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Synthesis of Linear Ranking Functions
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Closed-Form Upper Bounds in Static Cost Analysis
Journal of Automated Reasoning
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ICLP'05 Proceedings of the 21st international conference on Logic Programming
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
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Termination of integer linear programs
CAV'06 Proceedings of the 18th international conference on Computer Aided Verification
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Ranking function synthesis for bit-vector relations
TACAS'10 Proceedings of the 16th international conference on Tools and Algorithms for the Construction and Analysis of Systems
On the termination of integer loops
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Verification of gap-order constraint abstractions of counter systems
VMCAI'12 Proceedings of the 13th international conference on Verification, Model Checking, and Abstract Interpretation
A new look at the automatic synthesis of linear ranking functions
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TACAS'12 Proceedings of the 18th international conference on Tools and Algorithms for the Construction and Analysis of Systems
On the linear ranking problem for integer linear-constraint loops
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On the linear ranking problem for integer linear-constraint loops
POPL '13 Proceedings of the 40th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages
Eventual linear ranking functions
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CAV'13 Proceedings of the 25th international conference on Computer Aided Verification
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In this paper we study the complexity of the Linear Ranking problem: given a loop, described by linear constraints over a finite set of integer variables, is there a linear ranking function for this loop? While existence of such a function implies termination, this problem is not equivalent to termination. When the variables range over the rationals or reals, the Linear Ranking problem is known to be PTIME decidable. However, when they range over the integers, whether for single-path or multipath loops, the complexity of the Linear Ranking problem has not yet been determined. We show that it is coNP-complete. However, we point out some special cases of importance of PTIME complexity. We also present complete algorithms for synthesizing linear ranking functions, both for the general case and the special PTIME cases.