Partial Cylindrical Algebraic Decomposition for quantifier elimination
Journal of Symbolic Computation
REDLOG: computer algebra meets computer logic
ACM SIGSAM Bulletin
Computing in the field of complex algebraic numbers
Journal of Symbolic Computation - Special issue: validated numerical methods and computer algebra
The size-change principle for program termination
POPL '01 Proceedings of the 28th ACM SIGPLAN-SIGACT symposium on Principles of programming languages
Deciding stability and mortality of piecewise affine dynamical systems
Theoretical Computer Science
Synthesis of Linear Ranking Functions
TACAS 2001 Proceedings of the 7th International Conference on Tools and Algorithms for the Construction and Analysis of Systems
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
Discovering non-linear ranking functions by solving semi-algebraic systems
ICTAC'07 Proceedings of the 4th international conference on Theoretical aspects of computing
Symbolic decision procedure for termination of linear programs
Formal Aspects of Computing
Termination of polynomial programs
VMCAI'05 Proceedings of the 6th international conference on Verification, Model Checking, and Abstract Interpretation
Symbolic termination analysis of solvable loops
Journal of Symbolic Computation
Non-termination sets of simple linear loops
ICTAC'12 Proceedings of the 9th international conference on Theoretical Aspects of Computing
Automated Reasoning and Mathematics
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Tiwari (2004) proved that the termination problem of a class of linear programs (loops with linear loop conditions and updates) over the reals is decidable through Jordan forms and eigenvector computation. Braverman (2006) proved that it is also decidable over the integers. Following their work, we consider the termination problems of three more general classes of programs which are loops with linear updates and three kinds of polynomial loop conditions, i.e., strict constraints, non-strict constraints and both strict and non-strict constraints, respectively. First, we prove that the termination problems of such loops over the integers are all undecidable. Then, for each class we provide an algorithm to decide the termination of such programs over the reals. The algorithms are complete for those programs satisfying a property, Non-Zero Minimum.