3,491 research outputs found
Notions of Monad Strength
Over the past two decades the notion of a strong monad has found wide
applicability in computing. Arising out of a need to interpret products in
computational and semantic settings, different approaches to this concept have
arisen. In this paper we introduce and investigate the connections between
these approaches and also relate the results to monad composition. We also
introduce new methods for checking and using the required laws associated with
such compositions, as well as provide examples illustrating problems and issues
that arise.Comment: In Proceedings Festschrift for Dave Schmidt, arXiv:1309.455
Polynomial functors and polynomial monads
We study polynomial functors over locally cartesian closed categories. After
setting up the basic theory, we show how polynomial functors assemble into a
double category, in fact a framed bicategory. We show that the free monad on a
polynomial endofunctor is polynomial. The relationship with operads and other
related notions is explored.Comment: 41 pages, latex, 2 ps figures generated at runtime by the texdraw
package (does not compile with pdflatex). v2: removed assumptions on sums,
added short discussion of generalisation, and more details on tensorial
strength
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
On coalgebras with internal moves
In the first part of the paper we recall the coalgebraic approach to handling
the so-called invisible transitions that appear in different state-based
systems semantics. We claim that these transitions are always part of the unit
of a certain monad. Hence, coalgebras with internal moves are exactly
coalgebras over a monadic type. The rest of the paper is devoted to supporting
our claim by studying two important behavioural equivalences for state-based
systems with internal moves, namely: weak bisimulation and trace semantics.
We continue our research on weak bisimulations for coalgebras over order
enriched monads. The key notions used in this paper and proposed by us in our
previous work are the notions of an order saturation monad and a saturator. A
saturator operator can be intuitively understood as a reflexive, transitive
closure operator. There are two approaches towards defining saturators for
coalgebras with internal moves. Here, we give necessary conditions for them to
yield the same notion of weak bisimulation.
Finally, we propose a definition of trace semantics for coalgebras with
silent moves via a uniform fixed point operator. We compare strong and weak
bisimilation together with trace semantics for coalgebras with internal steps.Comment: Article: 23 pages, Appendix: 3 page
Behavioural equivalences for timed systems
Timed transition systems are behavioural models that include an explicit
treatment of time flow and are used to formalise the semantics of several
foundational process calculi and automata. Despite their relevance, a general
mathematical characterisation of timed transition systems and their behavioural
theory is still missing. We introduce the first uniform framework for timed
behavioural models that encompasses known behavioural equivalences such as
timed bisimulations, timed language equivalences as well as their weak and
time-abstract counterparts. All these notions of equivalences are naturally
organised by their discriminating power in a spectrum. We prove that this
result does not depend on the type of the systems under scrutiny: it holds for
any generalisation of timed transition system. We instantiate our framework to
timed transition systems and their quantitative extensions such as timed
probabilistic systems
Commutativity
We describe a general framework for notions of commutativity based on
enriched category theory. We extend Eilenberg and Kelly's tensor product for
categories enriched over a symmetric monoidal base to a tensor product for
categories enriched over a normal duoidal category; using this, we re-find
notions such as the commutativity of a finitary algebraic theory or a strong
monad, the commuting tensor product of two theories, and the Boardman-Vogt
tensor product of symmetric operads.Comment: 48 pages; final journal versio
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