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Compositional Invariant Generation via Linear Recurrence Analysis [article]

Azadeh Farzan, Zachary Kincaid
2015 arXiv   pre-print
This paper presents a new method for automatically generating numerical invariants for imperative programs.  ...  Our experiments demonstrate that on verification tasks, our method is competitive with leading invariant generation and verification tools.  ...  Non-linear invariant generation is not the main focus of our work, but it is sometimes a necessary intermediate step for proving linear invariants in a compositional setting: since our analysis cannot  ... 
arXiv:1502.00138v1 fatcat:vlyjzek2pzflrhruwg6jytdr2q

Algebraic Program Analysis [chapter]

Zachary Kincaid, Thomas Reps, John Cyphert
2021 Lecture Notes in Computer Science  
It explains the foundations of algebraic program analysis, its strengths and limitations, and gives examples of algebraic program analyses for numerical invariant generation and termination analysis.  ...  AbstractThis paper is a tutorial on algebraic program analysis.  ...  The class of linear recurrence inequations considered in this example can be generalized in various ways to yield more powerful invariant generation procedures.  ... 
doi:10.1007/978-3-030-81685-8_3 fatcat:375brjonbbgfdfwpp72juwnljq

Templates and Recurrences: Better Together [article]

Jason Breck, Thomas Reps
2020 arXiv   pre-print
This disadvantage is particularly prominent for non-linear invariant generation, because the user must supply maximum degrees on polynomials, bases for exponents, etc.  ...  However, a disadvantage of past recurrence-based invariant-generation methods is that they are primarily loop-based analyses: they use recurrences to relate the pre-state and post-state of a loop, so it  ...  This technique is used in the recurrence-based invariant generator ICRA to handle linear recursion [24] .  ... 
arXiv:2003.13515v1 fatcat:7nyzynmogzfyhdx32upifjlyy4

Non-linear reasoning for invariant synthesis

Zachary Kincaid, John Cyphert, Jason Breck, Thomas Reps
2017 Proceedings of the ACM on Programming Languages  
Automatic generation of non-linear loop invariants is a long-standing challenge in program analysis, with many applications.  ...  Compositional recurrence analysis (CRA) [Farzan and Kincaid 2015] is another line of work, which focuses on over-approximate analysis of general loops rather than precise analysis of syntactically restricted  ...  The following program illustrates the relevance of non-linear invariant generation to resource-bound analysis.  ... 
doi:10.1145/3158142 dblp:journals/pacmpl/KincaidCBR18 fatcat:rzj7q45oi5aihjxhduy7e6suqu

Page 392 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 63, Issue 6 [page]

1957 American Mathematical Society. Bulletin of the American Mathematical Society  
Results: solution of the long-standing problem of Hopf; connections between r.t. and electromagnetic theory via generalized harmonic analysis; connections be- tween r.t. and Mueller phenomenological optics  ...  The principle of relativity has far reaching consequences in the form of invariance requirements for general con- stitutive equations, i.e. equations describing the mechanical properties of general classes  ... 

Index

2004 Journal of symbolic computation  
Sylvester sums for subresultants and multi-Schur functions, 689 LEMAIRE, F., An orderly linear PDE system with analytic initial conditions with a nonanalytic solution, 487 Linear recurrent sequences and  ...  II, 153 MIGNOTTE, M. and Ş TEFȂNESCU, D., Linear recurrent sequences and polynomial roots, 637 Modular algorithms for computing Gröbner bases, 403 MORA, T. and SALA, M., On the Gröbner bases of some symmetric  ... 
doi:10.1016/s0747-7171(03)00152-4 fatcat:gnptdsf7grf75aoiujuxvve46a

Differential Invariants for Lie Pseudo-groups [chapter]

Peter Olver, Juha Pohjanpelto, Markus Rosenkranz, Dongming Wang
2007 Gröbner Bases in Symbolic Analysis  
As a consequence, the recurrence formulas (5.11) for the non-phantom invariants do not in general directly yield the leading order normalized differential invariants in terms of lower order invariants  ...  obtained via Lie's algorithm [51] .  ...  The monomials in N |z (∞) index the complete system of functionally independent basic differential invariants, whereas the complementary monomials in K |z (∞) index the constant phantom differential invariants  ... 
doi:10.1515/9783110922752.217 fatcat:z2ui2d52zrg7ba56cczza2odjy

Koopman Operator Dynamical Models: Learning, Analysis and Control [article]

Petar Bevanda and Stefan Sosnowski and Sandra Hirche
2021 arXiv   pre-print
The Koopman operator allows for handling nonlinear systems through a (globally) linear representation.  ...  In general, the operator is infinite-dimensional - necessitating finite approximations - for which there is no overarching framework.  ...  Representations via the Koopman paradigm generalize the notion of mode analysis from linear to nonlinear systems allowing for amendable relevance determination of the constituents of the full dynamics  ... 
arXiv:2102.02522v1 fatcat:wtjh3w6xmvbkvg4lz2cncmehyy

An Algebraic Framework for Compositional Program Analysis [article]

Azadeh Farzan, Zachary Kincaid
2013 arXiv   pre-print
This style of compositional loop analysis can yield interesting ways of computing loop invariants that cannot be defined iteratively.  ...  A compositional program analysis accomplishes this task with a divide-and-conquer strategy: the meaning of a program is computed by dividing it into sub-programs, computing their meaning, and then combining  ...  Linear Recurrence Analysis Let us refer to our implementation of the linear recurrence analysis as LRA.  ... 
arXiv:1310.3481v1 fatcat:42sj26xl5jbe5iasdgczyi4vpe

Umbral Calculus and Cancellative Semigroup Algebras [article]

Vladimir V. Kisil
2000 arXiv   pre-print
We describe some connections between three different fields: combinatorics (umbral calculus), functional analysis (linear functionals and operators) and harmonic analysis (convolutions on group-like structures  ...  Keywords: cancellative semigroups, umbral calculus, harmonic analysis, token, convolution algebra, integral transform  ...  Let we have a shift invariant operator S. Clearly that its composition lf = ǫSf = [Sf ](e) with the augmentation is a linear functional.  ... 
arXiv:funct-an/9704001v2 fatcat:qq6fjy4f5vg3bjeejipfw3hwku

Kernel-Based Approaches for Sequence Modeling: Connections to Neural Methods [article]

Kevin J Liang, Guoyin Wang, Yitong Li, Ricardo Henao, Lawrence Carin
2019 arXiv   pre-print
We investigate time-dependent data analysis from the perspective of recurrent kernel machines, from which models with hidden units and gated memory cells arise naturally.  ...  Our analysis provides a new perspective on the LSTM, while also extending it to n-gram convolutional filters.  ...  However, the recurrent neural network (RNN) has also received renewed interest for analysis of sequential data.  ... 
arXiv:1910.04233v1 fatcat:2nk2lqjv5rgvtbpdq6lftbaq6u

Foreword to the Special Focus on Advances in Symbolic and Numeric Computation III

Amélia Loja, Paulo Vasconcelos, Joaquim Infante Barbosa, José Alberto Rodrigues
2020 Mathematics in Computer Science  
On a 2-Orthogonal Polynomial Sequence via Quadratic Decomposition Teresa Augusta Mesquita In this work a symbolic approach of the general quadratic decomposition of polynomial sequences is applied to  ...  To this purpose, this paper introduces a fractional order SIRS epidemic model with non-linear incidence rate.  ... 
doi:10.1007/s11786-020-00487-9 fatcat:e6zckweekncalpr6imbyfredjm

Umbral Calculus and Cancellative Semigroup Algebras

V.V. Kisil
2000 Zeitschrift für Analysis und ihre Anwendungen  
We use them in Section 4 to describe principal combinatorial objects like delta families, generating functions, and recurrence operators.  ...  We describe some connections between three different fields: combinatorics (umbral calculus), functional analysis (linear functionals and operators) and harmonic analysis (convolutions on group-like structures  ...  The essential part of the paper was written while author's stay at the Department of Mathematical Analysis, University of Ghent, as a Visiting Postdoctoral Fellow of the FWO-Vlaanderen (Fund of Scientific  ... 
doi:10.4171/zaa/953 fatcat:sd4ds5vohzbgbkwwrtx6u7475m

Linear Convergence of Comparison-based Step-size Adaptive Randomized Search via Stability of Markov Chains [article]

Anne Auger
2016 arXiv   pre-print
Harrisrecurrence, geometric ergodicity) and thus define a general methodologyfor proving global linear convergence of CB-SARS algorithms onscaling-invariant functions.  ...  On scaling-invariant functions, we show the existence of ahomogeneous Markov chain, as a consequence of natural invarianceproperties of CB-SARS (essentially scale-invariance and invariance tostrictly increasing  ...  We believe that the author's analysis however is difficult to generalize to our context.  ... 
arXiv:1310.7697v6 fatcat:kokudy6yjjhqlfaadjs6vwzkwe

Moving Frames for Lie Pseudo--Groups

Peter J. Olver, Juha Pohjanpelto
2008 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
The moving frame provides an effective means for determining complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, and solving  ...  generators obtained via Lie's algorithm, [13, 14, 37, 38, 47] .  ...  Infinitesimal Generators. Our subsequent analysis will rely heavily on the infinitesimal generators of the pseudogroup action.  ... 
doi:10.4153/cjm-2008-057-0 fatcat:rtyh47funrch3jwxzu73cfxqfq
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