262 research outputs found
Nominal Abstraction
Recursive relational specifications are commonly used to describe the
computational structure of formal systems. Recent research in proof theory has
identified two features that facilitate direct, logic-based reasoning about
such descriptions: the interpretation of atomic judgments through recursive
definitions and an encoding of binding constructs via generic judgments.
However, logics encompassing these two features do not currently allow for the
definition of relations that embody dynamic aspects related to binding, a
capability needed in many reasoning tasks. We propose a new relation between
terms called nominal abstraction as a means for overcoming this deficiency. We
incorporate nominal abstraction into a rich logic also including definitions,
generic quantification, induction, and co-induction that we then prove to be
consistent. We present examples to show that this logic can provide elegant
treatments of binding contexts that appear in many proofs, such as those
establishing properties of typing calculi and of arbitrarily cascading
substitutions that play a role in reducibility arguments.Comment: To appear in the Journal of Information and Computatio
A Framework for Specifying, Prototyping, and Reasoning about Computational Systems
This thesis concerns the development of a framework that facilitates the
design and analysis of formal systems. Specifically, this framework provides a
specification language which supports the concise and direct description of
formal systems, a mechanism for animating the specification language thereby
producing prototypes of encoded systems, and a logic for proving properties of
specifications and therefore of the systems they encode. A defining
characteristic of the proposed framework is that it is based on two separate
but closely intertwined logics: a specification logic that facilitates the
description of computational structure and another logic that exploits the
special characteristics of the specification logic to support reasoning about
the computational behavior of systems that are described using it. Both logics
embody a natural treatment of binding structure by using the lambda-calculus as
a means for representing objects and by incorporating special mechanisms for
working with such structure. By using this technique, they lift the treatment
of binding from the object language into the domain of the relevant meta logic,
thereby allowing the specification or analysis components to focus on the more
essential logical aspects of the systems that are encoded. The primary
contributions of these thesis are the development of a rich meta-logic called G
with capabilities for sophisticated reasoning that includes induction and
co-induction over high-level specifications of computations and with an
associated cut-elimination result; an interactive reasoning system called
Abella based on G; and several reasoning examples which demonstrate the
expressiveness and naturalness of both G and Abella.Comment: PhD Thesis submitted September, 200
Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness
We give a semantics for the lambda-calculus based on a topological duality
theorem in nominal sets. A novel interpretation of lambda is given in terms of
adjoints, and lambda-terms are interpreted absolutely as sets (no valuation is
necessary)
A Strong Call-By-Need Calculus
We present a call-by-need ?-calculus that enables strong reduction (that is, reduction inside the body of abstractions) and guarantees that arguments are only evaluated if needed and at most once. This calculus uses explicit substitutions and subsumes the existing strong-call-by-need strategy, but allows for more reduction sequences, and often shorter ones, while preserving the neededness.
The calculus is shown to be normalizing in a strong sense: Whenever a ?-term t admits a normal form n in the ?-calculus, then any reduction sequence from t in the calculus eventually reaches a representative of the normal form n. We also exhibit a restriction of this calculus that has the diamond property and that only performs reduction sequences of minimal length, which makes it systematically better than the existing strategy. We have used the Abella proof assistant to formalize part of this calculus, and discuss how this experiment affected its design
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