2,417 research outputs found
Datalog and Constraint Satisfaction with Infinite Templates
On finite structures, there is a well-known connection between the expressive
power of Datalog, finite variable logics, the existential pebble game, and
bounded hypertree duality. We study this connection for infinite structures.
This has applications for constraint satisfaction with infinite templates. If
the template Gamma is omega-categorical, we present various equivalent
characterizations of those Gamma such that the constraint satisfaction problem
(CSP) for Gamma can be solved by a Datalog program. We also show that
CSP(Gamma) can be solved in polynomial time for arbitrary omega-categorical
structures Gamma if the input is restricted to instances of bounded treewidth.
Finally, we characterize those omega-categorical templates whose CSP has
Datalog width 1, and those whose CSP has strict Datalog width k.Comment: 28 pages. This is an extended long version of a conference paper that
appeared at STACS'06. In the third version in the arxiv we have revised the
presentation again and added a section that relates our results to
formalizations of CSPs using relation algebra
Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory
We describe categorical models of a circuit-based (quantum) functional
programming language. We show that enriched categories play a crucial role.
Following earlier work on QWire by Paykin et al., we consider both a simple
first-order linear language for circuits, and a more powerful host language,
such that the circuit language is embedded inside the host language. Our
categorical semantics for the host language is standard, and involves cartesian
closed categories and monads. We interpret the circuit language not in an
ordinary category, but in a category that is enriched in the host category. We
show that this structure is also related to linear/non-linear models. As an
extended example, we recall an earlier result that the category of W*-algebras
is dcpo-enriched, and we use this model to extend the circuit language with
some recursive types
New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic
Intuitionistic logic, in which the double negation law not-not-P = P fails,
is dominant in categorical logic, notably in topos theory. This paper follows a
different direction in which double negation does hold. The algebraic notions
of effect algebra/module that emerged in theoretical physics form the
cornerstone. It is shown that under mild conditions on a category, its maps of
the form X -> 1+1 carry such effect module structure, and can be used as
predicates. Predicates are identified in many different situations, and capture
for instance ordinary subsets, fuzzy predicates in a probabilistic setting,
idempotents in a ring, and effects (positive elements below the unit) in a
C*-algebra or Hilbert space. In quantum foundations the duality between states
and effects plays an important role. It appears here in the form of an
adjunction, where we use maps 1 -> X as states. For such a state s and a
predicate p, the validity probability s |= p is defined, as an abstract Born
rule. It captures many forms of (Boolean or probabilistic) validity known from
the literature. Measurement from quantum mechanics is formalised categorically
in terms of `instruments', using L\"uders rule in the quantum case. These
instruments are special maps associated with predicates (more generally, with
tests), which perform the act of measurement and may have a side-effect that
disturbs the system under observation. This abstract description of
side-effects is one of the main achievements of the current approach. It is
shown that in the special case of C*-algebras, side-effect appear exclusively
in the non-commutative case. Also, these instruments are used for test
operators in a dynamic logic that can be used for reasoning about quantum
programs/protocols. The paper describes four successive assumptions, towards a
categorical axiomatisation of quantitative logic for probabilistic and quantum
systems
An Algebraic Characterisation of Concurrent Composition
We give an algebraic characterization of a form of synchronized parallel
composition allowing for true concurrency, using ideas based on Peter Landin's
"Program-Machine Symmetric Automata Theory".Comment: This is an old technical report from 1981. I submitted it to a
special issue of HOSC in honour of Peter Landin, as explained in the Prelude,
added in 2008. However, at an advanced stage, the handling editor became
unresponsive, and the paper was never published. I am making it available via
the arXiv for the same reasons given in the Prelud
A Recipe for State-and-Effect Triangles
In the semantics of programming languages one can view programs as state
transformers, or as predicate transformers. Recently the author has introduced
state-and-effect triangles which capture this situation categorically,
involving an adjunction between state- and predicate-transformers. The current
paper exploits a classical result in category theory, part of Jon Beck's
monadicity theorem, to systematically construct such a state-and-effect
triangle from an adjunction. The power of this construction is illustrated in
many examples, covering many monads occurring in program semantics, including
(probabilistic) power domains
Condition/Decision Duality and the Internal Logic of Extensive Restriction Categories
In flowchart languages, predicates play an interesting double role. In the
textual representation, they are often presented as conditions, i.e.,
expressions which are easily combined with other conditions (often via Boolean
combinators) to form new conditions, though they only play a supporting role in
aiding branching statements choose a branch to follow. On the other hand, in
the graphical representation they are typically presented as decisions,
intrinsically capable of directing control flow yet mostly oblivious to Boolean
combination. While categorical treatments of flowchart languages are abundant,
none of them provide a treatment of this dual nature of predicates. In the
present paper, we argue that extensive restriction categories are precisely
categories that capture such a condition/decision duality, by means of
morphisms which, coincidentally, are also called decisions. Further, we show
that having these categorical decisions amounts to having an internal logic:
Analogous to how subobjects of an object in a topos form a Heyting algebra, we
show that decisions on an object in an extensive restriction category form a De
Morgan quasilattice, the algebraic structure associated with the (three-valued)
weak Kleene logic . Full classical propositional logic can be
recovered by restricting to total decisions, yielding extensive categories in
the usual sense, and confirming (from a different direction) a result from
effectus theory that predicates on objects in extensive categories form Boolean
algebras. As an application, since (categorical) decisions are partial
isomorphisms, this approach provides naturally reversible models of classical
propositional logic and weak Kleene logic.Comment: 19 pages, including 6 page appendix of proofs. Accepted for MFPS XXX
The Expectation Monad in Quantum Foundations
The expectation monad is introduced abstractly via two composable
adjunctions, but concretely captures measures. It turns out to sit in between
known monads: on the one hand the distribution and ultrafilter monad, and on
the other hand the continuation monad. This expectation monad is used in two
probabilistic analogues of fundamental results of Manes and Gelfand for the
ultrafilter monad: algebras of the expectation monad are convex compact
Hausdorff spaces, and are dually equivalent to so-called Banach effect
algebras. These structures capture states and effects in quantum foundations,
and also the duality between them. Moreover, the approach leads to a new
re-formulation of Gleason's theorem, expressing that effects on a Hilbert space
are free effect modules on projections, obtained via tensoring with the unit
interval.Comment: In Proceedings QPL 2011, arXiv:1210.029
Spectral C*-categories and Fell bundles with path-lifting
Following Crane's suggestion that categorification should be of fundamental
importance in quantising gravity, we show that finite dimensional even
-real spectral triples over \bbc are already nothing more than full
C*-categories together with a self-adjoint section of their domain and range
maps, while the latter are equivalent to unital saturated Fell bundles over
pair groupoids equipped with a path-lifting operator given by a normaliser.
Interpretations can be made in the direction of quantum Higgs gravity. These
geometries are automatically quantum geometries and we reconstruct the
classical limit, that is, general relativity on a Riemannian spin manifold.Comment: 20 pages, 1 figur
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