1,580 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
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
Term Graph Representations for Cyclic Lambda-Terms
We study various representations for cyclic lambda-terms as higher-order or
as first-order term graphs. We focus on the relation between
`lambda-higher-order term graphs' (lambda-ho-term-graphs), which are
first-order term graphs endowed with a well-behaved scope function, and their
representations as `lambda-term-graphs', which are plain first-order term
graphs with scope-delimiter vertices that meet certain scoping requirements.
Specifically we tackle the question: Which class of first-order term graphs
admits a faithful embedding of lambda-ho-term-graphs in the sense that: (i) the
homomorphism-based sharing-order on lambda-ho-term-graphs is preserved and
reflected, and (ii) the image of the embedding corresponds closely to a natural
class (of lambda-term-graphs) that is closed under homomorphism?
We systematically examine whether a number of classes of lambda-term-graphs
have this property, and we find a particular class of lambda-term-graphs that
satisfies this criterion. Term graphs of this class are built from application,
abstraction, variable, and scope-delimiter vertices, and have the
characteristic feature that the latter two kinds of vertices have back-links to
the corresponding abstraction.
This result puts a handle on the concept of subterm sharing for higher-order
term graphs, both theoretically and algorithmically: We obtain an easily
implementable method for obtaining the maximally shared form of
lambda-ho-term-graphs. Also, we open up the possibility to pull back properties
from first-order term graphs to lambda-ho-term-graphs. In fact we prove this
for the property of the sharing-order successors of a given term graph to be a
complete lattice with respect to the sharing order.
This report extends the paper with the same title
(http://arxiv.org/abs/1302.6338v1) in the proceedings of the workshop TERMGRAPH
2013.Comment: 35 pages. report extending proceedings article on arXiv:1302.6338
(changes with respect to version v2: added section 8, modified Proposition
2.4, added Remark 2.5, added Corollary 7.11, modified figures in the
conclusion
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