The [lambda]-calculus provides a simple, well-established framework for research in functional programming languages that readily lends itself to the use offormal methods--that is, the use of mathematically sound techniques and supporting tools--to describe and verify properties of programming languages, as well. This is no coincidence. After all, the [lambda]-calculus formalizes the concept of effective computability, for all computable functions are definable in the untyped [lambda]-calculus, making it expressively equivalent torecursive functions. In software, the expressiveness of functional languages is considereda strength. Functional approaches to language design, however, needn't be limited to soft-ware. In hardware, the expressiveness of functional languages becomes a major obstacle to successful hardware synthesis, for the reason that such languages are usually capable of expressing general recursion. The presence of general recursion makes it possible to generate expressions that run forever, never producing a well-defined value. In this dissertation, we study two novel variants of the simply typed [lambda]-calculus, representing fragments of functional hardware description languages. The first variant extends the type system, using natural numbers representing time. This addition, though simple, is non-trivial. We prove that this calculus possesses bounded variants of type-safety and strong normalization. That is to say, we show that all well-typed expressions evaluate to values within a bound determined by the natural number index of their corresponding types. The second variant is a computational [lambda]-calculus that formalizes the core fragment of the hardware description language known as ReWire. We prove that the language has type-safety and is strongly normalizing -- the proof of strong normalizationis the first mechanized proof of its kind. We define an equational theory with respect to this language. This allows us to prove that the language has desirable security properties by construction. This work supports a full-edged, formal methodology for producing high assurance hardware.
Constructing high-assurance, secure hardware remains a challenge, because to do so relies on both a verifiable means of hardware description and implementation. However, production hardware description languages (HDL) lack the formal underpinnings required by formal methods in security. Still, there is no such thing as high-assurance systems without high-assurance hardware. We present a core calculus of secure hardware description with its formal semantics, security type system, and mechanization in Coq. This calculus is the core of the functional HDL, ReWire, shown in previous work to have useful applications in reconfigurable computing. This work supports a full-fledged, formal methodology for producing high-assurance hardware.
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Abstract. We extend the relational algebra of Chin and Tarski so that it is multisorted or, as we prefer, typed. Each type supports a local Boolean algebra outfitted with a converse operator. From Lyndon, we know that relation algebras cannot be represented as proper relation algebras where a proper relation algebra has binary relations as elements and the algebra is singly-typed. Here, the intensional conjunction, which was to represent relational composition in Chin and Tarski, spans three different local algebras, thus the term distributed in the title. Since we do not rely on proper relation algebras, we are free to re-express the algebras as typed. In doing so, we allow many different intensional conjunction operators.We construct a typed logic over these algebras, also known as heterogeneous algebras of Birkhoff and Lipson. The logic can be seen as a form of relevance logic with a classical negation connective where the Routley-Meyer star operator is reified as a converse connective in the logic. Relevance logic itself is not typed but our work shows how it can be made so. Some of the properties of classical relevance logic are weakened from Routley-Meyer's version which is too strong for a logic over relation algebras.
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