## Thesis Proposal: The logical basis of evaluation order (2007)

### BibTeX

@MISC{Zeilberger07thesisproposal:,

author = {Noam Zeilberger and Peter Lee and Robert Harper and Paul-andré Melliès and Université Paris},

title = {Thesis Proposal: The logical basis of evaluation order},

year = {2007}

}

### OpenURL

### Abstract

Most type systems are agnostic regarding the evaluation strategy for the underlying languages, with the value restriction for ML which is absent in Haskell as a notable exception. As type systems become more precise, however, detailed properties of the underlying operational semantics may become visible because properties captured by the types may be sound under one strategy but not the other. To give an example, intersection types distinguish between call-by-name and call-by-value functions because the subtyping rule (A → B) ∩ (A → C) ≤ A → (B ∩ C) is valid for the former but not the latter in the presence of effects. I propose to develop a unified, proof-theoretic approach to analyzing the interaction of types with evaluation order, based on the notion of polarity. Polarity was discovered and developed through linear logic, but I seek a fresh origin in Dummett’s program of justifying the logical laws through alternative “meaning-theories, ” essentially hypotheses as to whether the verification or use of a proposition has a canonical form. In my preliminary work, I showed how a careful judgmental analysis of Dummett’s ideas may be used to define a system of proofs and refutations, with a Curry-Howard interpretation as a single programming language in which the duality between call-by-value and call-by-name is realized as one of types. After extending its type system with (both positive and negative) union and intersection operators and a derived subtyping relationship, I found that many operationally-sensitive typing phenomena (e.g., alternative CBV/CBN subtyping distributivity principles, value and “covalue” restrictions) could be logically reconstructed. Here I give the technical details of this work, and present a plan for addressing open questions and extensions.

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Citation Context ..., would be a polarized dissection of the dependent quantifiers Π and Σ. Doing so in CU might give (via focusing) a non-degenerate interpretation of dependent types for classical logic, in response to =-=[Her05]-=-. It may also shed some light on CLF, an extension of LF with linear connectives that makes essential use of a polarity distinction—but in a slightly different way from that described here [WCPW02]. 6... |

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