6,025 research outputs found
Actions on permutations and unimodality of descent polynomials
We study a group action on permutations due to Foata and Strehl and use it to
prove that the descent generating polynomial of certain sets of permutations
has a nonnegative expansion in the basis ,
. This property implies symmetry and unimodality. We
prove that the action is invariant under stack-sorting which strengthens recent
unimodality results of B\'ona. We prove that the generalized permutation
patterns and are invariant under the action and use this to
prove unimodality properties for a -analog of the Eulerian numbers recently
studied by Corteel, Postnikov, Steingr\'{\i}msson and Williams.
We also extend the action to linear extensions of sign-graded posets to give
a new proof of the unimodality of the -Eulerian polynomials of
sign-graded posets and a combinatorial interpretations (in terms of
Stembridge's peak polynomials) of the corresponding coefficients when expanded
in the above basis.
Finally, we prove that the statistic defined as the number of vertices of
even height in the unordered decreasing tree of a permutation has the same
distribution as the number of descents on any set of permutations invariant
under the action. When restricted to the set of stack-sortable permutations we
recover a result of Kreweras.Comment: 19 pages, revised version to appear in Europ. J. Combi
Weighted logics for artificial intelligence : an introductory discussion
International audienceBefore presenting the contents of the special issue, we propose a structured introductory overview of a landscape of the weighted logics (in a general sense) that can be found in the Artificial Intelligence literature, highlighting their fundamental differences and their application areas
Challenges and Successes of the Transition to Online Format of a Lower Division Aerospace Engineering Class during COVID-19
During the Spring 2020 semester, after the transition to online modality due to the COVID-19 pandemic, student engagement and participation level dropped significantly and student performance suffered in âComputer Programing for Aerospace Engineersâ (AE 30), a lower division computer programming course in Aerospace Engineering. Cognitive empathy, metacognition, and zyBooks (specific brand of interactive online course material) are known to improve student engagement, participation, and performance. Thus, a cognitive empathy ice-breaker activity, a metacognition exam reflection exercise, and interactive zyBook exercises were incorporated and implemented in AE 30 to help mitigate the effects of the pandemic in the new online environment. The current investigation presents the assessment of the activities and exercises as effective means of improving student engagement, participation, and performance in an online modality amid a pandemic during the Spring 2020 semester. Instructor observations revealed that the cognitive empathy ice-breaker was a powerful way to allow students to share difficult emotions but created a distracting and intimidating atmosphere. However, after the cognitive empathy ice-breaker, students were more engaged and participative than on other days. The metacognition exam reflection and interactive zyBook exercises were found to be moderately correlated to improved student performance
Functional Ownership through Fractional Uniqueness
Ownership and borrowing systems, designed to enforce safe memory management
without the need for garbage collection, have been brought to the fore by the
Rust programming language. Rust also aims to bring some guarantees offered by
functional programming into the realm of performant systems code, but the type
system is largely separate from the ownership model, with type and borrow
checking happening in separate compilation phases. Recent models such as
RustBelt and Oxide aim to formalise Rust in depth, but there is less focus on
integrating the basic ideas into more traditional type systems. An approach
designed to expose an essential core for ownership and borrowing would open the
door for functional languages to borrow concepts found in Rust and other
ownership frameworks, so that more programmers can enjoy their benefits.
One strategy for managing memory in a functional setting is through
uniqueness types, but these offer a coarse-grained view: either a value has
exactly one reference, and can be mutated safely, or it cannot, since other
references may exist. Recent work demonstrates that linear and uniqueness types
can be combined in a single system to offer restrictions on program behaviour
and guarantees about memory usage. We develop this connection further, showing
that just as graded type systems like those of Granule and Idris generalise
linearity, Rust's ownership model arises as a graded generalisation of
uniqueness. We combine fractional permissions with grading to give the first
account of ownership and borrowing that smoothly integrates into a standard
type system alongside linearity and graded types, and extend Granule
accordingly with these ideas.Comment: 23 pages + references. In submissio
Quantitative Equality in Substructural Logic via Lipschitz Doctrines
Substructural logics naturally support a quantitative interpretation of
formulas, as they are seen as consumable resources. Distances are the
quantitative counterpart of equivalence relations: they measure how much two
objects are similar, rather than just saying whether they are equivalent or
not. Hence, they provide the natural choice for modelling equality in a
substructural setting. In this paper, we develop this idea, using the
categorical language of Lawvere's doctrines. We work in a minimal fragment of
Linear Logic enriched by graded modalities, which are needed to write a
resource sensitive substitution rule for equality, enabling its quantitative
interpretation as a distance. We introduce both a deductive calculus and the
notion of Lipschitz doctrine to give it a sound and complete categorical
semantics. The study of 2-categorical properties of Lipschitz doctrines
provides us with a universal construction, which generates examples based for
instance on metric spaces and quantitative realisability. Finally, we show how
to smoothly extend our results to richer substructural logics, up to full
Linear Logic with quantifiers
Combining Effects and Coeffects via Grading
This is the author accepted manuscript. It is currently under an indefinite embargo pending publication by the Association for Computing Machinery. and are two general, complementary aspects of program behaviour. They roughly correspond to computations which change the execution context (effects) versus computations which make demands on the context (coeffects). Effectful features include partiality, non-determinism, input-output, state, and exceptions. Coeffectful features include resource demands, variable access, notions of linearity, and data input requirements.
The effectful or coeffectful behaviour of a program can be captured and described via type-based analyses, with fine grained information provided by monoidal effect annotations and semiring coeffects. Various recent work has proposed models for such typed calculi in terms of for effects and for coeffects.
Effects and coeffects have been studied separately so far, but in practice many computations are both effectful and coeffectful, e.g., possibly throwing exceptions but with resource requirements. To remedy this, we introduce a new general calculus with a combined . This can describe both the and that a program has on its context, as well as interactions between these effectful and coeffectful features of computation. The effect-coeffect system has a denotational model in terms of effect-graded monads and coeffect-graded comonads where interaction is expressed via the novel concept of . This graded semantics unifies the syntactic type theory with the denotational model. We show that our calculus can be instantiated to describe in a natural way various different kinds of interaction between a program and its evaluation context.Orchard was supported by EPSRC grant EP/M026124/1 and EP/K011715/1 (whilst previously at Imperial College London), Katsumata by JSPS KAKENHI grant JP15K00014, Uustalu by Estonian Min. of Educ. and Res. grant IUT33-13 and Estonian Sci. Found. grant 9475. Gaboardiâs work was done in part while at the University of Dundee, UK supported by EPSRC grant EP/M022358/1
Epistemic Logic Programs with World View Constraints
An epistemic logic program is a set of rules written in the language of Epistemic Specifications, an extension of the language of answer set programming that provides for more powerful introspective reasoning through the use of modal operators K and M. We propose adding a new construct to Epistemic Specifications called a world view constraint that provides a universal device for expressing global constraints in the various versions of the language. We further propose the use of subjective literals (literals preceded by K or M) in rule heads as syntactic sugar for world view constraints. Additionally, we provide an algorithm for finding the world views of such programs
Characterizing and Extending Answer Set Semantics using Possibility Theory
Answer Set Programming (ASP) is a popular framework for modeling
combinatorial problems. However, ASP cannot easily be used for reasoning about
uncertain information. Possibilistic ASP (PASP) is an extension of ASP that
combines possibilistic logic and ASP. In PASP a weight is associated with each
rule, where this weight is interpreted as the certainty with which the
conclusion can be established when the body is known to hold. As such, it
allows us to model and reason about uncertain information in an intuitive way.
In this paper we present new semantics for PASP, in which rules are interpreted
as constraints on possibility distributions. Special models of these
constraints are then identified as possibilistic answer sets. In addition,
since ASP is a special case of PASP in which all the rules are entirely
certain, we obtain a new characterization of ASP in terms of constraints on
possibility distributions. This allows us to uncover a new form of disjunction,
called weak disjunction, that has not been previously considered in the
literature. In addition to introducing and motivating the semantics of weak
disjunction, we also pinpoint its computational complexity. In particular,
while the complexity of most reasoning tasks coincides with standard
disjunctive ASP, we find that brave reasoning for programs with weak
disjunctions is easier.Comment: 39 pages and 16 pages appendix with proofs. This article has been
accepted for publication in Theory and Practice of Logic Programming,
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