Collapsing non-idempotent intersection types

In this article, we develop a new and somewhat unexpected connection between higher-order model-checking and linear logic. Our starting point is the observation that once embedded in the relational semantics of linear logic, the Church encoding of any higher-order recursion scheme (HORS) comes together with a dual Church encoding of an alternating tree automata (ATA) of the same signature. Moreover, the interaction between the relational interpretations of the HORS and of the ATA identifies the set of accepting states of the tree automaton against the infinite tree generated by the recursion scheme. We show how to extend this result to alternating parity automata (APT) by introducing a parametric version of the exponential modality of linear logic, capturing the formal properties of colors (or priorities) in higher-order model-checking. We show in particular how to reunderstand in this way the type-theoretic approach to higher-order model-checking developed by Kobayashi and Ong. We briefly explain in the end of the paper how this analysis driven by linear logic results in a new and purely semantic proof of decidability of the formulas of the monadic second-order logic for higher-order recursion schemes.

Finiteness spaces constitute a categorical model of Linear Logic (LL) whose objects can be seen as linearly topologised spaces, (a class of topological vector spaces introduced by Lefschetz in 1942) and morphisms as continuous linear maps. First, we recall definitions of finiteness spaces and describe their basic properties deduced from the general theory of linearly topologised spaces. Then we give an interpretation of LL based on linear algebra. Second, thanks to separation properties, we can introduce an algebraic notion of totality candidate in the framework of linearly topologised spaces: a totality candidate is a closed affine subspace which does not contain 0. We show that finiteness spaces with totality candidates constitute a model of classical LL. Finally, we give a barycentric simply typed lambda-calculus, with booleans \({\mathcal{B}}\) and a conditional operator, which can be interpreted in this model. We prove completeness at type \({\mathcal{B}}^n\to{\mathcal{B}}\) for every n by an algebraic method.KeywordsTopological Vector SpaceLinear LogicProduct TopologyContinuous Linear FunctionCompact Open TopologyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We present a functional characterization of deterministic logspace-computable predicates based on a variant (although not a subsystem) of propositional linear logic, which we call parsimonious logic. The resulting calculus is simply-typed and contains no primitive besides those provided by the underlying logical system, which makes it one of the simplest higher-order languages capturing logspace currently known. Completeness of the calculus uses the descriptive complexity characterization of logspace (we encode first-order logic with deterministic closure), whereas soundness is established by executing terms on a token machine (using the geometry of interaction).

We derive tight expressions for the maximum number of k-faces, k=0,...,d-1, of the Minkowski sum, P_1+...+P_r, of r convex d-polytopes P_1,...,P_r in R^d, where d >= 2 and r < d, as a (recursively defined) function on the number of vertices of the polytopes. Our results coincide with those recently proved by Adiprasito and Sanyal [1]. In contrast to Adiprasito and Sanyal's approach, which uses tools from Combinatorial Commutative Algebra, our approach is purely geometric and uses basic notions such as f- and h-vector calculus, stellar subdivisions and shellings, and generalizes the methodology used in [10] and [9] for proving upper bounds on the f-vector of the Minkowski sum of two and three convex polytopes, respectively. The key idea behind our approach is to express the Minkowski sum P_1+...+P_r as a section of the Cayley polytope C of the summands; bounding the k-faces of P_1+...+P_r reduces to bounding the subset of the (k+r-1)-faces of C that contain vertices from each of the r polytopes. We end our paper with a sketch of an explicit construction that establishes the tightness of the upper bounds.

The multiset-based relational model of linear logic induces a semantics of the untyped λ-calculus, which corresponds with a non-idempotent intersection type system, System R. We prove that, in System R, the size of type derivations and the size of types are closely related to the execution time of λ-terms in a particular environment machine, Krivine's machine.

In this thesis, it is examined the issue of equivalence between linear terms in higher order languages, that is, in languages which allow to use functions as variables, and where variables which appear in the terms must be used exactly once. The work is developed focusing on the bisimulation method, with the purpose to compare this technique with that which has become the standard for the comparison between the terms of a language, i.e. the context equivalence. The thesis is divided into three parts: in the first one, the introduction of the bisimulation and context equivalence techniques takes place within a deterministic linear and typed language. In the second part, the same techniques are reformulated for a language that, while preserving the linearity, loses the deterministic connotation, allowing the terms to evaluate to a set of values each one having a certain probability to appear in the end of calculation. In the last part, a quantum language is examined, discussing the advantages of quantum computation, which allows to speed-up many of the algorithms of computation. Here one gives the concept of quantum program, which is inextricably linked to the (quantum) register where the qubits used in the computation are stored, entailing a more complex notion of equivalence between terms. The techniques to demonstrate that bisimulation is a congruence are not standard and have been used for the first time by Howe for untyped languages: within the thesis, one shows that bisimulation is a congruence in all considered languages but it coincides with the context equivalence relation only for the deterministic one. Indeed, extending the techniques already used by Howe to the probabilistic and quantum environment, it is shown, as non trivial result, that in probabilistic and quantum linear languages the bisimulation is contained in context equivalence relation.

We present three different functional interpretations of intuitionistic linear logic ILL and show how these correspond to well-known functional interpretations of intuitionistic logic IL via embeddings of IL into ILL. The main difference from previous work of the second author is that in intuitionistic linear logic (as opposed to classical linear logic) the interpretations of !A are simpler and simultaneous quantifiers are no longer needed for the characterisation of the interpretations. We then compare our approach in developing these three proof interpretations with the one of de Paiva around the Dialectica category model of linear logic.

This paper is the fourth of a series exposing a systematic combinatorial approach to Girard's Geometry of Interaction (GoI) program. The GoI program aims at obtaining particular realisability models for linear logic that accounts for the dynamics of cut-elimination. This fourth paper tackles the complex issue of defining exponential connectives in this framework. For that purpose, we use the notion of \emph{graphings}, a generalisation of graphs which was defined in earlier work. We explain how to define a GoI for Elementary Linear Logic (ELL) with second-order quantification, a sub-system of linear logic that captures the class of elementary time computable functions.

Dear SCPE Reader, We present a selection of papers which are extensions of papers presented at the 7-th International Symposium on Parallel and Distributed Computing, 1–5 July 2008, in Krakow, Poland. The motivation for publishing the selection in the SCPE Journal was, on the one hand, to present the flavour of the research reported at the conference and on the other hand to present some of the most relevant topics currently focused on the research on parallel and distributed computing in general. The selection contains only 6 papers out of about 60 presented at the conference, and thus, is far from covering all relevant topics represented at the ISPDC 2008. This is because not all of the invited authors were patient enough to accept a fairly long paper publishing process. Nevertheless, we hope that the presented papers will bring you closer to the research covered by the ISPDC conferences and will encourage you to participate in future ISPDC editions. The first paper ``The Impact of Workload Variability on Load Balancing Algorithms'' is by Marta Beltran and Antonio Guzman from King Juan Carlos University in Spain. It concerns an important topic of load balancing in cluster systems, namely adaptativity of the load balancing algorithms to changes of the workload in the system. Adequate accounting for additional load in the hosting system is of great relevance for correct optimization effects. The paper presents a thorough formal analysis of the workload variability metrics and their influence on the quality of load balancing algorithms. Four basic activities appearing in load balancing algorithms are identified, and based on them some algorithmic solutions are proposed to correctly deal with workload variability in system load balancing. The problem of dynamic load balancing algorithms robustness has been discussed. Two different robustness metrics sensitive to the applied type of opimization: local task-oriented or a global one enable selecting task remote execution or migration as load balancing operations. The proposed approach is illustrated with experiments. The second paper ``Model-Driven Engineering and Formal Validation of High-Performance Embedded Systems'' is by Abdoulaye Gamatie, Eric Rutten, Huafeng Yu, Pierre Boulet, Jean-Luc Dekeyser, from University of Lille and INRIA in France. The paper is concerned with a very advanced methodology of designing correct parallel embedded systems for intensive data-parallel computing. In their previous research, the authors of the paper designed the GASPARD embedded system design framework. It is based on the hardware/software co-design approach through model-driven engineering. The framework is based on an UML-like model specification language in which hardware and software elements are modelled using a component approach with special mechanisms for repetitive structures. This paper tries to combine the modelling framework of GASPARD with the mechanisms of synchronous languages to achieve design verifiability provided for such languages. The paper shows how GASPARD models can be translated into synchronous models based on data flow equations in order to formally check their correctness. The proposed approach is illustrated with an example of a video processing system. The third paper ``Relations Between Several Parallel Computational Models'' is by Stephan Bruda and Yuanqiao Zhang from Bishop’s University in Canada. The paper is concerned with theoretical aspects of shared memory systems described by the parallel random access machine PRAM model and aims in studying performance properties of different types of PRAM systems. The attention is focussed on analysing the computational power of two more sophisticated PRAM models (Combining CRCW and Broadcast Selective Reduction), which include data reduction in case of concurrent writes. The paper shows that these two models have equivalent computational power, which is a new result comparing the existing literature. The performance of both models applied to reconfigurable multiple bus machines was studied as a possible architectural solution for current VLSI processor implementations. It was shown that in such systems under reasonable assumptions concurrent-write does not enhance performance comparing the exclusive-write model. Another result important for the VLSI technology is that the Combining CRCW PRAM model (in which data of concurrent writes are arthmetically or logically combined before write) and the exclusive-write on directed reconfigurable busses perform in equivalent way under strong real-time requirements. The fourth paper ``Experiences with Mesh-Like Computations Using Prediction Binary Trees'' is by Gennaro Cordasco, Biagio Cosenza, Rosario de Chiara, Ugo Erra and Vittorio Scarano from the University ``degli Studi'' of Salerno and the University ``degli Studi della Biasilicata'' of Potenza in Italy. The paper concerns optimization methods for mesh-like computations in clusters of processors. The computations are perfomed assuming a phase-like program execution control using a tiling approach which reduces inter-processor communication. A temporal coherence is also assumed, which means that task sizes provide similar execution times in consecutive phases. Temporary coherent computations are structured in a Prediction Binary Tree, in which leaves represent computing tiles to be mapped to processors. A phase-by-phase semi-static load balancing is introduced to the scheduling algorithm. The scheduling algorithm is equipped with a predictor, which estimates the computation time of next phase tiles based on previous execution times and modifies the tiles to achieve balanced execution in phases. For this, two heuristics are used to leverage on data locality in processors. The proposed approach is illustrated by the example of interactive rendering with Parallel Ray Tracing algorithm. The fifth paper ``The Influence of the IBM pSeries Servers Virtualization Mechanism on Dynamic Resource Allocation in AIX 5L'' is by Maciej Mlynski from ASpartner Limited in Poland. The paper concerns a very up-to-date problem of system virtualization and presents the results of research carried on IBM pSeries servers. IBM is strongly developing the virtualization technique especially on IBM pSeries servers enabling an improved and flexible sharing of system resources between applications. The paper investigates novel facilities for dynamic resource management such as micro-partitioning and partition load manager. They enable dynamic creation of workload logical partitions of system resources and their dynamic mangement. It includes run-time resource re-alocation between logical partitions including setting of sharing specifications as well as run-time adding/removing/setting parameters of resources in the system. It remains an open question how to properly tune parameters of the operating system using the provided virtualization facilities to obtain the best efficiency for a given application program. The paper presents the results of experiments which study the effects of tuning the disk subsystem parameters under the IBM AIX 5L operating system with the use of the provided virtualization facilities on the resulting application execution performance. The results show that even small deterioration in the resource pool status requires an immediate adaptation of the operating system parameters to maintain the required performance. The sixth paper ``HeteroPBLAS: A Set of Parallel Basic Linear Algebra Subprograms Optimized for Heterogeneous Computational Clusters'' is by Ravi Reddy, Alexey Lastovetsky and Pedro Alonso from University College Dublin in Ireland and Polytechnic University of Valencia in Spain. The paper concerns the methodology for parallelization of linear algebra computations for execution in heterogeneous cluster environments. The design of the HeteroPBLAS library (Parallel Basic Linear Algebra Subprograms) for heterogeneous computational clusters is presented. The main contribution of the paper is the automation of the parallelization and optimization of the PBLAS, which is done by means of a special user interface and the underlying set of functions. An important element is here a performance model that is based on program code instrumentation, which determines parameters of the application and the executive heterogeneous platform relevant for execution performance of parallel code. The parameter values specified for or returned by execution of the performance model functions are next used for generation and optimal mapping of the parallel code of the library subroutines. The proposed approach is illustrated by experimental results of execution of optimized HeteroPBLAS programs on homogeneous and heterogeneous computing clusters. Marek Tudruj

Abramsky’s affine combinatory algebras are models of affine combinatory logic, which refines standard combinatory logic in the direction of Linear Logic. Abramsky introduced various universal models of computation based on affine combinatory algebras, consisting of partial involutions over a suitable formal language {of moves}, in order to discuss reversible computation in a Geometry of Interaction setting. We investigate partial involutions from the point of view of the model theory of lambda!-calculus. The latter is a refinement of the standard lambda-calculus, corresponding to affine combinatory logic. We introduce intersection type systems for the lambda!-calculus, by extending standard intersection types with a !_u-operator. These induce affine combinatory algebras, and, via suitable quotients, models of the lambda!-calculus. In particular, we introduce an intersection type system for assigning principal types to lambda!-terms, and we state a correspondence between the partial involution interpreting a combinator and the principal type of the corresponding lambda!-term. This analogy allows for explaining as unification between principal types the somewhat awkward linear application of involutions arising from Geometry of Interaction.

This article shows how individual Petri nets form models of Girard's intuitionistic linear logic. It explores questions of expressiveness and completeness of linear logic with respect to this interpretation. An aim is to use Petri nets to give an understanding of linear logic and give some appraisal of the value of linear logic as a specification logic for Petri nets. This article might serve as a tutorial, providing one in-road into Girard's linear logic via Petri nets. With this in mind we have added several exercises and their solutions. We have made no attempt to be exhaustive in our treatment, dedicating our treatment to one semantics of intuitionistic linear logic.Completeness is shown for several versions of Girard's linear logic with respect to Petri nets as the class of models. The strongest logic considered is intuitionistic linear logic, with ⊗, ⊸, &, ⊕ and the exponential ! (“of course”), and forms of quantification. This logic is shown sound and complete with respect to atomic nets (these include nets in which every transition leads to a nonempty multiset of places). The logic is remarkably expressive, enabling descriptions of the kinds of properties one might wish to show of nets; in particular, negative properties, asserting the impossibility of an assertion, can also be expressed. A start is made on decidability issues.

Many dynamically-typed languages (including JavaScript, Ruby, Python or Racket) support first-class classes, or related concepts such as first-class traits and/or mixins. In those languages classes are first-class values and, like any other values, they can be passed as an argument, or returned from a function. Furthermore first-class classes support dynamic inheritance: i.e. they can inherit from other classes at runtime, enabling programmers to abstract over the inheritance hierarchy. In contrast, type system limitations prevent most statically-typed languages from having first-class classes and dynamic inheritance. This paper shows the design of SEDEL: a polymorphic statically-typed language with first-class traits, supporting dynamic inheritance as well as conventional OO features such as dynamic dispatching and abstract methods. To address the challenges of type-checking first-class traits, SEDEL employs a type system based on the recent work on disjoint intersection types and disjoint polymorphism. The novelty of SEDEL over core disjoint intersection calculi are source level features for practical OO programming, including first-class traits with dynamic inheritance, dynamic dispatching and abstract methods. Inspired by Cook and Palsberg's work on the denotational semantics for inheritance, we show how to design a source language that can be elaborated into Alpuim et al.'s F_{i} (a core polymorphic calculus with records supporting disjoint polymorphism). We illustrate the applicability of SEDEL with several example uses for first-class traits, and a case study that modularizes programming language interpreters using a highly modular form of visitors.

This paper shows how PILLY(Polymorphic Intuitionistic/Linear Lambda calculus with a fixed point combinatorY) with parametric polymorphism can be used as a metalanguage for domain theory, as originally suggested by Plotkin more than a decade ago. Using Plotkin's encodings of recursive types in PILLY, we show how parametric models of PILLYgive rise to models of FPC, which is a simply typed lambda calculus with recursive types and an operational call-by-value semantics, reflecting a classical result from domain theory. Essentially, this interpretation is an interpretation of intuitionistic logic into linear logic first discovered by Girard, which in this paper is extended to deal with recursive types. Of particular interest is a model based on ‘admissible’ pers over a reflexive domain, the theory of which can be seen as a domain theory for (impredicative) polymorphism. We show how this model gives rise to a parametric and computationally adequate model of PolyFPC, an extension of FPC with impredicative polymorphism. This is to the author's knowledge the first denotational model of a non-linear language with parametric polymorphism and recursive types.

How Should We Evaluate and Use Evidence to Improve Population Oral Health?

Social prescribing involves trusted individuals in clinical and community settings identifying non-medical, health-related social needs and connecting people to community-based supports through a collaboratively developed social prescription. Social prescribing operates within a dynamic, complex adaptive system, making evaluation challenging. This study presents two Australian case studies of co-designed social prescribing programs—Connect Local in Melbourne and Spark in Adelaide—to examine how evaluation can be conceptualized and implemented. Rather than focusing on program delivery, these case studies are used to interrogate the processes, methodological challenges, and system conditions that shape how impact is understood. The evaluation challenges for both initiatives included shared complexities: the need to balance meaningful data collection with individual and community preferences; measuring impact to meet the needs of interest holders; and the evolution of the contexts in which the programs are delivered and their influence on what constitutes ‘success.’ Analyzing the two case studies against assumptions of linear, simple systems highlighted that a shift in how evaluation is conceptualized and undertaken is required. Impacts were not static, discrete, measurable outputs, but dynamic processes shaped by relationships, shared meaning-making, and adaptive capacity. Conventional evaluation frameworks centered on linear logic models and fixed indicators do not effectively capture impacts driven by relationships, community capacity, and adaptive change. Therefore, program success must be reframed as an emergent presence in which outcomes unfold through interactions between individuals, organizations, and wider systems. This study argues for a shift from milestone-based models to ongoing stewardship-oriented approaches that prioritize monitoring patterns, relationships, and adaptive responses. Indicators may need to shift from static quantitative measures to relational indicators that reflect relationship alignment, coherence of working practices, and growth within the networks and relationships. The question this research poses is: How can evaluators identify and track indicators that remain meaningful when both the context and intervention are evolving, and thereby the outcomes are also changing? By examining the evaluation journeys of Connect Local and Spark, this study demonstrates the need for methodological approaches that align with complexity, center on community voice, and explain the emergent, co-constructed nature of social connection impacts.