62 research outputs found
Relating Nominal and Higher-order Abstract Syntax Specifications
Nominal abstract syntax and higher-order abstract syntax provide a means for
describing binding structure which is higher-level than traditional techniques.
These approaches have spawned two different communities which have developed
along similar lines but with subtle differences that make them difficult to
relate. The nominal abstract syntax community has devices like names,
freshness, name-abstractions with variable capture, and the new-quantifier,
whereas the higher-order abstract syntax community has devices like
lambda-binders, lambda-conversion, raising, and the nabla-quantifier. This
paper aims to unify these communities and provide a concrete correspondence
between their different devices. In particular, we develop a
semantics-preserving translation from alpha-Prolog, a nominal abstract syntax
based logic programming language, to G-, a higher-order abstract syntax based
logic programming language. We also discuss higher-order judgments, a common
and powerful tool for specifications with higher-order abstract syntax, and we
show how these can be incorporated into G-. This establishes G- as a language
with the power of higher-order abstract syntax, the fine-grained variable
control of nominal specifications, and the desirable properties of higher-order
judgments.Comment: To appear in PPDP 201
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
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