10 research outputs found
An algebraic generalization of Kripke structures
The Kripke semantics of classical propositional normal modal logic is made
algebraic via an embedding of Kripke structures into the larger class of
pointed stably supported quantales. This algebraic semantics subsumes the
traditional algebraic semantics based on lattices with unary operators, and it
suggests natural interpretations of modal logic, of possible interest in the
applications, in structures that arise in geometry and analysis, such as
foliated manifolds and operator algebras, via topological groupoids and inverse
semigroups. We study completeness properties of the quantale based semantics
for the systems K, T, K4, S4, and S5, in particular obtaining an axiomatization
for S5 which does not use negation or the modal necessity operator. As
additional examples we describe intuitionistic propositional modal logic, the
logic of programs PDL, and the ramified temporal logic CTL.Comment: 39 page
A topos for algebraic quantum theory
The aim of this paper is to relate algebraic quantum mechanics to topos
theory, so as to construct new foundations for quantum logic and quantum
spaces. Motivated by Bohr's idea that the empirical content of quantum physics
is accessible only through classical physics, we show how a C*-algebra of
observables A induces a topos T(A) in which the amalgamation of all of its
commutative subalgebras comprises a single commutative C*-algebra. According to
the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter
has an internal spectrum S(A) in T(A), which in our approach plays the role of
a quantum phase space of the system. Thus we associate a locale (which is the
topos-theoretical notion of a space and which intrinsically carries the
intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which
is the noncommutative notion of a space). In this setting, states on A become
probability measures (more precisely, valuations) on S(A), and self-adjoint
elements of A define continuous functions (more precisely, locale maps) from
S(A) to Scott's interval domain. Noting that open subsets of S(A) correspond to
propositions about the system, the pairing map that assigns a (generalized)
truth value to a state and a proposition assumes an extremely simple
categorical form. Formulated in this way, the quantum theory defined by A is
essentially turned into a classical theory, internal to the topos T(A).Comment: 52 pages, final version, to appear in Communications in Mathematical
Physic
Bohrification
New foundations for quantum logic and quantum spaces are constructed by
merging algebraic quantum theory and topos theory. Interpreting Bohr's
"doctrine of classical concepts" mathematically, given a quantum theory
described by a noncommutative C*-algebra A, we construct a topos T(A), which
contains the "Bohrification" B of A as an internal commutative C*-algebra. Then
B has a spectrum, a locale internal to T(A), the external description S(A) of
which we interpret as the "Bohrified" phase space of the physical system. As in
classical physics, the open subsets of S(A) correspond to (atomic)
propositions, so that the "Bohrified" quantum logic of A is given by the
Heyting algebra structure of S(A). The key difference between this logic and
its classical counterpart is that the former does not satisfy the law of the
excluded middle, and hence is intuitionistic. When A contains sufficiently many
projections (e.g. when A is a von Neumann algebra, or, more generally, a
Rickart C*-algebra), the intuitionistic quantum logic S(A) of A may also be
compared with the traditional quantum logic, i.e. the orthomodular lattice of
projections in A. This time, the main difference is that the former is
distributive (even when A is noncommutative), while the latter is not.
This chapter is a streamlined synthesis of 0709.4364, 0902.3201, 0905.2275.Comment: 44 pages; a chapter of the first author's PhD thesis, to appear in
"Deep Beauty" (ed. H. Halvorson
The Quantum Monadology
The modern theory of functional programming languages uses monads for
encoding computational side-effects and side-contexts, beyond bare-bone program
logic. Even though quantum computing is intrinsically side-effectful (as in
quantum measurement) and context-dependent (as on mixed ancillary states),
little of this monadic paradigm has previously been brought to bear on quantum
programming languages.
Here we systematically analyze the (co)monads on categories of parameterized
module spectra which are induced by Grothendieck's "motivic yoga of operations"
-- for the present purpose specialized to HC-modules and further to set-indexed
complex vector spaces. Interpreting an indexed vector space as a collection of
alternative possible quantum state spaces parameterized by quantum measurement
results, as familiar from Proto-Quipper-semantics, we find that these
(co)monads provide a comprehensive natural language for functional quantum
programming with classical control and with "dynamic lifting" of quantum
measurement results back into classical contexts.
We close by indicating a domain-specific quantum programming language (QS)
expressing these monadic quantum effects in transparent do-notation, embeddable
into the recently constructed Linear Homotopy Type Theory (LHoTT) which
interprets into parameterized module spectra. Once embedded into LHoTT, this
should make for formally verifiable universal quantum programming with linear
quantum types, classical control, dynamic lifting, and notably also with
topological effects.Comment: 120 pages, various figure
Components as coalgebras
In the tradition of mathematical modelling in physics and chemistry, constructive formal specification methods are based on the notion of a software model, understood as a state-based abstract machine which persists and evolves in time, according to a behavioural model capturing, for example, partiality or (different degrees of) nondeterminism. This can be identified with the more prosaic notion of a software component advocated by the software industry as ‘building block’ of large, often distributed, systems. Such a component typically encapsulates a number of services through a public interface which provides a limited access to a private state space, paying tribute to the nowadays widespread object-oriented programming principles.
The tradition of communicating systems formal design, by contrast, has developed the notion of a process as an abstraction of the behavioural patterns of a computing system, deliberately ignoring the data and state aspects of software systems.
Both processes and components are among the broad group of computing phenomena which are hardly definable (or simply not definable) algebraically, i.e., in terms of a complete set of constructors. Their semantics is essentially observational, in the sense that all that can be traced of their evolution is their interaction with the environment. Therefore, coalgebras, whose theory has recently witnessed remarkable
developments, appear as a suitable modelling tool.
The basic observation of category theory that universal constructions always come in pairs, has motivated research on the duality between algebras and coalgebras, which provides a bridge between models of static (constructive, data-oriented) and dynamical (observational, behaviour-oriented) systems. At the programming level, the intuitive symmetry between data and behaviour provides evidence of such a duality,
in its canonical initial-final specialisation.
This line of thought entails both definitional and proof principles, i.e., a basis for the development of program calculi directly based on (actually driven by) type specifications. Moreover, such properties can be expressed in terms of generic programming combinators which are used, not only to calculate programs, but also to program with.
Framed in this context, this thesis addresses the following main themes:
The investigation of a semantic model for (state-based) software components. These are regarded as concrete coalgebras for some Set endofunctors,
with specified initial conditions, and organise themselves in a bicategorical setting. The model is able to capture both behavioural issues, which
are usually left implicit in state-based specification methods, and interaction through structured data, which is usually a minor concern on process calculi. Two basic cases are considered entailing, respectively, a ‘functional’ and an ‘object-oriented’ shape for components. Both cases are parametrized by a
model of behaviour, introduced as a strong (usually commutative) monad.
The development of corresponding component calculi, also parametric on the behaviour model, which adds to the genericity of the approach.
The study of processes and the ‘reconstruction’ of classical (CCS-like) process calculi on top of their representation as inhabitants of (the carriers of) final coalgebras, in an essentially pointfree, calculational style.
An overall concern for genericity, in the sense that models and calculi for both components and processes are parametric on the behaviour model and the interaction discipline, respectively.
The animation of both processes and components in CHARITY, a functional programming language entirely based on inductive and coinductive categorical data types. In particular this leads to the development of a process calculi interpreter parametric on the interaction discipline.PRAXIS XXI - Projecto LOGCAMP; POO11/IC-PME/II/S -Projecto KARMA; Fundação para a Ciência e Tecnologia; ALGORITMI Research Center