Meta-programming with names and necessity

Mathematical Structures in Computer Science, 1993

The study of type theory may offer a uniform language for modular programming, structured specification and logical reasoning. We develop an approach to program specification and data refinement in a type theory with a strong logical power and nice structural mechanisms to show that it provides an adequate formalism for modular development of programs and specifications. Specification of abstract data types is considered, and a notion of abstract implementation between specifications is defined in the type theory and studied as a basis for correct and modular development of programs by stepwise refinement. The higher-order structural mechanisms in the type theory provide useful and flexible tools (specification operations and parameterized specifications) for modular design and structured specification. Refinement maps (programs and design decisions) and proofs of implementation correctness can be developed by means of the existing proof development systems based on type theories.

Proceedings of the 2003 workshop on Mechanized reasoning about languages with variable binding - MERLIN '03, 2003

We report on work in progress regarding a foundation for the notion of meta-variable in logical frameworks and type theories. Our proposal is to treat meta-variables as modal variables in a modal type theory, which is logically clean and justifies several low-level implementation techniques for meta-variables. We also speculate on other logical extensions of our modal type theory, at present without clear applications.

2008

In this paper, we show how Miquel’s Implicit Calculus of Constructions (ICC) can be used as a programming language featuring dependent types. Since this system has an undecidable type-checking, we introduce a more verbose variant, called ICC* which fixes this issue. Datatypes and program specifications are enriched with logical assertions (such as preconditions, postconditions, invariants) and programs are decorated with proofs of those assertions. The point of using ICC* rather than the Calculus of Constructions (the core formalism of the Coq proof assistant) is that all of the static information (types and proof objects) is transparent, in the sense that it does not affect the computational behavior. This is concretized by a built-in extraction procedure that removes this static information. We also illustrate the main features of ICC* on classical examples of dependently typed programs.

2008

We show how programming language semantics and definitions of their corresponding type systems can both be written in a single framework amenable to proofs of soundness. The framework is based on full rewriting logic (not to be confused with context reduction or term rewriting), where rules can match anywhere in a term (or configuration).

Meseguer and Rosu [MR04,MR07] proposed rewriting logic semantics (RLS) as a programing language definitional framework that unifies operational and algebraic denotational semantics. Once a language is defined as an RLS theory, many generic tools are immediately available for use with no additional cost to the designer. These include a formal inductive theorem proving environment, an efficient interpreter, a state space explorer, and even a model checker. RLS has already been used to define a series of didactic and real languages [MR04, MR07], but its benefits in connection with defining and reasoning about type systems have not been fully investigated yet. This paper shows how the same RLS style employed for giving formal definitions of languages can be used to define type systems. The same term-rewriting mechanism used to execute RLS language definitions can now be used to execute type systems, giving type checkers or type inferencers. Since both the language and its type system ar...

Lecture Notes in Computer Science, 2009

Rewriting logic semantics (RLS) was proposed as a programing language definitional framework that unifies operational and algebraic denotational semantics; see and the references there. Once a language is defined as an RLS theory, many generic tools are immediately available for use with no additional cost to the designer. These include a formal inductive theorem proving environment, an efficient interpreter, a state space explorer, and even a model checker. RLS has already been used to define a series of didactic and real languages .

Proceedings of the 12th international ACM SIGPLAN symposium on Principles and practice of declarative programming - PPDP '10, 2010

Dependently typed λ-calculi such as the Logical Framework (LF) can encode relationships between terms in types and can naturally capture correspondences between formulas and their proofs. Such calculi can also be given a logic programming interpretation: the Twelf system is based on such an interpretation of LF. We consider here whether a conventional logic programming language can provide the benefits of a Twelf-like system for encoding type and proof-and-formula dependencies. In particular, we present a simple mapping from LF specifications to a set of formulas in the higherorder hereditary Harrop (hohh) language, that relates derivations and proof-search between the two frameworks. We then show that this encoding can be improved by exploiting knowledge of the wellformedness of the original LF specifications to elide much redundant type-checking information. The resulting logic program has a structure that closely resembles the original specification, thereby allowing LF specifications to be viewed as hohh meta-programs. Using the Teyjus implementation of λProlog, we show that our translation provides an efficient means for executing LF specifications, complementing the ability that the Twelf system provides for reasoning about them.

2000

Writing (meta-)programs that manipulate other (object-) programs poses significant technical problems when the objectlanguage itself has a notion of binders and variable occurrences. Higher-order abstract syntax is a representation of object programs that has recently been the focus of several studies. This paper points out a number of limitations of using higher order syntax in a functional context, and argues that DALI, a language based on a simple and elegant proposal made by Dale Miller ten years ago can provide superior support for manipulating such object-languages. Miller's original proposal, however, did not provide any formal treatment. To fill this gap, we present both a big-step and a reduction semantics for DALI, and summarize the results of our extensive study of the semantics, including the rather involved proof of the soundness of the reduction semantics with respect to the big-step semantics. Because our formal development is carried out for the untyped version of the language, we hope it will serve as a solid basis for investigating type system(s) for DALI.

1996

This paper addresses the issue of giving a formal semantics to an object-oriented programming and specification language. Object-oriented constructs considered are objects with attributes and methods, encapsulation of attributes, subtyping, bounded type parameters, classes, and inheritance. Classes are distinguished from object types. Besides usual imperative statements, specification statements are included. Specification statements allow changes of variables to be described by a predicate. They are abstract in the sense that they are non-executable. Specification statements may appear in method bodies of classes, leading to abstract classes. The motivation for this approach is that abstract classes can be used for problem-oriented specification in early stages and later refined to efficient implementations. Various refinement calculi provide laws for procedural and data refinement, which can be used here for class refinement. This paper, however, focuses on the semantics of object-oriented programs and specifications and gives some examples of abstract and concrete classes. The semantics is given by a translation of the constructs into the type system F≤, an extension of the simple typed λ-calculus by subtyping and parametric polymorphism: The state of a program is represented by a record. A state predicate is a Boolean valued function from states. Statements, both abstract and concrete, are represented by predicate transformers, i. e. higher order functions mapping state predicates (postconditions) to state predicates (preconditions). Objects are represented by records of statements (the methods) operating on a record of attributes, where the attributes are hidden by existential quantification. Classes are understood as templates for the creation of objects. Classes are represented by records. Inheritance amounts to record overwriting. Subtyping and parametric polymorphism, e. g. for the parameterization of classes by types, are already present in F≤. The advantage of this semantic by translation is that it builds on the features already provided by F≤ (which are all used). Hence no direct reference to the model underlying F≤ needs to be made; a summary of the syntax and rules of F≤ is given.

Journal of Applied Logic, 2013

The modal logic S4 can be used via a Curry-Howard style correspondence to obtain a λcalculus. Modal (boxed) types are intuitively interpreted as 'closed syntax of the calculus'. This λ-calculus is called modal type theory -this is the basic case of a more general contextual modal type theory, or CMTT.