arXiv subjects

Roman Kuznets

Publications and source records attributed to Roman Kuznets.

At least 19 recordsLinked to original sources

arXiv2026

Interpolation in Proof Theory

This chapter provides a comprehensive overview of proof-theoretic methods for establishing interpolation properties across a range of logics, including classical, intuitionistic, modal, and substructural logics. Central to the discussion are two foundational techniques: Maehara's method for Craig interpolation and Pitts' method for uniform interpolation. The chapter demonstrates how these methods lead to results on the existence of well-behaved proof systems in the contemporary framework of universal proof theory and how they provide a road map for constructing interpolation proofs using modern proof formalisms. The emphasis of the chapter is on constructive, modular, and syntax-driven techniques that illuminate deeper connections between interpolation properties and proof systems.

cs.LO↗

arXiv2025

Agent Interpolation for Knowledge

We define a new type of proof formalism for multi-agent modal logics with S5-type modalities. This novel formalism combines the features of hypersequents to represent S5 modalities with nested sequents to represent the T-like modality alternations. We show that the calculus is sound and complete, cut-free, and terminating and yields decidability and the finite model property for multi-agent S5. We also use it to prove the Lyndon (and hence Craig) interpolation property for multi-agent S5, considering not only propositional atoms but also agents to be part of the common language. Finally, we discuss the difficulties on the way to extending these results to the logic of distributed knowledge and to deductive interpolation.

cs.LO↗

arXiv2024

Bisimulation for Impure Simplicial Complexes

As an alternative to Kripke models, simplicial complexes are a versatile semantic primitive on which to interpret epistemic logic. Given a set of vertices, a simplicial complex is a downward closed set of subsets, called simplexes, of the vertex set. A maximal simplex is called a facet. Impure simplicial complexes represent that some agents (processes) are dead. It is known that impure simplicial complexes categorically correspond to so-called partial epistemic (Kripke) models. In this contribution, we define a notion of bisimulation to compare impure simplicial complexes and show that it has the Hennessy-Milner property. These results are for a logical language including atoms that express whether agents are alive or dead. Without these atoms no reasonable standard notion of bisimulation exists, as we amply justify by counterexamples, because such a restricted language is insufficiently expressive.

cs.LO↗

arXiv2024

Consistent Update Synthesis via Privatized Beliefs

Kripke models are an effective and widely used tool for representing epistemic attitudes of agents in multi-agent systems, including distributed systems. Dynamic Epistemic Logic (DEL) adds communication in the form of model transforming updates. Private communication is key in distributed systems as processes exchanging (potentially corrupted) information about their private local state should not be detectable by any other processes. This focus on privacy clashes with the standard DEL assumption for which updates are applied to the whole Kripke model, which is usually commonly known by all agents, potentially leading to information leakage. In addition, a commonly known model cannot minimize the corruption of agents' local states due to fault information dissemination. The contribution of this paper is twofold: (I) To represent leak-free agent-to-agent communication, we introduce a way to synthesize an action model which stratifies a pointed Kripke model into private agent-clusters, each representing the local knowledge of the processes: Given a goal formula $\varphi$ representing the effect of private communication, we provide a procedure to construct an action model that (a) makes the goal formula true, (b) maintain consistency of agents' beliefs, if possible, without causing "unrelated" beliefs (minimal change) thus minimizing the corruption of local states in case of inconsistent information. (II) We introduce a new operation between pointed Kripke models and pointed action models called pointed updates which, unlike the product update operation of DEL, maintain only the subset of the world-event pairs that are reachable from the point, without unnecessarily blowing up the model size.

cs.LO↗

arXiv2024

Communication Modalities

Epistemic analysis of distributed systems is one of the biggest successes among applications of logic in computer science. The reason for that is that agents' actions are necessarily guided by their knowledge. Thus, epistemic modal logic, with its knowledge and belief modalities (and group versions thereof), has played a vital role in establishing both impossibility results and necessary conditions for solvable distributed tasks. In distributed systems, knowledge is largely attained via communication. It has been standard in both distributed systems and dynamic epistemic logic to treat incoming messages as trustworthy, thus, creating difficulties in the epistemic analysis of byzantine distributed systems where faulty agents may lie. In this paper, we argue that handling such communication scenarios calls for additional modalities representing the informational content of messages that should not be taken at face value. We present two such modalities: hope for the case of fully byzantine agents and creed for non-uniform communication protocols in general.

cs.DC↗

arXiv2024

A Logic for Repair and State Recovery in Byzantine Fault-tolerant Multi-agent Systems

We provide an epistemic logical language and semantics for the modeling and analysis of byzantine fault-tolerant multi-agent systems. This not only facilitates reasoning about the agents' fault status but also supports model updates for implementing repair and state recovery. For each agent, besides the standard knowledge modality our logic provides an additional modality called hope, which is capable of expressing that the agent is correct (not faulty), and also dynamic modalities enabling change of the agents' correctness status. These dynamic modalities are interpreted as model updates that come in three flavours: fully public, more private, or involving factual change. We provide complete axiomatizations for all these variants in the form of reduction systems: formulas with dynamic modalities are equivalent to formulas without. Therefore, they have the same expressivity as the logic of knowledge and hope. Multiple examples are provided to demonstrate the utility and flexibility of our logic for modeling a wide range of repair and state recovery techniques that have been implemented in the context of fault-detection, isolation, and recovery (FDIR) approaches in fault-tolerant distributed computing with byzantine agents.

cs.DC↗

arXiv2023

A priori Belief Updates as a Method for Agent Self-Recovery

Standard epistemic logic is concerned with describing agents' epistemic attitudes given the current set of alternatives the agents consider possible. While distributed systems can (and often are) discussed without mentioning epistemics, it has been well established that epistemic phenomena lie at the heart of what agents, or processes, can and cannot do. Dynamic epistemic logic (DEL) aims to describe how epistemic attitudes of the agents/processes change based on the new information they receive, e.g., based on their observations of events and actions in a distributed system. In a broader philosophical view, this appeals to an a posteriori kind of reasoning, where agents update the set of alternatives considered possible based on their "experiences." Until recently, there was little incentive to formalize a priori reasoning, which plays a role in designing and maintaining distributed systems, e.g., in determining which states must be considered possible by agents in order to solve the distributed task at hand, and consequently in updating these states when unforeseen situations arise during runtime. With systems becoming more and more complex and large, the task of fixing design errors "on the fly" is shifted to individual agents, such as in the increasingly popular self-adaptive and self-organizing (SASO) systems. Rather than updating agents' a posteriori beliefs, this requires modifying their a priori beliefs about the system's global design and parameters. The goal of this paper is to provide a formalization of such a priori reasoning by using standard epistemic semantic tools, including Kripke models and DEL-style updates, and provide heuristics that would pave the way to streamlining this inherently non-deterministic and ad hoc process for SASO systems.

cs.DC↗

arXiv2023

On Two- and Three-valued Semantics for Impure Simplicial Complexes

Simplicial complexes are a convenient semantic primitive to reason about processes (agents) communicating with each other in synchronous and asynchronous computation. Impure simplicial complexes distinguish active processes from crashed ones, in other words, agents that are alive from agents that are dead. In order to rule out that dead agents reason about themselves and about other agents, three-valued epistemic semantics have been proposed where, in addition to the usual values true and false, the third value stands for undefined: the knowledge of dead agents is undefined and so are the propositional variables describing their local state. Other semantics for impure complexes are two-valued where a dead agent knows everything. Different choices in designing a semantics produce different three-valued semantics, and also different two-valued semantics. In this work, we categorize the available choices by discounting the bad ones, identifying the equivalent ones, and connecting the non-equivalent ones via a translation. The main result of the paper is identifying the main relevant distinction to be the number of truth values and bridging this difference by means of a novel embedding from three- into two-valued semantics. This translation also enables us to highlight quite fundamental modeling differences underpinning various two- and three-valued approaches in this area of combinatorial topology. In particular, pure complexes can be defined as those invariant under the translation.

cs.DC↗

arXiv2023

Extensions of K5: Proof Theory and Uniform Lyndon Interpolation

We introduce a Gentzen-style framework, called layered sequent calculi, for modal logic K5 and its extensions KD5, K45, KD45, KB5, and S5 with the goal to investigate the uniform Lyndon interpolation property (ULIP), which implies both the uniform interpolation property and the Lyndon interpolation property. We obtain complexity-optimal decision procedures for all logics and present a constructive proof of the ULIP for K5, which to the best of our knowledge, is the first such syntactic proof. To prove that the interpolant is correct, we use model-theoretic methods, especially bisimulation modulo literals.

cs.LO↗

arXiv2023

Intuitionistic S4 is decidable

In this paper we demonstrate decidability for the intuitionistic modal logic S4 first formulated by Fischer Servi. This solves a problem that has been open for almost thirty years since it had been posed in Simpson's PhD thesis in 1994. We obtain this result by performing proof search in a labelled deductive system that, instead of using only one binary relation on the labels, employs two: one corresponding to the accessibility relation of modal logic and the other corresponding to the order relation of intuitionistic Kripke frames. Our search algorithm outputs either a proof or a finite counter-model, thus, additionally establishing the finite model property for intuitionistic S4, which has been another long-standing open problem in the area.

cs.LO↗

arXiv2022

Impure Simplicial Complexes: Complete Axiomatization

Combinatorial topology is used in distributed computing to model concurrency and asynchrony. The basic structure in combinatorial topology is the simplicial complex, a collection of subsets called simplices of a set of vertices, closed under containment. Pure simplicial complexes describe message passing in asynchronous systems where all processes (agents) are alive, whereas impure simplicial complexes describe message passing in synchronous systems where processes may be dead (have crashed). Properties of impure simplicial complexes can be described in a three-valued multi-agent epistemic logic where the third value represents formulae that are undefined, e.g., the knowledge and local propositions of dead agents. In this work we present an axiomatization for the logic of the class of impure complexes and show soundness and completeness. The completeness proof involves the novel construction of the canonical simplicial model and requires a careful manipulation of undefined formulae.

cs.LO↗

arXiv2021

Fire!

In this paper, we provide an epistemic analysis of a simple variant of the fundamental consistent broadcasting primitive for byzantine fault-tolerant asynchronous distributed systems. Our Firing Rebels with Relay (FRR) primitive enables agents with a local preference for acting/not acting to trigger an action (FIRE) at all correct agents, in an all-or-nothing fashion. By using the epistemic reasoning framework for byzantine multi-agent systems introduced in our TARK'19 paper, we develop the necessary and sufficient state of knowledge that needs to be acquired by the agents in order to FIRE. It involves eventual common hope (a modality related to belief), which we show to be attained already by achieving eventual mutual hope in the case of FRR. We also identify subtle variations of the necessary and sufficient state of knowledge for FRR for different assumptions on the local preferences.

cs.DC↗

arXiv2021

Uniform interpolation via nested sequents and hypersequents

A modular proof-theoretic framework was recently developed to prove Craig interpolation for normal modal logics based on generalizations of sequent calculi (e.g., nested sequents, hypersequents, and labelled sequents). In this paper, we turn to uniform interpolation, which is stronger than Craig interpolation. We develop a constructive method for proving uniform interpolation via nested sequents and apply it to reprove the uniform interpolation property for normal modal logics $\mathsf{K}$, $\mathsf{D}$, and $\mathsf{T}$. We then use the know-how developed for nested sequents to apply the same method to hypersequents and obtain the first direct proof of uniform interpolation for $\mathsf{S5}$ via a cut-free sequent-like calculus. While our method is proof-theoretic, the definition of uniform interpolation for nested sequents and hypersequents also uses semantic notions, including bisimulation modulo an atomic proposition.

cs.LO↗

arXiv2021

Wanted Dead or Alive : Epistemic logic for impure simplicial complexes

We propose a logic of knowledge for impure simplicial complexes. Impure simplicial complexes represent synchronous distributed systems under uncertainty over which processes are still active (are alive) and which processes have failed or crashed (are dead). Our work generalizes the logic of knowledge for pure simplicial complexes, where all processes are alive, by Goubault et al. In our semantics, given a designated face in a complex, a formula can only be true or false there if it is defined. The following are undefined: dead processes cannot know or be ignorant of any proposition, and live processes cannot know or be ignorant of factual propositions involving processes they know to be dead. The semantics are therefore three-valued, with undefined as the third value. We propose an axiomatization that is a version of the modal logic S5. We also show that impure simplicial complexes correspond to certain Kripke models where agents' accessibility relations are equivalence relations on a subset of the domain only. This work extends a WoLLIC 21 conference publication with the same title.

cs.DC↗

arXiv2020

The Persistence of False Memory: Brain in a Vat Despite Perfect Clocks

Recently, a detailed epistemic reasoning framework for multi-agent systems with byzantine faulty asynchronous agents and possibly unreliable communication was introduced. We have developed a modular extension framework implemented on top of it, which allows to encode and safely combine additional system assumptions commonly used in the modeling and analysis of fault-tolerant distributed systems, like reliable communication, time-bounded communication, multicasting, synchronous and lock-step synchronous agents and even agents with coordinated actions. We use this extension framework for analyzing basic properties of synchronous and lock-step synchronous agents, such as the agents' local and global fault detection abilities. Moreover, we show that even the perfectly synchronized clocks available in lock-step synchronous systems cannot be used to avoid "brain-in-a-vat" scenarios.

cs.MA↗

arXiv2019

Causality and Epistemic Reasoning in Byzantine Multi-Agent Systems

Causality is an important concept both for proving impossibility results and for synthesizing efficient protocols in distributed computing. For asynchronous agents communicating over unreliable channels, causality is well studied and understood. This understanding, however, relies heavily on the assumption that agents themselves are correct and reliable. We provide the first epistemic analysis of causality in the presence of byzantine agents, i.e., agents that can deviate from their protocol and, thus, cannot be relied upon. Using our new framework for epistemic reasoning in fault-tolerant multi-agent systems, we determine the byzantine analog of the causal cone and describe a communication structure, which we call a multipede, necessary for verifying preconditions for actions in this setting.

cs.MA↗

arXiv2016

Proving Craig and Lyndon Interpolation Using Labelled Sequent Calculi

We have recently presented a general method of proving the fundamental logical properties of Craig and Lyndon Interpolation (IPs) by induction on derivations in a wide class of internal sequent calculi, including sequents, hypersequents, and nested sequents. Here we adapt the method to a more general external formalism of labelled sequents and provide sufficient criteria on the Kripke-frame characterization of a logic that guarantee the IPs. In particular, we show that classes of frames definable by quantifier-free Horn formulas correspond to logics with the IPs. These criteria capture the modal cube and the infinite family of transitive Geach logics.

cs.LO↗

arXiv2015

Grafting Hypersequents onto Nested Sequents

We introduce a new Gentzen-style framework of grafted hypersequents that combines the formalism of nested sequents with that of hypersequents. To illustrate the potential of the framework, we present novel calculi for the modal logics $\mathsf{K5}$ and $\mathsf{KD5}$, as well as for extensions of the modal logics $\mathsf{K}$ and $\mathsf{KD}$ with the axiom for shift reflexivity. The latter of these extensions is also known as $\mathsf{SDL}^+$ in the context of deontic logic. All our calculi enjoy syntactic cut elimination and can be used in backwards proof search procedures of optimal complexity. The tableaufication of the calculi for $\mathsf{K5}$ and $\mathsf{KD5}$ yields simplified prefixed tableau calculi for these logic reminiscent of the simplified tableau system for $\mathsf{S5}$, which might be of independent interest.

cs.LO↗