784 research outputs found
Reduction Monads and Their Signatures
International audienc
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
Introducing a Calculus of Effects and Handlers for Natural Language Semantics
In compositional model-theoretic semantics, researchers assemble
truth-conditions or other kinds of denotations using the lambda calculus. It
was previously observed that the lambda terms and/or the denotations studied
tend to follow the same pattern: they are instances of a monad. In this paper,
we present an extension of the simply-typed lambda calculus that exploits this
uniformity using the recently discovered technique of effect handlers. We prove
that our calculus exhibits some of the key formal properties of the lambda
calculus and we use it to construct a modular semantics for a small fragment
that involves multiple distinct semantic phenomena
Heterogeneous substitution systems revisited
Matthes and Uustalu (TCS 327(1-2):155-174, 2004) presented a categorical
description of substitution systems capable of capturing syntax involving
binding which is independent of whether the syntax is made up from least or
greatest fixed points. We extend this work in two directions: we continue the
analysis by creating more categorical structure, in particular by organizing
substitution systems into a category and studying its properties, and we
develop the proofs of the results of the cited paper and our new ones in
UniMath, a recent library of univalent mathematics formalized in the Coq
theorem prover.Comment: 24 page
Gluing together proof environments: Canonical extensions of LF type theories featuring locks
© F. Honsell, L. Liquori, P. Maksimovic, I. Scagnetto This work is licensed under the Creative Commons Attribution License.We present two extensions of the LF Constructive Type Theory featuring monadic locks. A lock is a monadic type construct that captures the effect of an external call to an oracle. Such calls are the basic tool for gluing together diverse Type Theories and proof development environments. The oracle can be invoked either to check that a constraint holds or to provide a suitable witness. The systems are presented in the canonical style developed by the CMU School. The first system, CLLF/p,is the canonical version of the system LLF p, presented earlier by the authors. The second system, CLLF p?, features the possibility of invoking the oracle to obtain a witness satisfying a given constraint. We discuss encodings of Fitch-Prawitz Set theory, call-by-value λ-calculi, and systems of Light Linear Logic. Finally, we show how to use Fitch-Prawitz Set Theory to define a type system that types precisely the strongly normalizing terms
Nested Term Graphs (Work In Progress)
We report on work in progress on 'nested term graphs' for formalizing
higher-order terms (e.g. finite or infinite lambda-terms), including those
expressing recursion (e.g. terms in the lambda-calculus with letrec). The idea
is to represent the nested scope structure of a higher-order term by a nested
structure of term graphs.
Based on a signature that is partitioned into atomic and nested function
symbols, we define nested term graphs both in a functional representation, as
tree-like recursive graph specifications that associate nested symbols with
usual term graphs, and in a structural representation, as enriched term graph
structures. These definitions induce corresponding notions of bisimulation
between nested term graphs. Our main result states that nested term graphs can
be implemented faithfully by first-order term graphs.
keywords: higher-order term graphs, context-free grammars, cyclic
lambda-terms, higher-order rewrite systemsComment: In Proceedings TERMGRAPH 2014, arXiv:1505.0681
Modules over monads and operational semantics
This paper is a contribution to the search for efficient and high-level
mathematical tools to specify and reason about (abstract) programming languages
or calculi. Generalising the reduction monads of Ahrens et al., we introduce
transition monads, thus covering new applications such as
lambda-bar-mu-calculus, pi-calculus, Positive GSOS specifications, differential
lambda-calculus, and the big-step, simply-typed, call-by-value lambda-calculus.
Moreover, we design a suitable notion of signature for transition monads
Modules over Monads and Operational Semantics
This paper is a contribution to the search for efficient and high-level mathematical tools to specify and reason about (abstract) programming languages or calculi. Generalising the reduction monads of Ahrens et al., we introduce transition monads, thus covering new applications such as ???-calculus, ?-calculus, Positive GSOS specifications, differential ?-calculus, and the big-step, simply-typed, call-by-value ?-calculus. Finally, we design a suitable notion of signature for transition monads
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