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Feasible Interpolation for Polynomial Calculus and Sums-Of-Squares

Tuomas Hakoniemi, Emanuela Merelli, Anuj Dawar, Artur Czumaj
2020 International Colloquium on Automata, Languages and Programming  
We prove that both Polynomial Calculus and Sums-of-Squares proof systems admit a strong form of feasible interpolation property for sets of polynomial equality constraints.  ...  For Sums-of-Squares we rely on the use of Boolean axioms, but for Polynomial Calculus we do not assume their presence.  ...  A L P 2 0 2 0 63:4 Feasible Interpolation for Polynomial Calculus and Sums-Of-Squares where r i , t q , u i and v i are arbitrary real polynomials.  ... 
doi:10.4230/lipics.icalp.2020.63 dblp:conf/icalp/Hakoniemi20 fatcat:uyt7va4gwbdhxfk2krivqpnddy

subject index volumes 141-150

2003 Journal of Computational and Applied Mathematics  
scheme 145 459 Differentiability 146 343 Differential approximants 149 457 Differential calculus for p-norms 145 425 Differential calculus for vector norms 145 425 Differential equation  ...  large numbers 142 27 Least squares 146 179 Least squares 147 53 Left-definite self-adjoint operator 148 213 Left-definite Sobolev space 148 213 Legendre polynomials 148 213 Legendre  ...  147 349 Toeplitz operators 142 441 Total ranking method 146 155 Transformation 143 81 Traveling wave fronts 146 309 Tree 142 61 Triangulation 145 505 Trie 142 27 Trigonometric interpolation  ... 
doi:10.1016/s0377-0427(02)00844-0 fatcat:tsx7p6iw3jbxda3sdee3cdvooq

Sum-of-squares optimization without semidefinite programming [article]

Dávid Papp, Sercan Yıldız
2018 arXiv   pre-print
We propose a homogeneous primal-dual interior-point method to solve sum-of-squares optimization problems by combining non-symmetric conic optimization techniques and polynomial interpolation.  ...  The approach optimizes directly over the sum-of-squares cone and its dual, circumventing the semidefinite programming (SDP) reformulation which requires a large number of auxiliary variables.  ...  The authors would like to thank the Associate Editor and the Referees for their constructive feedback.  ... 
arXiv:1712.01792v3 fatcat:f2rnseczuvci7cc7abjgtct5ba

On the Automatizability of Resolution and Related Propositional Proof Systems [chapter]

Albert Atserias, María Luisa Bonet
2002 Lecture Notes in Computer Science  
This gives a better lower bound for the monotone interpolation of Res(2) and a separation from Resolution as a byproduct.  ...  After studying the robustness of the definition, we prove the equivalence between: (i) Resolution is weakly automatizable, (ii) Res(¤ ) is weakly automatizable and (iii) Res(¤ ) has feasible interpolation  ...  We are grateful to Pavel Pudlák for stimulating discussions on the idea of Theorem 11.  ... 
doi:10.1007/3-540-45793-3_38 fatcat:vxg4gu2sw5aglizzinrjp6mj74

On the automatizability of resolution and related propositional proof systems

Albert Atserias, Marı́a Luisa Bonet
2004 Information and Computation  
This gives a better lower bound for the monotone interpolation of Res(2) and a separation from Resolution as a byproduct.  ...  After studying the robustness of the definition, we prove the equivalence between: (i) Resolution is weakly automatizable, (ii) Res(¤ ) is weakly automatizable and (iii) Res(¤ ) has feasible interpolation  ...  We are grateful to Pavel Pudlák for stimulating discussions on the idea of Theorem 11.  ... 
doi:10.1016/j.ic.2003.10.004 fatcat:sbfnlio72zdonmh2leevz7gveu

The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs [chapter]

Albert Atserias
2013 Lecture Notes in Computer Science  
The inference rules are typically some more or less obvious, non-interesting, and polynomially checkable ways of producing some logical consequence of the hypotheses.  ...  disjunctions and conjunctions, general propositional formulas, polynomial equations over some ring, polynomial inequalities over some ordered ring, disjunctions of those, decision trees branching on variables  ...  Acknowledgments We thank the comments of Allen Van Gelder and an anonymous referee on the preliminary draft of this paper.  ... 
doi:10.1007/978-3-642-39071-5_1 fatcat:4vpknc2xqrbe3fu7hw3nzi7t5m

Proof Complexity (Dagstuhl Seminar 18051)

Albert Atserias, Jakob Nordström, Pavel Pudlák, Rahul Santhanam, Michael Wagner
2018 Dagstuhl Reports  
NP problem, and in the ensuing decades many powerful techniques have been discovered for analyzing different proof systems.  ...  While many longstanding open problems from the 1980s and 1990s still remain unsolved, recent progress gives hope that the area may be ripe for decisive breakthroughs.  ...  -Proof Complexity Russell Impagliazzo, Pavel Pudlák, and Jiří Sgall. Lower bounds for the polynomial calculus and the Gröbner basis algorithm. Computational Complexity, 8(2):127-144, 1999.  ... 
doi:10.4230/dagrep.8.1.124 dblp:journals/dagstuhl-reports/AtseriasNPS18 fatcat:5ksfbo2ehfhspcbuw4ppcuyaqu

Sums of Squares, Moment Matrices and Optimization Over Polynomials [chapter]

Monique Laurent
2008 IMA Volumes in Mathematics and its Applications  
Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory of sums of squares of polynomials.  ...  We review the mathematical tools underlying these properties, in particular, some sums of squares representation results for positive polynomials, some results about moment matrices (in particular, of  ...  I am very grateful to Markus Schweighofer for discussion and his help with some proofs in the survey, and to Hayato Waki for his careful reading and for his useful questions and suggestions.  ... 
doi:10.1007/978-0-387-09686-5_7 fatcat:buio7ftpcbfdbdobnaqopag5nu

Behavior Formula Extraction for Object Trajectory using Curve Fitting Method

Israa Hadi, Mustafa Sabah
2014 International Journal of Computer Applications  
Curve fitting, also known as regression analysis, is used to find the "best fit" line or curve for a series of data points.  ...  One of the main goals of computer vision is to enable computers to replicate the basic functions of human vision such as motion perception and scene understanding.  ...  the minimum sums of squares for the linear equation do not give minimum sums of squares for the un linear, or original form of the equation.  ... 
doi:10.5120/18177-9068 fatcat:cv723gdqzfcwdgjb73p6ljjqoa

Interpolant Synthesis for Quadratic Polynomial Inequalities and Combination with EUF [chapter]

Ting Gan, Liyun Dai, Bican Xia, Naijun Zhan, Deepak Kapur, Mingshuai Chen
2016 Lecture Notes in Computer Science  
An algorithm for generating interpolants for formulas which are conjunctions of quadratic polynomial inequalities (both strict and nonstrict) is proposed.  ...  theory of concave quadratic polynomial inequalities and the equality theory over uninterpreted functions (EUF).  ...  Under a technical condition that if no nonpositive constant combination of nonstrict inequalities is a sum of squares polynomials, then such an interpolant can be generated essentially using the linearization  ... 
doi:10.1007/978-3-319-40229-1_14 fatcat:bf2mwiy5tvdz3o7jjmdxhatwyi

Complexity of Semialgebraic Proofs

D. Grigoriev, E. Hirsch, D. Pasechnik
2002 Moscow Mathematical Journal  
calculus proofs), and Tseitin's tautologies (which are hard for many known proof systems).  ...  It is a known approach to translate propositional formulas into systems of polynomial inequalities and consider proof systems for the latter.  ...  RM1-2409-ST-02 of the U. S. Civilian Research & Development Foundation for the Independent States of the Former Soviet Union (CRDF).  ... 
doi:10.17323/1609-4514-2002-2-4-647-679 fatcat:yw73z7hxo5gf5b7fmgf4ijoo7u

subject index 10 volumes

2005 Journal of Computational and Applied Mathematics  
180 345 Mesh equidistribution 176 131 Method of feasible direction 175 447 Method of lines 176 293 Method of moments 175 77 m-Function 171 93, 411 Mittag-Leffler functions 171 39 Mixed boundary  ...  180 461 Indefinite weight 171 93 Indeterminate moment problems 178 91, 235 Index of determinacy 178 169 Inequalities 172 289 Inequalities of Hardy, Sobolev and Laptev/Weidl 171 59 Inequality constrained  ... 
doi:10.1016/s0377-0427(05)00274-8 fatcat:lnl66qx5lngarjw7i3ducv7qta

Reviews and Descriptions of Tables and Books

1966 Mathematics of Computation  
the Leverrier-Faddeev method), least squares, difference tables, polynomial interpolation and numerical differentiation, and finally a brief chapter on ordinary differential equations presenting some  ...  Walsh on simultaneous approximation and interpolation by polynomials, and the Stone-Weierstrass theorem.  ...  This is a self-instructional manual for FORTRAN II as designed for the IBM 700 series of computers. The material is presented at a very elementary level (high-  ... 
doi:10.1090/s0025-5718-66-99938-8 fatcat:qoyz4x4e2rdndlhtwi3hse4squ

Galerkin Optimal Control

Randy Boucher, Wei Kang, Qi Gong
2016 Journal of Optimization Theory and Applications  
Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington headquarters Services, Directorate for  ...  is estimated to average 1 hour per response, including the time for reviewing instruction, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the  ...  As with the calculus of variations both necessary and sufficient conditions for optimality may be established.  ... 
doi:10.1007/s10957-016-0918-x fatcat:ppa5viqtn5gwrkebxh5aseetkm

Least-squares image resizing using finite differences

A. Munoz, T. Blu, M. Unser
2001 IEEE Transactions on Image Processing  
For a given choice of basis functions, the results of our method are consistently better than those of the standard interpolation procedure; the present scheme achieves a reduction of artifacts such as  ...  The method can be generalized to include other classes of piecewise polynomial functions, expressed as linear combinations of B-splines and their derivatives.  ...  Thévenaz, of the Swiss Federal Institute of Technology, for his careful proofreading of this paper. The authors thank also the reviewers for their insightful comments.  ... 
doi:10.1109/83.941860 pmid:18255551 fatcat:gferpkmuobf6zeugnzw7x2g764
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