A logic road from special relativity to general relativity

We present a streamlined axiom system of special relativity in first-order logic. From this axiom system we "derive" an axiom system of general relativity in two natural steps. We will also see how the axioms of special relativity transform into those of general relativity. This way we hope to make general relativity more accessible for the non-specialist

Determining What Growers Need to Comply with the Food Safety Modernization Act Produce Safety Rule

Extension educators have been enlisted to assist farmers in meeting requirements of the Food Safety Modernization Act Produce Safety Rule (PSR). Although food safety is a familiar topic for Extension educators, helping farmers learn how to prepare for PSR regulations is new. In this article, we describe a needs assessment conducted in the north central United States according to a modified Delphi approach. Results revealed unique characteristics of farmers in the region, least understood components of the PSR, preferences regarding educational tools, and the need for materials for varied audiences. Our process can be adapted for the purpose of determining how to assist growers in other regions in complying with the PSR

Adding an Abstraction Barrier to ZF Set Theory

Much mathematical writing exists that is, explicitly or implicitly, based on set theory, often Zermelo-Fraenkel set theory (ZF) or one of its variants. In ZF, the domain of discourse contains only sets, and hence every mathematical object must be a set. Consequently, in ZF, with the usual encoding of an ordered pair ${\langle a, b\rangle}$, formulas like ${\{a\} \in \langle a, b \rangle}$ have truth values, and operations like ${\mathcal P (\langle a, b\rangle)}$ have results that are sets. Such 'accidental theorems' do not match how people think about the mathematics and also cause practical difficulties when using set theory in machine-assisted theorem proving. In contrast, in a number of proof assistants, mathematical objects and concepts can be built of type-theoretic stuff so that many mathematical objects can be, in essence, terms of an extended typed ${\lambda}$-calculus. However, dilemmas and frustration arise when formalizing mathematics in type theory. Motivated by problems of formalizing mathematics with (1) purely set-theoretic and (2) type-theoretic approaches, we explore an option with much of the flexibility of set theory and some of the useful features of type theory. We present ZFP: a modification of ZF that has ordered pairs as primitive, non-set objects. ZFP has a more natural and abstract axiomatic definition of ordered pairs free of any notion of representation. This paper presents axioms for ZFP, and a proof in ZF (machine-checked in Isabelle/ZF) of the existence of a model for ZFP, which implies that ZFP is consistent if ZF is. We discuss the approach used to add this abstraction barrier to ZF

The diagonalization method in quantum recursion theory

As quantum parallelism allows the effective co-representation of classical mutually exclusive states, the diagonalization method of classical recursion theory has to be modified. Quantum diagonalization involves unitary operators whose eigenvalues are different from one.Comment: 15 pages, completely rewritte

Enhancing Symbolic Execution of Heap-based Programs with Separation Logic for Test Input Generation

Symbolic execution is a well established method for test input generation. Despite of having achieved tremendous success over numerical domains, existing symbolic execution techniques for heap-based programs are limited due to the lack of a succinct and precise description for symbolic values over unbounded heaps. In this work, we present a new symbolic execution method for heap-based programs based on separation logic. The essence of our proposal is context-sensitive lazy initialization, a novel approach for efficient test input generation. Our approach differs from existing approaches in two ways. Firstly, our approach is based on separation logic, which allows us to precisely capture preconditions of heap-based programs so that we avoid generating invalid test inputs. Secondly, we generate only fully initialized test inputs, which are more useful in practice compared to those partially initialized test inputs generated by the state-of-the-art tools. We have implemented our approach as a tool, called Java StarFinder, and evaluated it on a set of programs with complex heap inputs. The results show that our approach significantly reduces the number of invalid test inputs and improves the test coverage

Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

Introducing LoCo, a Logic for Configuration Problems

In this paper we present the core of LoCo, a logic-based high-level representation language for expressing configuration problems. LoCo shall allow to model these problems in an intuitive and declarative way, the dynamic aspects of configuration notwithstanding. Our logic enforces that configurations contain only finitely many components and reasoning can be reduced to the task of model construction.Comment: In Proceedings LoCoCo 2011, arXiv:1108.609

Quantum value indefiniteness

The indeterministic outcome of a measurement of an individual quantum is certified by the impossibility of the simultaneous, definite, deterministic pre-existence of all conceivable observables from physical conditions of that quantum alone. We discuss possible interpretations and consequences for quantum oracles.Comment: 19 pages, 2 tables, 2 figures; contribution to PC0

On the Formal Semantics of IF-Like Logics

In classical logics, the meaning of a formula is invariant with respect to the renaming of bound variables. This property, normally taken for granted, has been shown not to hold in the case of Information Friendly (IF) logics. In this paper we argue that this is not an inherent characteristic of these logics but a defect in the way in which the compositional semantics given by Hodges for the regular fragment was generalized to arbitrary formulas. We fix this by proposing an alternative formalization, based on a variation of the classical notion of valuation. Basic metatheoretical results are proven. We present these results for Hodges' slash logic (from which these can be easily transferred to other IF-like logics) and we also consider the flattening operator, for which we give novel game-theoretical semantics