47,789 research outputs found
(HO)RPO Revisited
The notion of computability closure has been introduced for proving the
termination of the combination of higher-order rewriting and beta-reduction. It
is also used for strengthening the higher-order recursive path ordering. In the
present paper, we study in more details the relations between the computability
closure and the (higher-order) recursive path ordering. We show that the
first-order recursive path ordering is equal to an ordering naturally defined
from the computability closure. In the higher-order case, we get an ordering
containing the higher-order recursive path ordering whose well-foundedness
relies on the correctness of the computability closure. This provides a simple
way to extend the higher-order recursive path ordering to richer type systems
The computability path ordering
This paper aims at carrying out termination proofs for simply typed
higher-order calculi automatically by using ordering comparisons. To this end,
we introduce the computability path ordering (CPO), a recursive relation on
terms obtained by lifting a precedence on function symbols. A first version,
core CPO, is essentially obtained from the higher-order recursive path ordering
(HORPO) by eliminating type checks from some recursive calls and by
incorporating the treatment of bound variables as in the com-putability
closure. The well-foundedness proof shows that core CPO captures the essence of
computability arguments \'a la Tait and Girard, therefore explaining its name.
We further show that no further type check can be eliminated from its recursive
calls without loosing well-foundedness, but for one for which we found no
counterexample yet. Two extensions of core CPO are then introduced which allow
one to consider: the first, higher-order inductive types; the second, a
precedence in which some function symbols are smaller than application and
abstraction
Computability Closure: Ten Years Later
The notion of computability closure has been introduced for proving the
termination of higher-order rewriting with first-order matching by Jean-Pierre
Jouannaud and Mitsuhiro Okada in a 1997 draft which later served as a basis for
the author's PhD. In this paper, we show how this notion can also be used for
dealing with beta-normalized rewriting with matching modulo beta-eta (on
patterns \`a la Miller), rewriting with matching modulo some equational theory,
and higher-order data types (types with constructors having functional
recursive arguments). Finally, we show how the computability closure can easily
be turned into a reduction ordering which, in the higher-order case, contains
Jean-Pierre Jouannaud and Albert Rubio's higher-order recursive path ordering
and, in the first-order case, is equal to the usual first-order recursive path
ordering
Termination proofs by multiset path orderings imply primitive recursive derivation lengths
AbstractIt is shown that a termination proof for a term-rewriting system using multiset path orderings (i.e. recursive path orderings with multiset status only) yields a primitive recursive bound on the length of derivations, measured in the size of the starting term, confirming a conjecture of Plaisted (1978). This result holds for a great variety of path orderings, including path of subterms ordering, recursive decomposition ordering, and the path ordering of Kapur (1985) if lexicographic status is not incorporated. The result is essentially optimal as such derivation lengths can be found in each level of the Grzegorczyk hierarchy, even for string-rewriting systems
Automated Synthesis of a Finite Complexity Ordering for Saturation
We present in this paper a new procedure to saturate a set of clauses with
respect to a well-founded ordering on ground atoms such that A < B implies
Var(A) {\subseteq} Var(B) for every atoms A and B. This condition is satisfied
by any atom ordering compatible with a lexicographic, recursive, or multiset
path ordering on terms. Our saturation procedure is based on a priori ordered
resolution and its main novelty is the on-the-fly construction of a finite
complexity atom ordering. In contrast with the usual redundancy, we give a new
redundancy notion and we prove that during the saturation a non-redundant
inference by a priori ordered resolution is also an inference by a posteriori
ordered resolution. We also prove that if a set S of clauses is saturated with
respect to an atom ordering as described above then the problem of whether a
clause C is entailed from S is decidable
The computability path ordering: the end of a quest
In this paper, we first briefly survey automated termination proof methods
for higher-order calculi. We then concentrate on the higher-order recursive
path ordering, for which we provide an improved definition, the Computability
Path Ordering. This new definition appears indeed to capture the essence of
computability arguments \`a la Tait and Girard, therefore explaining the name
of the improved ordering.Comment: Dans CSL'08 (2008
Ordering constraints on trees
We survey recent results about ordering constraints on trees and discuss their applications. Our main interest lies in the family of recursive path orderings which enjoy the properties of being total, well-founded and compatible with the tree constructors. The paper includes some new results, in particular the undecidability of the theory of lexicographic path orderings in case of a non-unary signature
Termination orderings for associative-commutative rewriting systems
In this paper we describe a new class of orderings—associative path orderings—for proving termination of associative-commutative term rewriting systems .These orderings are based on the concept of simplification orderings and extend the well-known recursive path orderings to E - congruence classes, where E is an equational theory consisting of associativity and commutativity axioms. Associative path orderings are applicable to term rewriting systems for which a precedence ordering on the set of operator symbols can be defined that satisfies a certain condition,the associative path condition. The precedence ordering can often be derived from the structure of the reduction rules. We include termination proofs for various term rewriting systems (for rings,boolean algebra,etc.) and, in addition, point out ways to handle situations where the associative path condition is too restrictive
On the formalization of termination techniques based on multiset orderings
Multiset orderings are a key ingredient in certain termination techniques like the recursive path ordering and a variant of size-change termination. In order to integrate these techniques in a certifier for termination proofs, we have added them to the Isabelle Formalization of Rewriting. To this end, it was required to extend the existing formalization on multiset orderings towards a generalized multiset ordering. Afterwards, the soundness proofs of both techniques have been established, although only after fixing some definitions. Concerning efficiency, it is known that the search for suitable parameters for both techniques is NP-hard. We show that checking the correct application of the techniques-where all parameters are provided-is also NP-hard, since the problem of deciding the generalized multiset ordering is NP-hard. © René Thiemann, Guillaume Allais, and JulianNagele
Higher-Order Termination: from Kruskal to Computability
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..
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