Substructural logic

In logic, a substructural logic is a logic lacking one of the usual structural rules (e.g. of classical and intuitionistic logic), such as weakening, contraction, exchange or associativity. Two of the more significant substructural logics are relevant logic and linear logic.

In a sequent calculus, one writes each line of a proof as

Here the structural rules are rules for rewriting the LHS of the sequent, denoted Γ, initially conceived of as a string (sequence) of propositions. The standard interpretation of this string is as conjunction: we expect to read

as the sequent notation for

Here we are taking the RHS Σ to be a single proposition C (which is the intuitionistic style of sequent); but everything applies equally to the general case, since all the manipulations are taking place to the left of the turnstile symbol .

Since conjunction is a commutative and associative operation, the formal setting-up of sequent theory normally includes structural rules for rewriting the sequent Γ accordingly - for example for deducing

Concurrent computing

Concurrent computing is a form of computing in which several computations are executing during overlapping time periods—concurrently—instead of sequentially (one completing before the next starts). This is a property of a system—this may be an individual program, a computer, or a network—and there is a separate execution point or "thread of control" for each computation ("process"). A concurrent system is one where a computation can make progress without waiting for all other computations to complete—where more than one computation can make progress at "the same time".

As a programming paradigm, concurrent computing is a form of modular programming, namely factoring an overall computation into subcomputations that may be executed concurrently. Pioneers in the field of concurrent computing include Edsger Dijkstra, Per Brinch Hansen, and C.A.R. Hoare.

Introduction

Concurrent computing is related to but distinct from parallel computing, though these concepts are frequently confused, and both can be described as "multiple processes executing during the same period of time". In parallel computing, execution literally occurs at the same instant, for example on separate processors of a multi-processor machine, with the goal of speeding up computations—parallel computing is impossible on a (single-core) single processor, as only one computation can occur at any instant (during any single clock cycle). By contrast, concurrent computing consists of process lifetimes overlapping, but execution need not happen at the same instant. The goal here is to model processes in the outside world that happen concurrently, such as multiple clients accessing a server at the same time. Structuring software systems as composed of multiple concurrent, communicating parts can be useful for tackling complexity, regardless of whether the parts can be executed in parallel.

Frank Pfenning

Frank Pfenning is a professor of computer science, adjunct professor in the department of philosophy, and head of the Computer Science Department at Carnegie Mellon University. He received his Ph.D. from the Carnegie Mellon University Department of Mathematics in 1987, for his dissertation entitled Proof Transformations in Higher-Order Logic. He was a student of Peter B. Andrews.

His research includes work in the area of programming languages, logic and type theory, logical frameworks, automated deduction, and trustworthy computing. He is one of the principal authors of the Twelf system. He also developed Carnegie Mellon's introductory imperative programming course for undergraduates and the C0 programming language used in this course.

In 2015 he was named a fellow of the Association for Computing Machinery "for contributions to the logical foundations of automatic theorem proving and types for programming languages."

He is a competitive squash player, ranked in the top three of the university's squash ladder.

Programming language

A programming language is a formal constructed language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to create programs to control the behavior of a machine or to express algorithms.

The earliest known programmable machine preceded the invention of the digital computer and is the automatic flute player described in the 9th century by the brothers Musa in Baghdad, at the time a major centre of knowledge. From the early 1800s, "programs" were used to direct the behavior of machines such as Jacquard looms and player pianos. Thousands of different programming languages have been created, mainly in the computer field, and many more still are being created every year. Many programming languages require computation to be specified in an imperative form (i.e., as a sequence of operations to perform), while other languages use other forms of program specification such as the declarative form (i.e. the desired result is specified, not how to achieve it).

Summer school

Summer school (or summer university) is a school, or a program generally sponsored by a school or a school district, or provided by a private company, that provides lessons and activities during the summer vacation. Participation in summer schools has been shown to have substantial beneficial effects on educational progress.

Summer schools in North America

In elementary and middle school, these programs are often non-academic, though some are used for remedial instruction.

In high school, college or university, students can enroll in classes for credit to be taken into account in their grade point average or their transcript. Generally, this credit is in one of two categories: remediation or advancement. For remediation, the summer school is used to make up credits lost through absence or failure. For advancement, the summer school is used to obtain credit for classes to accelerate progress toward a degree or to lessen the load of courses during the regular school year. Many universities offer short-term summer courses to attract both local and international students, and these programs are often surrounded by social activities.