Virtual Evidence: A Constructive Semantics for Classical Logics

This article presents a computational semantics for classical logic using constructive type theory. Such semantics seems impossible because classical logic allows the Law of Excluded Middle (LEM), not accepted in constructive logic since it does not have computational meaning. However, the apparently oracular powers expressed in the LEM, that for any proposition P either it or its negation, not P, is true can also be explained in terms of constructive evidence that does not refer to "oracles for truth." Types with virtual evidence and the constructive impossibility of negative evidence provide sufficient semantic grounds for classical truth and have a simple computational meaning. This idea is formalized using refinement types, a concept of constructive type theory used since 1984 and explained here. A new axiom creating virtual evidence fully retains the constructive meaning of the logical operators in classical contexts. Key Words: classical logic, constructive logic, intuitionistic logic, propositions-as-types, constructive type theory, refinement types, double negation translation, computational content, virtual evidenc

Geomagnetic field models incorporating physical constraints on the secular variation

This proposal has been concerned with methods for constructing geomagnetic field models that incorporate physical constraints on the secular variation. The principle goal that has been accomplished is the development of flexible algorithms designed to test whether the frozen flux approximation is adequate to describe the available geomagnetic data and their secular variation throughout this century. These have been applied to geomagnetic data from both the early and middle part of this century and convincingly demonstrate that there is no need to invoke violations of the frozen flux hypothesis in order to satisfy the available geomagnetic data

Open Bar - a Brouwerian Intuitionistic Logic with a Pinch of Excluded Middle

One of the differences between Brouwerian intuitionistic logic and classical logic is their treatment of time. In classical logic truth is atemporal, whereas in intuitionistic logic it is time-relative. Thus, in intuitionistic logic it is possible to acquire new knowledge as time progresses, whereas the classical Law of Excluded Middle (LEM) is essentially flattening the notion of time stating that it is possible to decide whether or not some knowledge will ever be acquired. This paper demonstrates that, nonetheless, the two approaches are not necessarily incompatible by introducing an intuitionistic type theory along with a Beth-like model for it that provide some middle ground. On one hand they incorporate a notion of progressing time and include evolving mathematical entities in the form of choice sequences, and on the other hand they are consistent with a variant of the classical LEM. Accordingly, this new type theory provides the basis for a more classically inclined Brouwerian intuitionistic type theory

Logic of Events, a framework to reason about distributed systems

We present a logical framework to reason about distributed systems called the Logic of Events. This logic has been formalized in Nuprl. We developed a suite of tools and tactics in Nuprl to reason about event classes. We also developed a programming language called EventML which allows programmers to write specifications of distributed protocols

Non-Abelian BIonic Brane Intersections

We study "fuzzy funnel" solutions to the non-Abelian equations of motion of the D-string. Our funnel describes n^6/360 coincident D-strings ending on n^3/6 D7-branes, in terms of a fuzzy six-sphere which expands along the string. We also provide a dual description of this configuration in terms of the world volume theory of the D7-branes. Our work makes use of an interesting non-linear higher dimensional generalization of the instanton equations.Comment: 17 pages uses harvmac; v2: small typos corrected, refs adde

Constructively formalizing automata theory

We present a constructive formalization of the Myhill-Nerode the-orem on the minimization of nite automata that follows the account in Hopcroft and Ullman's book Formal Languages and Their Relation to Automata. We chose to formalize this theorem because it illustrates many points critical to formalization of computational mathematics, especially the extraction of an important algorithm from a proof as a method of knowing that the algorithm is correct. It also gave us an opportunity to experiment with a constructive implementation of quotient sets. We carried out the formalization in Nuprl, an interactive theorem prover based on constructive type theory. Nuprl borrows an imple-mentation of the ML language from the LCF system of Milner, Gordon, and Wadsworth, and makes heavy use of the notion of tactic pioneered by Milner in LCF. We are interested in the pedagogical value of electronic formal mathematical texts and have put our formalization on the World Wide Web. Readers are invited to judge whether the formalization adds value in comparison to a careful informal account. Key Words and Phrases: automata, constructivity, congruence, equivalence relation, formal languages, foundational logic, LCF, logic, Martin-Lof semantics, Myhill-Nerode theorem, Nuprl, program extrac

Fluctuating Fuzzy Funnels

It is well known that a D-string ending on a D3, D5 or D7 brane is described in terms of a non-commutative fuzzy funnel geometry. In this article, we give a numerical study of the fluctuations about this leading geometry. This allows us to investigate issues related to the stability and moduli space of these solutions. We comment on the comparison to the linearized fluctuations in supergravity.Comment: 24 pages, 3 figures; v2 references added and correcte

Recursive Definitions of Monadic Functions

Using standard domain-theoretic fixed-points, we present an approach for defining recursive functions that are formulated in monadic style. The method works both in the simple option monad and the state-exception monad of Isabelle/HOL's imperative programming extension, which results in a convenient definition principle for imperative programs, which were previously hard to define. For such monadic functions, the recursion equation can always be derived without preconditions, even if the function is partial. The construction is easy to automate, and convenient induction principles can be derived automatically.Comment: In Proceedings PAR 2010, arXiv:1012.455

Step-Indexed Normalization for a Language with General Recursion

The Trellys project has produced several designs for practical dependently typed languages. These languages are broken into two fragments-a_logical_fragment where every term normalizes and which is consistent when interpreted as a logic, and a_programmatic_fragment with general recursion and other convenient but unsound features. In this paper, we present a small example language in this style. Our design allows the programmer to explicitly mention and pass information between the two fragments. We show that this feature substantially complicates the metatheory and present a new technique, combining the traditional Girard-Tait method with step-indexed logical relations, which we use to show normalization for the logical fragment.Comment: In Proceedings MSFP 2012, arXiv:1202.240

Penrose Limits, Deformed pp-Waves and the String Duals of N=1 Large n Gauge Theory

A certain conformally invariant N=1 supersymmetric SU(n) gauge theory has a description as an infra-red fixed point obtained by deforming the N=4 supersymmetric Yang-Mills theory by giving a mass to one of its N=1 chiral multiplets. We study the Penrose limit of the supergravity dual of the large n limit of this N=1 gauge theory. The limit gives a pp-wave with R-R five-form flux and both R-R and NS-NS three-form flux. We discover that this new solution preserves twenty supercharges and that, in the light-cone gauge, string theory on this background is exactly solvable. Correspondingly, this latter is the stringy dual of a particular large charge limit of the large n gauge theory. We are able to identify which operators in the field theory survive the limit to form the string's ground state and some of the spacetime excitations. The full string model, which we exhibit, contains a family of non-trivial predictions for the properties of the gauge theory operators which survive the limit.Comment: 39 pages, Late