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Confluence properties of Weak and Strong Calculi of Explicit Substitutions
 JOURNAL OF THE ACM
, 1996
"... Categorical combinators [12, 21, 43] and more recently oecalculus [1, 23], have been introduced to provide an explicit treatment of substitutions in the calculus. We reintroduce here the ingredients of these calculi in a selfcontained and stepwise way, with a special emphasis on confluence prope ..."
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Cited by 120 (7 self)
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Categorical combinators [12, 21, 43] and more recently oecalculus [1, 23], have been introduced to provide an explicit treatment of substitutions in the calculus. We reintroduce here the ingredients of these calculi in a selfcontained and stepwise way, with a special emphasis on confluence properties. The main new results of the paper w.r.t. [12, 21, 1, 23] are the following: 1. We present a confluent weak calculus of substitutions, where no variable clashes can be feared. 2. We solve a conjecture raised in [1]: oecalculus is not confluent (it is confluent on ground terms only). This unfortunate result is "repaired" by presenting a confluent version of oecalculus, named the Envcalculus in [23], called here the confluent oecalculus.
Lexicographic Path Induction
"... Abstract. Programming languages theory is full of problems that reduce to proving the consistency of a logic, such as the normalization of typed lambdacalculi, the decidability of equality in type theory, equivalence testing of traces in security, etc. Although the principle of transfinite inductio ..."
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Abstract. Programming languages theory is full of problems that reduce to proving the consistency of a logic, such as the normalization of typed lambdacalculi, the decidability of equality in type theory, equivalence testing of traces in security, etc. Although the principle of transfinite induction is routinely employed by logicians in proving such theorems, it is rarely used by programming languages researchers who often prefer alternatives such as proofs by logical relations and model theoretic constructions. In this paper we harness the wellfoundedness of the lexicographic path ordering to derive an induction principle that combines the comfort of structural induction with the expressive strength of transfinite induction. Using lexicographic path induction, we give a consistency proof of MartinLöf’s intuitionistic theory of inductive definitions. The consistency of Heyting arithmetic follows directly, and weak normalization for Gödel’s T follows indirectly; both have been formalized in a prototypical extension of Twelf. 1
Abstract Syntactic Finitism in the Metatheory of Programming Languages
, 2010
"... One of the central goals of programminglanguage research is to develop mathematically sound formal methods for precisely specifying and reasoning about the behavior of programs. However, just as software developers sometimes make mistakes when programming, researchers sometimes make mistakes when p ..."
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One of the central goals of programminglanguage research is to develop mathematically sound formal methods for precisely specifying and reasoning about the behavior of programs. However, just as software developers sometimes make mistakes when programming, researchers sometimes make mistakes when proving that a formal method is mathematically sound. As the field of programminglanguage research has grown, these proofs have become larger and more complex, and thus harder to verify on paper. This phenomenon has motivated a great deal of research into the development of logical systems that provide an automated means to apply— and verify the application of—trusted reasoning principles to concrete proofs. The boundary between trusted and untrusted reasoning principles is inherently blurry, and different researchers draw the line in different places. However, just as certain principles are widely recognized to allow the proofs of contradictory statements, others are so uncontroversially ubiquitous in practice that they can be considered beyond reproach. We posit the following questions: (1) what are these principles and (2) how much can we do with them?