Type Theory

Variable types for meaning assembly: a logical syntax for generic noun phrases introduced by "most"

by Christian Retoré

To appear in Recherches linguistiques de Vincennes

We propose a model which computes semantic representations viewed as formulae of higher order multisorted logic by... more We propose a model which computes semantic representations viewed as formulae of higher order multisorted logic by assembling them in type theory (second order lambda calculus), in particular for sentences involving generics introduced by “most”.

Ottimalità dell'ottimalità - Complessità della riduzione su grafi di condivisione

by Marco Solieri

Master thesis in computer science, Alma Mater Studiorum, Università di Bologna, 2011.

Supervisor: Simone Martini
Cosupervisor: Stefano Guerrini

Italian language.

Trent’anni or sono il concetto di ottimalità venne formulato in senso teorico da Lévy, ma solo un decennio dopo... more Trent’anni or sono il concetto di ottimalità venne formulato in senso teorico da Lévy, ma solo un decennio dopo Lamping riesce a darne elegante implementazione algoritmica. Realizza un sistema di riduzione su grafi che si scoprirà poi avere interessanti analogie con la logica lineare presentata nello stesso periodo da Girard.

Ma l’ottimalità è davvero ottimale? In altre parole, l’implementazione ottimale del λ calcolo realizzata attraverso i grafi di condivisione, è davvero la migliore strategia di riduzione, in termini di complessità?

Dopo anni di infondati dubbi e di immeritato oblìo, alla conferenza LICS del 2007, Baillot, Coppola e Dal Lago, danno una prima risposta positiva, seppur parziale. Considerano infatti il caso particolare delle logiche affini elementare e leggera, che possiedono interessanti proprietà a livello di complessità intrinseca e semplificano l’arduo problema.

La prima parte di questa tesi presenta, in sintesi, la teoria dell’ottimalità e la sua implementazione condivisa.
La seconda parte affronta il tema della sua complessità, a cominciare da una panoramica dei più importanti risultati ad essa legati. La successiva introduzione alle logiche affini, e alle relative caratteristiche, costituisce la necessaria premessa ai due capitoli successivi, che presentano una dimostrazione alternativa ed originale degli ultimi risultati basati appunto su EAL e LAL. Nel primo dei due capitoli viene definito un sistema intermedio fra le reti di prova delle logiche e la riduzione dei grafi, nel secondo sono dimostrate correttezza ed ottimalità dell’implementazione condivisa per mezzo di una simulazione.

Lungo la trattazione sono offerti alcuni spunti di riflessione sulla dinamica interna della β-riduzione riduzione e sui suoi legami con le reti di prova della logica lineare.

Typing OpenDLib Repository Service: Strengths of an Information Object Type

by Paolo Manghi

Gradual Information Flow Typing

by Tim Disney

2011, Tim Disney and Cormac Flanagan, STOP 2011

We present a method to support the gradual evolution of secure scripts by formalizing an extension of the simply-typed... more We present a method to support the gradual evolution of secure scripts by formalizing an extension of the simply-typed lambda calculus that provides information flow constructs. These constructs allow initially insecure programs to evolve via targeted refactoring and to provide dynamic information flow guarantees via casts, as well as static information flow guarantees via labeled types.

Plausibility Revision in Higher-order Logic with an Application in Two-Dimensional Semantics

by Erich Rast

published in Arrazola, Xabier and Ponte, María (eds.): LogKCA-10 - Proceedings of the Second ILCLI International Workshop on Logic and Philosophy of Knowledge, Communication and Action. San Sebastian/Donostia: University of the Basque Country Press/ILCLI 2010, pp. 387-403.

The printed version unfortunately contains some nasty and embarrasing errors, most of them caused by hitting the Emacs key for "downcase region" just before the final deadline. :O

In this article, a qualitative notion of subjective plausibility and its revision based on a preorder relation are... more In this article, a qualitative notion of subjective plausibility and its revision based on a preorder relation are implemented in higher-order logic. This notion of plausibility is used for modeling pragmatic aspects of communication on top of traditional two-dimensional semantic representations.

On contextual domain restriction in categorial grammar

by Erich Rast

forthcoming in Synthese, online first: DOI 10.1007/s11229-011-9960-2.

Quantifier domain restriction (QDR) and two versions of nominal restriction (NR) are implemented as restrictions that... more Quantifier domain restriction (QDR) and two versions of nominal restriction (NR) are implemented as restrictions that depend on a previously introduced interpreter and interpretation time in a two-dimensional semantic framework on the basis of simple type theory and categorial grammar. Against Stanley (2002) it is argued that a suitable version of QDR can deal with superlatives like tallest. However, it is shown that NR is needed to account for utterances when the speaker intends to convey different restrictions for multiple uses of the same quantifying determiner. We argue that NR generally fares better with such examples but also observe that examples like Every sailor waves at every sailor might be pragmatically anomalous. An account of contextual domain restriction is proposed that (i) excludes these anomalous readings (but it is shown how they could be included), (ii) makes it possible to express different contextual domain restrictions as long-range dependencies on an interpreter and an interpretation time, and (iii) additionally models restrictions based on locative constructions as general mereological constraints introduced by shifting the index.

Un calcul de termes typés pour la pragmatique lexicale: chemins et voyageurs fictifs dans un corpus de récits de voyages

by Christian Retoré

(avec Richard Moot et Laurent Prévot)
papier court TALN2011 Juin 2011 Montpellier vol. 2 pages 161--166.

A discursive analysis of itineraries in an historical and regional corpus of travels:syntax, semantics, and pragmatics in a unified type theoretical framework

by Christian Retoré

(with Richard Moot and Laurent Prévot)
In Contraints in Discourse 2011

ON THE SYSTEM F AS A GLUE LANGUAGE FOR NATURAL-LANGUAGE COMPOSITIONAL-SEMANTICS

by Christian Retoré

Some arguments supporting the quantification over types that we use in many papers in compositional and lexical samentics and pragmatics.
To be included into a paper, to become a squib... don't know yet.

In order to model in a compositional framework some phenomena of lexical pragmatics
and in particular the ones... more In order to model in a compositional framework some phenomena of lexical pragmatics
and in particular the ones studied by Nicholas Asher
several contributions developed in our team did use the system F of Jean-Yves Girard
to construct logical formulae expressing the meaning of sentences
--- while other authors prefer to use Per Martin-Löf's type theory.
In this note we explain the motivations supporting our preference for system F.

Categories and Types for Axiomatic Domain Theory

by Adam Eppendahl

PhD Thesis

Domain Theory provides a denotational semantics for programming languages and calculi containing fixed point... more Domain Theory provides a denotational semantics for programming languages and calculi containing fixed point combinators and other so-called paradoxical combinators. This dissertation presents results in the category theory and type theory of Axiomatic Domain Theory.
Prompted by the adjunctions of Domain Theory, we extend Benton's linear/nonlinear dual-sequent calculus to include recursive linear types. and define a class of models by adding Freyd's notion of algebraic compactness to the monoidal adjunctions that model Benton's calculus.
We observe that algebraic compactness is better behaved in the context of categories with structural actions than in the usual context of enriched categories. We establish a theory of structural algebraic compactness that allows us to describe our models without reference to enrichment. We develop a 2-categorical perspective on structural actions, including a presentation of monoidal categories that leads directly to Kelly's reduced coherence conditions.
We observe that Benton's adjoint type constructors can be treated individually, semantically as well as syntactically, using free representations of distributors.
We type various of fixed point combinators using recursive types and function types, which we consider the core types of such calculi, together with the adjoint types. We use the idioms of these typings, which include oblique function spaces, to give a translation of the core of Levy's Call-By-Push-Value. The translation induces call-by-value and call-by-name translations of the core of Plotkin's Fixed Point Calculus.
Following Freyd, we construct a canonical fixed point operation from the algebras provided by the algebraic compactness of our models. Our analysis of Freyd's construction exposes a remarkable property of morphisms from coalgebras to algebras: morphisms from Gp to s correspond one-for-one to morphisms from p to Hs, where p is a coalgebra for HG and s an algebra for GH. We give an application of this property to the transposition of recursive coalgebras in Taylor's categorical theory of recursion where G is not left adjoint to H.
We develop a theory of parametric transformations corresponding to the uniformity property characterizing canonical fixed points and use this to derive abstract conditions on categories of domains which ensure that the interpretation of fixed point combinators coincides with the canonical fixed point operation.

Isabelle: the next 700 theorem provers

by Lawrence Paulson

In: P. Odifreddi (editor), Logic and Computer Science (Academic Press, 1990), 361–386.

The theorem prover Isabelle is described briefly and informally. Its historical development is traced from Edinburgh... more The theorem prover Isabelle is described briefly and informally. Its historical development is traced from Edinburgh LCF to the present day. The main issues are unification, quantifiers, and the representation of inference rules. The Edinburgh Logical Framework is also described, for a comparison with Isabelle. An appendix presents several Isabelle logics, including set theory and Constructive Type Theory, with examples of theorems.

A multi-modal type system and its procedural semantics for safe distributed programming

by Giuseppe Primiero

Draft. To be presented at Intuitionistic Modal Logic and Applications Workshop (IMLA11), co-located with 14th CLMPS.

Type-Theoretical Dynamics. Exploring Belief Revision in a Constructive Framework

by Giuseppe Primiero

Preprint. Final version forthcoming in "The Realism-Antirealism Debate in the Age of Alternative Logics". Springer 2011.

A Modular Hierarchy of Logical Frameworks

by Robin Adams

In Stefano Berardi, Mario Coppo, Ferruccio Damiani (Eds.): Types for Proofs and Programs, International Workshop, TYPES 2003, Torino, Italy, April 30 - May 4, 2003, Revised Selected Papers. LNCS 3085. Springer, 2004. 1-16.

We present a method for defining logical frameworks as a collection of features which are defined and behave... more We present a method for defining logical frameworks as a collection of features which are defined and behave independently of one another.  Each feature is a set of grammar clauses and rules of deduction such that the result of adding the feature to a framework is a conservative extension of the framework itself.  We show how several existing logical frameworks can be so built, and how several much weaker frameworks defined in this manner are adequate for expressing a wide variety of object logics.

Pure Type Systems with Judgemental Equality

by Robin Adams

Journal of Functional Programming 16 (2): 219-246, 2006

In a typing system, there are two approaches that may be taken to the
notion of equality.  One can use some... more In a typing system, there are two approaches that may be taken to the
notion of equality.  One can use some external relation of
convertibility defined on the terms of the grammar, such as
beta-convertibility or beta-eta-convertiblity; or one can
introduce a judgement form for equality into the rules of the typing
system itself.

For quite some time, it has been an open problem whether the two systems
produced by these two choices are equivalent.  This problem is
essentially the problem of proving the Subject Reduction property
holds in the system with judgemental equality.

In this paper, we shall
prove that the equivalence holds for all functional Pure Type
Systems (PTSs).  The proof essentially consists of proving the
Church-Rosser Theorem for a typed version of parallel one-step
reduction.  This method should generalise easily to many typing
systems which satisfy the Uniqueness of Types property.

Formalized Metatheory with Terms Represented by an Indexed Family of Types

by Robin Adams

In Filiatre, Paulin-Mohring and Werner (eds.), Types for Proofs and Programs, International Workshop, TYPES 2004, Jouy-en-Josas, France, December 15-18, 2004, Revised Selected Papers. LNCS 3839. Springer, 2006. 1-16.

We describe a recent formalization of several results from the metatheory of Pure Type Systems (PTSs) in Coq,... more We describe a recent formalization of several results from the metatheory of Pure Type Systems (PTSs) in Coq, including Subject Reduction, Uniqueness of Types in a functional PTS, and the difficult proof of Strengthening. The terms of the PTS are represented by an inductive family of types: "term n" is the type of all terms with at most n free variables. This representation of terms has often been used to study syntax and substitution, but not the metatheory of a formal system. We show how it requires many metatheorems to be stated in a somewhat unfamiliar "big-step" form, but then allows for very elegant and direct proofs.

Structural Subtyping for Inductive Types with Functorial Equality Rules

by Robin Adams

Co-authored with Zhaohui Luo.  Mathematical Structures in Computer Science 18 (2008), 931-972

Subtyping for inductive types in dependent type theories is studied in the framework of
coercive subtyping.... more Subtyping for inductive types in dependent type theories is studied in the framework of
coercive subtyping. General structural subtyping rules for parameterised inductive types
are formulated based on the notion of inductive schemata. Certain extensional equality
rules play an important role in proving some of the crucial properties of the type system
with these subtyping rules. In particular, it is shown that the structural subtyping rules
are coherent and that transitivity is admissible in the presence of the functorial rules of
computational equality.

Weyl's Predicative Classical Mathematics as a Logic-Enriched Type Theory

by Robin Adams

Co-authored with Zhaohui Luo.  ACM TOCL 11(2), 2010.

We construct a logic-enriched type theory LTTW that corresponds closely to the predicative system of foundations... more We construct a logic-enriched type theory LTTW that corresponds closely to the predicative system of foundations presented by Hermann Weyl in Das Kontinuum. We formalise many results from that book in LTTW, including Weyl's definition of the cardinality of a set and several results from real analysis, using the proof assistant Plastic that implements the logical framework LF. This case study shows how type theory can be used to represent a non-constructive foundation for mathematics.

Coercive Subtyping in Lambda-free Logical Frameworks

by Robin Adams

In Proceedings of the Fourth international Workshop on Logical Frameworks and Meta-Languages: theory and Practice (Montreal, Quebec, Canada, August 02 - 02, 2009). LFMTP '09. ACM, New York, NY, 30-39.  2009

We show how coercive subtyping may be added to a lambda-free
logical framework, by constructing the logical... more We show how coercive subtyping may be added to a lambda-free
logical framework, by constructing the logical framework TF<, an
extension of the lambda-free logical framework TF with coercive
subtyping. Instead of coercive application, TF< makes use of a
typecasting operation. We develop the metatheory of the resulting
framework, including providing some general conditions under
which typecasting in an object theory with coercive subtyping is
decidable. We show how TF< may be embedded in the logical
framework LF, and hence how results about LF may be deduced
from results about TF<.

Lambda-free Logical Frameworks

by Robin Adams

Submitted for publication in Annals of Pure and Applied Logic

We present the definition of the logical framework TF, the Type Framework. TF is a lambda-free logical framework; it... more We present the definition of the logical framework TF, the Type Framework. TF is a lambda-free logical framework; it does not include lambda-abstraction or product kinds. We give formal proofs of several results in the metatheory of TF, and show how it can be conservatively embedded in the logical framework LF: its judgements can be seen as the judgements of LF that are in beta-normal, eta-long normal form. We show how several properties, such as adequacy theorems for object theories and the injectivity of constants, can be proven more easily in TF, and then `lifted' to LF.