Automatizing termination proofs of recursively defined functions

Manoury, Pascal; Simonot, Marianne

doi:10.1016/0304-3975(94)00021-2

Cited by 14 publications

(8 citation statements)

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“…To our knowledge, there is no complete account of the meta-theoretical properties of pattern-matching in dependent type theory. Ongoing work on checking the termination of recursive function definitions in functional languages (see, for example, Telford and Turner (1997), Abel and Altenkirch (2002), Giesl et al (1998), and Manoury and Simonot (1994)) is relevant for this direction of type-theoretic developments. Giménez has remarked (Giménez 1996) that in a typing system with dependent pattern matching the computation rule used in this article for corecursive definitions only satisfies a weak form of the subject reduction property.…”

Section: Comparison With Martin-löf (1971) and Other Work On Traditiomentioning

confidence: 99%

Type-based termination of recursive definitions

Barthe

Frade

Giménez

et al. 2004

Math. Struct. Comp. Sci.

114

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This paper introduces λ , a simply typed lambda calculus supporting inductive types and recursive function definitions with termination ensured by types. The system is shown to enjoy subject reduction, strong normalisation of typable terms and to be stronger than a related system λ G in which termination is ensured by a syntactic guard condition. The system can, at will, be extended to support coinductive types and corecursive function definitions also.Type-based termination of recursive definitions 103 2 ι-reduction → ι is defined as the compatible closure of the rulewhere # a = ar(c i ). 3 µ-reduction → µ is defined as the compatible closure of the ruleRemark 2.4. Our formulation of the β-and µ-reduction rules relies on a variable convention: in the β-rule, the bound variables of e are assumed to be different from the free variables of e ; in the µ-rule, the bound and the free variables of e are assumed to be different.The mechanics of the reduction calculus is illustrated by the following example. Example 2.5. Consider the inductive type of natural numbers Nat with C(Nat) = {o, s}. Let plus ≡ (letrec plus = λx. λy. case x of {o ⇒ y | s ⇒ λx . s (plus x y)}). The following is a reduction sequence that computes one plus two, where, as usual, β denotes the reflexive and transitive closure of → β . plus (s o) (s (s o)) → µ (λx. λy. case x of {o ⇒ y | s ⇒ λx . s (plus x y)}) (s o) (s (s o)) β case s o of {o ⇒ s (s o) | s ⇒ λx . s (plus x (s (s o)))} → ι (λx . s (plus x (s (s o)))) o → β s (plus o (s (s o))) → µ s ((λx. λy. case x of {o ⇒ y | s ⇒ λx . s (plus x y)}) o (s(s o))) β s (case o of {o ⇒ s (s o) | s ⇒ λx . s (plus x (s (s o)))}) → ι s (s (s o)) Types and typing systemWe now assume we are given two denumerable sets V T of type variables and V S of stage variables, and adopt the naming convention that α, α , α i , β, δ, . . . range over V T , and ı, , . . . range over V S . Proceeding from these, we define stage and type expressions. Stage expressions are built of stage variables, a symbol for the successor function on stages, and a symbol for the limit stage. A type expression is either a type variable, a function type expression or a datatype approximation expression. Definition 2.6 (Stages and types).1 The set S of stage expressions is given by the abstract syntax: s, r ::= ı | ∞ | s.

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Section: Comparison With Martin-löf (1971) and Other Work On Traditiomentioning

confidence: 99%

Type-based termination of recursive definitions

Barthe

Frade

Giménez

et al. 2004

Math. Struct. Comp. Sci.

114

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“…The latter method comes from the analysis of another approach, the formal approach, that relies on formal termination proofs [21] built in a natural deduction style. The formal approach is essentially used to allow the extraction of A-terms that compute the functions as codes of programs (see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…It has been shown in [22,10] that it is possible to extract well-founded orderings, namely ordinal measures, from the above formal proofs devised in the fully automated system ProPre [21,20], such that the arguments in each recursive call of the given function are smaller than the initially given input (i.e., the calls decrease wrt the well-founded ordering). The extraction of a special class of measures, called ramified measures, appears as a useful way to find out new orderings as these measures are in particular not limited to lexicographic orderings.…”

Section: Introductionmentioning

confidence: 99%

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An Extension of an Automated Termination Method of Recursive Functions

Kamareddine¹,

Monin

2002

Int. J. Found. Comput. Sci.

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Inductive proofs are commonly used in automated deduction systems or functional programming, such as for instance the ProPre system, for establishing the termination of recursively defined functions. Such proofs deal with the structural orderings of the term algebras that define the domain of the functions. However there exists other interesting functions whose termination requires different underlying orderings. To treat a class of such functions that are not taken into account by systems such as ProPre, we develop termination properties that can be shown automatically. In contrast with the ProPre system that builds formal trees based on inductive proofs, we generate measures that satisfy an extended termination property and well-founded orderings which ensure the termination of the functions.

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“…Formal methods can successfully enhance the quality of software [3,5,23,25,29] and therefore increase the confidence in the system behavior. Unfortunately, formal methods are often neglected in practice: in industrial projects software verification usually relies on techniques such as code inspection and testing, while correctness is very seldom formally proved.…”

Section: Introductionmentioning

confidence: 99%

Using symbolic execution for verifying safety-critical systems

Coen‐Porisini

DenaroGiovanni

GhezziCarlo

et al. 2001

SIGSOFT Softw. Eng. Notes

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Safety critical systems require to be highly reliable and thus special care is taken when verifying them in order to increase the confidence in their behavior. This paper addresses the problem of formal verification of safety critical systems by providing empirical evidence of the practical applicability of symbolic execution and of its usefulness for checking safety-related properties. In this paper, symbolic execution is used for building an operational model of the software on which safety properties, expressed by means of a Path Description Language (PDL), can be assessed.

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Automatizing termination proofs of recursively defined functions

Cited by 14 publications

References 1 publication

Type-based termination of recursive definitions

Type-based termination of recursive definitions

An Extension of an Automated Termination Method of Recursive Functions

Using symbolic execution for verifying safety-critical systems

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