Type theory

Information Processing Letters, 2012

We describe a derivational approach to proving the equivalence of different representations of a type system. Different ways of representing type assignments are convenient for particular applications such as reasoning or implementation, but some kind of correspondence between them should be proven. In this paper we address two such semantics for type checking: one, due to Kuan et al., in the form of a term rewriting system and the other in the form of a traditional set of derivation rules. By employing a set of techniques investigated by Danvy et al., we mechanically derive the correspondence between a reductionbased semantics for type-checking and a traditional one in the form of derivation rules, implemented as a recursive descent. The correspondence is established through a series of semantics-preserving functional program transformations.

ACM SIGPLAN Notices, 2003

We develop an explicit two level system that allows programmers to reason about the behavior of effectful programs. The first level is an ordinary ML-style type system, which confers standard properties on program behavior. The second level is a conservative extension of the first that uses a logic of type refinements to check more precise properties of program behavior. Our logic is a fragment of intuitionistic linear logic, which gives programmers the ability to reason locally about changes of program state. We provide a generic resource semantics for our logic as well as a sound, decidable, syntactic refinement-checking system. We also prove that refinements give rise to an optimization principle for programs. Finally, we illustrate the power of our system through a number of examples.

1998

While visiting the Computer Science Department of the University of Torino in November/December 1997. vi The evaluation rules for PCF are a subset of the ones for PCFP (in Figure 2.2), since the rules APP 2 and PROJ i (i ∈ {1, 2}) are not needed. All the results in the previous sections apply to PCF as well. 2.6 PCFP with primitive recursion In this section we introduce an extension of PCFP, called PCFP T , which is more suitable for expressing programs extracted from formal proofs. The language PCFP T is obtained from PCFP by adding a program constructor for primitive recursion over natural numbers (rec). There are also two constructors for specifying simplified uses of primitive recursion: the constructor it (for iteration) and the constructor case. PCFP T , which is a variant of Gödel's system T , is the language considered in Part III of this thesis. The term formation rules for the new constructors are in Fig. 2.4, and the evaluation rules are in Fig. 2.5. Since for every PCFP T term there is an equivalent (w.r.t. the interpretation in all the closed term model described in Section 2.4) PCFP term (see Fact 2.10 below), we have that all the results for PCFP presented in the previous sections apply to PCFP T as well.

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 .

ACM SIGPLAN Notices, 2017

While type soundness proofs are taught in every graduate PL class, the gap between realistic languages and what is accessible to formal proofs is large. In the case of Scala, it has been shown that its formal model, the Dependent Object Types (DOT) calculus, cannot simultaneously support key metatheoretic properties such as environment narrowing and subtyping transitivity, which are usually required for a type soundness proof. Moreover, Scala and many other realistic languages lack a general substitution property. The first contribution of this paper is to demonstrate how type soundness proofs for advanced, polymorphic, type systems can be carried out with an operational semantics based on high-level, definitional interpreters, implemented in Coq. We present the first mechanized soundness proofs in this style for System F and several extensions, including mutable references. Our proofs use only straightforward induction, which is significant, as the combination of big-step semantics...

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.

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...

Anales de la 25 …, 1996

We propose a series of work areas related to type theory and functional programming. By type theory we mean the formulation of Martin-L of's set theory using the theory of types as logical framework, extended with record types and subtyping. The areas presented are: the ...

2007

The decidability of equality is proved for Martin-Löf type theory with a universeá la Russell and typed betaeta-equality judgements. A corollary of this result is that the constructor for dependent function types is injective, a property which is crucial for establishing the correctness of the type-checking algorithm. The decision procedure uses normalization by evaluation, an algorithm which first interprets terms in a domain with untyped semantic elements and then extracts normal forms. The correctness of this algorithm is established using a PER-model and a logical relation between syntax and semantics. * Research partially supported by the EU coordination action TYPES (510996). † Research partially supported by VR-project Typed Lambda Calculus and Applications. sional type theories such as Coq, Agda, and Epigram, all rely on the decision algorithm for a : A using normalization: the user attempts to prove the theorem A by building a construction a, and the system checks that a is indeed a proof of A.

1994

This paper gives an introduction to type theory, focusing on its recent use as a logical framework for proofs and programs. The rst two sections give a background to type theory intended for the reader who is new to the subject. The following presents Martin-L of's monomorphic type theory and an implementation, ALF, of this theory. Finally, a few small tutorial examples in ALF are given. 3 This work has been done within the ESPRIT Basic Research Action \Types for proofs and programs." It has been funded by NUTEK and Chalmers. An earlier version was published in the EATCS bulletin no 52, February 1994.