Towards computing non algebraic cylindrical decompositions
ISSAC '91 Proceedings of the 1991 international symposium on Symbolic and algebraic computation
Partial Cylindrical Algebraic Decomposition for quantifier elimination
Journal of Symbolic Computation
An axiomatic basis for computer programming
Communications of the ACM
POPL '77 Proceedings of the 4th ACM SIGACT-SIGPLAN symposium on Principles of programming languages
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
QEPCAD B: a program for computing with semi-algebraic sets using CADs
ACM SIGSAM Bulletin
Automatic Generation of Polynomial Loop Invariants: Algebraic Foundations
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
LICS '04 Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science
Generating all polynomial invariants in simple loops
Journal of Symbolic Computation
Deciding polynomial-exponential problems
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Discovering non-linear ranking functions by solving semi-algebraic systems
ICTAC'07 Proceedings of the 4th international conference on Theoretical aspects of computing
Reasoning algebraically about P-solvable loops
TACAS'08/ETAPS'08 Proceedings of the Theory and practice of software, 14th international conference on Tools and algorithms for the construction and analysis of systems
Termination of linear programs with nonlinear constraints
Journal of Symbolic Computation
Symbolic decision procedure for termination of linear programs
Formal Aspects of Computing
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
VMCAI'05 Proceedings of the 6th international conference on Verification, Model Checking, and Abstract Interpretation
Termination of integer linear programs
CAV'06 Proceedings of the 18th international conference on Computer Aided Verification
CAV'06 Proceedings of the 18th international conference on Computer Aided Verification
Real root isolation of multi-exponential polynomials with application
WALCOM'10 Proceedings of the 4th international conference on Algorithms and Computation
Program analysis using quantifier-elimination heuristics
TAMC'12 Proceedings of the 9th Annual international conference on Theory and Applications of Models of Computation
Termination analysis with algorithmic learning
CAV'12 Proceedings of the 24th international conference on Computer Aided Verification
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Termination is an essential part of program correctness. For a class of regular programs, both automatically proving termination and constructing witnesses of nontermination are significant in theoretical computer science. Many traditional theorem-proving methods for analyzing termination are based on Presburger arithmetic or linear programming, so they are valid only for restricted linear problems. On the contrary, some newly-emerged algebraic methods are suitable for polynomial problems, and are promising in deciding termination of polynomial programs. In this paper, we investigate a large class of imperative programs, called solvable loops, whose guards are general polynomials and assignments are special polynomial mappings. We then propose some sufficient criteria for proving termination and nontermination of such loops in parallel. These criteria can further be translated to the quantifier elimination problem over the reals, and hence are computable. Finally, feasible sample points in the process for inferring nontermination are eventually nonterminating inputs, which can be used to generate witnesses of nontermination. Our decision procedure uses symbolic computation and is mechanically implementable in spite of considerably high complexity. Thereby a series of strong and exact results are established in analyzing termination of loops.