1,342 research outputs found
Initial Semantics for Reduction Rules
We give an algebraic characterization of the syntax and operational semantics
of a class of simply-typed languages, such as the language PCF: we characterize
simply-typed syntax with variable binding and equipped with reduction rules via
a universal property, namely as the initial object of some category of models.
For this purpose, we employ techniques developed in two previous works: in the
first work we model syntactic translations between languages over different
sets of types as initial morphisms in a category of models. In the second work
we characterize untyped syntax with reduction rules as initial object in a
category of models. In the present work, we combine the techniques used earlier
in order to characterize simply-typed syntax with reduction rules as initial
object in a category. The universal property yields an operator which allows to
specify translations---that are semantically faithful by construction---between
languages over possibly different sets of types.
As an example, we upgrade a translation from PCF to the untyped lambda
calculus, given in previous work, to account for reduction in the source and
target. Specifically, we specify a reduction semantics in the source and target
language through suitable rules. By equipping the untyped lambda calculus with
the structure of a model of PCF, initiality yields a translation from PCF to
the lambda calculus, that is faithful with respect to the reduction semantics
specified by the rules.
This paper is an extended version of an article published in the proceedings
of WoLLIC 2012.Comment: Extended version of arXiv:1206.4547, proves a variant of a result of
PhD thesis arXiv:1206.455
Reconciling positional and nominal binding
We define an extension of the simply-typed lambda calculus where two
different binding mechanisms, by position and by name, nicely coexist. In the
former, as in standard lambda calculus, the matching between parameter and
argument is done on a positional basis, hence alpha-equivalence holds, whereas
in the latter it is done on a nominal basis. The two mechanisms also
respectively correspond to static binding, where the existence and type
compatibility of the argument are checked at compile-time, and dynamic binding,
where they are checked at run-time.Comment: In Proceedings ITRS 2012, arXiv:1307.784
Extended Initiality for Typed Abstract Syntax
Initial Semantics aims at interpreting the syntax associated to a signature
as the initial object of some category of 'models', yielding induction and
recursion principles for abstract syntax. Zsid\'o proves an initiality result
for simply-typed syntax: given a signature S, the abstract syntax associated to
S constitutes the initial object in a category of models of S in monads.
However, the iteration principle her theorem provides only accounts for
translations between two languages over a fixed set of object types. We
generalize Zsid\'o's notion of model such that object types may vary, yielding
a larger category, while preserving initiality of the syntax therein. Thus we
obtain an extended initiality theorem for typed abstract syntax, in which
translations between terms over different types can be specified via the
associated category-theoretic iteration operator as an initial morphism. Our
definitions ensure that translations specified via initiality are type-safe,
i.e. compatible with the typing in the source and target language in the
obvious sense. Our main example is given via the propositions-as-types
paradigm: we specify propositions and inference rules of classical and
intuitionistic propositional logics through their respective typed signatures.
Afterwards we use the category--theoretic iteration operator to specify a
double negation translation from the former to the latter. A second example is
given by the signature of PCF. For this particular case, we formalize the
theorem in the proof assistant Coq. Afterwards we specify, via the
category-theoretic iteration operator, translations from PCF to the untyped
lambda calculus
Modular, Fully-abstract Compilation by Approximate Back-translation
A compiler is fully-abstract if the compilation from source language programs
to target language programs reflects and preserves behavioural equivalence.
Such compilers have important security benefits, as they limit the power of an
attacker interacting with the program in the target language to that of an
attacker interacting with the program in the source language. Proving compiler
full-abstraction is, however, rather complicated. A common proof technique is
based on the back-translation of target-level program contexts to
behaviourally-equivalent source-level contexts. However, constructing such a
back- translation is problematic when the source language is not strong enough
to embed an encoding of the target language. For instance, when compiling from
STLC to ULC, the lack of recursive types in the former prevents such a
back-translation.
We propose a general and elegant solution for this problem. The key insight
is that it suffices to construct an approximate back-translation. The
approximation is only accurate up to a certain number of steps and conservative
beyond that, in the sense that the context generated by the back-translation
may diverge when the original would not, but not vice versa. Based on this
insight, we describe a general technique for proving compiler full-abstraction
and demonstrate it on a compiler from STLC to ULC. The proof uses asymmetric
cross-language logical relations and makes innovative use of step-indexing to
express the relation between a context and its approximate back-translation.
The proof extends easily to common compiler patterns such as modular
compilation and it, to the best of our knowledge, it is the first compiler full
abstraction proof to have been fully mechanised in Coq. We believe this proof
technique can scale to challenging settings and enable simpler, more scalable
proofs of compiler full-abstraction
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