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Introduction
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Logic

Contextual modal type theory

Profile image of Aleksandar NanevskiAleksandar Nanevski

2008, ACM Transactions on Computational Logic

https://doi.org/10.1145/1352582.1352591
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60 pages

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Abstract

The intuitionistic modal logic of necessity is based on the judgmental notion of categorical truth. In this paper we investigate the consequences of relativizing these concepts to explicitly specified contexts. We obtain contextual modal logic and its type-theoretic analogue. Contextual modal type theory provides an elegant, uniform foundation for understanding meta-variables and explicit substitutions. We sketch some applications in functional programming and logical frameworks.

Figures (10)
We omit the remaining derivations. On the other hand, the last three are not derivable in general. We do not yet have the tools to prove this, but we may attempt the proof of (9) from the bottom up. We now turn to the derivation of (5). The proof of (5) shows that a proposition derived relative to an empty context is actually unconditionally true, and can thus be obtained in an arbitrary context.
We omit the remaining derivations. On the other hand, the last three are not derivable in general. We do not yet have the tools to prove this, but we may attempt the proof of (9) from the bottom up. We now turn to the derivation of (5). The proof of (5) shows that a proposition derived relative to an empty context is actually unconditionally true, and can thus be obtained in an arbitrary context.
Fig. 2. Sequent Calculus for ICML
Fig. 2. Sequent Calculus for ICML
We can now show that (9) is not derivable for an arbitrary A. In each sequent there is at most one applicable rule in the bottom-up construction of a potential deriva- tion, leaving us with the subgoal of verifying A in an empty context of hypotheses. This subgoal does not have a verification in general.
We can now show that (9) is not derivable for an arbitrary A. In each sequent there is at most one applicable rule in the bottom-up construction of a potential deriva- tion, leaving us with the subgoal of verifying A in an empty context of hypotheses. This subgoal does not have a verification in general.
The choice of the first or second injection in the translation of box is arbitrary. Under these translations, the following theorem is easy to show. We omit some properties pertaining to simultaneous substitutions that follow directly from the properties of terms. In the translation of terms (M)°, meta-variables are mapped to ordinary vari- ables that take additional arguments, and closures clo(u, (Mi/y1,..., M@m/Yym)) are mapped to corresponding sequences of applications ((u M1)... Mm). The letbox construct is translated to a case construct in order to preserve the commuting reductions. ‘he choice of the first or second injection in the translation of box is arbitrary.
The choice of the first or second injection in the translation of box is arbitrary. Under these translations, the following theorem is easy to show. We omit some properties pertaining to simultaneous substitutions that follow directly from the properties of terms. In the translation of terms (M)°, meta-variables are mapped to ordinary vari- ables that take additional arguments, and closures clo(u, (Mi/y1,..., M@m/Yym)) are mapped to corresponding sequences of applications ((u M1)... Mm). The letbox construct is translated to a case construct in order to preserve the commuting reductions. ‘he choice of the first or second injection in the translation of box is arbitrary.
returned as a result of substituting into the function part of an application. For example, if [M/x]"(app(R, N)) does not return an atomic term or a lambda expression, or if the returned approximation a, — ag is not a subexpression of a, then the substitution into app(R, N) is undefined. Shortly, we will prove that Q, — Q, will always be a subexpression of @ so that the check is superfluous. We nevertheless include the check here to make it obvious that the hereditary substitution is well-founded, because recursive appeals to substitutions take place on smaller terms with equal approximation, or on a smaller approximation.
returned as a result of substituting into the function part of an application. For example, if [M/x]"(app(R, N)) does not return an atomic term or a lambda expression, or if the returned approximation a, — ag is not a subexpression of a, then the substitution into app(R, N) is undefined. Shortly, we will prove that Q, — Q, will always be a subexpression of @ so that the check is superfluous. We nevertheless include the check here to make it obvious that the hereditary substitution is well-founded, because recursive appeals to substitutions take place on smaller terms with equal approximation, or on a smaller approximation.
In the subsequent rules, we omit the subscript © from the judgments since it is the same in all premises and conclusion.
In the subsequent rules, we omit the subscript © from the judgments since it is the same in all premises and conclusion.
ACM Transactions on Computational Logic, Vol. V, No. N, February 2007.
ACM Transactions on Computational Logic, Vol. V, No. N, February 2007.
ACM Transactions on Computational Logic, Vol. V, No. N, February 2007.
ACM Transactions on Computational Logic, Vol. V, No. N, February 2007.
ACM Transactions on Computational Logic, Vol. V, No. N, February 2007.
ACM Transactions on Computational Logic, Vol. V, No. N, February 2007.

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    Cyrus Omar

    Proceedings of the ACM on Programming Languages

    Live programming environments aim to provide programmers (and sometimes audiences) with continuous feedback about a program's dynamic behavior as it is being edited. The problem is that programming languages typically assign dynamic meaning only to programs that are complete, i.e. syntactically well-formed and free of type errors. Consequently, live feedback presented to the programmer exhibits temporal or perceptive gaps. This paper confronts this "gap problem" from type-theoretic first principles by developing a dynamic semantics for incomplete functional programs, starting from the static semantics for incomplete functional programs developed in recent work on Hazelnut. We model incomplete functional programs as expressions with holes, with empty holes standing for missing expressions or types, and non-empty holes operating as membranes around static and dynamic type inconsistencies. Rather than aborting when evaluation encounters any of these holes as in some existing systems, evaluation proceeds around holes, tracking the closure around each hole instance as it flows through the remainder of the program. Editor services can use the information in these hole closures to help the programmer develop and confirm their mental model of the behavior of the complete portions of the program as they decide how to fill the remaining holes. Hole closures also enable a fill-and-resume operation that avoids the need to restart evaluation after edits that amount to hole filling. Formally, the semantics borrows machinery from both gradual type theory (which supplies the basis for handling unfilled type holes) and contextual modal type theory (which supplies a logical basis for hole closures), combining these and developing additional machinery necessary to continue evaluation past holes while maintaining type safety. We have mechanized the metatheory of the core calculus, called Hazelnut Live, using the Agda proof assistant. We have also implemented these ideas into the Hazel programming environment. The implementation inserts holes automatically, following the Hazelnut edit action calculus, to guarantee that every editor state has some (possibly incomplete) type. Taken together with this paper's type safety property, the result is a proof-of-concept live programming environment where rich dynamic feedback is truly available without gaps, i.e. for every reachable editor state.

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    Contextual modal types for algebraic effects and handlers
    Aleksandar Nanevski

    Proceedings of the ACM on Programming Languages

    Programming languages with algebraic effects often track the computations’ effects using type-and-effect systems. In this paper, we propose to view an algebraic effect theory of a computation as a variable context; consequently, we propose to track algebraic effects of a computation with contextual modal types . We develop ECMTT, a novel calculus which tracks algebraic effects by a contextualized variant of the modal □ (necessity) operator, that it inherits from Contextual Modal Type Theory (CMTT). Whereas type-and-effect systems add effect annotations on top of a prior programming language, the effect annotations in ECMTT are inherent to the language, as they are managed by programming constructs corresponding to the logical introduction and elimination forms for the □ modality. Thus, the type-and-effect system of ECMTT is actually just a type system. Our design obtains the properties of local soundness and completeness, and determines the operational semantics solely by β-reductio...

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    Plugging-in proof development environments using Locks in LF
    Furio Honsell

    Mathematical Structures in Computer Science

    We present two extensions of the LF constructive type theory featuring monadic locks. A lock is a monadic type construct that captures the effect of an external call to an oracle. Such calls are the basic tool for plugging-in and gluing together, different metalanguages and proof development environments. Oracles can be invoked either to check that a constraint holds or to provide a witness. The systems are presented in the canonical style developed by the ‘CMU School.’ The first system, CLLF𝒫, is the canonical version of the system LLF𝒫, presented earlier by the authors. The second system, CLLF𝒫?, features the possibility of invoking the oracle to obtain also a witness satisfying a given constraint. In order to illustrate the advantages of our new frameworks, we show how to encode logical systems featuring rules that deeply constrain the shape of proofs. The locks mechanisms of CLLF𝒫 and CLLF𝒫? permit to factor out naturally the complexities arising from enforcing these ‘side condi...

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