A foundation for higher-order concurrent constraint programming

We present the gamma-calculus, a computational calculus for higher-order concurrent programming. The calculus can elegantly express higher-order functions (both eager and lazy) and concurrent objects with encapsulated state and multiple inheritance. The primitives of the gamma-calculus are logic variables, names, procedural abstraction, and cells. Cells provide a notion of state that is fully compatible with concurrency and constraints. Although it does not have a dedicated communication primitive, the gamma-calculus can elegantly express one-to-many and many-to-one communication. There is an interesting relationship between the gamma-calculus and the pi-calculus: The gamma-calculus is subsumed by a calculus obtained by extending the asynchronous and polyadic pi-calculus with logic variables. The gamma-calculus can be extended with primitives providing for constraint-based problem solving in the style of logic programming. A such extended gamma-calculus has the remarkable property that it combines first-order constraints with higher-order programming

An extension of the Deutsch-Jozsa algorithm to arbitrary qudits

Recent advances in quantum computational science promise substantial improvements in the speed with which certain classes of problems can be computed. Various algorithms that utilize the distinctively non-classical characteristics of quantum mechanics have been formulated to take advantage of this promising new approach to computation. One such algorithm was formulated by David Deutsch and Richard Jozsa. By measuring the output of a quantum network that implements this algorithm, it is possible to determine with N – 1 measurements certain global properties of a function f(x), where N is the number of network inputs. Classically, it may not be possible to determine these same properties without evaluating f(x) a number of times that rises exponentially as N increases. Hitherto, the potential power of this algorithm has been explored in the context of qubits, the quantum computational analogue of classical bits. However, just as one can conceive of classical computation in the context of non-binary logic, such as ternary or quaternary logic, so also can one conceive of corresponding higher-order quantum computational equivalents.This thesis investigates the behaviour of the Deutsch-Jozsa algorithm in the context of these higher-order quantum computational forms of logic and explores potential applications for this algorithm. An important conclusion reached is that, not only can the Deutsch-Jozsa algorithm’s known computational advantages be formulated in more general terms, but also a new algorithmic property is revealed with potential practical applications

Mechanizing Principia Logico-Metaphysica in Functional Type Theory

Principia Logico-Metaphysica contains a foundational logical theory for metaphysics, mathematics, and the sciences. It includes a canonical development of Abstract Object Theory [AOT], a metaphysical theory (inspired by ideas of Ernst Mally, formalized by Zalta) that distinguishes between ordinary and abstract objects. This article reports on recent work in which AOT has been successfully represented and partly automated in the proof assistant system Isabelle/HOL. Initial experiments within this framework reveal a crucial but overlooked fact: a deeply-rooted and known paradox is reintroduced in AOT when the logic of complex terms is simply adjoined to AOT's specially-formulated comprehension principle for relations. This result constitutes a new and important paradox, given how much expressive and analytic power is contributed by having the two kinds of complex terms in the system. Its discovery is the highlight of our joint project and provides strong evidence for a new kind of scientific practice in philosophy, namely, computational metaphysics. Our results were made technically possible by a suitable adaptation of Benzm\"uller's metalogical approach to universal reasoning by semantically embedding theories in classical higher-order logic. This approach enables one to reuse state-of-the-art higher-order proof assistants, such as Isabelle/HOL, for mechanizing and experimentally exploring challenging logics and theories such as AOT. Our results also provide a fresh perspective on the question of whether relational type theory or functional type theory better serves as a foundation for logic and metaphysics.Comment: 14 pages, 6 figures; preprint of article with same title to appear in The Review of Symbolic Logi

Divergences on Monads for Relational Program Logics

Several relational program logics have been introduced for integrating reasoning about relational properties of programs and measurement of quantitative difference between computational effects. Towards a general framework for such logics, in this paper, we formalize quantitative difference between computational effects as divergence on monad, then develop a relational program logic acRL that supports generic computational effects and divergences on them. To give a categorical semantics of acRL supporting divergences, we give a method to obtain graded strong relational liftings from divergences on monads. We derive two instantiations of acRL for the verification of 1) various differential privacy of higher-order functional probabilistic programs and 2) difference of distribution of costs between higher-order functional programs with probabilistic choice and cost counting operations.Comment: Preprin