3,953 research outputs found
Taylor expansion in linear logic is invertible
Each Multiplicative Exponential Linear Logic (MELL) proof-net can be expanded
into a differential net, which is its Taylor expansion. We prove that two
different MELL proof-nets have two different Taylor expansions. As a corollary,
we prove a completeness result for MELL: We show that the relational model is
injective for MELL proof-nets, i.e. the equality between MELL proof-nets in the
relational model is exactly axiomatized by cut-elimination
Glueability of Resource Proof-Structures: Inverting the Taylor Expansion
A Multiplicative-Exponential Linear Logic (MELL) proof-structure can be expanded into a set of resource proof-structures: its Taylor expansion. We introduce a new criterion characterizing those sets of resource proof-structures that are part of the Taylor expansion of some MELL proof-structure, through a rewriting system acting both on resource and MELL proof-structures
Well-Pointed Coalgebras
For endofunctors of varieties preserving intersections, a new description of
the final coalgebra and the initial algebra is presented: the former consists
of all well-pointed coalgebras. These are the pointed coalgebras having no
proper subobject and no proper quotient. The initial algebra consists of all
well-pointed coalgebras that are well-founded in the sense of Osius and Taylor.
And initial algebras are precisely the final well-founded coalgebras. Finally,
the initial iterative algebra consists of all finite well-pointed coalgebras.
Numerous examples are discussed e.g. automata, graphs, and labeled transition
systems
Topological and algebraic structures on the ring of Fermat reals
The ring of Fermat reals is an extension of the real field containing
nilpotent infinitesimals, and represents an alternative to Synthetic
Differential Geometry in classical logic. In the present paper, our first aim
is to study this ring from using standard topological and algebraic structures.
We present the Fermat topology, generated by a complete pseudo-metric, and the
omega topology, generated by a complete metric. The first one is closely
related to the differentiation of (non standard) smooth functions defined on
open sets of Fermat reals. The second one is connected to the differentiation
of smooth functions defined on infinitesimal sets. Subsequently, we prove that
every (proper) ideal is a set of infinitesimals whose order is less than or
equal to some real number. Finally, we define and study roots of
infinitesimals. A computer implementation as well as an application to
infinitesimal Taylor formulas with fractional derivatives are presented.Comment: 43 page
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