5,516 research outputs found
Strong normalisation for applied lambda calculi
We consider the untyped lambda calculus with constructors and recursively
defined constants. We construct a domain-theoretic model such that any term not
denoting bottom is strongly normalising provided all its `stratified
approximations' are. From this we derive a general normalisation theorem for
applied typed lambda-calculi: If all constants have a total value, then all
typeable terms are strongly normalising. We apply this result to extensions of
G\"odel's system T and system F extended by various forms of bar recursion for
which strong normalisation was hitherto unknown.Comment: 14 pages, paper acceptet at electronic journal LMC
A proof of strong normalisation using domain theory
Ulrich Berger presented a powerful proof of strong normalisation using
domains, in particular it simplifies significantly Tait's proof of strong
normalisation of Spector's bar recursion. The main contribution of this paper
is to show that, using ideas from intersection types and Martin-Lof's domain
interpretation of type theory one can in turn simplify further U. Berger's
argument. We build a domain model for an untyped programming language where U.
Berger has an interpretation only for typed terms or alternatively has an
interpretation for untyped terms but need an extra condition to deduce strong
normalisation. As a main application, we show that Martin-L\"{o}f dependent
type theory extended with a program for Spector double negation shift.Comment: 16 page
A Finite Semantics of Simply-Typed Lambda Terms for Infinite Runs of<br> Automata
Model checking properties are often described by means of finite automata.
Any particular such automaton divides the set of infinite trees into finitely
many classes, according to which state has an infinite run. Building the full
type hierarchy upon this interpretation of the base type gives a finite
semantics for simply-typed lambda-trees.
A calculus based on this semantics is proven sound and complete. In
particular, for regular infinite lambda-trees it is decidable whether a given
automaton has a run or not. As regular lambda-trees are precisely recursion
schemes, this decidability result holds for arbitrary recursion schemes of
arbitrary level, without any syntactical restriction.Comment: 23 page
Cyclic Datatypes modulo Bisimulation based on Second-Order Algebraic Theories
Cyclic data structures, such as cyclic lists, in functional programming are
tricky to handle because of their cyclicity. This paper presents an
investigation of categorical, algebraic, and computational foundations of
cyclic datatypes. Our framework of cyclic datatypes is based on second-order
algebraic theories of Fiore et al., which give a uniform setting for syntax,
types, and computation rules for describing and reasoning about cyclic
datatypes. We extract the "fold" computation rules from the categorical
semantics based on iteration categories of Bloom and Esik. Thereby, the rules
are correct by construction. We prove strong normalisation using the General
Schema criterion for second-order computation rules. Rather than the fixed
point law, we particularly choose Bekic law for computation, which is a key to
obtaining strong normalisation. We also prove the property of "Church-Rosser
modulo bisimulation" for the computation rules. Combining these results, we
have a remarkable decidability result of the equational theory of cyclic data
and fold.Comment: 38 page
On Isomorphism of "Functional" Intersection and Union Types
Type isomorphism is useful for retrieving library components, since a
function in a library can have a type different from, but isomorphic to, the
one expected by the user. Moreover type isomorphism gives for free the coercion
required to include the function in the user program with the right type. The
present paper faces the problem of type isomorphism in a system with
intersection and union types. In the presence of intersection and union,
isomorphism is not a congruence and cannot be characterised in an equational
way. A characterisation can still be given, quite complicated by the
interference between functional and non functional types. This drawback is
faced in the paper by interpreting each atomic type as the set of functions
mapping any argument into the interpretation of the type itself. This choice
has been suggested by the initial projection of Scott's inverse limit
lambda-model. The main result of this paper is a condition assuring type
isomorphism, based on an isomorphism preserving reduction.Comment: In Proceedings ITRS 2014, arXiv:1503.0437
Categorical Aspects of Parameter Learning
Parameter learning is the technique for obtaining the probabilistic
parameters in conditional probability tables in Bayesian networks from tables
with (observed) data --- where it is assumed that the underlying graphical
structure is known. There are basically two ways of doing so, referred to as
maximal likelihood estimation (MLE) and as Bayesian learning. This paper
provides a categorical analysis of these two techniques and describes them in
terms of basic properties of the multiset monad M, the distribution monad D and
the Giry monad G. In essence, learning is about the reltionships between
multisets (used for counting) on the one hand and probability distributions on
the other. These relationsips will be described as suitable natural
transformations
Compositional Morphology for Word Representations and Language Modelling
This paper presents a scalable method for integrating compositional
morphological representations into a vector-based probabilistic language model.
Our approach is evaluated in the context of log-bilinear language models,
rendered suitably efficient for implementation inside a machine translation
decoder by factoring the vocabulary. We perform both intrinsic and extrinsic
evaluations, presenting results on a range of languages which demonstrate that
our model learns morphological representations that both perform well on word
similarity tasks and lead to substantial reductions in perplexity. When used
for translation into morphologically rich languages with large vocabularies,
our models obtain improvements of up to 1.2 BLEU points relative to a baseline
system using back-off n-gram models.Comment: Proceedings of the 31st International Conference on Machine Learning
(ICML
A Survey on Retrieval of Mathematical Knowledge
We present a short survey of the literature on indexing and retrieval of
mathematical knowledge, with pointers to 72 papers and tentative taxonomies of
both retrieval problems and recurring techniques.Comment: CICM 2015, 20 page
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