Many object-oriented dynamic languages allow programmers to _extract methods_ from objects and treat them as functions. This allows for flexible programming patterns, but presents challenges for type systems. In particular, a simple treatment of method extraction would require methods to be contravariant in the receiver type, making overriding all-but-impossible. We present a detailed investigation of this problem, as well as an implemented and evaluated solution. Method extraction is a feature of many dynamically-typed and gradually-typed languages, ranging from Python and PHP to Flow and TypeScript. In these languages, the underlying representation of objects as records of procedures can be accessed, and the procedures that implement methods can be reified as functions that can be called independently. In many of these languages, the programmer can then explicitly specify the `this` value to be used when the method implementation is called. Unfortunately, as we show, existing gradual type systems such as TypeScript and Flow are unsound in the presence of method extraction. The problem for typing any such system is that the flexibility it allows must be tamed by requiring a connection between the object the method was extracted from, and the function value that is later called. In Racket, where a method extraction-like facility, dubbed "structure type properties", is fundamental to classes, generic methods, and other APIs, these same challenges arise, and must be solved to support this feature in Typed Racket. We show how to combine two existing type system features -- existential types and occurrence typing -- to produce a sound approach to typing method extraction. We formalize our design, extending an existing formal model of the Typed Racket type system, and prove that our extension is sound. Our design is also implemented in the released version of Racket, and is compatible with all existing Typed Racket packages, many of which already used a previous version of this feature.
In this paper we describe how to leverage higher-order unification to type check a dependently typed language with meta-variables. The literature usually presents the unification algorithm as a standalone component, however the need to check definitional equality of terms while type checking gives rise to a tight interplay between type checking and unification. This interplay is a major source of complexity in the type-checking algorithm for existing dependently typed programming languages. We propose an algorithm that encodes a type-checking problem entirely in the form of unification constra…
It is a strength of graph-based data formats, like RDF, that they are very flexible with representing data. To avoid run-time errors, program code that processes highly-flexible data representations exhibits the difficulty that it must always include the most general case, in which attributes might be set-valued or possibly not available. The Shapes Constraint Language (SHACL) has been devised to enforce constraints on otherwise random data structures. We present our approach, Type checking using SHACL (TyCuS), for type checking code that queries RDF data graphs validated by a SHACL shape grap…
Relational type systems have been designed for several applications including information flow, differential privacy, and cost analysis. In order to achieve the best results, these systems often use relational refinements and relational effects to maximally exploit the similarity in the structure of the two programs being compared. Relational type systems are appealing for relational properties because they deliver simpler and more precise verification than what could be derived from typing the two programs separately. However, relational type systems do not yet achieve the practical appeal of…
Harnessing the power of dependently typed languages can be difficult. Programmers must manually construct proofs to produce well-typed programs, which is not an easy task. In particular, migrating code to these languages is challenging. Gradual typing can make dependently-typed languages easier to use by mixing static and dynamic checking in a principled way. With gradual types, programmers can incrementally migrate code to a dependently typed language. However, adding gradual types to dependent types creates a new challenge: mixing decidable type-checking and incremental migration in a full-f…
A type system combining type application, constants as types, union types (associative, commutative and idempotent) and recursive types has recently been proposed for statically typing path polymorphism, the ability to define functions that can operate uniformly over recursively specified applicative data structures. A typical pattern such functions resort to is $x\,y$ which decomposes a compound, in other words any applicative tree structure, into its parts. We study type-checking for this type system in two stages. First we propose algorithms for checking type equivalence and subtyping based…
We study a type checking algorithm that is able to type check a nontrivial subclass of functional programs that use features such as higher-rank, impredicative and second-order types. The only place the algorithm requires type annotation is before each function declaration. We prove the soundness of the type checking algorithm with respect to System $\mathbf{F}_ω$, i.e. if the program is type checked, then the type checker will produce a well-typed annotated System $\mathbf{F}_ω$ term. We extend the basic algorithm to handle pattern matching and let-bindings. We implement a prototype type chec…
Recent constraint logic programming (CLP) languages, such as HAL and Mercury, require type, mode and determinism declarations for predicates. This information allows the generation of efficient target code and the detection of many errors at compile-time. Unfortunately, mode checking in such languages is difficult. One of the main reasons is that, for each predicate mode declaration, the compiler is required to appropriately re-order literals in the predicate's definition. The task is further complicated by the need to handle complex instantiations (which interact with type declarations and hi…
This dissertation introduces executable refinement types, which refine structural types by semi-decidable predicates, and establishes their metatheory and accompanying implementation techniques. These results are useful for undecidable type systems in general. Particular contributions include: (1) Type soundness and a logical relation for extensional equivalence for executable refinement types (though type checking is undecidable); (2) hybrid type checking for executable refinement types, which blends static and dynamic checks in a novel way, in some sense performing better statically than any…
We present a type checking algorithm for establishing a session-based discipline in the pi calculus of Milner, Parrow and Walker. Our session types are qualified as linear or unrestricted. Linearly typed communication channels are guaranteed to occur in exactly one thread, possibly multiple times; afterwards they evolve as unrestricted channels. Session protocols are described by a type constructor that denotes the two ends of one and the same communication channel. We ensure the soundness of the algorithm by showing that processes consuming all linear resources are accepted by a type system p…
XPath is a language for addressing parts of an XML document. We give an abstract interpretation of XPath expressions in terms of relations on document node types. Node-set-related XPath language constructs are mapped straightforwardly onto basic, well-understood and easily computable relational operations. Hence our interpretation gives both extremely concise type-level denotational semantics and a practical analysis tool for the node-set fragment of the XPath 1.0 language. This method is part of the TPath implementation of XPath. XSL-T is a pure functional language for transforming XML docume…
In this paper we present the design and implementation of Flow, a fast and precise type checker for JavaScript that is used by thousands of developers on millions of lines of code at Facebook every day. Flow uses sophisticated type inference to understand common JavaScript idioms precisely. This helps it find non-trivial bugs in code and provide code intelligence to editors without requiring significant rewriting or annotations from the developer. We formalize an important fragment of Flow's analysis and prove its soundness. Furthermore, Flow uses aggressive parallelization and incrementalizat…
Dynamic languages such as Ruby, Python, and JavaScript have many compelling benefits, but the lack of static types means subtle errors can remain latent in code for a long time. While many researchers have developed various systems to bring some of the benefits of static types to dynamic languages, prior approaches have trouble dealing with metaprogramming, which generates code as the program executes. In this paper, we propose Hummingbird, a new system that uses a novel technique, just-in-time static type checking, to type check Ruby code even in the presence of metaprogramming. In Hummingbir…
We present a manifest contract system PCFv$Δ$H with intersection types. A manifest contract system is a typed functional calculus in which software contracts are integrated into a refinement type system and consistency of contracts is checked by combination of compile- and run-time type checking. Intersection types naturally arise when a contract is expressed by a conjunction of smaller contracts. Run-time contract checking for conjunctive higher-order contracts in an untyped language has been studied but our typed setting poses an additional challenge due to the fact that an expression of an …
Higher-order modal fixpoint logic (HFL) is a higher-order extension of the modal mu-calculus, and strictly more expressive than the modal mu-calculus. It has recently been shown that various program verification problems can naturally be reduced to HFL model checking: the problem of whether a given finite state system satisfies a given HFL formula. In this paper, we propose a novel algorithm for HFL model checking: it is the first practical algorithm in that it runs fast for typical inputs, despite the hyper-exponential worst-case complexity of the HFL model checking problem. Our algorithm is …
Two natural decision problems regarding the XML query language XQuery are well-definedness and semantic type-checking. We study these problems in the setting of a relational fragment of XQuery. We show that well-definedness and semantic type-checking are undecidable, even in the positive-existential case. Nevertheless, for a ``pure'' variant of XQuery, in which no identification is made between an item and the singleton containing that item, the problems become decidable. We also consider the analogous problems in the setting of the nested relational calculus.
Concurrent and parallel programming is difficult due to the presence of memory side-effects, which may introduce data races. Type qualifiers, such as reference capabilities, can remove data races by restricting sharing of mutable data. Unfortunately, reference capability languages are an all-in or nothing game, i.e., all the types must be annotated with reference capabilities. In this work in progress, we propose to mix the ideas from the reference capability literature with gradual typing, leading to gradual reference capabilities.
Tensor shape mismatch is a common source of bugs in deep learning programs. We propose a new type-based approach to detect tensor shape mismatches. One of the main features of our approach is the best-effort shape inference. As the tensor shape inference problem is undecidable in general, we allow static type/shape inference to be performed only in a best-effort manner. If the static inference cannot guarantee the absence of the shape inconsistencies, dynamic checks are inserted into the program. Another main feature is gradual typing, where users can improve the precision of the inference by …
Graphs are a generalized concept that encompasses more complex data structures than trees, such as difference lists, doubly-linked lists, skip lists, and leaf-linked trees. Normally, these structures are handled with destructive assignments to heaps, which is opposed to a purely functional programming style and makes verification difficult. We propose a new purely functional language, $λ_{GT}$, that handles graphs as immutable, first-class data structures with a pattern matching mechanism based on Graph Transformation and developed a new type system, $F_{GT}$, for the language. Our approach is…
One form of type checking used in gradually typed language is transient type checking: whenever an object 'flows' through code with a type annotation, the object is dynamically checked to ensure it has the methods required by the annotation. Just-in-time compilation and optimisation in virtual machines can eliminate much of the overhead of run-time transient type checks. Unfortunately this optimisation is not uniform: some type checks will significantly decrease, or even increase, a program's performance. In this paper, we refine the so called "Takikawa" protocol, and use it to identify which …
Linear algebra is a major field of numerical computation and is widely applied. Most linear algebra libraries (in most programming languages) do not statically guarantee consistency of the dimensions of vectors and matrices, causing runtime errors. While advanced type systems--specifically, dependent types on natural numbers--can ensure consistency among the sizes of collections such as lists and arrays, such type systems generally require non-trivial changes to existing languages and application programs, or tricky type-level programming. We have developed a linear algebra library interface t…
In this paper, we propose a way of assigning static type information to unmarshalling functions and we describe a verification technique for unmarshalled data that preserves the execution safety provided by static type checking. This technique, whose correctness is proven, relies on singleton types whose values are transmitted to unmarshalling routines at runtime, and on an efficient checking algorithm able to deal with sharing and cycles.
We consider type inference in the Hindley/Milner system extended with type annotations and constraints with a particular focus on Haskell-style type classes. We observe that standard inference algorithms are incomplete in the presence of nested type annotations. To improve the situation we introduce a novel inference scheme for checking type annotations. Our inference scheme is also incomplete in general but improves over existing implementations as found e.g. in the Glasgow Haskell Compiler (GHC). For certain cases (e.g. Haskell 98) our inference scheme is complete. Our approach has been full…
In this paper we use pre existing language support for type modifiers and object capabilities to enable a system for sound runtime verification of invariants. Our system guarantees that class invariants hold for all objects involved in execution. Invariants are specified simply as methods whose execution is statically guaranteed to be deterministic and not access any externally mutable state. We automatically call such invariant methods only when objects are created or the state they refer to may have been mutated. Our design restricts the range of expressible invariants but improves upon the …
Since Val Tannen's pioneer work on the combination of simply-typed lambda-calculus and first-order rewriting (LICS'88), many authors have contributed to this subject by extending it to richer typed lambda-calculi and rewriting paradigms, culminating in calculi like the Calculus of Algebraic Constructions. These works provide theoretical foundations for type-theoretic proof assistants where functions and predicates are defined by oriented higher-order equations. This kind of definitions subsumes induction-based definitions, is easier to write and provides more automation. On the other hand, che…
We introduce regular graph constraints and explore their decidability properties. The motivation for regular graph constraints is 1) type checking of changing types of objects in the presence of linked data structures, 2) shape analysis techniques, and 3) generalization of similar constraints over trees and grids. We define a subclass of graphs called heaps as an abstraction of the data structures that a program constructs during its execution. We prove that determining the validity of implication for regular graph constraints over the class of heaps is undecidable. We show undecidability by e…
The Web of Linked Data is the cumulation of over a decade of work by the Web standards community in their effort to make data more Web-like. We provide an introduction to the Web of Linked Data from the perspective of a Web developer that would like to build an application using Linked Data. We identify a weakness in the development stack as being a lack of domain specific scripting languages for designing background processes that consume Linked Data. To address this weakness, we design a scripting language with a simple but appropriate type system. In our proposed architecture some data is c…
JSON is a popular data format used pervasively in web APIs, cloud computing, NoSQL databases, and increasingly also machine learning. JSON Schema is a language for declaring the structure of valid JSON data. There are validators that can decide whether a JSON document is valid with respect to a schema. Unfortunately, like all instance-based testing, these validators can only show the presence and never the absence of a bug. This paper presents a complementary technique: JSON subschema checking, which can be used for static type checking with JSON Schema. Deciding whether one schema is a subsch…
In this paper, we describe our experience incorporating gradual types in a statically typed functional language with Hindley-Milner style type inference. Where most gradually typed systems aim to improve static checking in a dynamically typed language, we approach it from the opposite perspective and promote dynamic checking in a statically typed language. Our approach provides a glimpse into how languages like SML and OCaml might handle gradual typing. We discuss our implementation and challenges faced -- specifically how gradual typing rules apply to our representation of composite and recur…
This paper presents a new type analysis for logic programs. The analysis is performed with a priori type definitions; and type expressions are formed from a fixed alphabet of type constructors. Non-discriminative union is used to join type information from different sources without loss of precision. An operation that is performed repeatedly during an analysis is to detect if a fixpoint has been reached. This is reduced to checking the emptiness of types. Due to the use of non-discriminative union, the fundamental problem of checking the emptiness of types is more complex in the proposed type …
Consumption of REST services has become a popular means of invoking code provided by third parties, particularly in web applications. Nowadays programmers of web applications can choose TypeScript over JavaScript to benefit from static type checking that enables validating calls to local functions or to those provided by libraries. Errors in calls to REST services, however, can only be found at run-time. In this paper, we present SafeRESTScript (SRS, for short) a language that extends the support of static analysis to calls to REST services, with the ability to statically find common errors su…
Semantic subtyping enables simple, set-theoretical reasoning about types by interpreting a type as the set of its values. Previously, semantic subtyping has been studied primarily in the context of statically typed languages with structural typing. In this paper, we explore the applicability of semantic subtyping in the context of a dynamic language with nominal types. Instead of static type checking, dynamic languages rely on run-time checking of type tags associated with values, so we propose using the tags for semantic subtyping. We base our work on a fragment of the Julia language and pres…
To warrant programmer productivity, type checker results should be correct and available quickly. Correctness can be provided when a type checker implementation corresponds to a declarative type system specification. Statix is a type system specification language which achieves this by automatically deriving type checker implementations from declarative typing rules. A key feature of Statix is that it uses scope graphs for declarative specification of name resolution. However, compared to hand-written type checkers, type checkers derived from Statix specifications have sub-optimal run time per…
A long-standing shortcoming of statically typed functional languages is that type checking does not rule out pattern-matching failures (run-time match exceptions). Refinement types distinguish different values of datatypes; if a program annotated with refinements passes type checking, pattern-matching failures become impossible. Unfortunately, refinement is a monolithic property of a type, exacerbating the difficulty of adding refinement types to nontrivial programs. Gradual typing has explored how to incrementally move between static typing and dynamic typing. We develop a type system of grad…
The Edinburgh Logical Framework (LF) is a dependently type lambda calculus that can be used to encode formal systems. The versatility of LF allows specifications to be constructed also about the encoded systems. The Twelf system exploits the correspondence between formulas and types to give specifications in LF a logic programming interpretation. By interpreting particular arguments as input and others as output, specifications can be seen as describing non-deterministic functions. If particular such functions can be shown to be total, they represent constructive proofs of meta-theorems of the…
This paper describes a tool suite for the ACL2 programming language which incorporates certain ideas from the Hindley-Milner paradigm of functional programming (as exemplified in popular languages like ML and Haskell), including a "typed" style of programming with the ability to define polymorphic types. These ideas are introduced via macros into the language of ACL2, taking advantage of ACL2's guard-checking mechanism to perform type checking on both function definitions and theorems. Finally, we discuss how these macros were used to implement features of Specware, a software specification an…
Parser generators generate translators from language specifications. In many cases, such specifications contain semantic actions written in the same language as the generated code. Since these actions are subject to little static checking, they are usually a source of errors which are discovered only when generated code is compiled. In this paper we propose a parser generator front-end which statically checks semantic actions for typing errors and prevents such errors from appearing in generated code. The type checking procedure is extensible to support many implementation languages. An extens…
Bidirectional typing combines two modes of typing: type checking, which checks that a program satisfies a known type, and type synthesis, which determines a type from the program. Using checking enables bidirectional typing to support features for which inference is undecidable; using synthesis enables bidirectional typing to avoid the large annotation burden of explicitly typed languages. In addition, bidirectional typing improves error locality. We highlight the design principles that underlie bidirectional type systems, survey the development of bidirectional typing from the prehistoric per…
Sized types are a modular and theoretically well-understood tool for checking termination of recursive and productivity of corecursive definitions. The essential idea is to track structural descent and guardedness in the type system to make termination checking robust and suitable for strong abstractions like higher-order functions and polymorphism. To study the application of sized types to proof assistants and programming languages based on dependent type theory, we have implemented a core language, MiniAgda, with explicit handling of sizes. New considerations were necessary to soundly integ…
We present a principled theoretical framework for inferring and checking the union types, and show its work in practice on JSON data structures. The framework poses a union type inference as a learning problem from multiple examples. The categorical framework is generic and easily extensible.
Situation calculus has been widely applied in Artificial Intelligence related fields. This formalism is considered as a dialect of logic programming language and mostly used in dynamic domain modeling. However, type systems are hardly deployed in situation calculus in the literature. To achieve a correct and sound typed program written in situation calculus, adding typing elements into the current situation calculus will be quite helpful. In this paper, we propose to add more typing mechanisms to the current version of situation calculus, especially for three basic elements in situation calcul…
