2,287 research outputs found
Towards Correctness of Program Transformations Through Unification and Critical Pair Computation
Correctness of program transformations in extended lambda calculi with a
contextual semantics is usually based on reasoning about the operational
semantics which is a rewrite semantics. A successful approach to proving
correctness is the combination of a context lemma with the computation of
overlaps between program transformations and the reduction rules, and then of
so-called complete sets of diagrams. The method is similar to the computation
of critical pairs for the completion of term rewriting systems. We explore
cases where the computation of these overlaps can be done in a first order way
by variants of critical pair computation that use unification algorithms. As a
case study we apply the method to a lambda calculus with recursive
let-expressions and describe an effective unification algorithm to determine
all overlaps of a set of transformations with all reduction rules. The
unification algorithm employs many-sorted terms, the equational theory of
left-commutativity modelling multi-sets, context variables of different kinds
and a mechanism for compactly representing binding chains in recursive
let-expressions.Comment: In Proceedings UNIF 2010, arXiv:1012.455
Theories of analytic monads
We characterize the equational theories and Lawvere theories that correspond
to the categories of analytic and polynomial monads on Set, and hence also the
categories of the symmetric and rigid operads in Set. We show that the category
of analytic monads is equivalent to the category of regular-linear theories.
The category of polynomial monads is equivalent to the category of rigid
theories, i.e. regular-linear theories satisfying an additional global
condition. This solves a problem A. Carboni and P. T. Johnstone. The Lawvere
theories corresponding to these monads are identified via some factorization
systems.Comment: 29 pages. v2: minor correction
Monads of regular theories
We characterize the category of monads on and the category of Lawvere
theories that are equivalent to the category of regular equational theories.Comment: 36 page
The Finite Basis Problem for Kiselman Monoids
In an earlier paper, the second-named author has described the identities
holding in the so-called Catalan monoids. Here we extend this description to a
certain family of Hecke--Kiselman monoids including the Kiselman monoids
. As a consequence, we conclude that the identities of
are nonfinitely based for every and exhibit a finite
identity basis for the identities of each of the monoids and
.
In the third version a question left open in the initial submission has beed
answered.Comment: 16 pages, 1 table, 1 figur
A parameterization process as a categorical construction
The parameterization process used in the symbolic computation systems Kenzo
and EAT is studied here as a general construction in a categorical framework.
This parameterization process starts from a given specification and builds a
parameterized specification by transforming some operations into parameterized
operations, which depend on one additional variable called the parameter. Given
a model of the parameterized specification, each interpretation of the
parameter, called an argument, provides a model of the given specification.
Moreover, under some relevant terminality assumption, this correspondence
between the arguments and the models of the given specification is a bijection.
It is proved in this paper that the parameterization process is provided by a
free functor and the subsequent parameter passing process by a natural
transformation. Various categorical notions are used, mainly adjoint functors,
pushouts and lax colimits
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