Deductive Synthesis of the Unification Algorithm

Manna, Zohar; Waldinger, Richard

doi:10.1007/978-94-009-7019-9_8

Cited by 20 publications

(7 citation statements)

References 16 publications

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“…The example norm 2 shows that this can be done without domains. Manna and Waldinger have studied a much more interesting nested recursive function, UNIFY, which performs unification [5]. They define a well-founded relation involving the structure of the expressions being unified and the number of distinct variables in those expressions.…”

Section: Discussionmentioning

confidence: 99%

“…In the If-If case it calls itself with a larger expression than it was given. One way of proving termination is to find a well-founded relation under which the argument 'goes down' in every recursive call [1,5]. Classically, a relation ≺ is well-founded if and only if it has no infinite descending chains • • • ≺ x 2 ≺ x 1 ≺ x 0 .…”

Section: A Normalization Functionmentioning

confidence: 99%

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Proving termination of normalization functions for conditional expressions

Paulson

1986

J Autom Reasoning

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Boyer and Moore have discussed a recursive function that puts conditional expressions into normal form [1]. It is difficult to prove that this function terminates on all inputs. Three termination proofs are compared: (1) using a measure function, (2) in domain theory using LCF, (3) showing that its recursion relation, defined by the pattern of recursive calls, is well-founded. The last two proofs are essentially the same though conducted in markedly different logical frameworks. An obviously total variant of the normalize function is presented as the 'computational meaning' of those two proofs.A related function makes nested recursive calls. The three termination proofs become more complex: termination and correctness must be proved simultaneously. The recursion relation approach seems flexible enough to handle subtle termination proofs where previously domain theory seemed essential.1980 Math Classification: 68E10 (computer software correctness)

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Section: Discussionmentioning

confidence: 99%

Section: A Normalization Functionmentioning

confidence: 99%

Proving termination of normalization functions for conditional expressions

Paulson

1986

J Autom Reasoning

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“…They have played a key role in the Boyer/Moore Theorem Prover since its early days [4]. Manna and Waldinger's work on deductive program synthesis [12] illustrates the power of well-founded relations; they justify the termination of a unification algorithm using a relation that takes into account the size of a term and the number of free variables it contains.…”

Section: Recursive Functionsmentioning

confidence: 99%

Set theory for verification. II: Induction and recursion

Paulson

1995

J Autom Reasoning

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Abstract.A theory of recursive definitions has been mechanized in Isabelle's Zermelo-Fraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning.Inductively defined sets are expressed as least fixedpoints, applying the Knaster-Tarski Theorem over a suitable set. Recursive functions are defined by well-founded recursion and its derivatives, such as transfinite recursion. Recursive data structures are expressed by applying the Knaster-Tarski Theorem to a set, such as Vω, that is closed under Cartesian product and disjoint sum.Worked examples include the transitive closure of a relation, lists, variable-branching trees and mutually recursive trees and forests. The Schröder-Bernstein Theorem and the soundness of propositional logic are proved in Isabelle sessions.

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“…Manna and Waldinger use w.f. induction on this relation for their l-expressions [12]. I was surprised to discover that w.f.…”

Section: The Less-than Ordering On the Natural Numbersmentioning

confidence: 99%

“…relations appears often, but sometimes in disguise. In deriving a unification algorithm, Manna and Waldinger [12] define the "unification ordering" ≺ un on pairs of expressions. Let vars(x ) denote the (finite) set of variables in an expression, and ≺ the (w.f.)…”

Section: Well-founded Relations In the Literaturementioning

confidence: 99%

Constructing recursion operators in intuitionistic type theory

Paulson¹

1986

Journal of Symbolic Computation

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Martin-Löf's Intuitionistic Theory of Types is becoming popular for formal reasoning about computer programs. To handle recursion schemes other than primitive recursion, a theory of well-founded relations is presented. Using primitive recursion over higher types, induction and recursion are formally derived for a large class of well-founded relations. Included are < on natural numbers, and relations formed by inverse images, addition, multiplication, and exponentiation of other relations. The constructions are given in full detail to allow their use in theorem provers for Type Theory, such as Nuprl. The theory is compared with work in the field of ordinal recursion over higher types.

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Deductive Synthesis of the Unification Algorithm

Cited by 20 publications

References 16 publications

Proving termination of normalization functions for conditional expressions

Proving termination of normalization functions for conditional expressions

Set theory for verification. II: Induction and recursion

Constructing recursion operators in intuitionistic type theory

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