Indistinguishability Relations Research Articles

Logical equivalences, homomorphism indistinguishability, and forbidden minors

Two graphs G and H are homomorphism indistinguishable over a graph class F if for all graphs F∈F the number of homomorphisms from F to G is equal to the number of homomorphisms from F to H. Many graph isomorphism relaxations such as (quantum) isomorphism, spectral, and logical equivalences can be characterised as homomorphism indistinguishability relations over certain graph classes.Abstracting from the wealth of such instances, we show in this paper that equivalences w.r.t. any self-complementarity logic admitting a characterisation as homomorphism indistinguishability relation can be characterised by homomorphism indistinguishability over a minor-closed graph class. Self-complementarity is a mild property satisfied by most well-studied logics.Furthermore, we classify all graph classes which are in a sense finite and satisfy the maximality condition of being homomorphism distinguishing closed, i.e. adding any graph to the class strictly refines its homomorphism indistinguishability relation. Thereby, we answer various questions raised by Roberson (2022) on properties of the homomorphism distinguishing closure.

Relaxed Indistinguishability Relations and Relaxed Metrics: The Aggregation Problem

The main purpose of this paper is to study the relationship between those functions that aggregate relaxed indistinguishability fuzzy relations with respect to a collection of t-norms and those functions that merge relaxed pseudo-metrics, extending the classical approach explored for pseudo-metrics and indistinguishability fuzzy relations. Special attention is paid to the distinguished class of SSI-relaxed indistinguishability fuzzy relations showing that functions merging this special type of relaxed indistinguishability fuzzy relations can be expressed through functions aggregating SSD-relaxed pseudo-metrics. Outstanding differences between those functions aggregating indistinguishability fuzzy relations and those that aggregate their counterpart separating points are shown.

A policy-aware epistemic framework for social networks

Abstract We provide a semantic framework to specify information propagation in social networks; our semantic framework features both the operational description of information propagation and the epistemic aspects in social networks. In our framework, based on annotated labelled transition systems, actions are decorated with function views to specify different types of announcements. Our function views enforce various common types of local privacy policies, i.e. those policies concerning a single action. Furthermore, we specify global privacy policies, those concerning multiple actions, using a combination of modal $\mu $-calculus and epistemic logic. To illustrate the applicability of our framework, we apply it to the specification of a real-world case study. As a fundamental property for the epistemic aspect of our semantic model, we prove that its indistinguishability relations are equivalence relations, namely they are reflexive, symmetric and transitive. We also study the complexity bounds for the model-checking problem concerning a subset of our logic and show that model checking is PSPACE-complete for the studied subset.

Observation and Distinction: Representing Information in Infinite Games

We compare two approaches for modelling imperfect information in infinite games by using finite-state automata. The first, more standard approach views information as the result of an observation process driven by a sequential Mealy machine. In contrast, the second approach features indistinguishability relations described by synchronous two-tape automata. The indistinguishability-relation model turns out to be strictly more expressive than the one based on observations. We present a characterisation of the indistinguishability relations that admit a representation as a finite-state observation function. We show that the characterisation is decidable, and give a procedure to construct a corresponding Mealy machine whenever one exists.

The aggregation of transitive fuzzy relations revisited

In this paper, we consider the problem of aggregating a collection of transitive fuzzy binary relations in such a way that the aggregation process preserves transitivity. Specifically, we focus our efforts on the characterization of those functions that aggregate a collection of fuzzy binary relations which are transitive with respect to a collection of t-norms preserving the transitivity. We characterize them in terms of triangular triplets. Further, the relationship between triangular triplets, the monotonicity of the aggregation function and an appropriate dominance notion is explicitly stated. Special attention is paid to a few classes of transitive fuzzy binary relations that are relevant in the literature. Concretely, we describe, in terms of triangular triplets, those functions that aggregate a collection of fuzzy pre-orders, fuzzy partial orders, relaxed indistinguishability relations, indistinguishability relations and equalities. A surprising relationship between functions that aggregate transitive fuzzy relations into a TM-transitive fuzzy relation and those that aggregate relaxed indistinguishability relations is shown. A few consequences of the new results are also provided for those cases in which all the t-norms of the given collection are the same. Some celebrated results are retrieved as a particular case from the exposed theory.

On the representation of local indistinguishability operators

This paper studies local indistinguishability operators, i.e., symmetric and transitive fuzzy relations that do not need to be reflexive. This is an important generalization of global indistinguishability relations (fuzzy relations satisfying the reflexivity property in addition) because there are interesting families of fuzzy relations that are non-reflexive. One case are decomposable relations, that are generated by a fuzzy subset and contains the t-norms as an important subfamily. Also the relations associated naturally to fuzzy subgroups are local indistinguishability operators. In this paper these relations will be studied stressing the way they can be generated. A representation theorem will be proved and related to the concepts of extensionality and of fuzzy rough set. Decomposable local indistinguishability operators will also be studied and related with one-dimensional ones in the sense of the previous representation theorem. The presence of these relations in the study of fuzzy subgroups will also be analyzed.

Constructing a classifier with patterns

The goal of pattern extraction is to find some regularities among the objects included in a data base. When these objects are labeled the patterns can be used for the classification of unseen objects. Most of the algorithms for pattern extraction are based on finding the most frequent structures (or patterns). However, classifiers formed by the set of the most frequent patterns are not necessarily good classifiers since frequent patterns are, most of times, presents in objects of all the classes. In the current paper we propose a method to find patterns that, with a simple post-process, will form an appropriate model for classification. This method resembles to the procedure to grow decision trees, but relevant feature extraction is performed using JADE, a method based on the minimization of the distance between two indistinguishability relations.

Simplicial complexes and closure systems induced by indistinguishability relations

In this paper, we develop in a more general mathematical context the notion of indistinguishability, which in graph theory has recently been investigated as a symmetry relation with respect to a fixed vertex subset. The starting point of our analysis is to consider a set Ω of functions defined on a universe set U and to define an equivalence relation ≡A on U for any subset A⊆Ω in the following way: u≡Au′ if a(u)=a(u′) for any function a∈A. By means of this family of relations, we introduce the indistinguishability relation ≈ on the power set P(Ω) as follows: for A,A′∈P(Ω), we set A≈A′ if the relations ≡A and ≡A′ coincide. We use then the indistinguishability relation ≈ to introduce several set families on Ω that have interesting order, matroidal and combinatorial properties. We call the above set families the indistinguishability structures of the function system(U,Ω). Furthermore, we obtain a closure system and an abstract simplicial complex interacting each other by means of three hypergraphs having relevance in both theoretical computer science and graph theory. The first part of this paper is devoted to investigate the basic mathematical properties of the indistinguishability structures for arbitrary function systems. The second part deals with some specific cases of study derived from simple undirected graphs and the usual Euclidean real line.

A probabilistic approach to rough set theory with modal logic perspective

We propose the notion of probabilistic information system (PIS) to capture situations where information regarding attributes of objects are not precise, but given in terms of probability. Notions of indistinguishability relations and corresponding approximation operators based on PISs are proposed and studied. It is shown that the deterministic information systems (DISs), incomplete information systems (IISs) and non-deterministic information systems (NISs) are all special instances of PISs. Moreover, the approximation operators defined on DIS (relative to indiscernibility), IISs and NISs (relative to similarity relations) are all originated from a single approximation operator defined on PISs. Further, a logic LPIS for PISs is proposed that can be used to reason about the proposed approximation operators. A sound and complete deductive system for the logic is given. Decidability of the logic is also proved.

Axiomatization of a Branching Time Logic with Indistinguishability Relations

Trees with indistinguishability relations provide a semantics for a temporal language “composed by” the Peircean tense operators and the Ockhamist modal operator. In this paper, a finite axiomatization with a non standard rule for this language interpreted over bundled trees with indistinguishability relations is given. This axiomatization is proved to be sound and strongly complete.

Logic of confidence

The article studies knowledge in multiagent systems where data available to the agents may have small errors. To reason about such uncertain knowledge, a formal semantics is introduced in which indistinguishability relations, commonly used in the semantics for epistemic logic S5, are replaced with metrics to capture how much two epistemic worlds are different from an agent’s point of view. The main result is a logical system sound and complete with respect to the proposed semantics.

Deciding knowledge in security protocols under some e-voting theories

In the last decade, formal methods have proved their interest when analyzing security protocols. Security protocols require in particular to reason about the attacker knowledge. Two standard notions are often considered in formal approaches: deducibility and indistinguishability relations. The first notion states whether an attacker can learn the value of a secret, while the latter states whether an attacker can notice some difference between protocol runs with different values of the secret. Several decision procedures have been developed so far for both notions but none of them can be applied in the context of e-voting protocols, which require dedicated cryptographic primitives. In this work, we show that both deduction and indistinguishability are decidable in polynomial time for two theories modeling the primitives of e-voting protocols.

Decidability and Combination Results for Two Notions of Knowledge in Security Protocols

In formal approaches, messages sent over a network are usually modeled by terms together with an equational theory, axiomatizing the properties of the cryptographic functions (encryption, exclusive or, ...). The analysis of cryptographic protocols requires a precise understanding of the attacker knowledge. Two standard notions are usually considered: deducibility and indistinguishability. Those notions are well-studied and several decidability results already exist to deal with a variety of equational theories. Most of the existing results are dedicated to specific equational theories and only few results, especially in the case of indistinguishability, have been obtained for equational theories with associative and commutative properties $(\textsf{AC})$ . In this paper, we show that existing decidability results can be easily combined for any disjoint equational theories: if the deducibility and indistinguishability relations are decidable for two disjoint theories, they are also decidable for their union. We also propose a general setting for solving deducibility and indistinguishability for an important class (called monoidal) of equational theories involving $\textsf{AC}$ operators. As a consequence of these two results, new decidability and complexity results can be obtained for many relevant equational theories.

Deciding knowledge in security protocols under equational theories

The analysis of security protocols requires precise formulations of the knowledge of protocol participants and attackers. In formal approaches, this knowledge is often treated in terms of message deducibility and indistinguishability relations. In this paper we study the decidability of these two relations. The messages in question may employ functions (encryption, decryption, etc.) axiomatized in an equational theory. One of our main positive results says that deducibility and indistinguishability are both decidable in polynomial time for a large class of equational theories. This class of equational theories is defined syntactically and includes, for example, theories for encryption, decryption, and digital signatures. We also establish general decidability theorems for an even larger class of theories. These theorems require only loose, abstract conditions, and apply to many other useful theories, for example with blind digital signatures, homomorphic encryption, XOR, and other associative–commutative functions.

Indistinguishability relations in Dempster–Shafer theory of evidence

Each theory or model implicitly defines its inherent notion of equality for the objects in question. In turn, this equality, and its counterpart, the mathematical concept of equivalence, provides the basis on which to establish classification mechanisms for the domain at hand. Nevertheless, equivalence relations have not been proved to be sufficiently suited to capturing the underlying structure when dealing with domains pervaded with uncertainty. Therefore, the need for a more flexible definition of equality brings the concept of T-indistinguishability operator on to the scene. In this paper, we study the notion of indistinguishability within the context of the Dempster–Shafer theory of evidence and provide effective definitions and procedures for computing the T-indistinguishability operator associated with a given body of evidence. We also show how these procedures could also be adapted in order to provide a new method for tackling the problem of belief function approximation based on the concept of T-preorder.

The duality principle in fuzzy set theory

Abstract A rigorous mathematical approach to extending nonfuzzy structures to fuzzy sets is presented. This approach is based on classical ideas of duality in analysis and inverse image in set theory. We apply these new techniques to characterizing indistinguishability relations, fuzzy interval orders, and fuzzy operations. Zadeh's extension principle is characterized in terms of the duality principle.