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Register a new userCall-by-name Gradual Type Theory
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Added by
Thorsten Wissmann
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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We present gradual type theory, a logic and type theory for call-by-name gradual typing. We define the central constructions of gradual typing (the dynamic type, type casts and type error) in a novel way, by universal properties relative to new judgments for gradual type and term dynamism, which were developed in blame calculi and to state the gradual guarantee theorem of gradual typing. Combined with the ordinary extensionality ($eta$) principles that type theory provides, we show that most of the standard operational behavior of casts is uniquely determined by the gradual guarantee. This provides a semantic justification for the definitions of casts, and shows that non-standard definitions of casts must violate these principles. Our type theory is the internal language of a certain class of preorder categories called equipments. We give a general construction of an equipment interpreting gradual type theory from a 2-category representing non-gradual types and programs, which is a semantic analogue of Findler and Felleisens definitions of contracts, and use it to build some concrete domain-theoretic models of gradual typing.
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Gradually typed languages are designed to support both dynamically typed and statically typed programming styles while preserving the benefits of each. While existing gradual type soundness theorems for these languages aim to show that type-based reasoning is preserved when moving from the fully static setting to a gradual one, these theorems do not imply that correctness of type-based refactorings and optimizations is preserved. Establishing correctness of program transformations is technically difficult, and is often neglected in the metatheory of gradual languages. In this paper, we propose an axiomatic account of program equivalence in a gradual cast calculus, which we formalize in a logic we call gradual type theory (GTT). Based on Levys call-by-push-value, GTT gives an axiomatic account of both call-by-value and call-by-name gradual languages. We then prove theorems that justify optimizations and refactorings in gradually typed languages. For example, uniqueness principles for gradual type connectives show that if the $betaeta$ laws hold for a connective, then casts between that connective must be equivalent to the lazy cast semantics. Contrapositively, this shows that eager cast semantics violates the extensionality of function types. As another example, we show that gradual upcasts are pure and dually, gradual downcasts are strict. We show the consistency and applicability of our theory by proving that an implementation using the lazy cast semantics gives a logical relations model of our type theory, where equivalence in GTT implies contextual equivalence of the programs. Since GTT also axiomatizes the dynamic gradual guarantee, our model also establishes this central theorem of gradual typing. The model is parametrized by the implementation of the dynamic types, and so gives a family of implementations that validate type-based optimization and the gradual guarantee.
Gradually typed languages allow programmers to mix statically and dynamically typed code, enabling them to incrementally reap the benefits of static typing as they add type annotations to their code. However, this type migration process is typically a manual effort with limited tool support. This paper examines the problem of emph{automated type migration}: given a dynamic program, infer additional or improved type annotations. Existing type migration algorithms prioritize different goals, such as maximizing type precision, maintaining compatibility with unmigrated code, and preserving the semantics of the original program. We argue that the type migration problem involves fundamental compromises: optimizing for a single goal often comes at the expense of others. Ideally, a type migration tool would flexibly accommodate a range of user priorities. We present TypeWhich, a new approach to automated type migration for the gradually-typed lambda calculus with some extensions. Unlike prior work, which relies on custom solvers, TypeWhich produces constraints for an off-the-shelf MaxSMT solver. This allows us to easily express objectives, such as minimizing the number of necessary syntactic coercions, and constraining the type of the migration to be compatible with unmigrated code. We present the first comprehensive evaluation of GTLC type migration algorithms, and compare TypeWhich to four other tools from the literature. Our evaluation uses prior benchmarks, and a new set of ``challenge problems. Moreover, we design a new evaluation methodology that highlights the subtleties of gradual type migration. In addition, we apply TypeWhich to a suite of benchmarks for Grift, a programming language based on the GTLC. TypeWhich is able to reconstruct all human-written annotations on all but one program.
We examine the relationship between the algebraic lambda-calculus, a fragment of the differential lambda-calculus and the linear-algebraic lambda-calculus, a candidate lambda-calculus for quantum computation. Both calculi are algebraic: each one is equipped with an additive and a scalar-multiplicative structure, and their set of terms is closed under linear combinations. However, the two languages were built using different approaches: the former is a call-by-name language whereas the latter is call-by-value; the former considers algebraic equalities whereas the latter approaches them through rewrite rules. In this paper, we analyse how these different approaches relate to one another. To this end, we propose four canonical languages based on each of the possible choices: call-by-name versus call-by-value, algebraic equality versus algebraic rewriting. We show that the various languages simulate one another. Due to subtle interaction between beta-reduction and algebraic rewriting, to make the languages consistent some additional hypotheses such as confluence or normalisation might be required. We carefully devise the required properties for each proof, making them general enough to be valid for any sub-language satisfying the corresponding properties.
In each variant of the lambda-calculus, factorization and normalization are two key-properties that show how results are computed. Instead of proving factorization/normalization for the call-by-name (CbN) and call-by-value (CbV) variants separately, we prove them only once, for the bang calculus (an extension of the lambda-calculus inspired by linear logic and subsuming CbN and CbV), and then we transfer the result via translations, obtaining factorization/normalization for CbN and CbV. The approach is robust: it still holds when extending the calculi with operators and extra rules to model some additional computational features.
Garcia and Cimini study a type inference problem for the ITGL, an implicitly and gradually typed language with let-polymorphism, and develop a sound and complete inference algorithm for it. Soundness and completeness mean that, if the algorithm succeeds, the input term can be translated to a well-typed term of an explicitly typed blame calculus by cast insertion and vice versa. However, in general, there are many possible translations depending on how type variables that were left undecided by static type inference are instantiated with concrete static types. Worse, the translated terms may behave differently---some evaluate to values but others raise blame. In this paper, we propose and formalize a new blame calculus $lambda^{textsf{DTI}}_{textsf{B}}$ that avoids such divergence as an intermediate language for the ITGL. A main idea is to allow a term to contain type variables (that have not been instantiated during static type inference) and defer instantiation of these type variables to run time. We introduce dynamic type inference (DTI) into the semantics of $lambda^{textsf{DTI}}_{textsf{B}}$ so that type variables are instantiated along reduction. The DTI-based semantics not only avoids the divergence described above but also is sound and complete with respect to the semantics of fully instantiated terms in the following sense: if the evaluation of a term succeeds (i.e., terminates with a value) in the DTI-based semantics, then there is a fully instantiated version of the term that also succeeds in the explicitly typed blame calculus and vice versa. Finally, we prove the gradual guarantee, which is an important correctness criterion of a gradually typed language, for the ITGL.
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