Showing papers in "Mathematical Structures in Computer Science in 2009"

Correctness of high-level transformation systems relative to nested conditions†

Abstract: In this paper we introduce the notions of nested constraints and application conditions, short nested conditions. For a category associated with a graphical representation such as graphs, conditions are a graphical and intuitive, yet precise, formalism that is well suited to describing structural properties. We show that nested graph conditions are expressively equivalent to first-order graph formulas. A part of the proof includes transformations between two satisfiability notions of conditions, namely -satisfiability and -satisfiability. We consider a number of transformations on conditions that can be composed to construct constraint-guaranteeing and constraint-preserving application conditions, weakest preconditions and strongest postconditions. The restriction of rule applications by conditions can be used to correct transformation systems by pruning transitions leading to states violating given constraints. Weakest preconditions and strongest postconditions can be used to verify the correctness of transformation systems with respect to pre-and postconditions.

A quantitative computational model for complete partial metric spaces via formal balls

Abstract: Given a partial metric space (X, p), we use (BX, ⊑dp) to denote the poset of formal balls of the associated quasi-metric space (X, dp). We obtain characterisations of complete partial metric spaces and sup-separable complete partial metric spaces in terms of domain-theoretic properties of (BX, ⊑dp). In particular, we prove that a partial metric space (X, p) is complete if and only if the poset (BX, ⊑dp) is a domain. Furthermore, for any complete partial metric space (X, p), we construct a Smyth complete quasi-metric q on BX that extends the quasi-metric dp such that both the Scott topology and the partial order ⊑dp are induced by q. This is done using the partial quasi-metric concept recently introduced and discussed by H. P. Kunzi, H. Pajoohesh and M. P. Schellekens (Kunzi et al. 2006). Our approach, which is inspired by methods due to A. Edalat and R. Heckmann (Edalat and Heckmann 1998), generalises to partial metric spaces the constructions given by R. Heckmann (Heckmann 1999) and J. J. M. M. Rutten (Rutten 1998) for metric spaces.

The algebraic lambda calculus

Abstract: We introduce an extension of the pure lambda calculus by endowing the set of terms with the structure of a vector space, or, more generally, of a module, over a fixed set of scalars. Moreover, terms are subject to identities similar to the usual pointwise definition of linear combinations of functions with values in a vector space. We then study a natural extension of beta reduction in this setting: we prove it is confluent, then discuss consistency and conservativity over the ordinary lambda calculus. We also provide normalisation results for a simple type system.

Mobile petri nets

Abstract: We add mobility to Place-Transition Petri nets: tokens are names for places, and an input token of a transition can be used in its postset to specify a destination. Mobile Petri nets are then further extended to dynamic nets by adding the possibility of creating new nets during the firing of a transition. In this way, starting from Petri nets, we define a simple hierarchy of nets with increasing degrees of dynamicity. For each class in this hierarchy, we provide its encoding in the former class. Our work was largely inspired by the join-calculus of Fournet and Gonthier, which turns out to be a (well-motivated) particular case of dynamic Petri nets. The main difference is that, in the preset of a transition, we allow both non-linear patterns (name unification) and (locally) free names for input places (that is, we remove the locality constraint, and preserve reflexion).

Algebra and logic for resource-based systems modelling

Abstract: Mathematical modelling is one of the fundamental tools of science and engineering. Very often, models are required to be executable, as a simulation, on a computer. In this paper, we present some contributions to the process-theoretic and logical foundations of discrete-event modelling with resources and processes. We present a process calculus with an explicit representation of resources in which processes and resources co-evolve. The calculus is closely connected to a logic that may be used as a specification language for properties of models. The logic is strong enough to allow requirements that a system has a certain structure: for example, that it is a parallel composite of subsystems. This work consolidates, extends and improves upon aspects of earlier work of ours in this area. An extended example, consisting of a semantics for a simple parallel programming language, indicates a connection with separating logics for concurrency.

On the expressive power of recursion, replication and iteration in process calculi

Abstract: In this paper we investigate the expressive power of three alternative approaches to the definition of infinite behaviours in process calculi, namely, recursive definitions, replication and iteration. We prove several results discriminating between the calculi obtained from a core CCS by adding the three mechanisms mentioned above. These results are derived by considering the decidability of four basic properties: termination (that is, all computations are finite); convergence (that is, the existence of a finite computation); barb (that is, the ability to perform an action on a given channel) and weak bisimulation. Our results, which are summarised in Table 1, show that the three calculi form a strict expressiveness hierarchy in that: all the properties mentioned are undecidable in CCS with recursion; only termination and barb are decidable in CCS with replication; all the properties are decidable in CCS with iteration. As a corollary, we also obtain a strict expressiveness hierarchy with respect to weak bisimulation, since there exist weak bisimulation preserving encodings of iteration in replication and of replication in recursion, whereas there are no weak bisimulation preserving encodings in the other directions.

Two-dimensional models of type theory

Abstract: We describe a non-extensional variant of Martin-Lof type theory, which we call two-dimensional type theory, and equip it with a sound and complete semantics valued in 2-categories.

A theory of contracts for strong service compliance

Abstract: We investigate, in a process algebraic setting, a new notion of correctness for service compositions, which we call strong service compliance: composed services are strong compliant if their composition is both deadlock and livelock free (this is the traditional notion of compliance), and whenever a message can be sent to invoke a service, it is guranteed to be ready to serve the invocation. We also define a new notion of refinement, called strong subcontract pre-order, suitable for strong compliance: given a composition of strong compliant services, we can replace any service with any other service in subcontract relation while preserving the overall strong compliance. Finally, we present a characterisation of the strong subcontract pre-order by resorting to the theory of a (should) testing pre-order.

On traced monoidal closed categories

Abstract: The structure theorem of Joyal, Street and Verity says that every traced monoidal category arises as a monoidal full subcategory of the tortile monoidal category Int . In this paper we focus on a simple observation that a traced monoidal category is closed if and only if the canonical inclusion from into Int has a right adjoint. Thus, every traced monoidal closed category arises as a monoidal co-reflexive full subcategory of a tortile monoidal category. From this, we derive a series of facts for traced models of linear logic, and some for models of fixed-point computation. To make the paper more self-contained, we also include various background results for traced monoidal categories.

Structural non-interference in elementary and trace nets

Abstract: Several notions of non-interference have been proposed in the literature for studying the problem of confidentiality in concurrent systems. The common feature of these non-interference properties is that they are all defined as extensional properties based on some notion of behavioural equivalence on systems. Here, instead, we address the problem of defining non-interference by looking at the structure of the systems under investigation. We use a simple class of Petri nets, namely, contact-free elementary net systems, as the system model and define structural non-interference properties based on the absence of particular places in the net: such places show that a suitable causality or conflict relation is present between a high-level transition and a low-level one. We characterise one structural property, called PBNI+, which we show to be equivalent to the well-known behavioural property SBNDC. It essentially captures all the positive information flows (that is, a low-level user can deduce that some high-level action has occurred). We start by providing a characterisation of PBNI+ on contact-free elementary net systems, then extend the definition to cope with the richer class of trace nets.

Complexity of propositional projection temporal logic with star

Abstract: This paper investigates the complexity of Propositional Projection Temporal Logic with Star (PPTL*). To this end, Propositional Projection Temporal Logic (PPTL) is first extended to include projection star. Then, by reducing the emptiness problem of star-free expressions to the problem of the satisfiability of PPTL* formulas, the lower bound of the complexity for the satisfiability of PPTL* formulas is proved to be non-elementary. Then, to prove the decidability of PPTL*, the normal form, normal form graph (NFG) and labelled normal form graph (LNFG) for PPTL* are defined. Also, algorithms for transforming a formula to its normal form and LNFG are presented. Finally, a decision algorithm for checking the satisfiability of PPTL* formulas is formalised using LNFGs.

Exploring the relation between intuitionistic bi and boolean bi: An unexpected embedding

Abstract: The logic of Bunched Implications, through both its intuitionistic version ( BI ) and one of its classical versions, called Boolean BI ( BBI ), serves as a logical basis to spatial or separation logic frameworks. In BI , the logical implication is interpreted intuitionistically whereas it is generally interpreted classically in spatial or separation logics, as in BBI . In this paper, we aim to give some new insights into the semantic relations between BI and BBI . Then we propose a sound and complete syntactic constraints based framework for the Kripke semantics of both BI and BBI , a sound labelled tableau proof system for BBI , and a representation theorem relating the syntactic models of BI to those of BBI . Finally, we deduce as our main, and unexpected, result, a sound and faithful embedding of BI into BBI .

The space of formal balls and models of quasi-metric spaces

Abstract: In this paper we study quasi-metric spaces using domain theory. Our main objective in this paper is to study the maximal point space problem for quasi-metric spaces. Here we prove that quasi-metric spaces that satisfy certain completeness properties, such as Yoneda and Smyth completeness, can be modelled by continuous dcpo's. To achieve this goal, we first study the partially ordered set of formal balls (BX, ⊑) of a quasi-metric space (X, d). Following Edalat and Heckmann, we prove that the order properties of (BX, ⊑) are tightly connected to topological properties of (X, d). In particular, we prove that (BX, ⊑) is a continuous dcpo if (X, d) is algebraic Yoneda complete. Furthermore, we show that this construction gives a model for Smyth-complete quasi-metric spaces. Then, for a given quasi-metric space (X, d), we introduce the partially ordered set of abstract formal balls (BX, ⊑, ≺). We prove that if the conjugate space (X, d−1) of a quasi-metric space (X, d) is right K-complete, then the ideal completion of (BX, ⊑, ≺) is a model for (X, d). This construction provides a model for any Yoneda-complete quasi-metric space (X, d), as well as the Sorgenfrey line, Kofner plane and Michael line.

On a measurement-free quantum lambda calculus with classical control

Abstract: We study a measurement-free, untyped λ-calculus with quantum data and classical control. This work arises from previous proposals by Selinger and Valiron, and Van Tonder. We focus on operational and expressiveness issues, rather than (denotational) semantics. We prove subject reduction and confluence, and a standardisation theorem. Moreover, we prove the computational equivalence of the proposed calculus with a suitable class of quantum circuit families.

The dedekind reals in abstract stone duality

Abstract: Stone Duality (ASD) is a direct axiomatisation of general topology, in contrast to the traditional and all other contemporary approaches, which rely on a prior notion of discrete set, type or object of a topos. ASD reconciles mathematical and computational viewpoints, providing an inherently computable calculus that does not sacrifice key properties of real analysis such as compactness of the closed interval. Previous theories of recursive analysis failed to do this because they were based on points; ASD succeeds because, like locale theory and formal topology, it is founded on the algebra of open subspaces. ASD is presented as a lambda calculus, of which we provide a self-contained summary, as the foundational background has been investigated in earlier work. The core of the paper constructs the real line using two-sided Dedekind cuts. We show that the closed interval is compact and overt, where these concepts are defined using quantifiers. Further topics, such as the Intermediate Value Theorem, are presented in a separate paper that builds on this one. The interval domain plays an important foundational role. However, we see intervals as generalised Dedekind cuts, which underly the construction of the real line, not as sets or pairs of real numbers. We make a thorough study of arithmetic, in which our operations are more complicated than Moore's, because we work constructively, and we also consider back-to-front (Kaucher) intervals. Finally, we compare ASD with other systems of constructive and computable topology and analysis.

Predicate transformers for extended probability and non-determinism

Abstract: We investigate laws for predicate transformers for the combination of non-deterministic choice and (extended) probabilistic choice, where predicates are taken to be functions to the extended non-negative reals, or to closed intervals of such reals. These predicate transformers correspond to state transformers, which are functions to conical powerdomains, which are the appropriate powerdomains for the combined forms of non-determinism. As with standard powerdomains for non-deterministic choice, these come in three flavours – lower, upper and (order-)convex – so there are also three kinds of predicate transformers. In order to make the connection, the powerdomains are first characterised in terms of relevant classes of functionals. Much of the development is carried out at an abstract level, a kind of domain-theoretic functional analysis: one considers d-cones, which are dcpos equipped with a module structure over the non-negative extended reals, in place of topological vector spaces. Such a development still needs to be carried out for probabilistic choice per se; it would presumably be necessary to work with a notion of convex space rather than a cone.

Social processes, program verification and all that

Abstract: In a controversial paper (De Millo et al. 1979) at the end of the 1970's, R. A. De Millo, R. J. Lipton and A. J. Perlis argued against formal verifications of programs, mostly motivating their position by an analogy with proofs in mathematics, and, in particular, with the impracticality of a strictly formalist approach to this discipline. The recent, impressive achievements in the field of interactive theorem proving provide an interesting ground for a critical revisiting of their theses. We believe that the social nature of proof and program development is uncontroversial and ineluctable, but formal verification is not antithetical to it. Formal verification should strive not only to cope with, but to ease and enhance the collaborative, organic nature of this process, eventually helping us to master the growing complexity of scientific knowledge.

Boolean and classical restriction categories

Abstract: A restriction category is an abstract category of partial maps. A Boolean restriction category is a restriction category that supports classical (Boolean) reasoning. Such categories are models of loop-free dynamic logic that is deterministic in the sense that Q ⊂ [α]Q. Classical restriction categories are restriction categories with a locally Boolean structure: it is shown that they are precisely full subcategories of Boolean restriction categories. In particular, a Boolean restriction category may be characterised as a classical restriction category with finite coproducts in which all restriction idempotents split. Every restriction category admits a restriction embedding into a Boolean restriction category. Thus, every abstract category of partial maps admits a conservative extension that supports classical reasoning. An explicit construction of the classical completion of a restriction category is given.

An expressiveness study of priority in process calculi

Abstract: Priority is a frequently used feature of many computational systems. In this paper we study the expressiveness of two process algebras enriched with different priority mechanisms. In particular, we consider a finite (that is, recursion-free) fragment of asynchronous CCS with global priority (FAP, for short) and Phillips' CPG (CCS with local priority), and contrast their expressive power with that of two non-prioritised calculi, namely the π-calculus and its broadcast-based version, called bπ. We prove, by means of leader-election-based separation results, that, under certain conditions, there exists no encoding of FAP in π-Calculus or CPG. Moreover, we single out another problem in distributed computing, which we call the last man standing problem (LMS for short), that better reveals the gap between the two prioritised calculi above and the two non-prioritised ones, by proving that there exists no parallel-preserving encoding of the prioritised calculi in the non-prioritised calculi retaining any sincere (complete but partially correct, that is, admitting divergence or premature termination) semantics.

Editors' note: bibliometrics and the curators of orthodoxy

Abstract: Have you ever seen the Citation Indexes (CIs) for the year 1600? At that time, a very active community was working on the reconstruction of planetary movements by means of epicycles. In principle, any ellipse around the Sun may be approximated by sufficiently many epicycles around the Earth. This is a non-trivial geometrical task, especially given the lack of analytical tools (sums of series). And the books and papers of many talented geometers quoted one another. Scientific knowledge, however, was already taking other directions. Science has a certain ‘inertia’, it is prudent (at times, it has been exceedingly so, mostly for political or metaphysical reasons), but even under the best of conditions, we all know how difficult it is to accept new ideas, to let them blossom in time, away from short-term pressures.

On universal computably enumerable prefix codes

Abstract: We study computably enumerable (c.e.) prefix codes that are capable of coding all positive integers in an optimal way up to a fixed constant: these codes will be called universal. We prove various characterisations of these codes, including the following one: a c.e. prefix code is universal if and only if it contains the domain of a universal self-delimiting Turing machine. Finally, we study various properties of these codes from the points of view of computability, maximality and density.

Preface to special issue: Theory and applications of models of computation (tamc)

Abstract: Theory and Applications of Models of Computation (TAMC) is an international conference series with an interdisciplinary character bringing together researchers working in computer science, mathematics (especially logic) and the physical sciences. This interdisciplinary approach, with an emphasis on the theory of computation in a broad sense, gives the series its special appeal within China and internationally. At a time when the pressures are increasingly towards narrowly ad hoc research, and scientific fragmentation, meetings that reassert the importance of theory, fundamental concepts and a wider perspective have an important role to play.

Phase transition of multivariate polynomial systems

Abstract: A random multivariate polynomial system with more equations than variables is likely to be unsolvable. On the other hand, if there are more variables than equations, the system has at least one solution with high probability. In this paper we study in detail the phase transition between these two regimes, which occurs when the number of equations equals the number of variables. In particular, the limiting probability for no solution is 1/e at the phase transition, over a prime field. We also study the probability of having exactly s solutions, with s ≥ 1. In particular, the probability of a unique solution is asymptotically 1/e if the number of equations equals the number of variables. The probability decreases very rapidly if the number of equations increases or decreases. Our motivation is that many cryptographic systems can be expressed as large multivariate polynomial systems (usually quadratic) over a finite field. Since decoding is unique, the solution of the system must also be unique. Knowing the probability of having exactly one solution may help us to understand more about these cryptographic systems. For example, whether attacks should be evaluated by trying them against random systems depends very much on the likelihood of a unique solution.

Non-cupping, measure and computably enumerable splittings

Abstract: We show that there is a computably enumerable function f (that is, computably approximable from below) that dominates almost all functions, and f ⊕ W is incomplete for all incomplete computably enumerable sets W. Our main methodology is the LR equivalence relation on reals: A ≡LRB if and only if the notions of A-randomness and B-randomness coincide. We also show that there are c.e. sets that cannot be split into two c.e. sets of the same LR degree. Moreover, a c.e. set is low for random if and only if it computes no c.e. set with this property.

Unary automatic graphs: An algorithmic perspective

Abstract: This paper studies infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced using such operations are of finite degree and automatic over the unary alphabet (that is, they can be described by finite automata over the unary alphabet). We investigate algorithmic properties of such unfolded graphs given their finite presentations. In particular, we ask whether a given node belongs to an infinite component, whether two given nodes in the graph are reachable from one another and whether the graph is connected. We give polynomial-time algorithms for each of these questions. For a fixed input graph, the algorithm for the first question is in constant time and the second question is decided using an automaton that recognises the reachability relation in a uniform way. Hence, we improve on previous work, in which non-elementary or non-uniform algorithms were found.

A type system for discretionary access control

Abstract: Discretionary Access Control (DAC) systems provide powerful resource management mechanisms based on the selective distribution of capabilities to selected classes of principals. We study a type-based theory of DAC models for a process calculus that extends Cardelli, Ghelli and Gordon's pi-calculus with groups (Cardelli et al. 2005). In our theory, groups play the role of principals and form the unit of abstraction for our access control policies, and types allow the specification of fine-grained access control policies to govern the transmission of names, bound the (iterated) re-transmission of capabilities and predicate their use on the inability to pass them to third parties. The type system relies on subtyping to achieve a selective distribution of capabilities to the groups that control the communication channels. We show that the typing and subtyping relationships of the calculus are decidable. We also prove a type safety result, showing that in well-typed processes all names: (i) flow according to the access control policies specified by their types; and (ii) are received at the intended sites with the intended capabilities. We illustrate the expressive power and the flexibility of the typing system using several examples.

Process discovery and petri nets

Abstract: The aim of the research domain known as process mining is to use process discovery to construct a process model as an abstract representation of event logs. The goal is to build a model (in terms of a Petri net) that can reproduce the logs under consideration, and does not allow different behaviours compared with those shown in the logs. In particular, process mining aims to verify the accuracy of the model design (represented as a Petri net), basically checking whether the same net can be rediscovered. However, the main mining methods proposed in the literature have some drawbacks: the classical α-algorithm is unable to rediscover various nets, while the region-based approach, which can mine them correctly, is too complex. In this paper, we compare different approaches and propose some ideas to counter the weaknesses of the region-based approach.

Quantum multiparty communication complexity and circuit lower bounds

Abstract: We define a quantum model for multiparty communication complexity and prove a simulation theorem between the classical and quantum models. As a result, we show that if the quantum k-party communication complexity of a function f is Ω(n/2k), its classical k-party communication is Ω(n/2k/2). Finding such an f would allow us to prove strong classical lower bounds for k ≥ log n players and make progress towards solving a major open question about symmetric circuits.

Automata theory based on unsharp quantum logic

Abstract: By studying two unsharp quantum structures, namely extended lattice ordered effect algebras and lattice ordered QMV algebras, we obtain some characteristic theorems of MV algebras. We go on to discuss automata theory based on these two unsharp quantum structures. In particular, we prove that an extended lattice ordered effect algebra (or a lattice ordered QMV algebra) is an MV algebra if and only if a certain kind of distributive law holds for the sum operation. We introduce the notions of (quantum) finite automata based on these two unsharp quantum structures, and discuss closure properties of languages and the subset construction of automata. We show that the universal validity of some important properties (such as sum, concatenation and subset constructions) depend heavily on the above distributive law. These generalise results about automata theory based on sharp quantum logic.

A note on the feasibility of generalised universal composability

Abstract: In this paper we study (interpret) the precise composability guarantee of the generalised universal composability (GUC) feasibility with global setups that was proposed in the recent paper Canetti et al. (2007) from the point of view of full universal composability (FUC), that is, composability with arbitrary protocols, which was the original security goal and motivation for UC. By observing a counter-intuitive phenomenon, we note that the GUC feasibility implicitly assumes that the adversary has limited access to arbitrary external protocols. We then clarify a general principle for achieving FUC security, and propose some approaches for fixing the GUC feasibility under the general principle. Finally, we discuss the relationship between GUC and FUC from both technical and philosophical points of view. This should be helpful in gaining a precise understanding of the GUC feasibility, and for preventing potential misinterpretations and/or misuses in practice.