66 research outputs found

    Σ1\Sigma_1 gaps as derived models and correctness of mice

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    Assume ZF + AD + V=L(R). Let [α,β][\alpha,\beta] be a Σ1\Sigma_1 gap with Jα(R)J_\alpha(R) admissible. We analyze Jβ(R)J_\beta(R) as a natural form of ``derived model'' of a premouse PP, where PP is found in a generic extension of VV. In particular, we will have P(R)Jβ(R)=P(R)D\mathcal{P}(R)\cap J_\beta(R)=\mathcal{P}(R)\cap D, and if Jβ(R)J_\beta(R)\models``Θ\Theta exists'', then Jβ(R)J_\beta(R) and DD 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 (L(R))M(L(R))^M, for ω\omega-small mice MM. We also establish some preliminary work toward this conjecture in the present paper.Comment: 128 page

    Erasure in dependently typed programming

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    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

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    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 SS has total planar tilings, which we denote TILETILE, or whether it has infinite connected but not necessarily total tilings, WTILEWTILE (short for `weakly tile'). We show that both TILEmILLmWTILETILE \equiv_m ILL \equiv_m WTILE, and thereby both TILETILE and WTILEWTILE are Σ11\Sigma^1_1-complete. We also show that the opposite problems, ¬TILE\neg TILE and SNTSNT (short for `Strongly Not Tile') are such that ¬TILEmWELLmSNT\neg TILE \equiv_m WELL \equiv_m SNT and so both ¬TILE\neg TILE and SNTSNT are both Π11\Pi^1_1-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 PTilePTile of periodic tilings, and ATileATile of aperiodic tilings. We then show that both of these sets are complete for the class of problems of the form (Σ11Π11)(\Sigma^1_1 \wedge \Pi^1_1). 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 CTCT as well as weaker principles of tiling, and show that there exist Weihrauch equivalences to closed choice on Baire space, CωωC_{\omega^\omega}. We also show that all Domino Problems that tile some infinite connected region are Weihrauch reducible to CωωC_{\omega^\omega}. 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

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    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

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    We identify a particular mouse, MldM^{\text{ld}}, the minimal ladder mouse, that sits in the mouse order just past MnM_n^{\sharp} for all nn, and we show that RMld=Qω+1\mathbb{R}\cap M^{\text{ld}} = Q_{\omega+1}, the set of reals that are Δω+11\Delta^1_{\omega+1} in a countable ordinal. Thus Qω+1Q_{\omega+1} is a mouse set. This is analogous to the fact that RM1=Q3\mathbb{R}\cap M^{\sharp}_1 = Q_3 where M1M^{\sharp}_1 is the the sharp for the minimal inner model with a Woodin cardinal, and Q3Q_3 is the set of reals that are Δ31\Delta^1_3 in a countable ordinal. More generally RM2n+1=Q2n+3\mathbb{R}\cap M^{\sharp}_{2n+1} = Q_{2n+3}. The mouse MldM^{\text{ld}} and the set Qω+1Q_{\omega+1} 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. RMldQω+1\mathbb{R}\cap M^{\text{ld}} \subseteq Q_{\omega+1} was known in the 1990's. But Qω+1MldQ_{\omega+1} \subseteq M^{\text{ld}} 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

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    The parameter-free part PA2\text{PA}_2^\ast of PA2\text{PA}_2, the 2nd order Peano arithmetic, is considered. We make use of a product/iterated Sacks forcing to define an ω\omega-model of PA2+CA(Σ21)\text{PA}_2^\ast + \text{CA}(\Sigma^1_2), in which an example of the full Comprehension schema CA\text{CA} fails. Using Cohen's forcing, we also define an ω\omega-model of PA2\text{PA}_2^\ast, in which not every set has its complement, and hence the full CA\text{CA} fails in a rather elementary way.Comment: 13 page

    Dualities in modal logic

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    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

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    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

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    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

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    We show that it is consistent relative to ZF, that there is no well-ordering of R\mathbb{R} 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 R\mathbb{R} 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 DCω1\mathsf{DC}_{\omega_1} holds, or countable choice for reals fails. Assuming the consistency of an inaccessible cardinal, "projective" can even be replaced with "L(R)L(\mathbb{R})". This vastly strengthens earlier consistency results in the literature.Comment: 16 page
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