13 research outputs found
Consistency and Completeness of Rewriting in the Calculus of Constructions
Adding rewriting to a proof assistant based on the Curry-Howard isomorphism,
such as Coq, may greatly improve usability of the tool. Unfortunately adding an
arbitrary set of rewrite rules may render the underlying formal system
undecidable and inconsistent. While ways to ensure termination and confluence,
and hence decidability of type-checking, have already been studied to some
extent, logical consistency has got little attention so far. In this paper we
show that consistency is a consequence of canonicity, which in turn follows
from the assumption that all functions defined by rewrite rules are complete.
We provide a sound and terminating, but necessarily incomplete algorithm to
verify this property. The algorithm accepts all definitions that follow
dependent pattern matching schemes presented by Coquand and studied by McBride
in his PhD thesis. It also accepts many definitions by rewriting, containing
rules which depart from standard pattern matching.Comment: 20 page
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..
New Equations for Neutral Terms: A Sound and Complete Decision Procedure, Formalized
The definitional equality of an intensional type theory is its test of type
compatibility. Today's systems rely on ordinary evaluation semantics to compare
expressions in types, frustrating users with type errors arising when
evaluation fails to identify two `obviously' equal terms. If only the machine
could decide a richer theory! We propose a way to decide theories which
supplement evaluation with `-rules', rearranging the neutral parts of
normal forms, and report a successful initial experiment.
We study a simple -calculus with primitive fold, map and append operations on
lists and develop in Agda a sound and complete decision procedure for an
equational theory enriched with monoid, functor and fusion laws
Inductive types in the Calculus of Algebraic Constructions
In a previous work, we proved that an important part of the Calculus of
Inductive Constructions (CIC), the basis of the Coq proof assistant, can be
seen as a Calculus of Algebraic Constructions (CAC), an extension of the
Calculus of Constructions with functions and predicates defined by higher-order
rewrite rules. In this paper, we prove that almost all CIC can be seen as a
CAC, and that it can be further extended with non-strictly positive types and
inductive-recursive types together with non-free constructors and
pattern-matching on defined symbols.Comment: Journal version of TLCA'0
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
Polymorphic Higher-Order Termination
We generalise the termination method of higher-order polynomial interpretations to a setting with impredicative polymorphism. Instead of using weakly monotonic functionals, we interpret terms in a suitable extension of System F_omega. This enables a direct interpretation of rewrite rules which make essential use of impredicative polymorphism. In addition, our generalisation eases the applicability of the method in the non-polymorphic setting by allowing for the encoding of inductive data types. As an illustration of the potential of our method, we prove termination of a substantial fragment of full intuitionistic second-order propositional logic with permutative conversions
E-Unification for Second-Order Abstract Syntax
Higher-order unification (HOU) concerns unification of (extensions of) ?-calculus and can be seen as an instance of equational unification (E-unification) modulo ??-equivalence of ?-terms. We study equational unification of terms in languages with arbitrary variable binding constructions modulo arbitrary second-order equational theories. Abstract syntax with general variable binding and parametrised metavariables allows us to work with arbitrary binders without committing to ?-calculus or use inconvenient and error-prone term encodings, leading to a more flexible framework. In this paper, we introduce E-unification for second-order abstract syntax and describe a unification procedure for such problems, merging ideas from both full HOU and general E-unification. We prove that the procedure is sound and complete
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
Normal Higher-Order Termination
International audienceWe extend the termination proof methods based on reduction orderings to higher-order rewriting systems based on higher-order pattern matching. We accommodate, on the one hand, a weakly polymorphic, algebraic extension of Church's simply typed λ-calculus, and on the other hand, any use of eta, as a reduction, as an expansion or as an equation. The user's rules may be of any type in this type system, either a base, functional, or weakly polymorphic type