15,864 research outputs found
Generating Property-Directed Potential Invariants By Backward Analysis
This paper addresses the issue of lemma generation in a k-induction-based
formal analysis of transition systems, in the linear real/integer arithmetic
fragment. A backward analysis, powered by quantifier elimination, is used to
output preimages of the negation of the proof objective, viewed as unauthorized
states, or gray states. Two heuristics are proposed to take advantage of this
source of information. First, a thorough exploration of the possible
partitionings of the gray state space discovers new relations between state
variables, representing potential invariants. Second, an inexact exploration
regroups and over-approximates disjoint areas of the gray state space, also to
discover new relations between state variables. k-induction is used to isolate
the invariants and check if they strengthen the proof objective. These
heuristics can be used on the first preimage of the backward exploration, and
each time a new one is output, refining the information on the gray states. In
our context of critical avionics embedded systems, we show that our approach is
able to outperform other academic or commercial tools on examples of interest
in our application field. The method is introduced and motivated through two
main examples, one of which was provided by Rockwell Collins, in a
collaborative formal verification framework.Comment: In Proceedings FTSCS 2012, arXiv:1212.657
Modeling of Phenomena and Dynamic Logic of Phenomena
Modeling of complex phenomena such as the mind presents tremendous
computational complexity challenges. Modeling field theory (MFT) addresses
these challenges in a non-traditional way. The main idea behind MFT is to match
levels of uncertainty of the model (also, problem or theory) with levels of
uncertainty of the evaluation criterion used to identify that model. When a
model becomes more certain, then the evaluation criterion is adjusted
dynamically to match that change to the model. This process is called the
Dynamic Logic of Phenomena (DLP) for model construction and it mimics processes
of the mind and natural evolution. This paper provides a formal description of
DLP by specifying its syntax, semantics, and reasoning system. We also outline
links between DLP and other logical approaches. Computational complexity issues
that motivate this work are presented using an example of polynomial models
From spreadsheets to relational databases and back
This paper presents techniques and tools to transform spreadsheets into relational databases and back. A set of data refinement rules is introduced to map a tabular datatype into a relational database schema. Having expressed the transformation of the two data models as data refinements, we obtain for free the functions that migrate the data. We use well-known relational database techniques to optimize and query the data. Because data refinements define bidirectional transformations we can map such database back to an optimized spreadsheet. We have implemented the data refinement rules and we have constructed tools to manipulate, optimize and refactor Excel-like spreadsheets.(undefined
Whittaker limits of difference spherical functions
We introduce the (global) q-Whittaker function as the limit at t=0 of the
q,t-spherical function extending the symmetric Macdonald polynomials to
arbitrary eigenvalues. The construction heavily depends on the technique of the
q-Gaussians developed by the author (and Stokman in the non-reduced case). In
this approach, the q-Whittaker function is given by a series convergent
everywhere, a kind of generating function for multi-dimensional q-Hermite
polynomials (closely related to the level 1 Demazure characters). One of the
applications is a q-version of the Shintani- Casselman- Shalika formula, which
appeared directly connected with q-Mehta- Macdonald identities in terms of the
Jackson integrals. This formula generalizes that of type A due to Gerasimov et
al. to arbitrary reduced root systems. At the end of the paper, we obtain a
q,t-counterpart of the Harish-Chandra asymptotic formula for the spherical
functions, including the Whittaker limit.Comment: V2: a discussion of the one-dimensional case was added. V3: Jackson
integration and growth estimates were added. V4: a q-variant of the
Harish-Chandra asymptotic formula for spherical functions was added. V5:
editing, some improvements, adding references. V6: General editin
Fock factorizations, and decompositions of the spaces over general Levy processes
We explicitly construct and study an isometry between the spaces of square
integrable functionals of an arbitrary Levy process and a vector-valued
Gaussian white noise. In particular, we obtain explicit formulas for this
isometry at the level of multiplicative functionals and at the level of
orthogonal decompositions, as well as find its kernel. We consider in detail
the central special case: the isometry between the spaces over a Poisson
process and the corresponding white noise. The key role in our considerations
is played by the notion of measure and Hilbert factorizations and related
notions of multiplicative and additive functionals and logarithm. The obtained
results allow us to introduce a canonical Fock structure (an analogue of the
Wiener--Ito decomposition) in the space over an arbitrary Levy process.
An application to the representation theory of current groups is considered. An
example of a non-Fock factorization is given.Comment: 35 pages; LaTeX; to appear in Russian Math. Survey
Purposive discovery of operations
The Generate, Prune & Prove (GPP) methodology for discovering definitions of mathematical operators is introduced. GPP is a task within the IL exploration discovery system. We developed GPP for use in the discovery of mathematical operators with a wider class of representations than was possible with the previous methods by Lenat and by Shen. GPP utilizes the purpose for which an operator is created to prune the possible definitions. The relevant search spaces are immense and there exists insufficient information for a complete evaluation of the purpose constraint, so it is necessary to perform a partial evaluation of the purpose (i.e., pruning) constraint. The constraint is first transformed so that it is operational with respect to the partial information, and then it is applied to examples in order to test the generated candidates for an operator's definition. In the GPP process, once a candidate definition survives this empirical prune, it is passed on to a theorem prover for formal verification. We describe the application of this methodology to the (re)discovery of the definition of multiplication for Conway numbers, a discovery which is difficult for human mathematicians. We successfully model this discovery process utilizing information which was reasonably available at the time of Conway's original discovery. As part of this discovery process, we reduce the size of the search space from a computationally intractable size to 3468 elements
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