Colloquium

Research seminars

VIGRE Working Groups

Clifford lectures

"A Hofmann-Mislove Theorem for Bitopological Spaces "

The Hofmann-Mislove Theorem asserts that the family of compact, saturated subsets of a sober space X, ordered by reverse containment, is isomorphic to the family of Scott-open filters of the lattice O(X) of open subsets of X. In this paper the authors devise an analogous result in the setting of bitopological, bisober spaces - sets endowed with two sober topologies. In the process of deriving this generalization, the authors present a Stone duality for bitopological spaces that is analogous to the duality between topological spaces and frames.

"A Hofmann-Mislove Theorem for Bitopological Spaces: Part II "

We will finish our discussion on the paper "A Hofmann-Mislove Theorem for bitopological spaces" by Andrew Moshier and Achim Jung. We will briefly review the main points from the previous talks, which led up to a logical interpretation of the original Hofmann-Mislove Theorem, and how this interpretation motivates looking at a similar theorem on bitopological spaces. Then we will build up an analogue of the theorem on bitopological spaces, and look at some examples of what the resulting objects look like.

"Probabilistic Input/Output Automata and Domains"

Input/Output automata are systems for modeling computational processes. They were initially devised by Nancy Lynch (MIT) and her students who also have pioneered their application to a wealth of areas. A probabilistic version - where the result of a process taking actions results in a probability distribution over the states of the process - were developed by Roberto Segala as part of his PhD, written under Nancy's direction. These automata have recently been applied to modeling cryptoprotocols. They are unique in that they allow reasoning about these abstract protocols while also supporting detailed analyses of the lower level mechanisms that underlie modern cryptography. In this talk, I will introduce these systems with examples, and show how domain theory provides a simple tool for analyzing their behavior.

"Probabilistic Input/Output Automata and Domains- Part II "

Input/Output automata are systems for modeling computational processes. They were initially devised by Nancy Lynch (MIT) and her students who also have pioneered their application to a wealth of areas. A probabilistic version - where the result of a process taking actions results in a probability distribution over the states of the process - were developed by Roberto Segala as part of his PhD, written under Nancy's direction. These automata have recently been applied to modeling cryptoprotocols. They are unique in that they allow reasoning about these abstract protocols while also supporting detailed analyses of the lower level mechanisms that underlie modern cryptography. In this talk, I will introduce these systems with examples, and show how domain theory provides a simple tool for analyzing their behavior.

"Computational Soundness of First Order Logic-based Symbolic
Analysis of Cryptographic Protocols"

In the past few years it has become a central question of cryptography how relevant purely symbolic analysis of security protocols are to reality. This question can be analyzed by linking the symbolic approach to the so-called computational approach as the latter is a more detailed description of reality, involving probability and complexity theory. Several attempts have been made to link these two approaches. Unfortunately, most of them are not done with sufficient mathematical rigor, and some of the results are questionable. This time, we look at the case when symbolic analysis is done via first order logic. We review some of the basic notions, look at problems with a recent attempt (Computational PCL by Datta et al.) to provide computational semantics to the syntax, and suggest a new way of defining the semantics to fix those problems.

In this talk, I'll describe how to represent discrete binary, memoryless channels as a domain with a monoid structure, and show that the capacity function is a measurement on this domain.