48,231 research outputs found
Hybrid - a definitional two-level approach to reasoning with higher-order abstract syntax
Combining higher-order abstract syntax and (co)-induction in a logical
framework is well known to be problematic.We describe the theory and the practice
of a tool called Hybrid, within Isabelle/HOL and Coq, which aims to address many
of these difficulties. It allows object logics to be represented using higher-order
abstract syntax, and reasoned about using tactical theorem proving and principles
of (co)induction. Moreover, it is definitional, which guarantees consistency within
a classical type theory. The idea is to have a de Bruijn representation of \u3bb-terms
providing a definitional layer that allows the user to represent object languages using
higher-order abstract syntax, while offering tools for reasoning about them at the
higher level. In this paper we describe how to use Hybrid in a multi-level reasoning
fashion, similar in spirit to other systems such as Twelf and Abella. By explicitly
referencing provability in a middle layer called a specification logic, we solve the
problem of reasoning by (co)induction in the presence of non-stratifiable hypothetical
judgments, which allow very elegant and succinct specifications of object logic
inference rules. We first demonstrate the method on a simple example, formally
proving type soundness (subject reduction) for a fragment of a pure functional
language, using a minimal intuitionistic logic as the specification logic. We then
prove an analogous result for a continuation-machine presentation of the operational semantics of the same language, encoded this time in an ordered linear logic that
serves as the specification layer. This example demonstrates the ease with which
we can incorporate new specification logics, and also illustrates a significantly more
complex object logic whose encoding is elegantly expressed using features of the new
specification logic
Bindings as bounded natural functors
We present a general framework for specifying and reasoning about syntax with bindings. Abstract binder types are modeled using a universe of functors on sets, subject to a number of operations that can be used to construct complex binding patterns and binding-aware datatypes, including non-well-founded and infinitely branching types, in a modular fashion. Despite not committing to any syntactic format, the framework is “concrete” enough to provide definitions of the fundamental operators on terms (free variables, alpha-equivalence, and capture-avoiding substitution) and reasoning and definition principles. This work is compatible with classical higher-order logic and has been formalized in the proof assistant Isabelle/HOL
Bindings as bounded natural functors
We present a general framework for specifying and reasoning about syntax with bindings. Abstract binder types are modeled using a universe of functors on sets, subject to a number of operations that can be used to construct complex binding patterns and binding-aware datatypes, including non-well-founded and infinitely branching types, in a modular fashion. Despite not committing to any syntactic format, the framework is “concrete” enough to provide definitions of the fundamental operators on terms (free variables, alpha-equivalence, and capture-avoiding substitution) and reasoning and definition principles. This work is compatible with classical higher-order logic and has been formalized in the proof assistant Isabelle/HOL
Constraint Handling Rules with Binders, Patterns and Generic Quantification
Constraint Handling Rules provide descriptions for constraint solvers.
However, they fall short when those constraints specify some binding structure,
like higher-rank types in a constraint-based type inference algorithm. In this
paper, the term syntax of constraints is replaced by -tree syntax, in
which binding is explicit; and a new generic quantifier is introduced,
which is used to create new fresh constants.Comment: Paper presented at the 33nd International Conference on Logic
Programming (ICLP 2017), Melbourne, Australia, August 28 to September 1, 2017
16 pages, LaTeX, no PDF figure
Variable binding, symmetric monoidal closed theories, and bigraphs
This paper investigates the use of symmetric monoidal closed (SMC) structure
for representing syntax with variable binding, in particular for languages with
linear aspects. In our setting, one first specifies an SMC theory T, which may
express binding operations, in a way reminiscent from higher-order abstract
syntax. This theory generates an SMC category S(T) whose morphisms are, in a
sense, terms in the desired syntax. We apply our approach to Jensen and
Milner's (abstract binding) bigraphs, which are linear w.r.t. processes. This
leads to an alternative category of bigraphs, which we compare to the original.Comment: An introduction to two more technical previous preprints. Accepted at
Concur '0
- …