1,292 research outputs found
Safe Recursion on Notation into a Light Logic by Levels
We embed Safe Recursion on Notation (SRN) into Light Affine Logic by Levels
(LALL), derived from the logic L4. LALL is an intuitionistic deductive system,
with a polynomial time cut elimination strategy.
The embedding allows to represent every term t of SRN as a family of proof
nets |t|^l in LALL. Every proof net |t|^l in the family simulates t on
arguments whose bit length is bounded by the integer l. The embedding is based
on two crucial features. One is the recursive type in LALL that encodes Scott
binary numerals, i.e. Scott words, as proof nets. Scott words represent the
arguments of t in place of the more standard Church binary numerals. Also, the
embedding exploits the "fuzzy" borders of paragraph boxes that LALL inherits
from L4 to "freely" duplicate the arguments, especially the safe ones, of t.
Finally, the type of |t|^l depends on the number of composition and recursion
schemes used to define t, namely the structural complexity of t. Moreover, the
size of |t|^l is a polynomial in l, whose degree depends on the structural
complexity of t.
So, this work makes closer both the predicative recursive theoretic
principles SRN relies on, and the proof theoretic one, called /stratification/,
at the base of Light Linear Logic
Equations for Hereditary Substitution in Leivant's Predicative System F: A Case Study
This paper presents a case study of formalizing a normalization proof for
Leivant's Predicative System F using the Equations package. Leivant's
Predicative System F is a stratified version of System F, where type
quantification is annotated with kinds representing universe levels. A weaker
variant of this system was studied by Stump & Eades, employing the hereditary
substitution method to show normalization. We improve on this result by showing
normalization for Leivant's original system using hereditary substitutions and
a novel multiset ordering on types. Our development is done in the Coq proof
assistant using the Equations package, which provides an interface to define
dependently-typed programs with well-founded recursion and full dependent
pattern- matching. Equations allows us to define explicitly the hereditary
substitution function, clarifying its algorithmic behavior in presence of term
and type substitutions. From this definition, consistency can easily be
derived. The algorithmic nature of our development is crucial to reflect
languages with type quantification, enlarging the class of languages on which
reflection methods can be used in the proof assistant.Comment: In Proceedings LFMTP 2015, arXiv:1507.07597. www:
http://equations-fpred.gforge.inria.fr
Proof-irrelevant model of CC with predicative induction and judgmental equality
We present a set-theoretic, proof-irrelevant model for Calculus of
Constructions (CC) with predicative induction and judgmental equality in
Zermelo-Fraenkel set theory with an axiom for countably many inaccessible
cardinals. We use Aczel's trace encoding which is universally defined for any
function type, regardless of being impredicative. Direct and concrete
interpretations of simultaneous induction and mutually recursive functions are
also provided by extending Dybjer's interpretations on the basis of Aczel's
rule sets. Our model can be regarded as a higher-order generalization of the
truth-table methods. We provide a relatively simple consistency proof of type
theory, which can be used as the basis for a theorem prover
On Sharing, Memoization, and Polynomial Time (Long Version)
We study how the adoption of an evaluation mechanism with sharing and
memoization impacts the class of functions which can be computed in polynomial
time. We first show how a natural cost model in which lookup for an already
computed value has no cost is indeed invariant. As a corollary, we then prove
that the most general notion of ramified recurrence is sound for polynomial
time, this way settling an open problem in implicit computational complexity
Weak function word shift
The fact that object shift only affects weak pronouns in mainland Scandinavian is seen as an instance of a more general observation that can be made in all Germanic languages: weak function words tend to avoid the edges of larger prosodic domains. This generalisation has been formulated within Optimality Theory in terms of alignment constraints on prosodic structure by Selkirk (1996) in explaining thedistribution of prosodically strong and weak forms of English functionwords, especially modal verbs, prepositions and pronouns. But a purely phonological account fails to integrate the syntactic licensing conditions for object shift in an appropriate way. The standard semantico-syntactic accounts of object shift, onthe other hand, fail to explain why it is only weak pronouns that undergo object shift. This paper develops an Optimality theoretic model of the syntax-phonology interface which is based on the interaction of syntactic and prosodic factors. The account can successfully be applied to further related phenomena in English and German
Classical Predicative Logic-Enriched Type Theories
A logic-enriched type theory (LTT) is a type theory extended with a primitive
mechanism for forming and proving propositions. We construct two LTTs, named
LTTO and LTTO*, which we claim correspond closely to the classical predicative
systems of second order arithmetic ACAO and ACA. We justify this claim by
translating each second-order system into the corresponding LTT, and proving
that these translations are conservative. This is part of an ongoing research
project to investigate how LTTs may be used to formalise different approaches
to the foundations of mathematics.
The two LTTs we construct are subsystems of the logic-enriched type theory
LTTW, which is intended to formalise the classical predicative foundation
presented by Herman Weyl in his monograph Das Kontinuum. The system ACAO has
also been claimed to correspond to Weyl's foundation. By casting ACAO and ACA
as LTTs, we are able to compare them with LTTW. It is a consequence of the work
in this paper that LTTW is strictly stronger than ACAO.
The conservativity proof makes use of a novel technique for proving one LTT
conservative over another, involving defining an interpretation of the stronger
system out of the expressions of the weaker. This technique should be
applicable in a wide variety of different cases outside the present work.Comment: 49 pages. Accepted for publication in special edition of Annals of
Pure and Applied Logic on Computation in Classical Logic. v2: Minor mistakes
correcte
Determiner spreading as DP-predication
Determiner Spreading (DS) occurs in adjectivally modified nominal phrases comprising more than one definite article, a phenomenon that has received considerable attention and has been extensively described in Greek. This paper discusses the syntactic properties of DS in detail and argues that DS structures are both arguments and predication configurations involving two DPs. This account successfully captures the word-order facts and the distinctive interpretation of DS, while also laying the groundwork towards unifying it with other structures linking two DPs in a predicative relation
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