Co-Contextual Type Systems: Contextless Deductive Reasoning for Correct Incremental Type Checking
This thesis proposes a novel way of performing type checking, whose results are incremental, depending on the provided local information. This new way of type checking is called co-contextual, where all context information of expressions, methods, classes, etc., is removed. Instead, we introduce corresponding structures using requirements. Standard type systems are translated to the co-contextual ones systematically using dualism as technique. Type systems play an important role to prevent execution errors from occurring during runtime. They are used to check programs statically for potential errors. Programs are type checked against a given set of rules. Depending on these rules programs are well-typed or not. The set of these rules is called typing rules. Each type rule associates types to the constructs of a program given a certain context. There can be different forms of contexts, depending on the features of the typed programming language. Functional languages use a typing context of variables and their types; object-oriented (OO) languages use additional class tables. Class tables are used for example to ensure that method and class declarations are well-typed. Type checking is performed top-down. While traversing the syntax tree of a program, typing contexts are extended with information on the expressions and their types. In case of OO, class tables are extended with clauses from class declarations, including the class members, i.e., fields, methods, or constructors. Contexts are passed through the nodes of the syntax tree in order to coordinate type checking between them. Therefore, while traversing the syntax tree top-down, the type checker creates dependencies between otherwise independent subexpressions. This way, it inhibits incrementalization and parallelization of type checking. That is, a change to a node of the syntax tree would require to redo the type check of the whole syntax tree. In this thesis a novel formulation of type systems is proposed, in order to remove dependencies between subexpressions. We propose a co-contextual formulation of typing rules that depends only on the local program constructs, e.g., expressions, methods, classes. The co-contextual typing rules have as conclusion a type and sets of requirements. That is, contexts and class tables are replaced by the dual concept of context and class table requirements. In addition, operations on contexts and class tables are replaced by new dual operations on requirements. The co-contextual type checker traverses a syntax tree bottom-up and merges context requirements of independently checked subexpressions. We describe a method for systematically constructing a co-contextual formulation of type rules from a regular context-based formulation and we show how co-contextual type rules give rise to incremental type checking. We derive co-contextual type checkers for functional and OO languages. As a representative of functional languages we consider PCF and extensions of it: records, parametric polymorphism, structural subtyping and let-polymorphism. Also, we investigate featherweight java (FJ) as the basis of OO languages and extensions of it: method overloading and generics. We build a1 co-contextual type checker for FJ enabling key features of OO languages: subtype polymorphism, nominal typing and implementation inheritance. The dualism between the co-contextual and contextual type systems preserves the correctness of the contextual calculus. That is, we prove the correctness of the co-contextual calculus via the equivalence between contextual type rules and their co-contextual formulations. We implemented an incremental type checker for PCF along with a performance evaluation showing that co-contextual type checking has performance comparable to standard context-based type checking, and incrementalization can improve performance significantly. Regarding FJ, we implemented a co-contextual type checker with incrementalization and compared its performance against javac on a number of realistic programs. Our performance evaluation shows significant speedups for the co-contextual type checker with incrementalization in comparison to javac.
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