Domain-theoretic Models Research Articles

Semantics for variational Quantum programming

We consider a programming language that can manipulate both classical and quantum information. Our language is type-safe and designed for variational quantum programming, which is a hybrid classical-quantum computational paradigm. The classical subsystem of the language is the Probabilistic FixPoint Calculus (PFPC), which is a lambda calculus with mixed-variance recursive types, term recursion and probabilistic choice. The quantum subsystem is a first-order linear type system that can manipulate quantum information. The two subsystems are related by mixed classical/quantum terms that specify how classical probabilistic effects are induced by quantum measurements, and conversely, how classical (probabilistic) programs can influence the quantum dynamics. We also describe a sound and computationally adequate denotational semantics for the language. Classical probabilistic effects are interpreted using a recently-described commutative probabilistic monad on DCPO. Quantum effects and resources are interpreted in a category of von Neumann algebras that we show is enriched over (continuous) domains. This strong sense of enrichment allows us to develop novel semantic methods that we use to interpret the relationship between the quantum and classical probabilistic effects. By doing so we provide a very detailed denotational analysis that relates domain-theoretic models of classical probabilistic programming to models of quantum programming.

A Domain-Theoretic Model Of Nominally-Typed Object-Oriented Programming

The majority of contemporary mainstream object-oriented (OO) software is written using nominally-typed OO programming languages. Extant domain-theoretic models of OOP developed to analyze OO type systems miss crucial features of these mainstream OO languages, such as nominality. This paper summarizes the construction of NOOP as a domain-theoretic model of OOP that includes nominal information found in nominally-typed mainstream OO software. Inclusion of nominal type information and asserting that type inheritance in statically-typed OO programming languages is an inherently nominal notion allow readily proving that inheritance and subtyping are completely identified in these languages. This conclusion is in full agreement with intuitions of OO developers using these languages, and contrary to the belief that “inheritance is not subtyping”, which came from assuming non-nominal structural models of OO type systems. NOOP, thus, provides a firmer semantic foundation for analyzing and progressing nominally-typed mainstream OO programming languages.

Computational Models of Certain Hyperspaces of Quasi-metric Spaces

In this paper, for a given sequentially Yoneda-complete T_1 quasi-metric space (X,d), the domain theoretic models of the hyperspace K_0(X) of nonempty compact subsets of (X,d) are studied. To this end, the $\omega$-Plotkin domain of the space of formal balls BX, denoted by CBX is considered. This domain is given as the chain completion of the set of all finite subsets of BX with respect to the Egli-Milner relation. Further, a map $\phi:K_0(X)\rightarrow CBX$ is established and proved that it is an embedding whenever K_0(X) is equipped with the Vietoris topology and respectively CBX with the Scott topology. Moreover, if any compact subset of (X,d) is d^{-1}-precompact, \phi is an embedding with respect to the topology of Hausdorff quasi-metric H_d on K_0(X). Therefore, it is concluded that (CBX,\sqsubseteq,\phi) is an $\omega$-computational model for the hyperspace K_0(X) endowed with the Vietoris and respectively the Hausdorff topology. Next, an algebraic sequentially Yoneda-complete quasi-metric D on CBX$ is introduced in such a way that the specialization order $\sqsubseteq_D$ is equivalent to the usual partial order of CBX and, furthermore, $\phi:({\cal K}_0(X),H_d)\rightarrow({\bf C}{\bf B}X,D)$ is an isometry. This shows that (CBX,\sqsubseteq,\phi,D) is a quantitative $\omega$-computational model for (K_(X),H_d).

Semantic Domains for Combining Probability and Non-Determinism

We present domain-theoretic models that support both probabilistic and nondeterministic choice. In [A. McIver and C. Morgan. Partial correctness for probablistic demonic programs. Theoretical Computer Science , 266:513–541, 2001], Morgan and McIver developed an ad hoc semantics for a simple imperative language with both probabilistic and nondeterministic choice operators over a discrete state space, using domain-theoretic tools. We present a model also using domain theory in the sense of D.S. Scott (see e.g. [G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, and D. S. Scott. Continuous Lattices and Domains , volume 93 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, 2003]), but built over quite general continuous domains instead of discrete state spaces. Our construction combines the well-known domains modelling nondeterminism – the lower, upper and convex powerdomains, with the probabilistic powerdomain of Jones and Plotkin [C. Jones and G. Plotkin. A probabilistic powerdomain of evaluations. In Proceedings of the Fourth Annual Symposium on Logic in Computer Science , pages 186–195. IEEE Computer Society Press, 1989] modelling probabilistic choice. The results are variants of the upper, lower and convex powerdomains over the probabilistic powerdomain (see Chapter 4). In order to prove the desired universal equational properties of these combined powerdomains, we develop sandwich and separation theorems of Hahn-Banach type for Scott-continuous linear, sub- and superlinear functionals on continuous directed complete partially ordered cones, endowed with their Scott topologies, in analogy to the corresponding theorems for topological vector spaces in functional analysis (see Chapter 3). In the end, we show that our semantic domains work well for the language used by Morgan and McIver.

On the completeness of order-theoretic models of the λ-calculus

Scott discovered his domain-theoretic models of the λ-calculus, isomorphic to their function space, in 1969. A natural completeness problem then arises: whether any two terms equal in all Scott models are convertible. There is also an analogous consistency problem: whether every equation between two terms, consistent with the λ-calculus, has a Scott model. We consider such questions for wider sets of sentences and wider classes of models, the pointed (completely) partially ordered ones. A negative result for a set of sentences shows the impossibility of finding Scott models for that class; a positive result gives evidence that there might be enough Scott models. We find, for example, that the order-extensional pointed ω-cpo models are complete for Π 1 -sentences with positive matrices, whereas the consistency question for Σ 1-sentences with equational matrices depends on the consistency of certain critical sentences asserting the existence of certain functions analogous to the generalized Mal’cev operators first considered in the context of the λ-calculus by Selinger.

A type assignment system for game semantics

We present a type assignment system that provides a finitary interpretation of lambda terms in a game semantics model. Traditionally, type assignment systems describe the semantic interpretation of terms in domain-theoretic models. Quite surprisingly, the type assignment system presented in this paper is very similar to the traditional ones, the main difference being the omission of the subtyping rules.

Synthetic Domain Theory and Models of Linear Abadi & Plotkin Logic

In a recent article [L. Birkedal, R. E. Møgelberg, and R. L. Petersen. Parametric domain-theoretic models of linear Abadi & Plotkin logic. Technical Report TR-2005-57, IT University of Copenhagen, February 2005] the first two authors and R.L. Petersen have defined a notion of parametric LAPL-structure. Such structures are parametric models of the equational theory PILLY, a polymorphic intuitionistic / linear type theory with fixed points, in which one can reason using parametricity and, for example, solve a large class of domain equations [L. Birkedal, R. E. Møgelberg, and R. L. Petersen. Parametric domain-theoretic models of linear Abadi & Plotkin logic. Technical Report TR-2005-57, IT University of Copenhagen, February 2005, L Birkedal, R. E. Møgelberg, and R. L. Petersen. Parametric domain-theoretic models of polymorphic intuitionistic / linear lambda calculus. In Proceedings of the Twenty-first Conference on the Mathematical Foundations of Programming Semantics, 2005. To appear].Based on recent work by Simpson and Rosolini [G. Rosolini and A. Simpson. Using synthetic domain theory to prove operational properties of a polymorphic programming language based on strictness. Manuscript, 2004] we construct a family of parametric LAPL-structures using synthetic domain theory and use the results of loc. cit. and results about LAPL-structures to prove operational consequences of parametricity for a strict version of the Lily programming language. In particular we can show that one can solve domain equations in the strict version of Lily up to ground contextual equivalence.

Computational adequacy for recursive types in models of intuitionistic set theory

This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domain-theoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown that the existence of solutions to recursive domain equations depends upon the strength of the set theory. We observe that the internal set theory of an elementary topos is not strong enough to guarantee their existence. In contrast, as our first main result, we establish that solutions to recursive domain equations do exist when the category of sets is a model of full intuitionistic Zermelo–Fraenkel set theory. We then apply this result to obtain a denotational interpretation of FPC, a recursively typed lambda-calculus with call-by-value operational semantics. By exploiting the intuitionistic logic of the ambient model of intuitionistic set theory, we analyse the relationship between operational and denotational semantics. We first prove an “internal” computational adequacy theorem: the model always believes that the operational and denotational notions of termination agree. This allows us to identify, as our second main result, a necessary and sufficient condition for genuine “external” computational adequacy to hold, i.e. for the operational and denotational notions of termination to coincide in the real world. The condition is formulated as a simple property of the internal logic, related to the logical notion of 1-consistency. We provide useful sufficient conditions for establishing that the logical property holds in practice. Finally, we outline how the methods of the paper may be applied to concrete models of FPC. In doing so, we obtain computational adequacy results for an extensive range of realizability and domain-theoretic models.

Local DCPOs, Local CPOs and Local Completions

We use a subfamily of the Scott-closed sets of a poset to form a local completion of the poset. This is simultaneously a topological analogue of the ideal completion of a poset and a generalization of the sobrification of a topological space. After we show that our construction is the object level of a left adjoint to the forgetful functor from the category of local cpos to the category of posets and Scott-continuous maps, we use this completion to show how local domains can play a role in the study of domain-theoretic models of topological spaces. Our main result shows that any topological space that is homeomorphic to the maximal elements of a continuous poset that is weak at the top also is homeomorphic to the maximal elements of a bounded complete local domain. The advantage is that continuous maps between such spaces extend to Scott-continuous maps between the modeling local domains.

Domain theoretic models of topological spaces

A model of a space X is simply a continuous dcpo D and a homeomorphism ∅: X → max D, where max D is given its inherited Scott topology. We show that a space has a coherent model iff it has a Scott domain model and investigate the topological structure of spaces which have Gδ models.

A Characterization of Finitary Bisimulation

Following a paradigm put forward by Milner and Plotkin, a primary criterion to judge the appropriateness of denotational models for programming and specification languages is that they be in agreement with operational intuition about program behaviour. Of the "good t" criteria for such models that have been<br />discussed in the literature, the most desirable one is that of full abstraction.<br />Intuitively, a fully abstract denotational model is guaranteed to relate exactly all those programs that are operationally indistinguishable with respect to some chosen notion of observation. <br />Because of its prominent role in process theory, bisimulation [12] has been a natural yardstick to assess the appropriateness of denotational models for several process description languages. In particular, when proving full abstraction<br />results for denotational semantics based on the Scott-Strachey approach for CCS-like languages, several preorders based on bisimulation have been considered; see, e.g., [6, 3, 4]. In this paper, we shall study one such bisimulationbased<br />preorder whose connections with domain-theoretic models are by now well understood, viz. the prebisimulation preorder . investigated in, e.g., [6, 3]. Intuitively, p < q holds of processes p and q if p and q can simulate each other's<br />behaviour, but at times the behaviour of p may be less specified than that of q. <br />A common problem in relating denotational semantics for process description<br />languages, based on Scott's theory of domains or on the theory of algebraic semantics, with behavioural semantics based on bisimulation is that the chosen behavioural theory is, in general, too concrete. The reason for this phenomenon is that two programs are related by a standard denotational interpretation if, in some precise sense, they afford the same finite observations. On the other hand, bisimulation can make distinctions between the behaviours of two processes<br />based on infinite observations. (Cf. the seminal study [1] for a detailed analysis of this phenomenon.) To overcome this mismatch between the denotational<br />and the behavioural theory, all the aforementioned full abstraction results are obtained with respect to the so-called finitely observable, or finitary, part of bisimulation. The finitary bisimulation is defined on any labelled transition system thus: p <^F q iff t < p implies t < q, for every finite synchronization tree t.<br /> An alternative characterization of the finitary bisimulation for arbitrary transition systems has been given by Abramsky in [1]. This characterization is couched in logical terms, and is an impressive byproduct of Abramsky's "theory<br />of domains in logical form" programme. More precisely, Abramsky shows that two processes in any transition system are equated by the finitary bisimulation<br />iff they satisfy the same formulae in the finitary version of the domain logic for transition systems. The existence of this logical view of the finitary bisimulation gives us a handle to work with this relation. However, an alternative,<br />behavioural view of the finitary bisimulation might be more useful when establishing results which are more readily shown on the behavioural, rather than on the logical, side. Examples of such results are complete axiomatizations for<br />the finitary bisimulation and full abstraction results. A behavioural characterization of the finitary bisimulation would also provide an easier way to establish when two processes in a transition system are related by it or not, thus giving more insight on the kind of identifications made by this relation. In this study, we offer a behavioural characterization of the finitary bisimulation<br />for arbitrary transition systems (cf. Thm. 3.5). This result may be seen as the behavioural counterpart of Abramsky's logical characterization theorem [1, Thm. 5.5.8]. Moreover, for the important class of sort-finite transition systems<br />we present a sharpened version of a behavioural characterization result first proven by Abramsky in [3, Propn. 6.13]. The interested reader may consult the unpublished report [5] for more results on the finitary bisimulation.

Domain theoretic models of polymorphism

We give an illustration of a construction useful in producing and describing models of Girard and Reynolds' polymorphic λ-calculus. The key unifying ideas are that of a Grothendieck fibration and the category of continuous sections associated with it, constructions used in indexed category theory; the universal types of the calculus are interpreted as the category of continuous sections of the fibration. As a major example a new model for the polymorphic λ-calculus is presented. In it a type is interpreted as a Scott domain. In fact, understanding universal types of the polymorphic λ-calculus as categories of continuous sections appears to be useful generally. For example, the technique also applies to the finitary projection model of Bruce and Longo, and a recent model of Girard. (Indeed the work here was inspired by Girard's and arose through trying to extend the construction of his model to Scott domains). It is hoped that by pin-pointing a key construction this paper will help towards a deeper understanding of models for the polymorphic λ-calculus and the relations between them.