1,177 research outputs found
Extensional and Intensional Strategies
This paper is a contribution to the theoretical foundations of strategies. We
first present a general definition of abstract strategies which is extensional
in the sense that a strategy is defined explicitly as a set of derivations of
an abstract reduction system. We then move to a more intensional definition
supporting the abstract view but more operational in the sense that it
describes a means for determining such a set. We characterize the class of
extensional strategies that can be defined intensionally. We also give some
hints towards a logical characterization of intensional strategies and propose
a few challenging perspectives
Intensional Models for the Theory of Types
In this paper we define intensional models for the classical theory of types,
thus arriving at an intensional type logic ITL. Intensional models generalize
Henkin's general models and have a natural definition. As a class they do not
validate the axiom of Extensionality. We give a cut-free sequent calculus for
type theory and show completeness of this calculus with respect to the class of
intensional models via a model existence theorem. After this we turn our
attention to applications. Firstly, it is argued that, since ITL is truly
intensional, it can be used to model ascriptions of propositional attitude
without predicting logical omniscience. In order to illustrate this a small
fragment of English is defined and provided with an ITL semantics. Secondly, it
is shown that ITL models contain certain objects that can be identified with
possible worlds. Essential elements of modal logic become available within
classical type theory once the axiom of Extensionality is given up.Comment: 25 page
An extensional Kleene realizability semantics for the Minimalist Foundation
We build a Kleene realizability semantics for the two-level Minimalist
Foundation MF, ideated by Maietti and Sambin in 2005 and completed by Maietti
in 2009. Thanks to this semantics we prove that both levels of MF are
consistent with the (Extended) formal Church Thesis CT. MF consists of two
levels, an intensional one, called mTT and an extensional one, called emTT,
based on versions of Martin-L\"of's type theory. Thanks to the link between the
two levels, it is enough to build a semantics for the intensional level to get
one also for the extensional level. Hence here we just build a realizability
semantics for the intensional level mTT. Such a semantics is a modification of
the realizability semantics in Beeson 1985 for extensional first order
Martin-L\"of's type theory with one universe. So it is formalised in Feferman's
classical arithmetic theory of inductive definitions. It is called extensional
Kleene realizability semantics since it validates extensional equality of
type-theoretic functions extFun, as in Beeson 1985. The main modification we
perform on Beeson's semantics is to interpret propositions, which are defined
primitively in MF, in a proof-irrelevant way. As a consequence, we gain the
validity of CT. Recalling that extFun+ CT+ AC are inconsistent over arithmetics
with finite types, we conclude that our semantics does not validate the full
Axiom of Choice AC. On the contrary, Beeson's semantics does validate AC, being
this a theorem of Martin-L\"of's theory, but it does not validate CT. The
semantics we present here appears to be the best Kleene realizability semantics
for the extensional level emTT of MF. Indeed Beeson's semantics is not an
option for emTT since the full AC added to it entails the excluded middle
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A general theory of action languages
We present a general theory of action-based languages as a paradigm, for the description, of those computational
systems which include elements of concurrency and networking, and extend this approach
to describe dist.ributed systems and also t,o describe the interaction of a system, with an environment.
As part of this approach we introduce the Action Language as a common model for the class of nondeterministic
concurrent programming languages and define its intensional and interaction semantics
in terrors of continuous transformation of environment behavior. This semantics i.s specialized for
programs with stores, and extended to describe distributed computations
Expressive Stream Reasoning with Laser
An increasing number of use cases require a timely extraction of non-trivial
knowledge from semantically annotated data streams, especially on the Web and
for the Internet of Things (IoT). Often, this extraction requires expressive
reasoning, which is challenging to compute on large streams. We propose Laser,
a new reasoner that supports a pragmatic, non-trivial fragment of the logic
LARS which extends Answer Set Programming (ASP) for streams. At its core, Laser
implements a novel evaluation procedure which annotates formulae to avoid the
re-computation of duplicates at multiple time points. This procedure, combined
with a judicious implementation of the LARS operators, is responsible for
significantly better runtimes than the ones of other state-of-the-art systems
like C-SPARQL and CQELS, or an implementation of LARS which runs on the ASP
solver Clingo. This enables the application of expressive logic-based reasoning
to large streams and opens the door to a wider range of stream reasoning use
cases.Comment: 19 pages, 5 figures. Extended version of accepted paper at ISWC 201
Elementary quotient completion
We extend the notion of exact completion on a weakly lex category to
elementary doctrines. We show how any such doctrine admits an elementary
quotient completion, which freely adds effective quotients and extensional
equality. We note that the elementary quotient completion can be obtained as
the composite of two free constructions: one adds effective quotients, and the
other forces extensionality of maps. We also prove that each construction
preserves comprehensions
A Lambda Term Representation Inspired by Linear Ordered Logic
We introduce a new nameless representation of lambda terms inspired by
ordered logic. At a lambda abstraction, number and relative position of all
occurrences of the bound variable are stored, and application carries the
additional information where to cut the variable context into function and
argument part. This way, complete information about free variable occurrence is
available at each subterm without requiring a traversal, and environments can
be kept exact such that they only assign values to variables that actually
occur in the associated term. Our approach avoids space leaks in interpreters
that build function closures.
In this article, we prove correctness of the new representation and present
an experimental evaluation of its performance in a proof checker for the
Edinburgh Logical Framework.
Keywords: representation of binders, explicit substitutions, ordered
contexts, space leaks, Logical Framework.Comment: In Proceedings LFMTP 2011, arXiv:1110.668
Quotient completion for the foundation of constructive mathematics
We apply some tools developed in categorical logic to give an abstract
description of constructions used to formalize constructive mathematics in
foundations based on intensional type theory. The key concept we employ is that
of a Lawvere hyperdoctrine for which we describe a notion of quotient
completion. That notion includes the exact completion on a category with weak
finite limits as an instance as well as examples from type theory that fall
apart from this.Comment: 32 page
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