4,571 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
Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness
We give a semantics for the lambda-calculus based on a topological duality
theorem in nominal sets. A novel interpretation of lambda is given in terms of
adjoints, and lambda-terms are interpreted absolutely as sets (no valuation is
necessary)
Self-Reference, Biologic and the Structure of Reproduction
This paper concentrates on relationships of formal systems with biology. The
paper is based on previous papers by the author. We have freely used texts of
those papers where the formulations are of use, and we have extended the
concepts and discussions herein considerably beyond the earlier work. We
concentrate on formal systems not only for the sake of showing how there is a
fundamental mathematical structure to biology, but also to consider and
reconsider philosophical and phenomenological points of view in relation to
natural science and mathematics. The relationship with phenomenology comes
about in the questions that arise about the nature of the observer in relation
to the observed that arise in philosophy, but also in science in the very act
of determining the context and models upon which it shall be based.We examine
the schema behind the reproduction of DNA. The DNA molecule consists of two
interwound strands, the Watson Strand (W) and the Crick Strand (C). The two
strands are bonded to each other via a backbone of base-pairings and these
bonds can be broken by certain enzymes present in the cell. In reproduction of
DNA the bonds between the two strands are broken and the two strands then
acquire the needed complementary base molecules from the cellular environment
to reconstitute each a separate copy of the DNA. At this level the situation
can be described by a symbolism like this. DNA = ------->
--------> = = DNA DNA. Here E stands for the
environment of the cell. The first arrow denotes the separation of the DNA into
the two strands. The second arrow denotes the action between the bare strands
and the environment that leads to the production of the two DNA molecules. The
paper considers and compares many formalisms for self-replication, including
aspects of quantum formalism and the Temperley-Lieb algebra.Comment: LaTeX document, 71 pages, 33 figures. arXiv admin note: substantial
text overlap with arXiv:quant-ph/020400
Introducing a Calculus of Effects and Handlers for Natural Language Semantics
In compositional model-theoretic semantics, researchers assemble
truth-conditions or other kinds of denotations using the lambda calculus. It
was previously observed that the lambda terms and/or the denotations studied
tend to follow the same pattern: they are instances of a monad. In this paper,
we present an extension of the simply-typed lambda calculus that exploits this
uniformity using the recently discovered technique of effect handlers. We prove
that our calculus exhibits some of the key formal properties of the lambda
calculus and we use it to construct a modular semantics for a small fragment
that involves multiple distinct semantic phenomena
High-level signatures and initial semantics
We present a device for specifying and reasoning about syntax for datatypes,
programming languages, and logic calculi. More precisely, we study a notion of
signature for specifying syntactic constructions.
In the spirit of Initial Semantics, we define the syntax generated by a
signature to be the initial object---if it exists---in a suitable category of
models. In our framework, the existence of an associated syntax to a signature
is not automatically guaranteed. We identify, via the notion of presentation of
a signature, a large class of signatures that do generate a syntax.
Our (presentable) signatures subsume classical algebraic signatures (i.e.,
signatures for languages with variable binding, such as the pure lambda
calculus) and extend them to include several other significant examples of
syntactic constructions.
One key feature of our notions of signature, syntax, and presentation is that
they are highly compositional, in the sense that complex examples can be
obtained by assembling simpler ones. Moreover, through the Initial Semantics
approach, our framework provides, beyond the desired algebra of terms, a
well-behaved substitution and the induction and recursion principles associated
to the syntax.
This paper builds upon ideas from a previous attempt by Hirschowitz-Maggesi,
which, in turn, was directly inspired by some earlier work of
Ghani-Uustalu-Hamana and Matthes-Uustalu.
The main results presented in the paper are computer-checked within the
UniMath system.Comment: v2: extended version of the article as published in CSL 2018
(http://dx.doi.org/10.4230/LIPIcs.CSL.2018.4); list of changes given in
Section 1.5 of the paper; v3: small corrections throughout the paper, no
major change
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