Quantum Bayesian networks [Henson et al., 2014] provide a mathematical formalism to describe causal relations, to analyse correlations, and to predict the probabilities of measurement outcomes, in systems involving both classical and quantum data. They generalize Pearl’s Bayesian networks [Pearl, 2009] - prominent graphical models for classical probabilistic reasoning and inference. The goal of this paper is to bring compositional principles and a typing discipline into this setting. A key feature of our compositional semantics is that when all causes are classical, it coincides with the standard factor-based semantics of Bayesian networks, while in the purely quantum case it reduces to tensor networks. We then propose a typed formalism based on linear logic proof-nets, where types ensure well-behaved composition of systems, and which we prove sound and complete with respect to quantum Bayesian networks.

Quantum lambda calculus has been studied mainly as an idealized programming language - the evaluation essentially corresponds to a deterministic abstract machine. Very little work has been done to develop a rewriting theory for quantum lambda calculus. Recent advances in the theory of probabilistic rewriting give us a way to tackle this task with tools unavailable a decade ago. Our primary focus are standardization and normalization results.

Abstract Variable Elimination ( $$\textsf{VE}$$ VE ) is a classical exact inference algorithm for probabilistic graphical models such as Bayesian Networks, computing the marginal distribution of a subset of the random variables in the model. Our goal is to understand Variable Elimination as an algorithm acting on programs in an idealized probabilistic functional language—a linear simply-typed $$\lambda $$ λ -calculus suffices for our purpose. Precisely, we express $$\textsf{VE}$$ VE as a term rewriting process, which transforms a global definition of a variable into a local definition, by swapping and nesting let-in expressions. We exploit in an essential way linear types.

Bayesian networks are graphical first-order probabilistic models that allow for a compact representation of large probability distributions, and for efficient inference, both exact and approximate. We introduce a higher-order programming language—in the idealized form of a λ -calculus—which we prove sound and complete w.r.t. Bayesian networks: each Bayesian network can be encoded as a term, and conversely each (possibly higher-order and recursive) program of ground type compiles into a Bayesian network. The language allows for the specification of recursive probability models and hierarchical structures. Moreover, we provide a compositional and cost-aware semantics which is based on factors, the standard mathematical tool used in Bayesian inference. Our results rely on advanced techniques rooted into linear logic, intersection types, rewriting theory, and Girard’s geometry of interaction, which are here combined in a novel way.

We consider an extension of multiplicative linear logic which encompasses bayesian networks and expresses samples sharing and marginalisation with the polarised rules of contraction and weakening. We introduce the necessary formalism to import exact inference algorithms from bayesian networks, giving the sum-product algorithm as an example of calculating the weighted relational semantics of a multiplicative proof-net improving runtime performance by storing intermediate results.

We present a technique to study normalizing strategies when termination is asymptotic, that is, it appears as a limit, as opposite to reaching a normal form in a finite number of steps. Asymptotic termination occurs in several settings, such as effectful, and in particular probabilistic computation -- where the limits are distributions over the possible outputs -- or infinitary lambda-calculi -- where the limits are infinitary normal forms such as Boehm trees. As a concrete application, we obtain a result which is of independent interest: a normalization theorem for Call-by-Value (and -- in a uniform way -- for Call-by-Name) probabilistic lambda-calculus.

While a mature body of work supports the study of rewriting systems, abstract tools for Probabilistic Rewriting are still limited. In this paper we study the question of uniqueness of the result (unique limit distribution), and develop a set of proof techniques to analyze and compare reduction strategies. The goal is to have tools to support the operational analysis of probabilistic calculi (such as probabilistic lambda-calculi) where evaluation allows for different reduction choices (hence different reduction paths).

Abstract We study the reduction in a $\lambda$ -calculus derived from Moggi’s computational one, which we call the computational core. The reduction relation consists of rules obtained by orienting three monadic laws. Such laws, in particular associativity and identity, introduce intricacies in the operational analysis. We investigate the central notions of returning a value versus having a normal form and address the question of normalizing strategies. Our analysis relies on factorization results.

Factorization -- a simple form of standardization -- is concerned with reduction strategies, i.e. how a result is computed. We present a new technique for proving factorization theorems for compound rewriting systems in a modular way, which is inspired by the Hindley-Rosen technique for confluence. Specifically, our technique is well adapted to deal with extensions of the call-by-name and call-by-value lambda-calculi. The technique is first developed abstractly. We isolate a sufficient condition (called linear swap) for lifting factorization from components to the compound system, and which is compatible with beta-reduction. We then closely analyze some common factorization schemas for the lambda-calculus. Concretely, we apply our technique to diverse extensions of the lambda-calculus, among which de' Liguoro and Piperno's non-deterministic lambda-calculus and -- for call-by-value -- Carraro and Guerrieri's shuffling calculus. For both calculi the literature contains factorization theorems. In both cases, we give a new proof which is neat, simpler than the original, and strikingly shorter.

Abstract In each variant of the $$\lambda $$ λ -calculus, factorization and normalization are two key properties that show how results are computed. Instead of proving factorization/normalization for the call-by-name (CbN) and call-by-value (CbV) variants separately, we prove them only once, for the bang calculus (an extension of the $$\lambda $$ λ -calculus inspired by linear logic and subsuming CbN and CbV), and then we transfer the result via translations, obtaining factorization/normalization for CbN and CbV. The approach is robust: it still holds when extending the calculi with operators and extra rules to model some additional computational features.

Motivated by the study of effectful programming languages and computations, we introduce a relational theory of monadic rewriting systems. The latter are rewriting systems whose notion of reduction is effectful, where effects are modelled as monads. Contrary to what happens in the ordinary operational semantics of monadic programming languages, defining meaningful notions of monadic rewriting turns out to problematic for several monads, including the distribution, powerset, reader, and global state monad. This raises the question of when monadic rewriting is possible. We answer that question by identifying a class of monads, known as weakly cartesian monads, that guarantee monadic rewriting to be well-behaved. In case monads are given as equational theories, as it is the case for algebraic effects, we also show that a sufficient condition to have a well-behaved notion of monadic rewriting is that all equations in the theory are linear. Finally, we apply the abstract theory of monadic rewriting systems to the call-by-value λ-calculus with algebraic effects, this way obtaining effectful (surface) standardisation and confluence theorems.

Randomized higher-order computation can be seen as being captured by a λ-calculus endowed with a single algebraic operation, namely a construct for binary probabilistic choice. What matters about such computations is the probability of obtaining any given result, rather than the possibility or the necessity of obtaining it, like in (non)deterministic computation. Termination, arguably the simplest kind of reachability problem, can be spelled out in at least two ways, depending on whether it talks about the probability of convergence or about the expected evaluation time, the second one providing a stronger guarantee. In this paper, we show that intersection types are capable of precisely characterizing both notions of termination inside a single system of types: the probability of convergence of any λ-term can be underapproximated by its type , while the underlying derivation’s weight gives a lower bound to the term’s expected number of steps to normal form. Noticeably, both approximations are tight—not only soundness but also completeness holds. The crucial ingredient is non-idempotency, without which it would be impossible to reason on the expected number of reduction steps which are necessary to completely evaluate any term. Besides, the kind of approximation we obtain is proved to be optimal recursion theoretically: no recursively enumerable formal system can do better than that.

The notion of solvability, crucial in the λ-calculus, is conservatively extended to a probabilistic setting, and a complete characterization of it is given. The employed technical tool is a type assignment system, based on non-idempotent intersection types, whose typable terms turn out to be precisely the terms which are solvable with nonnull probability. We also supply an operational characterization of solvable terms, through the notion of head normal form, and a denotational model of Λ_⊕, itself induced by the type system, which equates all the unsolvable terms.

We introduce two extensions of the λ -calculus with a probabilistic choice operator, Λ⊕cbvand Λ⊕cbn, modeling respectively call-by-value and call-by-name probabilistic computation. We prove that both enjoys confluence and standardization, in an extended way: we revisit these two fundamental notions to take into account the asymptotic behaviour of terms. The common root of the two calculi is a further calculus based on Linear Logic, Λ⊕!, which allows us to develop a unified, modular approach.

We introduce a Geometry of Interaction model for higher-order quantum computation, and prove its adequacy for a fully fledged quantum programming language in which entanglement, duplication, and recursion are all available.

We study multitoken interaction machines in the context of a very expressive linear logical system with exponentials, fix points and synchronization. The advantage of such machines is to provide models in the style of the Geometry of Interaction, i.e., An interactive semantics which is close to low-level implementation. On the one hand, we prove that despite the inherent complexity of the framework, interaction is guaranteed to be deadlock-free. On the other hand, the resulting logical system is powerful enough to embed PCF and to adequately model its behaviour, both when call-by-name and when call-by-value evaluation are considered. This is not the case for single-token stateless interactive machines.

We prove that is possible to extend Girard's Ludics so as to have repetitions (hence exponentials), and still have the results on semantical types which characterize Ludics in the panorama of Game Semantics. The results are obtained by using less structure than in the original paper; this has an interest on its own, and we hope that it will open the way to applying the approach of Ludics to a larger domain.

We introduce L-nets as a game model of concurrent interaction. L-nets, which correspond to strategies (in Games Semantics) or designs (in Ludics), are graphs, and the interactions (plays) result into partial orders, hence allowing for parallelism.

An increasing interest is directed at the extension of the \proof search as computation" paradigm, already successfully applied to Linear Logic, to a logic that is not only resource-aware but also order-sensitive.This paper is a contribution to proof search in Non-Commutative L o g i c .Our key result is to give a simple method for propagating the order structure during proof search.Such a method is general, in that it can be applied to n-ary connectives.This enables us to de ne a cluster calculus, which analyses clusters of synchronous and of asynchronous connectives in a single step, with a single n-ary rule.

Abstract We introduce a sequent calculusBfor a new logic, named basic logic. The aim of basic logic is to find a structure in the space of logics. Classical, intuitionistic. quantum and non-modal linear logics, are all obtained as extensions in a uniform way and in a single framework. We isolate three properties, which characterizeBpositively: reflection, symmetry and visibility. A logical constant obeys to the principle of reflection if it is characterized semantically by an equation binding it with a metalinguistic link between assertions, and if its syntactic inference rules are obtained by solving that equation. All connectives of basic logic satisfy reflection. To the control of weakening and contraction of linear logic, basic logic adds a strict control of contexts, by requiring that all active formulae in all rules are isolated, that is visible. From visibility, cut-elimination follows. The full, geometric symmetry of basic logic induces known symmetries of its extensions, and adds a symmetry among them, producing the structure of a cube.

No abstract available.