Formalizing Size-Optimal Sorting Networks: Extracting a Certified Proof Checker

Since the proof of the four color theorem in 1976, computer-generated proofs have become a reality in mathematics and computer science. During the last decade, we have seen formal proofs using verified proof assistants being used to verify the validity of such proofs. In this paper, we describe a formalized theory of size-optimal sorting networks. From this formalization we extract a certified checker that successfully verifies computer-generated proofs of optimality on up to 8 inputs. The checker relies on an untrusted oracle to shortcut the search for witnesses on more than 1.6 million NP-complete subproblems.

[1]  Pierre Courtieu,et al.  Automated Certified Proofs with CiME3 , 2011, RTA.

[2]  René Thiemann,et al.  The Certification Problem Format , 2014, UITP.

[3]  K. Appel,et al.  Every planar map is four colorable. Part I: Discharging , 1977 .

[4]  René Thiemann.  Formalizing Bounded Increase , 2013, ITP.

[5]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[6]  David Pichardie,et al.  Interactive Theorem Proving , 2013, Lecture Notes in Computer Science.

[7]  David Monniaux,et al.  Efficient Generation of Correctness Certificates for the Abstract Domain of Polyhedra , 2013, SAS.

[8]  F. Wiedijk,et al.  The challenge of computer mathematics , 2005, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[9]  Georges Gonthier,et al.  Formal Proof—The Four- Color Theorem , 2008 .

[10]  Michael Frank,et al.  Twenty-Five Comparators Is Optimal When Sorting Nine Inputs (and Twenty-Nine for Ten) , 2014, 2014 IEEE 26th International Conference on Tools with Artificial Intelligence.

[11]  Ian Parberry.  A computer-assisted optimal depth lower bound for nine-input sorting networks , 2005, Mathematical systems theory.

[12]  David C. van Voorhis.  Toward a Lower Bound for Sorting Networks , 1972, Complexity of Computer Computations.

[13]  K. Appel,et al.  Every planar map is four colorable. Part II: Reducibility , 1977 .

[14]  Robert W. Floyd,et al.  The Bose-Nelson Sorting Problem††The preparation of this report has been supported in part by the National Science Foundation, and in part by the Office of Naval Research. , 1970 .

[15]  K. Appel,et al.  Every Planar Map Is Four Colorable , 2019, Mathematical Solitaires & Games.

[16]  Donald E. Knuth,et al.  The Art of Computer Programming: Volume 3: Sorting and Searching , 1998 .

[17]  Peter Schneider-Kamp,et al.  Optimizing a Certified Proof Checker for a Large-Scale Computer-Generated Proof , 2015, CICM.

[18]  Alexei Lisitsa,et al.  A SAT Attack on the Erdős Discrepancy Conjecture , 2014, SAT.

[19]  Xavier Leroy,et al.  Formal verification of a realistic compiler , 2009, CACM.

[20]  Yann Régis-Gianas,et al.  Lightweight Proof by Reflection Using a Posteriori Simulation of Effectful Computation , 2013, ITP.