895 research outputs found
Syntactic Cut-Elimination for Intuitionistic Fuzzy Logic via Linear Nested Sequents
This paper employs the linear nested sequent framework to design a new
cut-free calculus LNIF for intuitionistic fuzzy logic--the first-order G\"odel
logic characterized by linear relational frames with constant domains. Linear
nested sequents--which are nested sequents restricted to linear
structures--prove to be a well-suited proof-theoretic formalism for
intuitionistic fuzzy logic. We show that the calculus LNIF possesses highly
desirable proof-theoretic properties such as invertibility of all rules,
admissibility of structural rules, and syntactic cut-elimination.Comment: Appended version of the paper "Syntactic Cut-Elimination for
Intuitionistic Fuzzy Logic via Linear Nested Sequents", accepted to the
International Symposium on Logical Foundations of Computer Science (LFCS
2020
Semi-simplicial Types in Logic-enriched Homotopy Type Theory
The problem of defining Semi-Simplicial Types (SSTs) in Homotopy Type Theory
(HoTT) has been recognized as important during the Year of Univalent
Foundations at the Institute of Advanced Study. According to the interpretation
of HoTT in Quillen model categories, SSTs are type-theoretic versions of Reedy
fibrant semi-simplicial objects in a model category and simplicial and
semi-simplicial objects play a crucial role in many constructions in homotopy
theory and higher category theory. Attempts to define SSTs in HoTT lead to some
difficulties such as the need of infinitary assumptions which are beyond HoTT
with only non-strict equality types.
Voevodsky proposed a definition of SSTs in Homotopy Type System (HTS), an
extension of HoTT with non-fibrant types, including an extensional strict
equality type. However, HTS does not have the desirable computational
properties such as decidability of type checking and strong normalization. In
this paper, we study a logic-enriched homotopy type theory, an alternative
extension of HoTT with equational logic based on the idea of logic-enriched
type theories. In contrast to Voevodskys HTS, all types in our system are
fibrant and it can be implemented in existing proof assistants. We show how
SSTs can be defined in our system and outline an implementation in the proof
assistant Plastic
Bibliography on Realizability
AbstractThis document is a bibliography on realizability and related matters. It has been collected by Lars Birkedal based on submissions from the participants in âA Workshop on Realizability Semantics and Its Applicationsâ, Trento, Italy, June 30âJuly 1, 1999. It is available in BibTEX format at the following URL: http://www.cs.cmu.edu./~birkedal/realizability-bib.html
Explicit Evidence Systems with Common Knowledge
Justification logics are epistemic logics that explicitly include
justifications for the agents' knowledge. We develop a multi-agent
justification logic with evidence terms for individual agents as well as for
common knowledge. We define a Kripke-style semantics that is similar to
Fitting's semantics for the Logic of Proofs LP. We show the soundness,
completeness, and finite model property of our multi-agent justification logic
with respect to this Kripke-style semantics. We demonstrate that our logic is a
conservative extension of Yavorskaya's minimal bimodal explicit evidence logic,
which is a two-agent version of LP. We discuss the relationship of our logic to
the multi-agent modal logic S4 with common knowledge. Finally, we give a brief
analysis of the coordinated attack problem in the newly developed language of
our logic
W-types in setoids
W-types and their categorical analogue, initial algebras for polynomial
endofunctors, are an important tool in predicative systems to replace
transfinite recursion on well-orderings. Current arguments to obtain W-types in
quotient completions rely on assumptions, like Uniqueness of Identity Proofs,
or on constructions that involve recursion into a universe, that limit their
applicability to a specific setting. We present an argument, verified in Coq,
that instead uses dependent W-types in the underlying type theory to construct
W-types in the setoid model. The immediate advantage is to have a proof more
type-theoretic in flavour, which directly uses recursion on the underlying
W-type to prove initiality. Furthermore, taking place in intensional type
theory and not requiring any recursion into a universe, it may be generalised
to various categorical quotient completions, with the aim of finding a uniform
construction of extensional W-types.Comment: 17 pages, formalised in Coq; v2: added reference to formalisatio
Bayesian Confirmation and Justifications
We introduce a family of probabilistic justification logics that feature Bayesian confirmations. Our logics include new justification terms representing evidence that make a proposition firm in the sense of making it more probable. We present syntax and semantics of our logic and establish soundness and strong completeness. Moreover, we show how to formalize in our logic the screening-off condition for transitivity of Bayesian confirmations
Extended ML: Past, present and future
An overview of past, present and future work on the Extended ML formal program development framework is given, with emphasis on two topics of current active research: the semantics of the Extended ML specification language, and tools to support formal program development
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