A Coalgebraic Semantics for Epistemic Programs

This paper gives a fresh look at my previous work on “epistemic actions” and information updates in distributed systems, from a coalgebraic perspective. I show that the “relational” semantics of epistemic programs, given in [BMS2] in terms of epistemic updates, can be understood in terms of functors on the category of coalgebras and natural transformations associated to them. Then, I introduce a new, alternative, more refined semantics for epistemic programs: programs as “epistemic coalgebras”. I argue for the advantages of this second semantics, from a semantic, heuristic, syntactical and proof-theoretic point of view. Finally, as a step towards a generalization, I show these concepts make sense for other functors, and that apparently unrelated concepts, such as Bayesian belief updates and process transformations, can be seen to arise in the same way as our “epistemic actions”.     

Towards coalgebraic behaviourism

In this paper we show that it is possible to model observable behaviour of coalgebras independently from their internal dynamics, but within the general framework of representing behaviour by a map into a “final” coalgebra.In the first part of the paper we characterise Set-endofunctors F with the property that bisimilarity of elements of F-coalgebras coincides with having the same observable behaviour. We show that such functors have the final coalgebra of a rather simple nature, and preserve some weak pullbacks. We also show that this is the case if and only if F-bisimilarity corresponds to logical equivalence in the finitary fragment of the coalgebraic logic.In the second part of the paper, we present a construction of a “final” coalgebra that captures the observable behaviour of F-coalgebras. We keep the word “final” quoted since the object we are going to construct need not belong to the original category. The construction is carried out for arbitrary Set-endofunctor F, throughout the construction we remain in Set, but the price to pay is the introduction of new morphisms. The paper concludes with a hint to a possible application to modelling weak bisimilarity for coalgebras.     

A modal proof theory for final polynomial coalgebras

An infinitary proof theory is developed for modal logics whose models are coalgebras of polynomial functors on the category of sets. The canonical model method from modal logic is adapted to construct a final coalgebra for any polynomial functor. The states of this final coalgebra are certain “maximal” sets of formulas that have natural syntactic closure properties.

The syntax of these logics extends that of previously developed modal languages for polynomial coalgebras by adding formulas that express the “termination” of certain functions induced by transition paths. A completeness theorem is proven for the logic of functors which have the Lindenbaum property that every consistent set of formulas has a maximal extension. This property is shown to hold if the deducibility relation is generated by countably many inference rules.

A counter-example to completeness is also given. This is a polynomial functor that is not Lindenbaum: it has an uncountable set of formulas that is deductively consistent but has no maximal extension and is unsatisfiable, even though all of its countable subsets are satisfiable.     

Extensional PERs

A class of Partial Equivalence Relations (PERs) is described such that the resulting full subcategory of the realizability universe has the expected properties of a good category of CPOs; that is, it is a Cartesian closed category and every endomorphism has a canonical fixed point. Moreover, the reflection functor into the subcategory of strict maps (usually called the “lifting operation”) yields a good notion of “partial map,” a necessary condition if one wishes to maintain a connection between strict maps and partial maps. It is shown that all functors that arise in practice have canonical invariant objects; hence a host of domain equations are guaranteed to have solutions.     

Simulations in Coalgebra

A new approach to simulations is proposed within the theory of coalgebras by taking a notion of order on a functor as primitive. Such an order forms a basic building block for a “lax relation lifting”, or “relator” as used by other authors. Simulations appear as coalgebras of this lifted functor, and similarity as greatest simulation. Two-way similarity is then similarity in both directions. In general, it is different from bisimilarity (in the usual coalgebraic sense), but a sufficient condition is formulated (and illustrated) to ensure that bisimilarity and two-way similarity coincide. Also, a distributive law is identified which ensures that similarity on a final coalgebra forms a dcpo structure.     

Reductions among number theoretic problems

Many number theoretic problems such as integer factorization and the discrete logarithm problem have defied all attempts to classify their complexities. Thirteen such problems are considered, none of which is known either to have a deterministic polynomial time solution, or to be complete for any natural complexity class. Failing this, the next best goal is to determine which among these are the “easiest” and which are the “hardest” problems. Toward this end, this paper gives an overview of reductions among the problems. Two reductions are new: a deterministic polynomial time reduction from squarefreeness to Euler's function φ(n), and a probabilistic polynomial time reduction from order modulo a prime power to discrete logarithm modulo a prime power.     

Dimension elevation formula for Chebyshevian blossoms

A given polynomial of degree less than or equal to n naturally “blossoms” into a function of n variables called its blossom. Considered as a polynomial function of degree less than or equal to (n+1) it “blossoms” into a “new” blossom which is now a function of (n+1) variables. A classical formula expresses any value of this new blossom as a strictly convex combination of (n+1) values of the initial one. We establish a similar formula for Chebyshevian blossoms.     

Comprehension for Coalgebras

The notion of an endofunctor having “greatest subcoalgebras” is introduced as a form of comprehension. This notion is shown to be instrumental in giving a systematic and abstract proof of the existence of limits for coalgebras—proved earlier by Worrell and by Gumm & Schröder. These insights, in dual form, are used to reinvestigate colimits for algebras in terms of “least quotient algebras”—leading to a uniform approach to limits of coalgebras and colimits of algebras. Finally, at an abstract level of fibrations, an equivalence is established between having greatest subcoalgebras (in a base category of types) and greatest invariants (in a total category of predicates).     

Error-Free and Best-Fit Extensions of Partially Defined Boolean Functions

In this paper, we address a fundamental problem related to the induction of Boolean logic: Given a set of data, represented as a set of binary “truen-vectors” (or “positive examples”) and a set of “falsen-vectors” (or “negative examples”), we establish a Boolean function (or an extension)f, so thatfis true (resp., false) in every given true (resp., false) vector. We shall further require that such an extension belongs to a certain specified class of functions, e.g., class of positive functions, class of Horn functions, and so on. The class of functions represents our a priori knowledge or hypothesis about the extensionf, which may be obtained from experience or from the analysis of mechanisms that may or may not cause the phenomena under consideration. The real-world data may contain errors, e.g., measurement and classification errors might come in when obtaining data, or there may be some other influential factors not represented as variables in the vectors. In such situations, we have to give up the goal of establishing an extension that is perfectly consistent with the given data, and we are satisfied with an extensionfhaving the minimum number of misclassifications. Both problems, i.e., the problem of finding an extension within a specified class of Boolean functions and the problem of finding a minimum error extension in that class, will be extensively studied in this paper. For certain classes we shall provide polynomial algorithms, and for other cases we prove their NP-hardness.     

Preprocessing of Intractable Problems

Some computationally hard problems, e.g., deduction in logical knowledge bases– are such that part of an instance is known well before the rest of it, and remains the same for several subsequent instances of the problem. In these cases, it is useful to preprocess off-line this known part so as to simplify the remaining on-line problem. In this paper we investigate such a technique in the context of intractable, i.e., NP-hard, problems. Recent results in the literature show that not all NP-hard problems behave in the same way: for some of them preprocessing yields polynomial-time on-line simplified problems (we call them compilable), while for other ones their compilability implies some consequences that are considered unlikely. Our primary goal is to provide a sound methodology that can be used to either prove or disprove that a problem is compilable. To this end, we define new models of computation, complexity classes, and reductions. We find complete problems for such classes, “completeness” meaning they are “the less likely to be compilable.” We also investigate preprocessing that does not yield polynomial-time on-line algorithms, but generically “decreases” complexity. This leads us to define “hierarchies of compilability,” that are the analog of the polynomial hierarchy. A detailed comparison of our framework to the idea of “parameterized tractability” shows the differences between the two approaches.     

No-Longer-Foreign: Teaching an ML compiler to speak C “natively”

We present a new foreign-function interface for SML/NJ. It is based on the idea of data-level interoperability—the ability of ML programs to inspect as well as manipulate C data structures directly.The core component of this work is an encoding of the almost² complete C type system in ML types. The encoding makes extensive use of a “folklore” typing trick, taking advantage of ML's polymorphism, its type constructors, its abstraction mechanisms, and even functors. A small low-level component which deals with C struct and union declarations as well as program linkage is hidden from the programmer's eye by a simple program-generator tool that translates C declarations to corresponding ML glue code.     

Validating for Liveness in Hidden Adversary Systems

Multi-stream interactive systems can be seen as “hidden adversary” systems (HAS), where the observable behaviour on any interaction channel is affected by interactions happening on other channels. One way of modelling HAS is in the form of a multi-process I/O automata, where each interacting process appears as a token in a shared state space. Constraints in the state space specify how the dynamics of one process affects other processes. We define the “liveness criterion” of each process as the end objective to be achieved by the process. The problem now for each process is to achieve this objective in the face of unforeseen interferences from other processes. In an earlier paper, it was proposed that this uncertainty can be mitigated by collaboration among the disparate processes. Two types of collaboration philosophies were also suggested: altruistic collaboration and pragmatic collaboration. This paper addresses the HAS validation problem where processes collaborate altruistically.     

Probabilistic π-Calculus and Event Structures

This paper proposes two semantics of a probabilistic variant of the π-calculus: an interleaving semantics in terms of Segala automata and a true concurrent semantics, in terms of probabilistic event structures. The key technical point is a use of types to identify a good class of non-deterministic probabilistic behaviours which can preserve a compositionality of the parallel operator in the event structures and the calculus. We show an operational correspondence between the two semantics. This allows us to prove a “probabilistic confluence” result, which generalises the confluence of the linearly typed π-calculus.     

Process expressions and Hoare's logic: Showing an irreconcilability of context-free recursion with Scott's induction rule

In this paper processes specifiable over a non-uniform language are considered. The language contains constants for a set of atomic actions and constructs for alternative and sequential composition. Furthermore it provides a mechanism for specifying processes recursively (including nested recursion). We consider processes as having a state: atomic actions are to be specified in terms of observable behaviour (relative to initial states) and state transformations. Any process having some initial state can be associated with a transition system representing all possible courses of execution. This leads to an operational semantics in the style of Plotkin. The partial correctness assertion {α} p{β} expresses that for any transition system associated with the process p and having some initial state satisfying α, its final states representing successful execution satisfy β. A logic in the style of Hoare, containing a proof system for deriving partial correctness assertions, is presented. This proof system is sound and relatively complete, so any partial correctness assertion can be evaluated by investigating its derivability. Included is a short discussion about the extension of the process language with “guarded recursion”. It appears that such an extension violates the completeness of the Hoare logic. This reveals a remarkable property of Scott's induction rule in the context of non-determinism: only regular recursion allows a completeness result.     

On the controllability of linear juggling mechanical systems

This paper deals with the controllability of a class of nonsmooth complementarity mechanical systems. Due to their particular structure they can be decomposed into an “object” and a “robot”, consequently they are named juggling systems. It is shown that the accessibility of the “object” can be characterized by nonlinear constrained equations, or generalized equations. Examples are presented, including a simple model of backlash. The main focus of the work is about linear jugglers.     

Control design for recycled, multi unit processes

An integrated multi-unit chemical plant presents a challenging control design problem due to the existence of recycling streams. In this paper, we develop a framework for analyzing the effects of recycling dynamics on closed-loop performance from which a systematic design of a decentralized control system for a recycled, multi-unit plant is established. In the proposed approach, the recycled streams are treated as unmodelled dynamics of the “unit” model so that their effects on closed-loop stability and performance can be analyzed using the robust control theory. As a result, two measures are produced: (1) the ν-gap metric, which quantifies the strength of recycling effects, and (2) the maximum stability margin of “unit” controller, which represents the ability of the “unit” controller to compensate for such effects. A simple rule for the “unit” control design is then established using the combined two measures in order to guarantee the attainment of good overall closed-loop performances. As illustrated by several design examples, the controllability of a recycled, multi unit process under a decentralized “unit” controller can be determined without requiring any detailed design of the “unit” controller because the simple rule is calculated from the open-loop information only.