1,421 research outputs found
Physics, Topology, Logic and Computation: A Rosetta Stone
In physics, Feynman diagrams are used to reason about quantum processes. In
the 1980s, it became clear that underlying these diagrams is a powerful analogy
between quantum physics and topology: namely, a linear operator behaves very
much like a "cobordism". Similar diagrams can be used to reason about logic,
where they represent proofs, and computation, where they represent programs.
With the rise of interest in quantum cryptography and quantum computation, it
became clear that there is extensive network of analogies between physics,
topology, logic and computation. In this expository paper, we make some of
these analogies precise using the concept of "closed symmetric monoidal
category". We assume no prior knowledge of category theory, proof theory or
computer science.Comment: 73 pages, 8 encapsulated postscript figure
A Frobenius Algebraic Analysis for Parasitic Gaps
The interpretation of parasitic gaps is an ostensible case of non-linearity
in natural language composition. Existing categorial analyses, both in the
typelogical and in the combinatory traditions, rely on explicit forms of
syntactic copying. We identify two types of parasitic gapping where the
duplication of semantic content can be confined to the lexicon. Parasitic gaps
in adjuncts are analysed as forms of generalized coordination with a
polymorphic type schema for the head of the adjunct phrase. For parasitic gaps
affecting arguments of the same predicate, the polymorphism is associated with
the lexical item that introduces the primary gap. Our analysis is formulated in
terms of Lambek calculus extended with structural control modalities. A
compositional translation relates syntactic types and derivations to the
interpreting compact closed category of finite dimensional vector spaces and
linear maps with Frobenius algebras over it. When interpreted over the
necessary semantic spaces, the Frobenius algebras provide the tools to model
the proposed instances of lexical polymorphism.Comment: SemSpace 2019, to appear in Journal of Applied Logic
Enriched Lawvere Theories for Operational Semantics
Enriched Lawvere theories are a generalization of Lawvere theories that allow
us to describe the operational semantics of formal systems. For example, a
graph enriched Lawvere theory describes structures that have a graph of
operations of each arity, where the vertices are operations and the edges are
rewrites between operations. Enriched theories can be used to equip systems
with operational semantics, and maps between enriching categories can serve to
translate between different forms of operational and denotational semantics.
The Grothendieck construction lets us study all models of all enriched theories
in all contexts in a single category. We illustrate these ideas with the
SKI-combinator calculus, a variable-free version of the lambda calculus.Comment: In Proceedings ACT 2019, arXiv:2009.0633
Max Dehn, Axel Thue, and the Undecidable
This is a short essay on the roles of Max Dehn and Axel Thue in the
formulation of the word problem for (semi)groups, and the story of the proofs
showing that the word problem is undecidable.Comment: Definition of undecidability and unsolvability improve
Family of 2-simplex cognitive tools and their application for decision-making and its justifications
Urgency of application and development of cognitive graphic tools for usage
in intelligent systems of data analysis, decision making and its justifications
is given. Cognitive graphic tool "2-simplex prism" and examples of its usage
are presented. Specificity of program realization of cognitive graphics tools
invariant to problem areas is described. Most significant results are given and
discussed. Future investigations are connected with usage of new approach to
rendering, cross-platform realization, cognitive features improving and
expanding of n-simplex family.Comment: 14 pages, 6 figures, conferenc
Certified Universal Gathering in for Oblivious Mobile Robots
We present a unified formal framework for expressing mobile robots models,
protocols, and proofs, and devise a protocol design/proof methodology dedicated
to mobile robots that takes advantage of this formal framework. As a case
study, we present the first formally certified protocol for oblivious mobile
robots evolving in a two-dimensional Euclidean space. In more details, we
provide a new algorithm for the problem of universal gathering mobile oblivious
robots (that is, starting from any initial configuration that is not bivalent,
using any number of robots, the robots reach in a finite number of steps the
same position, not known beforehand) without relying on a common orientation
nor chirality. We give very strong guaranties on the correctness of our
algorithm by proving formally that it is correct, using the COQ proof
assistant. This result demonstrates both the effectiveness of the approach to
obtain new algorithms that use as few assumptions as necessary, and its
manageability since the amount of developed code remains human readable.Comment: arXiv admin note: substantial text overlap with arXiv:1506.0160
- …