1,377 research outputs found
Semantics of a Typed Algebraic Lambda-Calculus
Algebraic lambda-calculi have been studied in various ways, but their
semantics remain mostly untouched. In this paper we propose a semantic analysis
of a general simply-typed lambda-calculus endowed with a structure of vector
space. We sketch the relation with two established vectorial lambda-calculi.
Then we study the problems arising from the addition of a fixed point
combinator and how to modify the equational theory to solve them. We sketch an
algebraic vectorial PCF and its possible denotational interpretations
Simulation in the Call-by-Need Lambda-Calculus with Letrec, Case, Constructors, and Seq
This paper shows equivalence of several versions of applicative similarity
and contextual approximation, and hence also of applicative bisimilarity and
contextual equivalence, in LR, the deterministic call-by-need lambda calculus
with letrec extended by data constructors, case-expressions and Haskell's
seq-operator. LR models an untyped version of the core language of Haskell. The
use of bisimilarities simplifies equivalence proofs in calculi and opens a way
for more convenient correctness proofs for program transformations. The proof
is by a fully abstract and surjective transfer into a call-by-name calculus,
which is an extension of Abramsky's lazy lambda calculus. In the latter
calculus equivalence of our similarities and contextual approximation can be
shown by Howe's method. Similarity is transferred back to LR on the basis of an
inductively defined similarity. The translation from the call-by-need letrec
calculus into the extended call-by-name lambda calculus is the composition of
two translations. The first translation replaces the call-by-need strategy by a
call-by-name strategy and its correctness is shown by exploiting infinite trees
which emerge by unfolding the letrec expressions. The second translation
encodes letrec-expressions by using multi-fixpoint combinators and its
correctness is shown syntactically by comparing reductions of both calculi. A
further result of this paper is an isomorphism between the mentioned calculi,
which is also an identity on letrec-free expressions.Comment: 50 pages, 11 figure
Reasoning about modular datatypes with Mendler induction
In functional programming, datatypes a la carte provide a convenient modular
representation of recursive datatypes, based on their initial algebra
semantics. Unfortunately it is highly challenging to implement this technique
in proof assistants that are based on type theory, like Coq. The reason is that
it involves type definitions, such as those of type-level fixpoint operators,
that are not strictly positive. The known work-around of impredicative
encodings is problematic, insofar as it impedes conventional inductive
reasoning. Weak induction principles can be used instead, but they considerably
complicate proofs.
This paper proposes a novel and simpler technique to reason inductively about
impredicative encodings, based on Mendler-style induction. This technique
involves dispensing with dependent induction, ensuring that datatypes can be
lifted to predicates and relying on relational formulations. A case study on
proving subject reduction for structural operational semantics illustrates that
the approach enables modular proofs, and that these proofs are essentially
similar to conventional ones.Comment: In Proceedings FICS 2015, arXiv:1509.0282
Using models to model-check recursive schemes
We propose a model-based approach to the model checking problem for recursive
schemes. Since simply typed lambda calculus with the fixpoint operator,
lambda-Y-calculus, is equivalent to schemes, we propose the use of a model of
lambda-Y-calculus to discriminate the terms that satisfy a given property. If a
model is finite in every type, this gives a decision procedure. We provide a
construction of such a model for every property expressed by automata with
trivial acceptance conditions and divergence testing. Such properties pose
already interesting challenges for model construction. Moreover, we argue that
having models capturing some class of properties has several other virtues in
addition to providing decidability of the model-checking problem. As an
illustration, we show a very simple construction transforming a scheme to a
scheme reflecting a property captured by a given model.Comment: Long version of a paper presented at TLCA 201
Relational semantics of linear logic and higher-order model-checking
In this article, we develop a new and somewhat unexpected connection between
higher-order model-checking and linear logic. Our starting point is the
observation that once embedded in the relational semantics of linear logic, the
Church encoding of any higher-order recursion scheme (HORS) comes together with
a dual Church encoding of an alternating tree automata (ATA) of the same
signature. Moreover, the interaction between the relational interpretations of
the HORS and of the ATA identifies the set of accepting states of the tree
automaton against the infinite tree generated by the recursion scheme. We show
how to extend this result to alternating parity automata (APT) by introducing a
parametric version of the exponential modality of linear logic, capturing the
formal properties of colors (or priorities) in higher-order model-checking. We
show in particular how to reunderstand in this way the type-theoretic approach
to higher-order model-checking developed by Kobayashi and Ong. We briefly
explain in the end of the paper how his analysis driven by linear logic results
in a new and purely semantic proof of decidability of the formulas of the
monadic second-order logic for higher-order recursion schemes.Comment: 24 pages. Submitte
The exp-log normal form of types
Lambda calculi with algebraic data types lie at the core of functional
programming languages and proof assistants, but conceal at least two
fundamental theoretical problems already in the presence of the simplest
non-trivial data type, the sum type. First, we do not know of an explicit and
implemented algorithm for deciding the beta-eta-equality of terms---and this in
spite of the first decidability results proven two decades ago. Second, it is
not clear how to decide when two types are essentially the same, i.e.
isomorphic, in spite of the meta-theoretic results on decidability of the
isomorphism.
In this paper, we present the exp-log normal form of types---derived from the
representation of exponential polynomials via the unary exponential and
logarithmic functions---that any type built from arrows, products, and sums,
can be isomorphically mapped to. The type normal form can be used as a simple
heuristic for deciding type isomorphism, thanks to the fact that it is a
systematic application of the high-school identities.
We then show that the type normal form allows to reduce the standard beta-eta
equational theory of the lambda calculus to a specialized version of itself,
while preserving the completeness of equality on terms. We end by describing an
alternative representation of normal terms of the lambda calculus with sums,
together with a Coq-implemented converter into/from our new term calculus. The
difference with the only other previously implemented heuristic for deciding
interesting instances of eta-equality by Balat, Di Cosmo, and Fiore, is that we
exploit the type information of terms substantially and this often allows us to
obtain a canonical representation of terms without performing sophisticated
term analyses
Guard Your Daggers and Traces: On The Equational Properties of Guarded (Co-)recursion
Motivated by the recent interest in models of guarded (co-)recursion we study
its equational properties. We formulate axioms for guarded fixpoint operators
generalizing the axioms of iteration theories of Bloom and Esik. Models of
these axioms include both standard (e.g., cpo-based) models of iteration
theories and models of guarded recursion such as complete metric spaces or the
topos of trees studied by Birkedal et al. We show that the standard result on
the satisfaction of all Conway axioms by a unique dagger operation generalizes
to the guarded setting. We also introduce the notion of guarded trace operator
on a category, and we prove that guarded trace and guarded fixpoint operators
are in one-to-one correspondence. Our results are intended as first steps
leading to the description of classifying theories for guarded recursion and
hence completeness results involving our axioms of guarded fixpoint operators
in future work.Comment: In Proceedings FICS 2013, arXiv:1308.589
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