A refutational approach to geometry theorem proving
Artificial Intelligence - Special issue on geometric reasoning
Programming: the derivation of algorithms
Programming: the derivation of algorithms
Computer algebra (2nd ed.): systems and algorithms for algebraic computation
Computer algebra (2nd ed.): systems and algorithms for algebraic computation
On the mechanical derivation of loop invariants
Journal of Symbolic Computation - Special issue on automatic programming
The synthesis of loop predicates
Communications of the ACM
Automatic discovery of linear restraints among variables of a program
POPL '78 Proceedings of the 5th ACM SIGACT-SIGPLAN symposium on Principles of programming languages
POPL '77 Proceedings of the 4th ACM SIGACT-SIGPLAN symposium on Principles of programming languages
A Discipline of Programming
Systematic design of program analysis frameworks
POPL '79 Proceedings of the 6th ACM SIGACT-SIGPLAN symposium on Principles of programming languages
Semantic Analysis of Interval Congruences
Proceedings of the International Conference on Formal Methods in Programming and Their Applications
Automatic synthesis of optimal invariant assertions: Mathematical foundations
Proceedings of the 1977 symposium on Artificial intelligence and programming languages
Mathematical Theory of Computation
Mathematical Theory of Computation
Constraint-Based Verification of Parameterized Cache Coherence Protocols
Formal Methods in System Design
Non-linear loop invariant generation using Gröbner bases
Proceedings of the 31st ACM SIGPLAN-SIGACT symposium on Principles of programming languages
Precise interprocedural analysis through linear algebra
Proceedings of the 31st ACM SIGPLAN-SIGACT symposium on Principles of programming languages
Automatic Generation of Polynomial Loop Invariants: Algebraic Foundations
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Computing polynomial program invariants
Information Processing Letters
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
An equational approach to theorem proving in first-order predicate calculus
IJCAI'85 Proceedings of the 9th international joint conference on Artificial intelligence - Volume 2
Program verification using automatic generation of invariants
ICTAC'04 Proceedings of the First international conference on Theoretical Aspects of Computing
Derivation of non-structural invariants of petri nets using abstract interpretation
ICATPN'05 Proceedings of the 26th international conference on Applications and Theory of Petri Nets
Generating all polynomial invariants in simple loops
Journal of Symbolic Computation
Analysing All Polynomial Equations in ${\mathbb Z_{2^w}}$
SAS '08 Proceedings of the 15th international symposium on Static Analysis
Endomorphisms for Non-trivial Non-linear Loop Invariant Generation
Proceedings of the 5th international colloquium on Theoretical Aspects of Computing
SubPolyhedra: A (More) Scalable Approach to Infer Linear Inequalities
VMCAI '09 Proceedings of the 10th International Conference on Verification, Model Checking, and Abstract Interpretation
Invariant generation for P-solvable loops with assignments
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
Polynomial invariants for linear loops
Cybernetics and Systems Analysis
Journal of Symbolic Computation
Mechanical inference of invariants for FOR-loops
Journal of Symbolic Computation
An iterative method for generating loop invariants
FAW-AAIM'11 Proceedings of the 5th joint international frontiers in algorithmics, and 7th international conference on Algorithmic aspects in information and management
A complete invariant generation approach for p-solvable loops
PSI'09 Proceedings of the 7th international Andrei Ershov Memorial conference on Perspectives of Systems Informatics
ESOP'10 Proceedings of the 19th European conference on Programming Languages and Systems
Inference of polynomial invariants for imperative programs: a farewell to gröbner bases
SAS'12 Proceedings of the 19th international conference on Static Analysis
A data driven approach for algebraic loop invariants
ESOP'13 Proceedings of the 22nd European conference on Programming Languages and Systems
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A method for generating polynomial invariants of imperative programs is presented using the abstract interpretation framework. It is shown that for programs with polynomial assignments, an invariant consisting of a conjunction of polynomial equalities can be automatically generated for each program point. The proposed approach takes into account tests in conditional statements as well as in loops, insofar as they can be abstracted into polynomial equalities and disequalities. The semantics of each program statement is given as a transformation on polynomial ideals. Merging of execution paths is defined as the intersection of the polynomial ideals associated with each path. For loop junctions, a family of widening operators based on selecting polynomials up to a certain degree is proposed. The presented method has been implemented and successfully tried on many programs. Heuristics employed in the implementation to improve its efficiency are discussed, and tables providing details about its performance are included.