Partial Relations Research Articles

Axiomatising an information flow logic based on partial equivalence relations

We present a relational program logic for reasoning about information flow properties formalised in an assertion language based on partial equivalence relations. We define and prove the soundness of the logic, a proof technique for precise, logic-based information flow properties. The logic extends Hoare logic and its unary state predicates to binary PER-based predicates for relating observationally equivalent states. A salient feature of the logic is that it is capable of reasoning about programs that test on secret data in a secure manner.

On difunctions

The notion of a difunction was introduced by Jacques Riguet in 1948. Since then it has played a prominent role in database theory, type theory, program specification and process theory. The theory of difunctions is, however, less known in computing than it perhaps should be. The main purpose of the current paper is to give an account of difunction theory in relation algebra, with the aim of making the topic more mainstream.As is common with many important concepts, there are several different but equivalent characterisations of difunctionality, each with its own strength and practical significance. This paper compares different proofs of the equivalence of the characterisations.A well-known property is that a difunction is a set of completely disjoint rectangles. This property suggests the introduction of the (general) notion of the “core” of a relation; we use this notion to give a novel and, we believe, illuminating characterisation of difunctionality as a bijection between the classes of certain partial equivalence relations.

Admissible Types-to-PERs Relativization in Higher-Order Logic

Relativizing statements in Higher-Order Logic (HOL) from types to sets is useful for improving productivity when working with HOL-based interactive theorem provers such as HOL4, HOL Light and Isabelle/HOL. This paper provides the first comprehensive definition and study of types-to-sets relativization in HOL, done in the more general form of types-to-PERs (partial equivalence relations). We prove that, for a large practical fragment of HOL which includes container types such as datatypes and codatatypes, types-to-PERs relativization is admissible, in that the provability of the original, type-based statement implies the provability of its relativized, PER-based counterpart. Our results also imply the admissibility of a previously proposed axiomatic extension of HOL with local type definitions. We have implemented types-to-PERs relativization as an Isabelle tool that performs relativization of HOL theorems on demand.

Categories of partial equivalence relations as localizations

We construct a category of fibrant objectsC〈P〉 in the sense of K. Brown from any indexed frame (a kind of indexed poset generalizing triposes) P, and show that its homotopy category is the Barr-exact category C[P] of partial equivalence relations and compatible functional relations. In particular this gives a presentation of realizability toposes as homotopy categories.We give criteria for the existence of left and right derived functors to functors C〈Φ〉:C〈P〉→C〈Q〉 induced by finite-meet-preserving transformations Φ:P→Q between indexed frames.

A Myhill-Nerode theorem for register automata and symbolic trace languages

We propose a new symbolic trace semantics for register automata (extended finite state machines) which records both the sequence of input symbols that occur during a run as well as the constraints on input parameters that are imposed by this run. Our main result is a generalization of the classical Myhill-Nerode theorem to this symbolic setting. Our generalization requires the use of three relations to capture the additional structure of register automata. Location equivalence ≡l captures that symbolic traces end in the same location, transition equivalence ≡t captures that they share the same final transition, and a partial equivalence relation ≡r captures that symbolic values v and v′ are stored in the same register after symbolic traces w and w′, respectively. A symbolic language is defined to be regular if relations ≡l, ≡t and ≡r exist that satisfy certain conditions, in particular, they all have finite index. We show that the symbolic language associated to a register automaton is regular, and we construct, for each regular symbolic language, a register automaton that accepts this language. Our result provides a foundation for grey-box learning algorithms in settings where the constraints on data parameters can be extracted from code using e.g. tools for symbolic/concolic execution or tainting. Moving to a grey-box setting may overcome the scalability problems of state-of-the-art black-box learning algorithms.

The endotopism semigroups of a partial equivalence relation

The endotopism semigroups of a partial equivalence relation

Endotypes of partial equivalence relations

Endotypes of partial equivalence relations

Cautious relational clustering: A thresholding approach

We propose in this article a new relational clustering method that can return a partial answer (i.e., a set of clusterings) in some cases. Starting from relational or similarity data, we determine a partial equivalence relation defined on the set of objects (two objects are linked if they belong to the same cluster): the key idea is to allow the method to abstain on some pairwise links because they cannot be determined with enough certainty from the data. This cautious equivalence relation represents a set of possible hard clusterings which can be obtained by completing the partial relation. This formalization makes it possible to easily detect ambiguous links and to identify subsets of objects with uncertain relationship. We illustrate the potential interest of our approach as a tool for exploratory data analysis of synthetic and real data sets.

Endotypes of certain partial equivalence relations

Endotypes of certain partial equivalence relations

The Algebra of Partial Equivalence Relations

Recent work by the author with Bonchi and Sobociński shows how PROPs of linear relations (subspaces) can be presented by generators and equations via a “cube construction”, based on letting very simple structures interact according to PROP operations of sum, fibered sum and composition via a distributive law. This paper shows how the same construction can be used in a cartesian setting to obtain presentations by generators and equations for the PROP of equivalence relations and of partial equivalence relations.

Abstract effects and proof-relevant logical relations

We give a denotational semantics for a region-based effect system that supports type abstraction in the sense that only externally visible effects need to be tracked: non-observable internal modifications, such as the reorganisation of a search tree or lazy initialisation, can count as 'pure' or 'read only'. This 'fictional purity' allows clients of a module to validate soundly more effect-based program equivalences than would be possible with previous semantics. Our semantics uses a novel variant of logical relations that maps types not merely to partial equivalence relations on values, as is commonly done, but rather to a proof-relevant generalisation thereof, namely setoids. The objects of a setoid establish that values inhabit semantic types, whilst its morphisms are understood as proofs of semantic equivalence. The transition to proof-relevance solves twoawkward problems caused by naïve use of existential quantification in Kripke logical relations, namely failure of admissibility and spurious functional dependencies.

Normalization by Evaluation and Algebraic Effects

We examine the interplay between computational effects and higher types. We do this by presenting a normalization by evaluation algorithm for a language with function types as well as computational effects. We use algebraic theories to treat the computational effects in the normalization algorithm in a modular way. Our algorithm is presented in terms of an interpretation in a category of presheaves equipped with partial equivalence relations. The normalization algorithm and its correctness proofs are formalized in dependent type theory (Agda).

Modification of Generalization of Equivalence Relation

In this paper contains description of certain relations called partial equivalence relations .Moreover ,it includes the formal algorithms for determining the order of relations like partial reflexive, partial symmetric and partial transitive relations .Some examples are also presented for the clarification of these concept.

Automorphism Groups of Countably Categorical Linear Orders are Extremely Amenable

We show that the automorphism groups of countably categorical linear orders are extremely amenable. Using methods of Kechris, Pestov, and Todorcevic, we use this fact to derive a structural Ramsey theorem for certain families of finite ordered structures with finitely many partial equivalence relations with convex classes.

Determinisation of relational substitutions in ordered categories with domain

Restating a unification problem as a single relational substitution instead of as multiple functional substitutions (or terms), a solution becomes a “determiniser” arrow and allows formalisation in the context of locally ordered categories with domain. This relies on the determinacy concept of “characterisation by domain” introduced by Desharnais and Möller for Kleene algebras with domain; this is here applied in the weakest possible setting. We show how “most general determinisers” can be seen as generalisation of quotient projections of partial equivalence relations, and show a characterisation that manages to avoid using converse or symmetry by employing restricted residuals instead.

Embedding mappings and splittings with applications

We present relation-algebraic specifications of injective embedding mappings and splittings of partial equivalence relations and show in each case that the axioms characterize these constructions up to isomorphism, i.e., in an essentially unique way. Based on the specifications, we develop a relational program for computing splitting and demonstrate some applications. The examples originate from a relation-algebraic treatment of processes, graph theory, and the decomposition of specific relations.

Bunched polymorphism

We describe a polymorphic, typed lambda calculus with substructural features. This calculus extends the first-order substructural lambda calculus αλ associated with bunched logic. A particular novelty of our new calculus is the substructural treatment of second-order variables. This is accomplished through the use of bunches of type variables in typing contexts. Both additive and multiplicative forms of polymorphic abstraction are then supported. The calculus has sensible proof-theoretic properties and a straightforward categorical semantics using indexed categories. We produce a model for additive polymorphism with first-order bunching based on partial equivalence relations. We consider additive and multiplicative existential quantifiers separately from the universal quantifiers.

Controlling information release in the π-calculus

We introduce a notion of controlled information release for a typed version of the π -calculus extended with declassification primitives; this property scales to noninterference when downgrading is not allowed. We provide various characterizations of controlled release, based on a typed behavioural equivalence relative to a security level σ , which captures the idea of external observers of level σ . First, we define our security property through a universal quantification over all the possible active attackers , i.e., malicious processes which interact with the system possibly leaking secret information. Then we characterize the controlled release property in terms of an unwinding condition, which deals with so-called passive attackers trying to infer confidential information just by observing the behaviour of the system. Furthermore, we express controlled information release in terms of partial equivalence relations ( per models, for short) in the style of a stream of similar studies for imperative and multi-threaded languages. We show that the controlled release property is compositional with respect to most operators of the language leading to efficient proof techniques for the verification and the construction of (compositional) secure systems.

Parametric Domain-theoretic Models of Polymorphic Intuitionistic / Linear Lambda Calculus

We present a formalization of a version of Abadi and Plotkin's logic for parametricity for a polymorphic dual intuitionistic / linear type theory with fixed points, and show, following Plotkin's suggestions, that it can be used to define a wide collection of types, including solutions to recursive domain equations. We further define a notion of parametric LAPL-structure and prove that it provides a sound and complete class of models for the logic, and conclude that such models have solutions for a wide class of recursive domain equations. Finally, we present a concrete parametric LAPL-structure based on suitable categories of partial equivalence relations over a universal model of the untyped lambda calculus.

Fast and loose reasoning is morally correct

Functional programmers often reason about programs as if they were written in a total language, expecting the results to carry over to non-total (partial) languages. We justify such reasoning.Two languages are defined, one total and one partial, with identical syntax. The semantics of the partial language includes partial and infinite values, and all types are lifted, including the function spaces. A partial equivalence relation (PER) is then defined, the domain of which is the total subset of the partial language. For types not containing function spaces the PER relates equal values, and functions are related if they map related values to related values.It is proved that if two closed terms have the same semantics in the total language, then they have related semantics in the partial language. It is also shown that the PER gives rise to a bicartesian closed category which can be used to reason about values in the domain of the relation.