Fred: An approach to generating real, correct, reusable programs from proofs

In this paper we describe our system for automatically extracting "correct" programs from proofs using a development of the Curry-Howard process. Although program extraction has been developed by many authors, our system has a number of novel features designed to make it very easy to use and as close as possible to ordinary mathematical terminology and practice. These features include 1. the use of Henkin's technique to reduce higher-order logic to many-sorted (first-order) logic; 2. the free use of new rules for induction subject to certain conditions; 3. the extensive use of previously programmed (total, recursive) functions; 4. the use of templates to make the reasoning much closer to normal mathematical proofs and 5. a conceptual distinction between the computational type theory (for representing programs)and the logical type theory (for reasoning about programs). As an example of our system we give a constructive proof of the well known theorem that every graph of even parity, which is non-trivial in the sense that it does not consist of isolated vertices, has a cycle. Given such a graph as input, the extracted program produces a cycle as promised.

Protocols between programs and proofs

In this paper we describe a new protocol that we call the Curry-Howard protocol between a theory and the programs extracted from it. This protocol leads to the expansion of the theory and the production of more powerful programs. The methodology we use for automatically extracting “correct” programs from proofs is a development of the well-known Curry-Howard process. Program extraction has been developed by many authors, but our presentation is ultimately aimed at a practical, usable system and has a number of novel features. These include 1. a very simple and natural mimicking of ordinary mathematical practice and likewise the use of established computer programs when we obtain programs from formal proofs, and 2. a conceptual distinction between programs on the one hand, and proofs of theorems that yield programs on the other. An implementation of our methodology is the Fred system. As an example of our protocol we describe a constructive proof of the well-known theorem that every graph of even parity can be decomposed into a list of disjoint cycles. Given such a graph as input, the extracted program produces a list of the (non-trivial) disjoint cycles as promised.

Extraction of structured programs from specification proofs

We present a method using an extended logical system for obtaining programs from specifications written in a sublanguage of CASL. These programs are “correct” in the sense that they satisfy their specifications. The technique we use is to extract programs from proofs in formal logic by techniques due to Curry and Howard. The logical calculus, however, is novel because it adds structural rules corresponding to the standard ways of modifying specifications: translating (renaming), taking unions, and hiding signatures. Although programs extracted by the Curry-Howard process can be very cumbersome, we use a number of simplifications that ensure that the programs extracted are in a language close to a standard high-level programming language. We use this to produce an executable refinement of a given specification and we then provide a method for producing a program module that maximally respects the original structure of the specification. Throughout the paper we demonstrate the technique with a simple example.

New Programs and Accreditations: Meaningful Measurement & Assessment

This article provides a blueprint for successful library strategies in support of accreditations for regional bodies as well as reviews of discipline-specific academic programs spanning a wide range of knowledge areas.

Administering Outcomes-Based Financial Aid Programs

As state and federal agencies increasingly condition institutional funding on student outcomes, university leaders are under pressure to develop innovative strategies to boost student retention and degree completion rates. This report examines public university initiatives that strategically leverage financial aid to support institutional retention and degree completion goals.

Creating High-Impact Workforce Development Programs

As communities across the nation continue to face persistent unemployment, many states are turning to community college workforce development programs to combat joblessness. However, state governments and community colleges alike are often unable to devote significant resources to building new workforce initiatives. This brief examines how community colleges in four states have worked with state governments and other partners to overcome budget constraints and create effective workforce development programs, highlighting funding models and innovative practices to make efficient use of limited resources.