189 research outputs found
Cyclic proof systems for modal fixpoint logics
This thesis is about cyclic and ill-founded proof systems for modal fixpoint logics, with and without explicit fixpoint quantifiers.Cyclic and ill-founded proof-theory allow proofs with infinite branches or paths, as long as they satisfy some correctness conditions ensuring the validity of the conclusion. In this dissertation we design a few cyclic and ill-founded systems: a cyclic one for the weak Grzegorczyk modal logic K4Grz, based on our explanation of the phenomenon of cyclic companionship; and ill-founded and cyclic ones for the full computation tree logic CTL* and the intuitionistic linear-time temporal logic iLTL. All systems are cut-free, and the cyclic ones for K4Grz and iLTL have fully finitary correctness conditions.Lastly, we use a cyclic system for the modal mu-calculus to obtain a proof of the uniform interpolation property for the logic which differs from the original, automata-based one
Bootstrapping extensionality
Intuitionistic type theory is a formal system designed by Per Martin-Loef to be a full-fledged foundation in which to develop constructive mathematics. One particular variant, intensional type theory (ITT), features nice computational properties like decidable type-checking, making it especially suitable for computer implementation. However, as traditionally defined, ITT lacks many vital extensionality principles, such as function extensionality. We would like to extend ITT with the desired extensionality principles while retaining its convenient computational behaviour. To do so, we must first understand the extent of its expressive power, from its strengths to its limitations.
The contents of this thesis are an investigation into intensional type theory, and in particular into its power to express extensional concepts. We begin, in the first part, by developing an extension to the strict setoid model of type theory with a universe of setoids. The model construction is carried out in a minimal intensional type theoretic metatheory, thus providing a way to bootstrap extensionality by ``compiling'' it down to a few building blocks such as inductive families and proof-irrelevance.
In the second part of the thesis we explore inductive-inductive types (ITTs) and their relation to simpler forms of induction in an intensional setting. We develop a general method to reduce a subclass of infinitary IITs to inductive families, via an encoding that can be expressed in ITT without any extensionality besides proof-irrelevance. Our results contribute to further understand IITs and the expressive power of intensional type theory, and can be of practical use when formalizing mathematics in proof assistants that do not natively support induction-induction
Enriched universal algebra
Following the classical approach of Birkhoff, we introduce enriched universal
algebra. Given a suitable base of enrichment , we define a language
to be a collection of -ary function symbols whose arities
are taken among the objects of . The class of -terms is
constructed recursively from the symbols of , the morphisms in
, and by incorporating the monoidal structure of .
Then, -structures and interpretations of terms are defined, leading
to enriched equational theories. In this framework we prove several fundamental
theorems of universal algebra, including the characterization of algebras for
finitary monads on as models of an equational theories, and
several Birkhoff-type theorems
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Normal Form Bisimulations By Value
Normal form bisimilarities are a natural form of program equivalence resting
on open terms, first introduced by Sangiorgi in call-by-name. The literature
contains a normal form bisimilarity for Plotkin's call-by-value
-calculus, Lassen's \emph{enf bisimilarity}, which validates all of
Moggi's monadic laws and can be extended to validate . It does not
validate, however, other relevant principles, such as the identification of
meaningless terms -- validated instead by Sangiorgi's bisimilarity -- or the
commutation of \letexps. These shortcomings are due to issues with open terms
of Plotkin's calculus. We introduce a new call-by-value normal form
bisimilarity, deemed \emph{net bisimilarity}, closer in spirit to Sangiorgi's
and satisfying the additional principles. We develop it on top of an existing
formalism designed for dealing with open terms in call-by-value. It turns out
that enf and net bisimilarities are \emph{incomparable}, as net bisimilarity
does not validate Moggi's laws nor . Moreover, there is no easy way to
merge them. To better understand the situation, we provide an analysis of the
rich range of possible call-by-value normal form bisimilarities, relating them
to Ehrhard's relational model.Comment: Rewritten version (deleted toy similarity and explained proof method
on naive similarity) -- Submitted to POPL2
Nominal Recursors as Epi-Recursors: Extended Technical Report
We study nominal recursors from the literature on syntax with bindings and
compare them with respect to expressiveness. The term "nominal" refers to the
fact that these recursors operate on a syntax representation where the names of
bound variables appear explicitly, as in nominal logic. We argue that nominal
recursors can be viewed as epi-recursors, a concept that captures abstractly
the distinction between the constructors on which one actually recurses, and
other operators and properties that further underpin recursion.We develop an
abstract framework for comparing epi-recursors and instantiate it to the
existing nominal recursors, and also to several recursors obtained from them by
cross-pollination. The resulted expressiveness hierarchies depend on how
strictly we perform this comparison, and bring insight into the relative merits
of different axiomatizations of syntax. We also apply our methodology to
produce an expressiveness hierarchy of nominal corecursors, which are
principles for defining functions targeting infinitary non-well-founded terms
(which underlie lambda-calculus semantics concepts such as B\"ohm trees). Our
results are validated with the Isabelle/HOL theorem prover
From Double Pushout Grammars to Hypergraph Lambek Grammars With and Without Exponential Modality
We study how to relate well-known hypergraph grammars based on the double
pushout (DPO) approach and grammars over the hypergraph Lambek calculus HL
(called HL-grammars). It turns out that DPO rules can be naturally encoded by
types of HL using methods similar to those used by Kanazawa for
multiplicative-exponential linear logic. In order to generalize his reasonings
we extend the hypergraph Lambek calculus by adding the exponential modality,
which results in a new calculus HMEL0; then we prove that any DPO grammar can
be converted into an equivalent HMEL0-grammar. We also define the conjunctive
Kleene star, which behaves similarly to this exponential modality, and
establish a similar result. If we add neither the exponential modality nor the
conjunctive Kleene star to HL, then we can still use the same encoding and show
that any DPO grammar with a linear restriction on the length of derivations can
be converted into an equivalent HL-grammar.Comment: In Proceedings TERMGRAPH 2022, arXiv:2303.1421
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