300 research outputs found
Positive Logic with Adjoint Modalities: Proof Theory, Semantics and Reasoning about Information
We consider a simple modal logic whose non-modal part has conjunction and
disjunction as connectives and whose modalities come in adjoint pairs, but are
not in general closure operators. Despite absence of negation and implication,
and of axioms corresponding to the characteristic axioms of (e.g.) T, S4 and
S5, such logics are useful, as shown in previous work by Baltag, Coecke and the
first author, for encoding and reasoning about information and misinformation
in multi-agent systems. For such a logic we present an algebraic semantics,
using lattices with agent-indexed families of adjoint pairs of operators, and a
cut-free sequent calculus. The calculus exploits operators on sequents, in the
style of "nested" or "tree-sequent" calculi; cut-admissibility is shown by
constructive syntactic methods. The applicability of the logic is illustrated
by reasoning about the muddy children puzzle, for which the calculus is
augmented with extra rules to express the facts of the muddy children scenario.Comment: This paper is the full version of the article that is to appear in
the ENTCS proceedings of the 25th conference on the Mathematical Foundations
of Programming Semantics (MFPS), April 2009, University of Oxfor
Two loop detection mechanisms: a comparison
In order to compare two loop detection mechanisms we describe two calculi for theorem proving in intuitionistic propositional logic. We call them both MJ Hist, and distinguish between them by description as `Swiss' or `Scottish'. These calculi combine in different ways the ideas on focused proof search of Herbelin and Dyckhoff & Pinto with the work of Heuerding emphet al on loop detection. The Scottish calculus detects loops earlier than the Swiss calculus but at the expense of modest extra storage in the history. A comparison of the two approaches is then given, both on a theoretic and on an implementational level
From Display to Labelled Proofs for Tense Logics
We introduce an effective translation from proofs in the display calculus to proofs in the labelled calculus in the context of tense logics. We identify the labelled calculus proofs in the image of this translation as those built from labelled sequents whose underlying directed graph possesses certain properties. For the basic normal tense logic Kt, the image is shown to be the set of all proofs in the labelled calculus G3Kt
Cut-elimination for the modal Grzegorczyk logic via non-well-founded proofs
We present a sequent calculus for the modal Grzegorczyk logic Grz allowing
non-well-founded proofs and obtain the cut-elimination theorem for it by
constructing a continuous cut-elimination mapping acting on these proofs.Comment: WOLLIC'17, 12 pages, 1 appendi
A Focused Sequent Calculus Framework for Proof Search in Pure Type Systems
Basic proof-search tactics in logic and type theory can be seen as the
root-first applications of rules in an appropriate sequent calculus, preferably
without the redundancies generated by permutation of rules. This paper
addresses the issues of defining such sequent calculi for Pure Type Systems
(PTS, which were originally presented in natural deduction style) and then
organizing their rules for effective proof-search. We introduce the idea of
Pure Type Sequent Calculus with meta-variables (PTSCalpha), by enriching the
syntax of a permutation-free sequent calculus for propositional logic due to
Herbelin, which is strongly related to natural deduction and already well
adapted to proof-search. The operational semantics is adapted from Herbelin's
and is defined by a system of local rewrite rules as in cut-elimination, using
explicit substitutions. We prove confluence for this system. Restricting our
attention to PTSC, a type system for the ground terms of this system, we obtain
the Subject Reduction property and show that each PTSC is logically equivalent
to its corresponding PTS, and the former is strongly normalising iff the latter
is. We show how to make the logical rules of PTSC into a syntax-directed system
PS for proof-search, by incorporating the conversion rules as in
syntax-directed presentations of the PTS rules for type-checking. Finally, we
consider how to use the explicitly scoped meta-variables of PTSCalpha to
represent partial proof-terms, and use them to analyse interactive proof
construction. This sets up a framework PE in which we are able to study
proof-search strategies, type inhabitant enumeration and (higher-order)
unification
<i>“We’re Seeking Relevance”</i>: Qualitative Perspectives on the Impact of Learning Analytics on Teaching and Learning
Whilst a significant body of learning analytics research tends to focus on impact from the perspective of usability or improved learning outcomes, this paper proposes an approach based on Affordance Theory to describe awareness and intention as a bridge between usability and impact. 10 educators at 3 European institutions participated in detailed interviews on the affordances they perceive in using learning analytics to support practice in education. Evidence illuminates connections between an educator’s epistemic beliefs about learning and the purpose of education, their perception of threats or resources in delivering a successful learning experience, and the types of data they would consider as evidence in recognising or regulating learning. This evidence can support the learning analytics community in considering the proximity to the student, the role of the educator, and their personal belief structure in developing robust analytics tools that educators may be more likely to use
Sequent Calculus in the Topos of Trees
Nakano's "later" modality, inspired by G\"{o}del-L\"{o}b provability logic,
has been applied in type systems and program logics to capture guarded
recursion. Birkedal et al modelled this modality via the internal logic of the
topos of trees. We show that the semantics of the propositional fragment of
this logic can be given by linear converse-well-founded intuitionistic Kripke
frames, so this logic is a marriage of the intuitionistic modal logic KM and
the intermediate logic LC. We therefore call this logic
. We give a sound and cut-free complete sequent
calculus for via a strategy that decomposes
implication into its static and irreflexive components. Our calculus provides
deterministic and terminating backward proof-search, yields decidability of the
logic and the coNP-completeness of its validity problem. Our calculus and
decision procedure can be restricted to drop linearity and hence capture KM.Comment: Extended version, with full proof details, of a paper accepted to
FoSSaCS 2015 (this version edited to fix some minor typos
Proving Craig and Lyndon Interpolation Using Labelled Sequent Calculi
We have recently presented a general method of proving the fundamental
logical properties of Craig and Lyndon Interpolation (IPs) by induction on
derivations in a wide class of internal sequent calculi, including sequents,
hypersequents, and nested sequents. Here we adapt the method to a more general
external formalism of labelled sequents and provide sufficient criteria on the
Kripke-frame characterization of a logic that guarantee the IPs. In particular,
we show that classes of frames definable by quantifier-free Horn formulas
correspond to logics with the IPs. These criteria capture the modal cube and
the infinite family of transitive Geach logics
Focused labeled proof systems for modal logic
International audienceFocused proofs are sequent calculus proofs that group inference rules into alternating positive and negative phases. These phases can then be used to define macro-level inference rules from Gentzen's original and tiny introduction and structural rules. We show here that the inference rules of labeled proof systems for modal logics can similarly be described as pairs of such phases within the LKF focused proof system for first-order classical logic. We consider the system G3K of Negri for the modal logic K and define a translation from labeled modal formulas into first-order polarized formulas and show a strict correspondence between derivations in the two systems, i.e., each rule application in G3K corresponds to a bipole—a pair of a positive and a negative phases—in LKF. Since geometric axioms (when properly polarized) induce bipoles, this strong correspondence holds for all modal logics whose Kripke frames are characterized by geometric properties. We extend these results to present a focused labeled proof system for this same class of modal logics and show its soundness and completeness. The resulting proof system allows one to define a rich set of normal forms of modal logic proofs
北海道における知的障がい者の就労支援に関する一考察
知的障がい者の就労について、北海道及び北海道教育委員会が進めている障が いのある人の就労支援の充実に向けた取組の状況を概観することに加えて、北海道内 の特別支援学校在籍者の約8割を占めている知的障がい特別支援学校の現状や就労支 援の取組について整理した。北海道において障がいある人の就労に大きな役割を果た してきた職親会の設立の経緯やなよろ地方職親会の障がい者雇用の状況やジョブコー チ養成研修の成果をまとめた。以上のことを踏まえて、知的障がい者の就労支援やキ ャリア教育の在り方について考察する
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