A theory of hyperfinite sets

We develop an axiomatic set theory -- the Theory of Hyperfinite Sets THS, which is based on the idea of existence of proper subclasses of big finite sets. We demonstrate how theorems of classical continuous mathematics can be transfered to THS, prove consistency of THS and present some applications.Comment: 28 page

Logical Characterizations of Behavioral Relations on Transition Systems of Probability Distributions

Probabilistic nondeterministic processes are commonly modeled as probabilistic LTSs (PLTSs). A number of logical characterizations of the main behavioral relations on PLTSs have been studied. In particular, Parma and Segala [2007] and Hermanns et al. [2011] define a probabilistic Hennessy-Milner logic interpreted over probability distributions, whose corresponding logical equivalence/preorder when restricted to Dirac distributions coincide with standard bisimulation/simulation between the states of a PLTS. This result is here extended by studying the full logical equivalence/preorder between (possibly non-Dirac) distributions in terms of a notion of bisimulation/simulation defined on a LTS whose states are distributions (dLTS). We show that the well-known spectrum of behavioral relations on nonprobabilistic LTSs as well as their corresponding logical characterizations in terms of Hennessy-Milner logic scales to the probabilistic setting when considering dLTSs

On Tarski's fixed point theorem

A concept of abstract inductive definition on a complete lattice is formulated and studied. As an application, a constructive and predicative version of Tarski's fixed point theorem is obtained.Comment: Proc. Amer. Math. Soc., to appea

Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness

We give a semantics for the lambda-calculus based on a topological duality theorem in nominal sets. A novel interpretation of lambda is given in terms of adjoints, and lambda-terms are interpreted absolutely as sets (no valuation is necessary)

Mechanizing Abstract Interpretation

It is important when developing software to verify the absence of undesirable behavior such as crashes, bugs and security vulnerabilities. Some settings require high assurance in verification results, e.g., for embedded software in automobiles or airplanes. To achieve high assurance in these verification results, formal methods are used to automatically construct or check proofs of their correctness. However, achieving high assurance for program analysis results is challenging, and current methods are ill suited for both complex critical domains and mainstream use. To verify the correctness of software we consider program analyzers---automated tools which detect software defects---and to achieve high assurance in verification results we consider mechanized verification---a rigorous process for establishing the correctness of program analyzers via computer-checked proofs. The key challenges to designing verified program analyzers are: (1) achieving an analyzer design for a given programming language and correctness property; (2) achieving an implementation for the design; and (3) achieving a mechanized verification that the implementation is correct w.r.t. the design. The state of the art in (1) and (2) is to use abstract interpretation: a guiding mathematical framework for systematically constructing analyzers directly from programming language semantics. However, achieving (3) in the presence of abstract interpretation has remained an open problem since the late 1990's. Furthermore, even the state-of-the art which achieves (3) in the absence of abstract interpretation suffers from the inability to be reused in the presence of new analyzer designs or programming language features. First, we solve the open problem which has prevented the combination of abstract interpretation (and in particular, calculational abstract interpretation) with mechanized verification, which advances the state of the art in designing, implementing, and verifying analyzers for critical software. We do this through a new mathematical framework Constructive Galois Connections which supports synthesizing specifications for program analyzers, calculating implementations from these induced specifications, and is amenable to mechanized verification. Finally, we introduce reusable components for implementing analyzers for a wide range of designs and semantics. We do this though two new frameworks Galois Transformers and Definitional Abstract Interpreters. These frameworks tightly couple analyzer design decisions, implementation fragments, and verification properties into compositional components which are (target) programming-language independent and amenable to mechanized verification. Variations in the analysis design are then recovered by simply re-assembling the combination of components. Using this framework, sophisticated program analyzers can be assembled by non-experts, and the result are guaranteed to be verified by construction

Generalized Strong Preservation by Abstract Interpretation

Standard abstract model checking relies on abstract Kripke structures which approximate concrete models by gluing together indistinguishable states, namely by a partition of the concrete state space. Strong preservation for a specification language L encodes the equivalence of concrete and abstract model checking of formulas in L. We show how abstract interpretation can be used to design abstract models that are more general than abstract Kripke structures. Accordingly, strong preservation is generalized to abstract interpretation-based models and precisely related to the concept of completeness in abstract interpretation. The problem of minimally refining an abstract model in order to make it strongly preserving for some language L can be formulated as a minimal domain refinement in abstract interpretation in order to get completeness w.r.t. the logical/temporal operators of L. It turns out that this refined strongly preserving abstract model always exists and can be characterized as a greatest fixed point. As a consequence, some well-known behavioural equivalences, like bisimulation, simulation and stuttering, and their corresponding partition refinement algorithms can be elegantly characterized in abstract interpretation as completeness properties and refinements

Large Cardinals and the Iterative Conception of Set

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles for the iterative conception, and assert that the length of the iterative stages is as long as possible. In this paper, we argue that whether or not large cardinal principles count as maximality principles depends on prior commitments concerning the richness of the subset forming operation. In particular we argue that there is a conception of maximality through absoluteness, that when given certain technical formulations, supports the idea that large cardinals are consistent, but false. On this picture, large cardinals are instead true in inner models and serve to restrict the subsets formed at successor stages