## 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.

### Citations

597 | Untersuchungen über das logische Schließen - Gentzen - 1935 |

344 | Logic programming with focusing proofs in linear logic
- Andreoli
- 1992
(Show Context)
Citation Context ... eliminate some of the crippling (but apparently needless) non-determinism that arises in doing proof search on the linear sequent calculus, through a restrictive yet still complete focusing strategy =-=[And92]-=-. However, my work seeks an origin for polarity that is not bound to linearity or the sequent calculus. In fact, I propose that polarity may be traced back to Michael Dummett’s examination (in the 197... |

340 |
Lambda-mu-calculus: An algorithmic interpretation of classical natural deduction
- Parigot
- 1992
(Show Context)
Citation Context ...st-class continuations. Filinski was inspired by categorical duality; logical studies of this duality have been done largely in the setting of classical logic, particularly based upon the λµ-calculus =-=[Par92]-=-. For example in [CH00], Curien and Herbelin define a programming language with control operators (the λµ˜µ-calculus) as a Curry-Howard interpretation of a sequent calculus for implicational classical... |

308 | Definitional interpreters for higher-order programming languages
- Reynolds
- 1972
(Show Context)
Citation Context ...essay by John Reynolds centers on the dangers of specifying the semantics of a language by means of a definitional interpreter, when the meaning of the defining language is itself potentially unclear =-=[Rey72]-=-. In a functional programming language, the result of function application can depend on whether evaluation order is call-by-value or call-by-name (due to the presence of nontermination and other side... |

300 | Dependent types in practical programming
- Xi, Pfenning
- 1999
(Show Context)
Citation Context ...der quantifiers, index refinements, and modal types. The same questions could be asked about the matrix of first-order quanti45sfiers. For one, this would address the interaction of index refinements =-=[XP99]-=- with evaluation order. Interestingly, this should also shed some light on modal types, given their Kripke reading as quantification over worlds. Since � corresponds to universal quantification, its s... |

290 | Computational interpretation of linear logic
- Abramsky
- 1990
(Show Context)
Citation Context ...ype isomorphisms [Lev06]. Polarity and focusing. The discovery of linear logic [Gir87] and its crisp symmetries led to some early attempts at explaining strict vs. lazy evaluation within linear logic =-=[Abr93]-=-. Focusing [And92] greatly improved the understanding of these symmetries, and sparked interest in the idea of polarity as a way of taming misbehaved logics [Gir91, Gir93]. It was reconstructed in the... |

252 | A formulae-as-types notion of control
- Griffin
- 1990
(Show Context)
Citation Context ...igure 6) we can now directly mix the two fragments in a controlled manner. Let us begin by considering callcc, which has long been associated with Curry-Howard interpretations of classical logic (cf. =-=[Gri90]-=-). Given a statement abstracting in a P-continuation, we can construct an expression of type ↑P: 21sCovalues C ::= x | [] | [C1, C2] | fst(C) | snd(C) | exp(E) Covalue patterns c ::= x | [] | [c1, c2]... |

232 | Natural Deduction: A Proof Theoretical Study. Almquist & Wiksell - Prawitz - 1965 |

221 |
A filter lambda model and the completeness of type assignment
- Barendregt, K
(Show Context)
Citation Context ...mming languages so as to capture more invariants, without changing the languages’ fundamental character [FP91]. Although this research drew from very old theoretical foundations on intersection types =-=[BCDC83]-=-, those foundations were found to be shaky in this more operational setting. Davies and Pfenning found examples showing the necessity of a value restriction on intersection introduction, as well as th... |

182 | Refinement types for ML
- Freeman, Pfenning
- 1991
(Show Context)
Citation Context ...out when clear from context (e.g., we write P ∩ Q rather than P � Q). 4.1 Preliminaries Much of the work on intersection and union types for practical programming has imposed a refinement restriction =-=[FP91]-=-, viz. that the extended type system is only used to verify richer properties of programs that are already typable in the base type system (e.g., ML’s), rather than to make additional programs typable... |

178 | The duality of computation
- Curien, Herbelin
- 2000
(Show Context)
Citation Context ...and co-intuitionistic reasoning, it may be seen as a focusing system for the classical sequent calculus. Suppose we had tried to include callcc directly in the language with a rule similar to that in =-=[CH00]-=-: Φ, u cnt : P ⊢ S stm : · Φ ⊢ µu.S : P callcc?? Given a statement abstracting in a P-continuation, this rule constructs a (is it a “value”?...“expression”?...let’s just say) term of type P. The fact ... |

172 | A new constructive logic: classical logic - Girard - 1991 |

167 | A judgmental reconstruction of modal logic
- Pfenning, Davies
(Show Context)
Citation Context ... weak, or the introduction rules too strong? Dummett proposes a test called stability—we omit his precise formulation here, but instead describe a simpler but similarly-motivated criterion taken from =-=[PD01]-=-. Suppose we start from a verificationist interpretation, and we wish to show that the elimination rules for a connective are not too weak, 6 .si.e., that any consequence of a proposition with that pr... |

133 |
On the meanings of the logical constants and the justifications of the logical laws
- Martin-Löf
- 1985
(Show Context)
Citation Context ...tion rules for a particular connective. This is what makes his analysis so closely related to focusing. Of course another famous investigation of the justification of the logical laws is Martin-Löf’s =-=[ML96]-=-. While his 1983 Siena Lectures lack some of the important ideas of Dummett’s earlier analysis, 14 they elucidate the very notion of judgment, which we have taken for granted. And also “Locus Solum,” ... |

110 | A new deconstructive logic: Linear logic
- Danos, Joinet, et al.
- 1997
(Show Context)
Citation Context ...s, and sparked interest in the idea of polarity as a way of taming misbehaved logics [Gir91, Gir93]. It was reconstructed in the “colour protocol” of Danos et al. for studying normalization of proofs =-=[DJS97]-=-, and used significantly in the dependent type theory CLF in order to maintain an appropriate notion of canonical forms [WCPW02]. Recently the theory of polarity has been developed in greater depth, b... |

109 |
Nuel D., ‘Display logic
- Belnap
- 1982
(Show Context)
Citation Context ... set of introduction and elimination rules satisfies some “harmony” conditions, it is self-justifying [Dum91, p. 251].) Belnap eventually refined and formalized his ideas in the form of Display Logic =-=[Bel82]-=-, which has seen a recent resurgence of interest. Part of its appeal is a general cut-admissibility theorem that Belnap proved for any displayed logic satisfying a particular set of (eight) conditions... |

98 | Simple imperative polymorphism - Wright - 1995 |

87 | On the Interpretation of Intuitionistic Number Theory - Kleene - 1945 |

78 |
Constructible falsity
- Nelson
- 1949
(Show Context)
Citation Context ...2: Canonical refutations and justified proofs-by-contradiction legitimate: consider the law of contradiction A × ¬A untrue. Instead what we have here is the stronger notion of “constructible falsity” =-=[Nel49]-=-, which in turn may be deconstructed to justify a co-intuitionistic notion of truth: a proposition is co-intuitionistically true just in case its falsehood entails a contradiction. Figure 2 presents a... |

78 | A concurrent logical framework I: Judgments and properties
- Watkins, Cervesato, et al.
- 2002
(Show Context)
Citation Context ... eventually yielding a canonical statement S such that Φ ⊢ S stm : ·. Terms of canonical CU are therefore always “η-expanded, β-reduced.” While this makes for an exceptionally simple type theory (see =-=[WCPW02]-=- for the benefits derived from this property in a logical framework), it is problematic from the point of view of a programming language. Forcing expansion means that programs may get much larger (con... |

71 |
Locus solum: From the rules of logic to the logic of rules
- Girard
- 2001
(Show Context)
Citation Context ... also not clear that this gains us anything—indeed, the syntactic separation ensures that there are no possible interactions between the two sets of rules! But to break this impasse we can (following =-=[Gir01]-=-) add a pair of mediating connectives: P, Q ::= · · · | ↓N M, N ::= · · · | ↑P With these operators (Girard’s “shift” operators), we may now interpret (the more permissive) co-intuitionistic truth as ... |

68 |
Iterated Inductive Definitions and Subsystems of Analysis
- Buchholz, Feferman, et al.
- 1981
(Show Context)
Citation Context ...ems to make sense for continuations to be defined by arbitrary, infinite sets of pattern branches. Essentially, the form of CU’s continuation-typing rule is highly reminiscent of the so called Ω-rule =-=[BFPS81]-=-. Certainly a restriction could be mandated so as to avoid including in the language continuations that may not even be computable—but is there a logical way to impose such a restriction? Is it reflec... |

66 | Intersection types and computational effects
- Davies, Pfenning
- 2000
(Show Context)
Citation Context ...oregoing preface about the refinement restriction, the introduction rules are completely standard—or are they? They impose a value restriction, which did not figure in intersection type systems until =-=[DP00]-=- found that one was necessary for effectful, call-by-value languages. Of course all of the introduction rules for positive types in CU are value-typing rules, so what does this “value restriction” rea... |

63 |
Intersection and Union Types: Syntax and Semantics
- Barbanera, Dezani-Ciancaglini, et al.
- 1995
(Show Context)
Citation Context ...oundness of the standard subtyping distributivity rule for function types [DP00]. The design of a refinement system with union types [DP04] uncovered further dissonance with prior theoretical studies =-=[BDCD95]-=-. Recent work by Vouillon and Melliès [VM04, MV05] attempts to address some of the issues involved in subtyping intersection and union types in an operational setting. In [Vou04], Vouillon proves that... |

57 |
Danvy and Andrzej Filinski. Abstracting control
- Olivier
(Show Context)
Citation Context ...e desirable from not only a practical but also a theoretical point-of-view. I conjecture that such an analysis could also lead (through duality) to a logical reconstruction of delimited continuations =-=[DF90]-=-. �(Co)Inductive types. This is another extension of obvious practical value. The main theoretical question is how inversion interacts with induction. For example, if inductive types are added to CU i... |

57 |
Call-by-value is dual to call-by-name
- Wadler
- 2003
(Show Context)
Citation Context ...[CH00], Curien and Herbelin define a programming language with control operators (the λµ˜µ-calculus) as a Curry-Howard interpretation of a sequent calculus for implicational classical logic, while in =-=[Wad03]-=-, Wadler reformulates this language as a “dual calculus” for propositional classical logic (with implication defined via De Morgan translation). Both papers analyze the duality between call-by-value a... |

55 | Classical logic, continuation semantics and abstract machines
- Streicher, Reus
- 1998
(Show Context)
Citation Context ...undamental to the definition of subtyping. From a more semantic front, Streicher and Reus give a domain-theoretic interpretation of call-by-name based on “negative domains” and continuation semantics =-=[SR98]-=-, an extension of earlier work with Lafont [LRS93], which was itself inspired by Girard’s polarized deconstruction of classical logic (see below). Perhaps closest in spirit to CU is Paul-Blain Levy’s ... |

52 | A symmetric modal lambda calculus for distributed computing
- Crary, Harper, et al.
- 2004
(Show Context)
Citation Context ...uld also shed some light on modal types, given their Kripke reading as quantification over worlds. Since � corresponds to universal quantification, its standard interpretation is lazy: for example in =-=[VCHP04]-=-, the S5 � is given a CurryHoward reading as a suspended computation that can be run at any node on a network. But what about universally valid values? A “�-like” type operator internalizing such valu... |

39 | Tridirectional typechecking
- Dunfield, Pfenning
- 2004
(Show Context)
Citation Context ...ion of this duality is the necessity of a covalue restriction for call-by-name. Essentially this reconstructs the fact that in effectful settings, unions may only be eliminated in evaluation position =-=[DP04]-=-, and whereas every call-by-value continuation corresponds to an evaluation context, not every call-by-name continuation does. The restriction is required, at least for empty unions, even when the onl... |

39 |
Etude de la polarisation en logique. Thèse de doctorat, Université Aix-Marseille
- Laurent
- 2002
(Show Context)
Citation Context ... order to maintain an appropriate notion of canonical forms [WCPW02]. Recently the theory of polarity has been developed in greater depth, both in Ludics [Gir01] and in Olivier Laurent’s dissertation =-=[Lau02]-=-. In [Lau05], Laurent pursues an analysis in various ways technically similar to ours, defining dual translations into polarized linear logic and then studying the different type isomorphisms induced ... |

37 |
Focussing and proof construction
- Andreoli
(Show Context)
Citation Context ...adings of the connectives, it is also possible to view them as a focusing strategy for the classical sequent calculus. For a full background on focusing, the reader is referred to Andreoli’s tutorial =-=[And01]-=-. The technique was originally invented as a way of guiding proof search in linear logic [And92], based on the observation that the order of application of rules need not be entirely unconstrained, bu... |

31 | Declarative continuations and categorical duality
- Filinski
- 1989
(Show Context)
Citation Context ...putations could be represented as ¬ v ¬ v P ⊗¬ v ¬ v Q. But using the shift operators, we also have a more direct encoding as ↑P�↑Q. As first observed by Filinski, there is a dual phenomenon for sums =-=[Fil89]-=-. Note that what we call lazy sums (following Filinski’s example) are not what most call-by-need languages (e.g., Haskell) call sums: upon pattern-matching, the latter are eagerly reduced down to a ta... |

30 |
Natural deduction with general elimination rules
- Plato
(Show Context)
Citation Context ... by Prawitz [Pra65, Pra74]. As with Belnap’s work, Prawitz’s has seen a burst of renewed interest, following von Plato’s analysis of natural deduction and the framework of “general elimination rules” =-=[Pla01]-=-. While Dummett bases his notion of proof-theoretic justification on Prawitz’s, it is a proper extension since he considers the justification of arbitrary inferences, rather than merely the eliminatio... |

25 | The Definition of Standard ML, Revised edition - Milner, Tofte, et al. - 1997 |

19 | The runabout inference ticket - Prior - 1960 |

18 | LJQ: a strongly focused calculus for intuitionistic logic
- Dyckhoff, Lengrand
- 2006
(Show Context)
Citation Context ...cent extension of Wadler’s calculus [Sha05]. In even more recent work, Dyckhoff and Lengrand define a positive focusing system and use the implication fragment to encode call-by-value lambda calculus =-=[DL06]-=-. The key technical concept that seems to be missing from all these works is the notion of inversion, which arises naturally from the judgmental interpretation, and as we have seen is fundamental to t... |

17 | On computational open-endedness in Martin-Löf’s type theory - Howe - 1991 |

15 | Cut-Elimination in the Strict Intersection Type Assignment System is Strongly Normalising. To appear in: Notre Dame Journal of Formal Logic
- Bakel
- 2004
(Show Context)
Citation Context ...ble by standard techniques—though it would be interesting if the discipline of focusing enables alternative proofs. Known techniques for proving strong normalization of intersection type systems (cf. =-=[Bak04]-=-) could also be applied to CU with unions and intersections, though such a result seems even more tangential, since my main interest is in refinement types for general-purpose programming languages. �... |

14 | Classical isomorphisms of types
- Laurent
- 2005
(Show Context)
Citation Context ...intain an appropriate notion of canonical forms [WCPW02]. Recently the theory of polarity has been developed in greater depth, both in Ludics [Gir01] and in Olivier Laurent’s dissertation [Lau02]. In =-=[Lau05]-=-, Laurent pursues an analysis in various ways technically similar to ours, defining dual translations into polarized linear logic and then studying the different type isomorphisms induced by η-expansi... |

14 |
Continuation semantics or expressing implication by negation
- Lafont, Reus, et al.
- 1993
(Show Context)
Citation Context ... more semantic front, Streicher and Reus give a domain-theoretic interpretation of call-by-name based on “negative domains” and continuation semantics [SR98], an extension of earlier work with Lafont =-=[LRS93]-=-, which was itself inspired by Girard’s polarized deconstruction of classical logic (see below). Perhaps closest in spirit to CU is Paul-Blain Levy’s call-by-push-value language [Lev01], which maintai... |

10 | A modal language for the safety of mobile values
- Park
- 2006
(Show Context)
Citation Context ...y node on a network. But what about universally valid values? A “�-like” type operator internalizing such values clearly has its own use, distinct from the standard �, and indeed Park proposes one in =-=[Par06]-=-. This very much looks like a polarity distinction. �Dependent types. Even more interesting, but more difficult, would be a polarized dissection of the dependent quantifiers Π and Σ. Doing so in CU mi... |

9 |
On the degeneracy of sigma-types in presence of computational classical logic
- Herbelin
- 2005
(Show Context)
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... |

9 | On the idea of a general proof theory - Prawitz - 1974 |

8 |
On the unity of logic. Annals of pure and applied logic
- Girard
- 1993
(Show Context)
Citation Context ...e this problem by adding a single closed statement representing successful termination. This may appear exotic from a logical perspective (it internalizes inconsistency), but from 7 With apologies to =-=[Gir93]-=-. 18san operational standpoint, all that is going on is we are specifying an observable result. We then finish this section with a simple statement and proof of type safety as a special case of cut-el... |

8 | Subtyping union types
- Vouillon
- 2004
(Show Context)
Citation Context ... theoretical studies [BDCD95]. Recent work by Vouillon and Melliès [VM04, MV05] attempts to address some of the issues involved in subtyping intersection and union types in an operational setting. In =-=[Vou04]-=-, Vouillon proves that a particular subtyping relationship for union types is sound and complete with respect to arbitrary evaluation order. While this result is interesting (and we adopted their eleg... |

7 | plonk and plink - Tonk - 1962 |

7 |
A Remark on Gentzen’s Calculus of Sequents
- Czermak
- 1977
(Show Context)
Citation Context ...l co-intuitionistic logic, wherein truth is defined as unfalse, has been explored in various formulations (alternatively called “anti-intuitionistic” or “dual intuitionistic” logic), first by Czermak =-=[Cze77]-=- and Goodman [Goo81]. In a recent paper (and very much in line with our pragmatist interpretation), Shramko argues that co-intuitionistic logic is the “logic of scientific research,” tieing it to Popp... |

6 | The Logical Basis of Metaphysics. The William James Lectures - Dummett - 1991 |

5 |
The Logic of Contradiction. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik
- Goodman
- 1981
(Show Context)
Citation Context ...logic, wherein truth is defined as unfalse, has been explored in various formulations (alternatively called “anti-intuitionistic” or “dual intuitionistic” logic), first by Czermak [Cze77] and Goodman =-=[Goo81]-=-. In a recent paper (and very much in line with our pragmatist interpretation), Shramko argues that co-intuitionistic logic is the “logic of scientific research,” tieing it to Popper’s conception of s... |

4 |
ML with callcc is unsound
- Harper, Lillibridge
- 1991
(Show Context)
Citation Context ... its distinctive and useful features, it has a troubled history: the original (standard) polymorphism introduction rule was unsound due to the presence of effects such as mutable storage 2sand callcc =-=[HL91]-=-, and prompted various workarounds, the simplest being the value restriction [Wri95, MTHM97]. It might be easy to dismiss this as an anomaly peculiar to polymorphism and ML, but recent studies of inte... |