66 research outputs found
gaps as derived models and correctness of mice
Assume ZF + AD + V=L(R). Let be a gap with
admissible. We analyze as a natural form of
``derived model'' of a premouse , where is found in a generic extension
of . In particular, we will have , and if ``
exists'', then and in fact have the same universe. This
analysis will be employed in further work, yet to appear, toward a resolution
of a conjecture of Rudominer and Steel on the nature of , for
-small mice . We also establish some preliminary work toward this
conjecture in the present paper.Comment: 128 page
Erasure in dependently typed programming
It is important to reduce the cost of correctness in programming. Dependent types
and related techniques, such as type-driven programming, offer ways to do so.
Some parts of dependently typed programs constitute evidence of their typecorrectness
and, once checked, are unnecessary for execution. These parts can easily
become asymptotically larger than the remaining runtime-useful computation, which
can cause linear-time algorithms run in exponential time, or worse. It would be
unnacceptable, and contradict our goal of reducing the cost of correctness, to make
programs run slower by only describing them more precisely.
Current systems cannot erase such computation satisfactorily. By modelling
erasure indirectly through type universes or irrelevance, they impose the limitations
of these means to erasure. Some useless computation then cannot be erased and
idiomatic programs remain asymptotically sub-optimal.
This dissertation explains why we need erasure, that it is different from other
concepts like irrelevance, and proposes two ways of erasing non-computational data.
One is an untyped flow-based useless variable elimination, adapted for dependently
typed languages, currently implemented in the Idris 1 compiler.
The other is the main contribution of the dissertation: a dependently typed core
calculus with erasure annotations, full dependent pattern matching, and an algorithm
that infers erasure annotations from unannotated (or partially annotated) programs.
I show that erasure in well-typed programs is sound in that it commutes with
single-step reduction. Assuming the Church-Rosser property of reduction, I show
that properties such as Subject Reduction hold, which extends the soundness result
to multi-step reduction. I also show that the presented erasure inference is sound
and complete with respect to the typing rules; that this approach can be extended
with various forms of erasure polymorphism; that it works well with monadic I/O
and foreign functions; and that it is effective in that it not only removes the runtime
overhead caused by dependent typing in the presented examples, but can also shorten
compilation times."This work was supported by the University of St Andrews (School of Computer
Science)." -- Acknowledgement
Computability and Tiling Problems
In this thesis we will present and discuss various results pertaining to
tiling problems and mathematical logic, specifically computability theory. We
focus on Wang prototiles, as defined in [32]. We begin by studying Domino
Problems, and do not restrict ourselves to the usual problems concerning finite
sets of prototiles. We first consider two domino problems: whether a given set
of prototiles has total planar tilings, which we denote , or whether
it has infinite connected but not necessarily total tilings, (short for
`weakly tile'). We show that both , and
thereby both and are -complete. We also show that
the opposite problems, and (short for `Strongly Not Tile')
are such that and so both
and are both -complete. Next we give some consideration to the
problem of whether a given (infinite) set of prototiles is periodic or
aperiodic. We study the sets of periodic tilings, and of
aperiodic tilings. We then show that both of these sets are complete for the
class of problems of the form . We also present
results for finite versions of these tiling problems. We then move on to
consider the Weihrauch reducibility for a general total tiling principle
as well as weaker principles of tiling, and show that there exist Weihrauch
equivalences to closed choice on Baire space, . We also show
that all Domino Problems that tile some infinite connected region are Weihrauch
reducible to . Finally, we give a prototile set of 15
prototiles that can encode any Elementary Cellular Automaton (ECA). We make use
of an unusual tile set, based on hexagons and lozenges that we have not see in
the literature before, in order to achieve this.Comment: PhD thesis. 179 pages, 13 figure
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established
The Mouse Set Theorem Just Past Projective
We identify a particular mouse, , the minimal ladder mouse,
that sits in the mouse order just past for all , and we show
that , the set of reals that are
in a countable ordinal. Thus is a mouse
set. This is analogous to the fact that
where is the the sharp for the minimal inner model with a Woodin
cardinal, and is the set of reals that are in a countable
ordinal. More generally . The
mouse and the set compose the next natural pair
to consider in this series of results. Thus we are proving the mouse set
theorem just past projective. Some of this is not new. was known in the 1990's. But
was open until Woodin found a proof in
2018. The main goal of this paper is to give Woodin's proof.Comment: 30 page
The parameterfree Comprehension does not imply the full Comprehension in the 2nd order Peano arithmetic
The parameter-free part of , the 2nd order
Peano arithmetic, is considered. We make use of a product/iterated Sacks
forcing to define an -model of , in which an example of the full Comprehension schema
fails. Using Cohen's forcing, we also define an -model of
, in which not every set has its complement, and hence the
full fails in a rather elementary way.Comment: 13 page
Dualities in modal logic
Categorical dualities are an important tool in the study of (modal) logics. They offer conceptual understanding and enable the transfer of results between the different semantics of a logic. As such, they play a central role in the proofs of completeness theorems, Sahlqvist theorems and Goldblatt-Thomason theorems. A common way to obtain dualities is by extending existing ones. For example, Jonsson-Tarski duality is an extension of Stone duality. A convenient formalism to carry out such extensions is given by the dual categorical notions of algebras and coalgebras. Intuitively, these allow one to isolate the new part of a duality from the existing part. In this thesis we will derive both existing and new dualities via this route, and we show how to use the dualities to investigate logics. However, not all (modal logical) paradigms fit the (co)algebraic perspective. In particular, modal intuitionistic logics do not enjoy a coalgebraic treatment, and there is a general lack of duality results for them. To remedy this, we use a generalisation of both algebras and coalgebras called dialgebras. Guided by the research field of coalgebraic logic, we introduce the framework of dialgebraic logic. We show how a large class of modal intuitionistic logics can be modelled as dialgebraic logics and we prove dualities for them. We use the dialgebraic framework to prove general completeness, Hennessy-Milner, representation and Goldblatt-Thomason theorems, and instantiate this to a wide variety of modal intuitionistic logics. Additionally, we use the dialgebraic perspective to investigate modal extensions of the meet-implication fragment of intuitionistic logic. We instantiate general dialgebraic results, and describe how modal meet-implication logics relate to modal intuitionistic logics
Academic Integrity in Canada
This open access book presents original contributions and thought leadership on academic integrity from a variety of Canadian scholars. It showcases how our understanding and support for academic integrity have progressed, while pointing out areas urgently requiring more attention. Firmly grounded in the scholarly literature globally, it engages with the experience of local practicioners. It presents aspects of academic integrity that is specific to Canada, such as the existence of an "honour culture", rather than relying on an "honour code". It also includes Indigenous voices and perspectives that challenge traditional understandings of intellectual property, as well as new understandings that have arisen as a consequence of Covid-19 and the significant shift to online and remote learning. This book will be of interest to senior university and college administrators who are interested in ensuring the integrity of their institutions. It will also be of interest to those implementing university and college policy, as well as those who support students in their scholarly work
Programming Languages and Systems
This open access book constitutes the proceedings of the 31st European Symposium on Programming, ESOP 2022, which was held during April 5-7, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 21 regular papers presented in this volume were carefully reviewed and selected from 64 submissions. They deal with fundamental issues in the specification, design, analysis, and implementation of programming languages and systems
Maximal sets without Choice
We show that it is consistent relative to ZF, that there is no well-ordering
of while a wide class of special sets of reals such as Hamel
bases, transcendence bases, Vitali sets or Bernstein sets exists. To be more
precise, we can assume that every projective hypergraph on has a
maximal independent set, among a few other things. For example, we get
transversals for all projective equivalence relations. Moreover, this is
possible while either holds, or countable choice for
reals fails. Assuming the consistency of an inaccessible cardinal, "projective"
can even be replaced with "". This vastly strengthens earlier
consistency results in the literature.Comment: 16 page
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