning frameworks is the design of inference techniques that operate on the level of the abstract model representation language, rather than on the level of ground, propositional instances of the model. Numerous approaches for such "lifted inference" techniques have been proposed. While it has been demonstrated that these techniques will lead to significantly more efficient inference on some specific models, there are only very recent and still quite restricted results that show the feasibility of lifted inference on certain syntactically defined classes of models. Lower complexity bounds that imply some limitations for the feasibility of lifted inference on more expressive model classes were established early on in (Jaeger 2000). However, it is not immediate that these results also apply to the type of modeling languages that currently receive the most attention, i.e., weighted, quantifier-free formulas. In this paper we extend these earlier results, and show that under the assumption that NETIME =/= ETIME, there is no polynomial lifted inference algorithm for knowledge bases of weighted, quantifier- and function-free formulas. Further strengthening earlier results, this is also shown to hold for approximate inference, and for knowledge bases not containing the equality predicate.Comment: To appear in Theory and Practice of Logic Programming (TPLP

Resolution-based decision procedures for subclasses of first-order logic

This thesis studies decidable fragments of first-order logic which are relevant to the field of nonclassical logic and knowledge representation. We show that refinements of resolution based on suitable liftable orderings provide decision procedures for the subclasses E+, K, and DK of first-order logic. By the use of semantics-based translation methods we can embed the description logic ALB and extensions of the basic modal logic K into fragments of first-order logic. We describe various decision procedures based on ordering refinements and selection functions for these fragments and show that a polynomial simulation of tableaux-based decision procedures for these logics is possible. In the final part of the thesis we develop a benchmark suite and perform an empirical analysis of various modal theorem provers.Diese Arbeit untersucht entscheidbare Fragmente der Logik erster Stufe, die mit nicht-klassischen Logiken und Wissensrepräsentationsformalismen im Zusammenhang stehen. Wir zeigen, daß Entscheidungsverfahren für die Teilklassen E+, K, und DK der Logik erster Stufe unter Verwendung von Resolution eingeschränkt durch geeignete liftbare Ordnungen realisiert werden können. Durch Anwendung von semantikbasierten Übersetzungsverfahren lassen sich die Beschreibungslogik ALB und Erweiterungen der Basismodallogik K in Teilklassen der Logik erster Stufe einbetten. Wir stellen eine Reihe von Entscheidungsverfahren auf der Basis von Resolution eingeschränkt durch liftbare Ordnungen und Selektionsfunktionen für diese Logiken vor und zeigen, daß eine polynomielle Simulation von tableaux-basierten Entscheidungsverfahren für diese Logiken möglich ist. Im abschließenden Teil der Arbeit führen wir eine empirische Untersuchung der Performanz verschiedener modallogischer Theorembeweiser durch