296 research outputs found
Algebraic invariants for homotopy types
We define inductively a sequence of purely algebraic invariants - namely,
classes in the Quillen cohomology of the Pi-algebra \pi_* X - for
distinguishing between different homotopy types of spaces. Another sequence of
such cohomology classes allows one to decide whether a given abstract
Pi-algebra can be realized as the homotopy Pi-algebra of a space in the first
place.
The paper is written for a relatively general "resolution model category", so
it also applies, for example, to rational homotopy types
Lessons from Formally Verified Deployed Software Systems (Extended version)
The technology of formal software verification has made spectacular advances,
but how much does it actually benefit the development of practical software?
Considerable disagreement remains about the practicality of building systems
with mechanically-checked proofs of correctness. Is this prospect confined to a
few expensive, life-critical projects, or can the idea be applied to a wide
segment of the software industry?
To help answer this question, the present survey examines a range of
projects, in various application areas, that have produced formally verified
systems and deployed them for actual use. It considers the technologies used,
the form of verification applied, the results obtained, and the lessons that
can be drawn for the software industry at large and its ability to benefit from
formal verification techniques and tools.
Note: a short version of this paper is also available, covering in detail
only a subset of the considered systems. The present version is intended for
full reference.Comment: arXiv admin note: text overlap with arXiv:1211.6186 by other author
Product structure of heat phase space and branching Brownian motion
A generical formalism for the discussion of Brownian processes with
non-constant particle number is developed, based on the observation that the
phase space of heat possesses a product structure that can be encoded in a
commutative unit ring. A single Brownian particle is discussed in a Hilbert
module theory, with the underlying ring structure seen to be intimately linked
to the non-differentiability of Brownian paths. Multi-particle systems with
interactions are explicitly constructed using a Fock space approach. The
resulting ring-valued quantum field theory is applied to binary branching
Brownian motion, whose Dyson-Schwinger equations can be exactly solved. The
presented formalism permits the application of the full machinery of quantum
field theory to Brownian processes.Comment: 32 pages, journal version. Annals of Physics, N.Y. (to appear
On surgery along Brunnian links in 3-manifolds
We consider surgery moves along (n+1)-component Brunnian links in compact
connected oriented 3-manifolds, where the framing of the each component is 1/k
for k in Z. We show that no finite type invariant of degree < 2n-2 can detect
such a surgery move. The case of two link-homotopic Brunnian links is also
considered. We relate finite type invariants of integral homology spheres
obtained by such operations to Goussarov-Vassiliev invariants of Brunnian
links.Comment: This is the version published by Algebraic & Geometric Topology on 13
December 200
Practical Subtyping for System F with Sized (Co-)Induction
We present a rich type system with subtyping for an extension of System F.
Our type constructors include sum and product types, universal and existential
quantifiers, inductive and coinductive types. The latter two size annotations
allowing the preservation of size invariants. For example it is possible to
derive the termination of the quicksort by showing that partitioning a list
does not increase its size. The system deals with complex programs involving
mixed induction and coinduction, or even mixed (co-)induction and polymorphism
(as for Scott-encoded datatypes). One of the key ideas is to completely
separate the induction on sizes from the notion of recursive programs. We use
the size change principle to check that the proof is well-founded, not that the
program terminates. Termination is obtained by a strong normalization proof.
Another key idea is the use symbolic witnesses to handle quantifiers of all
sorts. To demonstrate the practicality of our system, we provide an
implementation that accepts all the examples discussed in the paper and much
more
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