,
, =Piecewise[{{x(1-rn),nm<x<=nM},{x-as,-nm<=x<=nm},{x(1+rn)
, =Piecewise[{{x(1+rn),nm<x<=nM},{x+as,-nm<=x<=nm},{x(1-rn),-nM<=x<-nm}}] assum, pp.0-0
, We use Mathematica to assist us. Definitions in Appendice A are assumed
Multi-dimensional Rankings, Program Termination and Complexity Bounds of Flowchart Programs, Proc. of the 17th International Static Analysis Symposium (SAS'10). Ed. by Radhia Cousot and Matthieu Martel, vol.6337, pp.117-133, 2010. ,
URL : https://hal.archives-ouvertes.fr/inria-00523298
A New Look at the Automatic Synthesis of Linear Ranking Functions, Information and Computation, vol.215, pp.47-67, 2012. ,
URL : https://hal.archives-ouvertes.fr/hal-01186145
Size-Change Termination, Monotonicity Constraints and Ranking Functions, Proc. of the 21st International Conference on Computer Aided Verification (CAV'09), vol.5643, pp.109-123, 2009. ,
Monotonicity Constraints for Termination in the Integer Domain, Logical Methods in Computer Science, vol.7, issue.3, 2011. ,
Ranking Functions for Linear-Constraint Loops, Proc. of the 1st International Workshop on Verification and Program Transformation (VPT'13), vol.16, pp.1-8, 2013. ,
Mortality of Iterated Piecewise Affine Functions over the Integers: Decidability and complexity, pp.19-56 ,
Ranking Functions for LinearConstraint Loops, Journal of the ACM, vol.61, p.55, 2014. ,
Complexity of Bradley-MannaSipma Lexicographic Ranking Functions, Proc. of the 27th International Conference on Computer Aided Verification (CAV'15), vol.9207, pp.304-321, 2015. ,
On Multiphase-Linear Ranking Functions, Proc. of the 29th International Conference on Computer Aided Verification (CAV'17), Part II, vol.10427, pp.601-620, 2017. ,
On the Termination of Integer Loops, ACM Transactions on Programming Languages and Systems, vol.34, p.24, 2012. ,
Deciding Conditional Termination, Logical Methods in Computer Science, vol.10, 2014. ,
URL : https://hal.archives-ouvertes.fr/hal-01418906
, Proc. of the 18th International Conference on Computer Aided Verification (CAV'06), vol.4144, pp.3-540, 2006.
Principles of Model Checking, 2008. ,
Program Termination Analysis in Polynomial Time, ACM Transactions on Programming Languages and Systems, vol.29, p.37, 2007. ,
, Proc. of the 21st International Conference on Computer Aided Verification (CAV'09), vol.5643, 2009.
Eventual Linear Ranking Functions, Proc. of the 15th International Symposium on Principles and Practice of Declarative Programming (PPDP'13), pp.229-238, 2013. ,
URL : https://hal.archives-ouvertes.fr/hal-01451694
Boosting Local Consistency Algorithms over Floating-Point Numbers, Proc. of the 18th International Conference on Principles and Practice of Constraint Programming (CP'12). Ed. by Michela Milano, vol.7514, pp.127-140, 2012. ,
Linear Ranking with Reachability, Proc. of the 17th International Conference on Computer Aided Verification (CAV'05), vol.3576, pp.491-504, 2005. ,
Termination of Polynomial Programs, Proc. of the 6th International Conference on Verification, Model Checking, and Abstract Interpretation (VMCAI'05). Ed. by Radhia Cousot, vol.3385, pp.113-129, 2005. ,
Hardware-Independent Proofs of Numerical Programs, Proc. of the 2nd NASA Formal Methods Symposium (NFM'10). Ed. by César A. Muñoz. Vol. NASA/CP-2010-216215. NASA Conference Proceedings, pp.14-23, 2010. ,
URL : https://hal.archives-ouvertes.fr/inria-00534410
Proofs of Numerical Programs when the Compiler Optimizes, Innovations in Systems and Software Engineering, vol.7, pp.151-160, 2011. ,
URL : https://hal.archives-ouvertes.fr/hal-00777639
Stupid is as Stupid Does: Taking the Square Root of the Square of a Floating-Point Number, In: Electronic Notes in Theoretical Computer Science, vol.317, pp.27-32, 2015. ,
URL : https://hal.archives-ouvertes.fr/hal-01148409
Termination of Integer Linear Programs, Proc. of the 18th International Conference on Computer Aided Verification (CAV'06), vol.4144, pp.372-385, 2006. ,
DOI : 10.1007/11817963_34
URL : http://www.cs.toronto.edu/~mbraverm/CAV2006.pdf
,
An Intuitionistic Version of Ramsey's Theorem and its use in Program Termination, Annals of Pure and Applied Logic, vol.166, pp.1382-1406, 2015. ,
A Gentle Introduction to Formal Verification of Computer Systems by Abstract Interpretation, NATO Science for Peace and Security Series-D: Information and Communication Security, vol.25, pp.1-29, 2010. ,
URL : https://hal.archives-ouvertes.fr/inria-00543886
An Abstract Interpretation Framework for Termination, Proc. of the 39th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pp.245-258, 2012. ,
Abstract Interpretation: A Unified Lattice Model for Static Analysis of Programs by Construction or Approximation of Fixpoints, Proc. of the 4th ACM Symposium on Principles of Programming Languages (POPL'77), pp.238-252, 1977. ,
The Early Computers of Konrad Zuse, IEEE Annals of the History of Computing, vol.3, pp.241-262, 1935. ,
,
Termination Proofs for Linear Simple Loops, Software Tools for Technology Transfer, vol.17, pp.47-57, 2015. ,
A Note on the Entscheidungsproblem, The Journal of Symbolic Logic, vol.1, pp.40-41, 1936. ,
Testing for Termination with Monotonicity Constraints, Proc. of the 21st International Conference on Logic Programming (ICLP'05), vol.3668, pp.326-340, 2005. ,
, Proc. of the 17th International Static Analysis Symposium (SAS'10), vol.6337, 2010.
Proving Stabilization of Biological Systems, Proc. of the 12th International Conference on Verification, Model Checking, and Abstract Interpretation (VMCAI'11), pp.134-149, 2011. ,
,
Ranking Function Synthesis for Bit-vector Relations, Formal Methods in System Design, vol.43, pp.93-120, 2013. ,
, Terminator: Beyond Safety". In: Proc. of the 18th International Conference on Computer Aided Verification (CAV'06), vol.4144, pp.415-418, 2006.
Proving Program Termination, Communications of the ACM, vol.54, pp.88-98, 2011. ,
Ramsey vs. Lexicographic Termination Proving, Proc. of the 19th International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS'13) ,
, Lecture Notes in Computer Science, vol.7795, pp.47-61, 2013.
Efficiently Computing Static Single Assignment Form and the Control Dependence Graph, ACM Transactions on Programming Languages and Systems, vol.13, pp.451-490, 1991. ,
Decomposition of Tree-Automatic Structures, 2016. ,
Unrestricted Termination and Non-termination Arguments for Bit-Vector Programs, Proc. of the 24th European Symposium on Programming (ESOP'15). Ed. by Jan Vitek, vol.9032, pp.183-204, 2015. ,
DOI : 10.1007/978-3-662-46669-8_8
URL : http://arxiv.org/pdf/1410.5089
MPFR: A Multiple-precision Binary Floating-point Library with Correct Rounding, ACM Transactions on Mathematical Software, vol.33, p.13, 2007. ,
URL : https://hal.archives-ouvertes.fr/inria-00070266
The Zonotope Abstract Domain Taylor1+, Proc. of the 21st International Conference on Computer Aided Verification (CAV'09), vol.5643, pp.627-633, 2009. ,
DOI : 10.1007/978-3-642-02658-4_47
Automated Termination Proofs with AProVE, Proc. of the 15th International Conference on Rewriting Techniques and Applications (RTA'04). Ed. by Vincent van Oostrom, vol.3091, pp.210-220, 2004. ,
DOI : 10.1007/978-3-540-25979-4_15
URL : http://www-i2.informatik.rwth-aachen.de/giesl/papers/RTA04-distribute.pdf
Termination Competition (TermComp'15), Proc. of the 25th International Conference on Automated Deduction (CADE'15) ,
DOI : 10.1007/978-3-319-21401-6_6
URL : http://upcommons.upc.edu/bitstream/2117/85179/1/termCompCADE2015.pdf
, Lecture Notes in Computer Science, vol.9195, pp.105-108, 2015.
Analyzing Program Termination and Complexity Automatically with AProVE, Journal of Automated Reasoning, vol.58, pp.3-31, 2017. ,
DOI : 10.1007/s10817-016-9388-y
URL : http://eprints.bbk.ac.uk/16551/1/JAR-AProVE.pdf
On the Maximum Relative Error when Computing Integer Powers by Iterated Multiplications in Floating-point Arithmetic, Numerical Algorithms, vol.70, pp.1572-9265, 2015. ,
URL : https://hal.archives-ouvertes.fr/ensl-00945033
Synthesis of Ranking Functions using Extremal Counterexamples, Proc. of the 36th ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI'15), pp.608-618, 2015. ,
URL : https://hal.archives-ouvertes.fr/hal-01144622
What Every Computer Scientist Should Know About Floating-Point Arithmetic, ACM Computing Surveys, vol.23, pp.5-48, 1991. ,
Handling Floating-Point Exceptions in Numeric Programs, ACM Transactions on Programming Languages and Systems, vol.18, pp.139-174, 1996. ,
Size-Change Termination and Transition Invariants, Proc. of the 17th International Static Analysis Symposium (SAS'10). Ed. by Radhia Cousot and Matthieu Martel, vol.6337, pp.22-50, 2010. ,
, IEEE Standard for Binary Floating-Point Arithmetic, 1985.
, IEEE Standard for Floating-Point Arithmetic, IEEE Standard, pp.1-70, 2008.
Computability and Complexity-From a Programming Perspective. Foundations of computing series, pp.978-978, 1997. ,
On Relative Errors of Floating-point Operations: Optimal Bounds and Applications, Mathematics of Computation, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-00934443
How to Print Floating-point Numbers Accurately (with Retrospective), A Selection from 20 Years of the ACM SIGPLAN Conference on Programming Language Design and Implementation 1979-1999, pp.1-58113, 2004. ,
Idempotent Binary->Decimal->Binary Conversion, 2002. ,
Why do we Need a Floating-point Arithmetic Standard, 1981. ,
Polynomial Algorithms in Linear Programming", In: USSR Computational Mathematics and Mathematical Physics, vol.20, pp.53-72, 1980. ,
Second-Order SAT Solving using Program Synthesis, Computing Research Repository, 2014. ,
The Ultimate Challenge: the 3x + 1 Problem, pp.978-0821849408, 2011. ,
Formal Verification of a Realistic Compiler, Communications of the ACM 52, vol.7, pp.107-115, 2009. ,
URL : https://hal.archives-ouvertes.fr/inria-00415861
Ranking Templates for Linear Loops, Logical Methods in Computer Science, vol.11, issue.1, 2015. ,
The Size-change Principle for Program Termination, Proc. of the 28th ACM SIGPLANSIGACT Symposium on Principles of Programming Languages ,
, , pp.81-92, 2001.
Termination of Just/Fair Computations in Term Rewriting, Information and Computation, vol.206, pp.652-675, 2008. ,
TermiLog: A System for Checking Termination of Queries to Logic Programs, Proc. of the 9th International Conference on Computer Aided Verification (CAV'97) ,
, Lecture Notes in Computer Science, vol.1254
, , vol.isbn, pp.3-540, 1997.
Polynomials Over the Reals in Proofs of Termination: From Theory to Practice, Theoretical Informatics and Applications, vol.39, pp.547-586, 2005. ,
The L-Depth Eventual Linear Ranking Functions for Single-Path Linear Constraint Loops, Proc. of the 10th International Symposium on Theoretical Aspects of Software Engineering (TASE'16), pp.30-37, 2016. ,
In-and-out Conversions, Communications of the ACM, vol.11, issue.1, pp.47-50, 1968. ,
Relational Abstract Domains for the Detection of FloatingPoint Run-Time Errors, Computing Research Repository, 2007. ,
A Proof of Infinite Ramsey Theorem ,
On the Linear Ranking Problem for Simple Floating-Point Loops, Proc. of the 23rd International Static Analysis Symposium (SAS'16). Ed. by Xavier Rival, vol.9837, pp.300-316, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-01451688
Termination Analysis of Floating-Point Programs using Parameterizable Rational Approximations, Proc. of the 31st ACM Symposium on Applied Computing (SAC'16). Ed. by Sascha Ossowski, pp.1674-1679, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-01451687
Optimal Approximation for Efficient Termination Analysis of Floating-Point Loops, Proc. of the 1st IEEE Conference on on Next Generation Computing Applications (NextComp'17, 2017. ,
URL : https://hal.archives-ouvertes.fr/hal-01579791
Number of Solutions to (A 2 + B 2 = C 2 + C) in a Binade, 2003. ,
URL : https://hal.archives-ouvertes.fr/inria-00071634
The Pitfalls of Verifying Floating-Point Computations, ACM Transactions on Programming Languages and Systems, vol.30, 2008. ,
URL : https://hal.archives-ouvertes.fr/hal-00128124
On Proving Left Termination of Constraint Logic Programs, ACM Transactions on Computational Logic, vol.4, pp.207-259, 2003. ,
Handbook of Floating-Point Arithmetic. Birk-häuser, 2010. ,
URL : https://hal.archives-ouvertes.fr/ensl-00379167
On the Definition of ulp(x), p.16, 2005. ,
URL : https://hal.archives-ouvertes.fr/inria-00070503
Loop Detection in Term Rewriting using the Eliminating Unfoldings, Theoretical Computer Science, vol.403, pp.307-327, 2008. ,
URL : https://hal.archives-ouvertes.fr/hal-01186183
Nontermination Inference of Logic Programs, ACM Transactions on Programming Languages and Systems, vol.28, issue.2, pp.256-289, 2006. ,
URL : https://hal.archives-ouvertes.fr/hal-00129526
A Complete Method for the Synthesis of Linear Ranking Functions, Proc. of the 5th International Conference on Verification, Model Checking, and Abstract Interpretation, vol.2937, pp.239-251, 2004. ,
Transition Invariants, Proc. of the 19th IEEE Symposium on Logic in Computer Science (LICS' 04), pp.32-41, 2004. ,
DOI : 10.1109/lics.2004.1319598
Transition Predicate Abstraction and Fair Termination, ACM Transactions on Programming Languages and Systems, vol.29, 2007. ,
Transition Invariants and Transition Predicate Abstraction for Program Termination, Proc. of 17th International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS'11), vol.6605, pp.3-10, 2011. ,
Experiments with Non-Termination Analysis for Java Bytecode, In: Electronic Notes in Theoretical Computer Science, vol.253, pp.83-96, 2009. ,
URL : https://hal.archives-ouvertes.fr/hal-01188696
On a Problem of Formal Logic, Proc. of the London Mathematical Society, pp.264-286, 1930. ,
, Static Analysis-23rd International Symposium, vol.9837, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01388205
, , pp.978-981
Termination of Logic Programs: The Never-Ending Story, Journal of Logic Programming, vol.19, pp.90027-90029, 1994. ,
Introduction to the Theory of Computation, pp.978-978, 1997. ,
, 2LS for Program Analysis-(Competition Contribution)". In: Proc. of the 22nd International Conference on Tools and Algorithms for the Construction and Analysis of Systems. Ed. by Marsha Chechik and Jean-François Raskin, vol.9636, pp.905-907, 2016.
DOI : 10.1007/978-3-662-49674-9_56
A Termination Analyzer for Java Bytecode Based on Path-length, ACM Transactions on Programming Languages and Systems, vol.32, issue.3, 2010. ,
URL : https://hal.archives-ouvertes.fr/hal-01186167
, Proc. of the 23rd International Static Analysis Symposium (SAS'16). Ed. by Xavier Rival, vol.9837, pp.39-57, 2016.
Termination of Floating-Point Computations, Journal of Automated Reasoning, vol.34, pp.141-177, 2005. ,
Floating-Point Computation. Prentice-Hall series in automatic computation, 1974. ,
An Impossible Program, The Computer Journal, vol.7, p.313, 1965. ,
Termination of Linear Programs, Proc. of the 16th International Conference on Computer Aided Verification (CAV'04), vol.3114, pp.70-82, 2004. ,
On Computable Numbers, with an Application to the Entscheidungsproblem, Proceedings of the London Mathematical society, vol.2, pp.230-265, 1937. ,
Static Analysis by Abstract Interpretation of Functional Temporal Properties of Programs, 2015. ,
URL : https://hal.archives-ouvertes.fr/tel-01176641
Eventual Linear Ranking Functions for Multi-path Linear Loops, Proc. of the 2016 IEEE Conference on Information Technology, Networking, Electronic and Automation Control (ITNEC'16, pp.331-337, 2016. ,