6,634 research outputs found
Monoidal computer III: A coalgebraic view of computability and complexity
Monoidal computer is a categorical model of intensional computation, where
many different programs correspond to the same input-output behavior. The
upshot of yet another model of computation is that a categorical formalism
should provide a much needed high level language for theory of computation,
flexible enough to allow abstracting away the low level implementation details
when they are irrelevant, or taking them into account when they are genuinely
needed. A salient feature of the approach through monoidal categories is the
formal graphical language of string diagrams, which supports visual reasoning
about programs and computations.
In the present paper, we provide a coalgebraic characterization of monoidal
computer. It turns out that the availability of interpreters and specializers,
that make a monoidal category into a monoidal computer, is equivalent with the
existence of a *universal state space*, that carries a weakly final state
machine for any pair of input and output types. Being able to program state
machines in monoidal computers allows us to represent Turing machines, to
capture their execution, count their steps, as well as, e.g., the memory cells
that they use. The coalgebraic view of monoidal computer thus provides a
convenient diagrammatic language for studying computability and complexity.Comment: 34 pages, 24 figures; in this version: added the Appendi
Kolmogorov Complexity in perspective. Part II: Classification, Information Processing and Duality
We survey diverse approaches to the notion of information: from Shannon
entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov
complexity are presented: randomness and classification. The survey is divided
in two parts published in a same volume. Part II is dedicated to the relation
between logic and information system, within the scope of Kolmogorov
algorithmic information theory. We present a recent application of Kolmogorov
complexity: classification using compression, an idea with provocative
implementation by authors such as Bennett, Vitanyi and Cilibrasi. This stresses
how Kolmogorov complexity, besides being a foundation to randomness, is also
related to classification. Another approach to classification is also
considered: the so-called "Google classification". It uses another original and
attractive idea which is connected to the classification using compression and
to Kolmogorov complexity from a conceptual point of view. We present and unify
these different approaches to classification in terms of Bottom-Up versus
Top-Down operational modes, of which we point the fundamental principles and
the underlying duality. We look at the way these two dual modes are used in
different approaches to information system, particularly the relational model
for database introduced by Codd in the 70's. This allows to point out diverse
forms of a fundamental duality. These operational modes are also reinterpreted
in the context of the comprehension schema of axiomatic set theory ZF. This
leads us to develop how Kolmogorov's complexity is linked to intensionality,
abstraction, classification and information system.Comment: 43 page
Shaded Tangles for the Design and Verification of Quantum Programs (Extended Abstract)
We give a scheme for interpreting shaded tangles as quantum programs, with
the property that isotopic tangles yield equivalent programs. We analyze many
known quantum programs in this way -- including entanglement manipulation and
error correction -- and in each case present a fully-topological formal
verification, yielding in several cases substantial new insight into how the
program works. We also use our methods to identify several new or generalized
procedures.Comment: In Proceedings QPL 2017, arXiv:1802.0973
An Abstract Approach to Stratification in Linear Logic
We study the notion of stratification, as used in subsystems of linear logic
with low complexity bounds on the cut-elimination procedure (the so-called
light logics), from an abstract point of view, introducing a logical system in
which stratification is handled by a separate modality. This modality, which is
a generalization of the paragraph modality of Girard's light linear logic,
arises from a general categorical construction applicable to all models of
linear logic. We thus learn that stratification may be formulated independently
of exponential modalities; when it is forced to be connected to exponential
modalities, it yields interesting complexity properties. In particular, from
our analysis stem three alternative reformulations of Baillot and Mazza's
linear logic by levels: one geometric, one interactive, and one semantic
Bialgebraic Semantics for Logic Programming
Bialgebrae provide an abstract framework encompassing the semantics of
different kinds of computational models. In this paper we propose a bialgebraic
approach to the semantics of logic programming. Our methodology is to study
logic programs as reactive systems and exploit abstract techniques developed in
that setting. First we use saturation to model the operational semantics of
logic programs as coalgebrae on presheaves. Then, we make explicit the
underlying algebraic structure by using bialgebrae on presheaves. The resulting
semantics turns out to be compositional with respect to conjunction and term
substitution. Also, it encodes a parallel model of computation, whose soundness
is guaranteed by a built-in notion of synchronisation between different
threads
Extended Initiality for Typed Abstract Syntax
Initial Semantics aims at interpreting the syntax associated to a signature
as the initial object of some category of 'models', yielding induction and
recursion principles for abstract syntax. Zsid\'o proves an initiality result
for simply-typed syntax: given a signature S, the abstract syntax associated to
S constitutes the initial object in a category of models of S in monads.
However, the iteration principle her theorem provides only accounts for
translations between two languages over a fixed set of object types. We
generalize Zsid\'o's notion of model such that object types may vary, yielding
a larger category, while preserving initiality of the syntax therein. Thus we
obtain an extended initiality theorem for typed abstract syntax, in which
translations between terms over different types can be specified via the
associated category-theoretic iteration operator as an initial morphism. Our
definitions ensure that translations specified via initiality are type-safe,
i.e. compatible with the typing in the source and target language in the
obvious sense. Our main example is given via the propositions-as-types
paradigm: we specify propositions and inference rules of classical and
intuitionistic propositional logics through their respective typed signatures.
Afterwards we use the category--theoretic iteration operator to specify a
double negation translation from the former to the latter. A second example is
given by the signature of PCF. For this particular case, we formalize the
theorem in the proof assistant Coq. Afterwards we specify, via the
category-theoretic iteration operator, translations from PCF to the untyped
lambda calculus
Turing Automata and Graph Machines
Indexed monoidal algebras are introduced as an equivalent structure for
self-dual compact closed categories, and a coherence theorem is proved for the
category of such algebras. Turing automata and Turing graph machines are
defined by generalizing the classical Turing machine concept, so that the
collection of such machines becomes an indexed monoidal algebra. On the analogy
of the von Neumann data-flow computer architecture, Turing graph machines are
proposed as potentially reversible low-level universal computational devices,
and a truly reversible molecular size hardware model is presented as an
example
- …