11,497 research outputs found
From compositional to systematic semantics
We prove a theorem stating that any semantics can be encoded as a
compositional semantics, which means that, essentially, the standard definition
of compositionality is formally vacuous. We then show that when compositional
semantics is required to be "systematic" (that is, the meaning function cannot
be arbitrary, but must belong to some class), it is possible to distinguish
between compositional and non-compositional semantics. As a result, we believe
that the paper clarifies the concept of compositionality and opens a
possibility of making systematic formal comparisons of different systems of
grammars.Comment: 11 pp. Latex.
Strongly Complete Logics for Coalgebras
Coalgebras for a functor model different types of transition systems in a
uniform way. This paper focuses on a uniform account of finitary logics for
set-based coalgebras. In particular, a general construction of a logic from an
arbitrary set-functor is given and proven to be strongly complete under
additional assumptions. We proceed in three parts. Part I argues that sifted
colimit preserving functors are those functors that preserve universal
algebraic structure. Our main theorem here states that a functor preserves
sifted colimits if and only if it has a finitary presentation by operations and
equations. Moreover, the presentation of the category of algebras for the
functor is obtained compositionally from the presentations of the underlying
category and of the functor. Part II investigates algebras for a functor over
ind-completions and extends the theorem of J{\'o}nsson and Tarski on canonical
extensions of Boolean algebras with operators to this setting. Part III shows,
based on Part I, how to associate a finitary logic to any finite-sets
preserving functor T. Based on Part II we prove the logic to be strongly
complete under a reasonable condition on T
A Case Study on Logical Relations using Contextual Types
Proofs by logical relations play a key role to establish rich properties such
as normalization or contextual equivalence. They are also challenging to
mechanize. In this paper, we describe the completeness proof of algorithmic
equality for simply typed lambda-terms by Crary where we reason about logically
equivalent terms in the proof environment Beluga. There are three key aspects
we rely upon: 1) we encode lambda-terms together with their operational
semantics and algorithmic equality using higher-order abstract syntax 2) we
directly encode the corresponding logical equivalence of well-typed
lambda-terms using recursive types and higher-order functions 3) we exploit
Beluga's support for contexts and the equational theory of simultaneous
substitutions. This leads to a direct and compact mechanization, demonstrating
Beluga's strength at formalizing logical relations proofs.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759
Polynomial Invariants for Affine Programs
We exhibit an algorithm to compute the strongest polynomial (or algebraic)
invariants that hold at each location of a given affine program (i.e., a
program having only non-deterministic (as opposed to conditional) branching and
all of whose assignments are given by affine expressions). Our main tool is an
algebraic result of independent interest: given a finite set of rational square
matrices of the same dimension, we show how to compute the Zariski closure of
the semigroup that they generate
Layer by layer - Combining Monads
We develop a method to incrementally construct programming languages. Our
approach is categorical: each layer of the language is described as a monad.
Our method either (i) concretely builds a distributive law between two monads,
i.e. layers of the language, which then provides a monad structure to the
composition of layers, or (ii) identifies precisely the algebraic obstacles to
the existence of a distributive law and gives a best approximant language. The
running example will involve three layers: a basic imperative language enriched
first by adding non-determinism and then probabilistic choice. The first
extension works seamlessly, but the second encounters an obstacle, which
results in a best approximant language structurally very similar to the
probabilistic network specification language ProbNetKAT
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