Semantic Domains and Denotational Semantics

The theory of domains was established in order to have appropriate spaces on which to define semantic functions for the denotational approach to programming-language semantics. There were two needs: first, there had to be spaces of several different types available to mirror both the type distinctions in the languages and also to allow for different kinds of semantical constructs - especially in dealing with languages with side effects; and second, the theory had to account for computability properties of functions - if the theory was going to be realistic. The first need is complicated by the fact that types can be both compound (or made up from other types) and recursive (or self-referential), and that a high-level language of types and a suitable semantics of types is required to explain what is going on. The second need is complicated by these complications of the semantical definitions and the fact that it has to be checked that the level of abstraction reached still allows a precise definition of computability

A computable expression of closure to efficient causation

International audienceIn this paper, we propose a mathematical expression of closure to efficient causation in terms of lambda-calculus; we argue that this opens up the perspective of developing principled computer simulations of systems closed to efficient causation in an appropriate programming language. An important implication of our formulation is that, by exhibiting an expression in lambda-calculus, which is a paradigmatic formalism for computability and programming, we show that there are no conceptual or principled problems in realizing a computer simulation or model of closure to efficient causation. We conclude with a brief discussion of the question whether closure to efficient causation captures all relevant properties of living systems. We suggest that it might not be the case, and that more complex definitions could indeed create crucial some obstacles to computability

Semantics of Higher-Order Recursion Schemes

Higher-order recursion schemes are recursive equations defining new operations from given ones called "terminals". Every such recursion scheme is proved to have a least interpreted semantics in every Scott's model of \lambda-calculus in which the terminals are interpreted as continuous operations. For the uninterpreted semantics based on infinite \lambda-terms we follow the idea of Fiore, Plotkin and Turi and work in the category of sets in context, which are presheaves on the category of finite sets. Fiore et al showed how to capture the type of variable binding in \lambda-calculus by an endofunctor H\lambda and they explained simultaneous substitution of \lambda-terms by proving that the presheaf of \lambda-terms is an initial H\lambda-monoid. Here we work with the presheaf of rational infinite \lambda-terms and prove that this is an initial iterative H\lambda-monoid. We conclude that every guarded higher-order recursion scheme has a unique uninterpreted solution in this monoid

High-level signatures and initial semantics

We present a device for specifying and reasoning about syntax for datatypes, programming languages, and logic calculi. More precisely, we study a notion of signature for specifying syntactic constructions. In the spirit of Initial Semantics, we define the syntax generated by a signature to be the initial object---if it exists---in a suitable category of models. In our framework, the existence of an associated syntax to a signature is not automatically guaranteed. We identify, via the notion of presentation of a signature, a large class of signatures that do generate a syntax. Our (presentable) signatures subsume classical algebraic signatures (i.e., signatures for languages with variable binding, such as the pure lambda calculus) and extend them to include several other significant examples of syntactic constructions. One key feature of our notions of signature, syntax, and presentation is that they are highly compositional, in the sense that complex examples can be obtained by assembling simpler ones. Moreover, through the Initial Semantics approach, our framework provides, beyond the desired algebra of terms, a well-behaved substitution and the induction and recursion principles associated to the syntax. This paper builds upon ideas from a previous attempt by Hirschowitz-Maggesi, which, in turn, was directly inspired by some earlier work of Ghani-Uustalu-Hamana and Matthes-Uustalu. The main results presented in the paper are computer-checked within the UniMath system.Comment: v2: extended version of the article as published in CSL 2018 (http://dx.doi.org/10.4230/LIPIcs.CSL.2018.4); list of changes given in Section 1.5 of the paper; v3: small corrections throughout the paper, no major change

On Berry's conjectures about the stable order in PCF

PCF is a sequential simply typed lambda calculus language. There is a unique order-extensional fully abstract cpo model of PCF, built up from equivalence classes of terms. In 1979, G\'erard Berry defined the stable order in this model and proved that the extensional and the stable order together form a bicpo. He made the following two conjectures: 1) "Extensional and stable order form not only a bicpo, but a bidomain." We refute this conjecture by showing that the stable order is not bounded complete, already for finitary PCF of second-order types. 2) "The stable order of the model has the syntactic order as its image: If a is less than b in the stable order of the model, for finite a and b, then there are normal form terms A and B with the semantics a, resp. b, such that A is less than B in the syntactic order." We give counter-examples to this conjecture, again in finitary PCF of second-order types, and also refute an improved conjecture: There seems to be no simple syntactic characterization of the stable order. But we show that Berry's conjecture is true for unary PCF. For the preliminaries, we explain the basic fully abstract semantics of PCF in the general setting of (not-necessarily complete) partial order models (f-models.) And we restrict the syntax to "game terms", with a graphical representation.Comment: submitted to LMCS, 39 pages, 23 pstricks/pst-tree figures, main changes for this version: 4.1: proof of game term theorem corrected, 7.: the improved chain conjecture is made precise, more references adde

Denotational Semantics for Subtyping Between Recursive Types

Inheritance in the form of subtyping is considered in the framework of a polymorphic type discipline with records, variants, and recursive types. We give a denotational semantics based on the paradigm that interprets subtyping as explicit coercion. The main technical result gives a coherent interpretation for a strong rule for deriving inheritances between recursive types

Inheritance as Implicit Coercion

We present a method for providing semantic interpretations for languages with a type system featuring inheritance polymorphism. Our approach is illustrated on an extension of the language Fun of Cardelli and Wegner, which we interpret via a translation into an extended polymorphic lambda calculus. Our goal is to interpret inheritances in Fun via coercion functions which are definable in the target of the translation. Existing techniques in the theory of semantic domains can be then used to interpret the extended polymorphic lambda calculus, thus providing many models for the original language. This technique makes it possible to model a rich type discipline which includes parametric polymorphism and recursive types as well as inheritance. A central difficulty in providing interpretations for explicit type disciplines featuring inheritance in the sense discussed in this paper arises from the fact that programs can type-check in more than one way. Since interpretations follow the type-checking derivations, coherence theorems are required: that is, one must prove that the meaning of a program does not depend on the way it was type-checked. The proof of such theorems for our proposed interpretation are the basic technical results of this paper. Interestingly, proving coherence in the presence of recursive types, variants, and abstract types forced us to reexamine fundamental equational properties that arise in proof theory (in the form of commutative reductions) and domain theory (in the form of strict vs. non-strict functions)