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Michael Stay

Publications and source records attributed to Michael Stay.

10 recordsLinked to original sources
arXiv2022

Validating Labelled State Transition and Message Production Systems: A Theory for Modelling Faulty Distributed Systems

Modeling and formally reasoning about distributed systems with faults is a challenging task. To address this problem, we propose the theory of Validating Labeled State transition and Message production systems (VLSMs). The theory of VLSMs provides a general approach to describing and verifying properties of distributed protocols whose executions are subject to faults, supporting a correct-by-construction system development methodology. The central focus of our investigation is equivocation, a mode of faulty behavior that we formally model, reason about, and then show how to detect from durable evidence that may be available locally to system components. Equivocating components exhibit behavior that is inconsistent with single-trace system executions, while also only interacting with other components by sending and receiving valid messages. Components of system are called validators for that system if their validity constraints validate that the messages they receive are producible by the system. Our main result shows that for systems of validators, the effect that Byzantine components can have on honest validators is precisely identical to the effect that equivocating components can have on non-equivocating validators. Therefore, for distributed systems of potentially faulty validators, replacing Byzantine components with equivocating components has no material analytical consequences, and forms the basis of a sound alternative foundation to Byzantine fault tolerance analysis. All of the results and examples in the paper have been formalised and checked in the Coq proof assistant.

cs.DC↗
arXiv2021

Native Type Theory

Native type systems are those in which type constructors are derived from term constructors, as well as the constructors of predicate logic and intuitionistic type theory. We present a method to construct native type systems for a broad class of languages, lambda-theories with equality, by embedding such a theory into the internal language of its topos of presheaves. Native types provide total specification of the structure of terms; and by internalizing transition systems, native type systems serve to reason about structure and behavior simultaneously. The construction is functorial, thereby providing a shared framework of higher-order reasoning for many languages, including programming languages.

cs.LO↗
arXiv2018

Casanova

We introduce Casanova, a leaderless optimistic consensus protocol designed for a permissioned blockchain. Casanova produces blocks in a DAG rather than a chain, and combines voting rounds with block production by singling out individual conflicting transactions.

cs.CR↗
arXiv2017

Representing operational semantics with enriched Lawvere theories

Many term calculi, like lambda calculus or pi calculus, involve binders for names, and the mathematics of bound variable names is subtle. Schoenfinkel introduced the SKI combinator calculus in 1924 to clarify the role of quantified variables in intuitionistic logic by eliminating them. Yoshida demonstrated how to eliminate the bound names coming from the input prefix in the asynchronous pi calculus, but her combinators still depend on the new operator to bind names. Recently, Meredith and Stay showed how to modify Yoshida's combinators by replacing new and replication with reflective operators to provide the first combinator calculus with no bound names into which the asynchronous pi calculus has a faithful embedding. Here we provide an alternative set of combinators built from SKI plus reflection that also eliminates all nominal phenomena, yet provides a faithful embedding of a reflective higher-order pi calculus. We show that with the nominal features effectively eliminated as syntactic sugar, multisorted Lawvere theories enriched over graphs suffice to capture the operational semantics of the calculus.

cs.LO↗
arXiv2017

Name-free combinators for concurrency

Yoshida demonstrated how to eliminate the bound names coming from the input prefix in the asynchronous pi calculus, but her combinators still depend on the "new" operator to bind names. We modify Yoshida's combinators by replacing "new" and replication with reflective operators to provide the first combinator calculus with no bound names into which the asynchronous pi calculus has a faithful embedding. We also show that multisorted Lawvere theories enriched over graphs suffice to capture the operational semantics of the calculus.

cs.LO↗
arXiv2013

Compact Closed Bicategories

A compact closed bicategory is a symmetric monoidal bicategory where every object is equipped with a weak dual. The unit and counit satisfy the usual "zig-zag" identities of a compact closed category only up to natural isomorphism, and the isomorphism is subject to a coherence law. We give several examples of compact closed bicategories, then review previous work. In particular, Day and Street defined compact closed bicategories indirectly via Gray monoids and then appealed to a coherence theorem to extend the concept to bicategories; we restate the definition directly. We prove that given a 2-category T with finite products and weak pullbacks, the bicategory of objects of C, spans, and isomorphism classes of maps of spans is compact closed. As corollaries, the bicategory of spans of sets and certain bicategories of "resistor networks" are compact closed.

math.CT↗
arXiv2005

Generalized Number Derivatives

We generalize the concept of a number derivative, and examine one particular instance of a deformed number derivative for finite field elements. We find that the derivative is linear when the deformation is a Frobenius map and go on to examine some of its basic properties.

math.NT↗
arXiv2005

Very Simple Chaitin Machines for Concrete AIT

In 1975, Chaitin introduced his celebrated Omega number, the halting probability of a universal Chaitin machine, a universal Turing machine with a prefix-free domain. The Omega number's bits are {\em algorithmically random}--there is no reason the bits should be the way they are, if we define ``reason'' to be a computable explanation smaller than the data itself. Since that time, only {\em two} explicit universal Chaitin machines have been proposed, both by Chaitin himself. Concrete algorithmic information theory involves the study of particular universal Turing machines, about which one can state theorems with specific numerical bounds, rather than include terms like O(1). We present several new tiny Chaitin machines (those with a prefix-free domain) suitable for the study of concrete algorithmic information theory. One of the machines, which we call Keraia, is a binary encoding of lambda calculus based on a curried lambda operator. Source code is included in the appendices. We also give an algorithm for restricting the domain of blank-endmarker machines to a prefix-free domain over an alphabet that does not include the endmarker; this allows one to take many universal Turing machines and construct universal Chaitin machines from them.

cs.IT↗
arXiv2000

Artificial Orbitals and a Solution to Grover's Problem

By allowing measurements of observables other than the state of the qubits in a quantum computer, one can find eigenvectors very quickly. If a unitary operation U is implemented as a time-independent Hamiltonian, for instance, one can collapse the state of the computer to a nearby eigenvector of U with a measurement of the energy. We examine some recent proposals for quantum computation using time-independent Hamiltonians and show how to convert them into ``artificial orbitals'' whose energy eigenstates match those of U. This system can be used to find eigenvectors and eigenvalues with a single measurement. We apply this technique to Grover's algorithm and the continuous variant proposed by Farhi and Gutmann.

quant-ph↗