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

A Multiset Rewriting Model for Specifying and Verifying Timing Aspects of Security Protocols

Catherine Meadows has played an important role in the advancement of formal methods for protocol security verification. Her insights on the use of, for example, narrowing and rewriting logic has made possible the automated discovery of new attacks and the shaping of new protocols. Meadows has also investigated other security aspects, such as, distance-bounding protocols and denial of service attacks. We have been greatly inspired by her work. This paper describes the use of Multiset Rewriting for the specification and verification of timing aspects of protocols, such as network delays, timeouts, timed intruder models and distance-bounding properties. We detail these timed features with a number of examples and describe decidable fragments of related verification problems

Revisiting Enumerative Instantiation

International audienceFormal methods applications often rely on SMT solvers to automatically discharge proof obligations. SMT solvers handle quantified formulas using incomplete heuristic techniques like E-matching, and often resort to model-based quantifier instantiation (MBQI) when these techniques fail. This paper revisits enumerative instantiation, a technique that considers instantiations based on exhaustive enumeration of ground terms. Although simple, we argue that enumer-ative instantiation can supplement other instantiation techniques and be a viable alternative to MBQI for valid proof obligations. We first present a stronger Her-brand Theorem, better suited as a basis for the instantiation loop used in SMT solvers; it furthermore requires considering less instances than classical Herbrand instantiation. Based on this result, we present different strategies for combining enumerative instantiation with other instantiation techniques in an effective way. The experimental evaluation shows that the implementation of these new techniques in the SMT solver CVC4 leads to significant improvements in several benchmark libraries, including many stemming from verification efforts

Fregean Logical Graphs

In \u201cGedankengef\ufcge\u201d Frege says that any two sentences of the form \u201cA and B\u201d and \u201cB and A\u201d have the same sense. In a 1906 letter to Husserl he says that sentences with the same sense should be represented in a perfect notation by one and the same formula. Frege\u2019s own notation, just like any linear notation for sentential logic, is not perfect in this sense, because in it \u201cA and B\u201d and \u201cB and A\u201d are represented by distinct formulas, as is any pair of logically equivalent compound conditionals. A notation for the sentential calculus that meets Frege\u2019s worries about conjunction, and indeed about any symmetric relation that there may be occasion to symbolize, is Peirce\u2019s Alpha graphs