ISSAC '89 Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation
Polymorphic rewriting conserves algebraic strong normalization
Selected papers of the 16th international colloquium on Automata, languages, and programming
Handbook of theoretical computer science (vol. B)
Theoretical Computer Science - Special issue: algebraic development techniques
Theoretical Computer Science - Special issue on theories of types and proofs
A Monotonic Higher-Order Semantic Path Ordering
LPAR '01 Proceedings of the Artificial Intelligence on Logic for Programming
Inductive Definitions in the system Coq - Rules and Properties
TLCA '93 Proceedings of the International Conference on Typed Lambda Calculi and Applications
The Calculus of algebraic Constructions
RtA '99 Proceedings of the 10th International Conference on Rewriting Techniques and Applications
Adding Algebraic Rewriting to the Calculus of Constructions: Strong Normalization Preserved
Proceedings of the 2nd International CTRS Workshop on Conditional and Typed Rewriting Systems
The Higher-Order Recursive Path Ordering
LICS '99 Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science
Definitions by rewriting in the Calculus of Constructions
Mathematical Structures in Computer Science
On Dependency Pair Method for Proving Termination of Higher-Order Rewrite Systems
IEICE - Transactions on Information and Systems
Polymorphic higher-order recursive path orderings
Journal of the ACM (JACM)
Inductive types in the Calculus of Algebraic Constructions
Fundamenta Informaticae - Typed Lambda Calculi and Applications 2003, Selected Papers
Rewriting modulo in deduction modulo
RTA'03 Proceedings of the 14th international conference on Rewriting techniques and applications
LPAR'06 Proceedings of the 13th international conference on Logic for Programming, Artificial Intelligence, and Reasoning
Polymorphic higher-order recursive path orderings
Journal of the ACM (JACM)
The Computability Path Ordering: The End of a Quest
CSL '08 Proceedings of the 22nd international workshop on Computer Science Logic
HORPO with computability closure: a reconstruction
LPAR'07 Proceedings of the 14th international conference on Logic for programming, artificial intelligence and reasoning
Typing termination in a higher-order concurrent imperative language
Information and Computation
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Termination is a major question in both logic and computer science. In logic, termination is at the heart of proof theory where it is usually called strong normalization (of cut elimination). In computer science, termination has always been an important issue for showing programs correct. In the early days of logic, strong normalization was usually shown by assigning ordinals to expressions in such a way that eliminating a cut would yield an expression with a smaller ordinal. In the early days of verification, computer scientists used similar ideas, interpreting the arguments of a program call by a natural number, such as their size. Showing the size of the arguments to decrease for each recursive call gives a termination proof of the program, which is however rather weak since it can only yield quite small ordinals. In the sixties, Tait invented a new method for showing cut elimination of natural deduction, based on a predicate over the set of terms, such that the membership of an expression to the predicate implied the strong normalization property for that expression. The predicate being defined by induction on types, or even as a fixpoint, this method could yield much larger ordinals. Later generalized by Girard under the name of reducibility or computability candidates, it showed very effective in proving the strong normalization property of typed lambda-calculi with polymorphic types, dependent types, inductive types, and finally a cumulative hierarchy of universes. On the programming side, research on termination shifted from programming to executable specification languages based on rewriting, and concentrated on automatable methods based on the construction on well-founded orderings of the set of terms. The milestone here is Dershowitz's recursive path ordering (RPO), in the late seventies, whose well-foundedness proof is based on a powerful combinatorial argument, Kruskal's tree theorem, which also yields rather large ordinals. While the computability predicates must be defined for each particular case, and their properties proved by hand, the recursive path ordering can be effectively automated.