3,461 research outputs found
Second-Order Algebraic Theories
Fiore and Hur recently introduced a conservative extension of universal
algebra and equational logic from first to second order. Second-order universal
algebra and second-order equational logic respectively provide a model theory
and a formal deductive system for languages with variable binding and
parameterised metavariables. This work completes the foundations of the subject
from the viewpoint of categorical algebra. Specifically, the paper introduces
the notion of second-order algebraic theory and develops its basic theory. Two
categorical equivalences are established: at the syntactic level, that of
second-order equational presentations and second-order algebraic theories; at
the semantic level, that of second-order algebras and second-order functorial
models. Our development includes a mathematical definition of syntactic
translation between second-order equational presentations. This gives the first
formalisation of notions such as encodings and transforms in the context of
languages with variable binding
Diagrammatic Inference
Diagrammatic logics were introduced in 2002, with emphasis on the notions of
specifications and models. In this paper we improve the description of the
inference process, which is seen as a Yoneda functor on a bicategory of
fractions. A diagrammatic logic is defined from a morphism of limit sketches
(called a propagator) which gives rise to an adjunction, which in turn
determines a bicategory of fractions. The propagator, the adjunction and the
bicategory provide respectively the syntax, the models and the inference
process for the logic. Then diagrammatic logics and their morphisms are applied
to the semantics of side effects in computer languages.Comment: 16 page
Cyclic Datatypes modulo Bisimulation based on Second-Order Algebraic Theories
Cyclic data structures, such as cyclic lists, in functional programming are
tricky to handle because of their cyclicity. This paper presents an
investigation of categorical, algebraic, and computational foundations of
cyclic datatypes. Our framework of cyclic datatypes is based on second-order
algebraic theories of Fiore et al., which give a uniform setting for syntax,
types, and computation rules for describing and reasoning about cyclic
datatypes. We extract the "fold" computation rules from the categorical
semantics based on iteration categories of Bloom and Esik. Thereby, the rules
are correct by construction. We prove strong normalisation using the General
Schema criterion for second-order computation rules. Rather than the fixed
point law, we particularly choose Bekic law for computation, which is a key to
obtaining strong normalisation. We also prove the property of "Church-Rosser
modulo bisimulation" for the computation rules. Combining these results, we
have a remarkable decidability result of the equational theory of cyclic data
and fold.Comment: 38 page
On the mathematical synthesis of equational logics
We provide a mathematical theory and methodology for synthesising equational
logics from algebraic metatheories. We illustrate our methodology by means of
two applications: a rational reconstruction of Birkhoff's Equational Logic and
a new equational logic for reasoning about algebraic structure with
name-binding operators.Comment: Final version for publication in Logical Methods in Computer Scienc
Partial Horn logic and cartesian categories
A logic is developed in which function symbols are allowed to represent partial functions. It has the usual rules of logic (in the form of a sequent calculus) except that the substitution rule has to be modified. It is developed here in its minimal form, with equality and conjunction, as “partial Horn logic”.
Various kinds of logical theory are equivalent: partial Horn theories, “quasi-equational” theories (partial Horn theories without predicate symbols), cartesian theories and essentially algebraic theories.
The logic is sound and complete with respect to models in , and sound with respect to models in any cartesian (finite limit) category.
The simplicity of the quasi-equational form allows an easy predicative constructive proof of the free partial model theorem for cartesian theories: that if a theory morphism is given from one cartesian theory to another, then the forgetful (reduct) functor from one model category to the other has a left adjoint.
Various examples of quasi-equational theory are studied, including those of cartesian categories and of other classes of categories. For each quasi-equational theory another, , is constructed, whose models are cartesian categories equipped with models of . Its initial model, the “classifying category” for , has properties similar to those of the syntactic category, but more precise with respect to strict cartesian functors
Patterns for computational effects arising from a monad or a comonad
This paper presents equational-based logics for proving first order
properties of programming languages involving effects. We propose two dual
inference system patterns that can be instanciated with monads or comonads in
order to be used for proving properties of different effects. The first pattern
provides inference rules which can be interpreted in the Kleisli category of a
monad and the coKleisli category of the associated comonad. In a dual way, the
second pattern provides inference rules which can be interpreted in the
coKleisli category of a comonad and the Kleisli category of the associated
monad. The logics combine a 3-tier effect system for terms consisting of pure
terms and two other kinds of effects called 'constructors/observers' and
'modifiers', and a 2-tier system for 'up-to-effects' and 'strong' equations.
Each pattern provides generic rules for dealing with any monad (respectively
comonad), and it can be extended with specific rules for each effect. The paper
presents two use cases: a language with exceptions (using the standard monadic
semantics), and a language with state (using the less standard comonadic
semantics). Finally, we prove that the obtained inference system for states is
Hilbert-Post complete
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
Algebraic Theories over Nominal Sets
We investigate the foundations of a theory of algebraic data types with
variable binding inside classical universal algebra. In the first part, a
category-theoretic study of monads over the nominal sets of Gabbay and Pitts
leads us to introduce new notions of finitary based monads and uniform monads.
In a second part we spell out these notions in the language of universal
algebra, show how to recover the logics of Gabbay-Mathijssen and
Clouston-Pitts, and apply classical results from universal algebra.Comment: 16 page
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