arXiv subjects

Sonja Smets

Publications and source records attributed to Sonja Smets.

17 recordsLinked to original sources

arXiv2026

The Logic of Data Access and Data Exchanges

We investigate a new logic that extends Dynamic Epistemic Logic (DEL), by combining standard epistemic modalities for (individual and distributed) propositional knowledge with operators for (conditional) non-propositional knowledge of a number (in which an agent or a group have knowledge of the value of some variable x, conditional on some additional information). We also generalize these operators, by considering formulas that express the fact that an agent or group can (conditionally) narrow down the possible values of the variable x to at most N possibilities (for some natural number N). In order to name and compare such hypothetical values, we extend the logic further with definite descriptions based on minimization operators, denoting the least of the N possible values of x (according to some fixed order) that are considered possible by the agent or group. On this static base, we consider DEL-style extensions with dynamic modalities for general 'data-exchange events' (covering private and public propositional announcements, but also secret hacking of a private database, or public sharing of one's data via open-source repositories, etc.). In such scenarios, whole 'chunks' of information may be exchanged or modified: once access to a given source is gained, all the 'data' stored at that specific location becomes available. We give complete axiomatizations for the resulting logics, and prove their decidability and co-expressivity.

cs.LO↗

arXiv2025

Group Knowledge of Hypothetical Values

In recent years, epistemic logics have been extended with operators K_ax for knowledge of (the value of) a variable x (by an agent a). We study dynamic versions of these logics, enriched with modalities for semi-public data-exchange events (e.g., public announcements, data-sharing within a subgroup, or changing the value of a variable). To obtain a complete axiomatization of data-exchange events, in the presence of equality x = y and K_ax, one needs to extend the logic further: first, with an operator for distributed knowledge K_Ax of the value (by a group of agents A); next, with a conditional version of this: distributed knowledge K^P_A x (of the value by a group) given some hypothetical condition (expressed by some proposition P); then, with definite descriptions x^P_A , denoting the 'hypothetical' value of x according to A's (distributed) knowledge given condition P. In order to deal with common knowledge in the presence of semi-public data exchanges, we also need to add a novel conditional version of the recent concept of common distributed knowledge. We investigate the resulting logic, giving examples and presenting a complete axiomatization and a decidability proof.

cs.LO↗

arXiv2024

Logic meets Wigner's Friend (and their Friends)

We take a fresh look at Wigner's Friend thought-experiment and some of its more recent variants and extensions, such as the Frauchiger-Renner (FR) Paradox. We discuss various solutions proposed in the literature, focusing on a few questions: What is the correct epistemic interpretation of the multiplicity of state assignments in these scenarios? Under which conditions can one include classical observers into the quantum state descriptions, in a way that is still compatible with traditional Quantum Mechanics? Under which conditions can one system be admitted as an additional 'observer' from the perspective of another background observer? When can the standard axioms of multi-agent Epistemic Logic (that allow "knowledge transfer" between agents) be applied to quantum-physical observers? In the last part of the paper, we propose a new answer to these questions, sketch a particular formal implementation of this answer, and apply it to obtain a principled solution to Wigner Friend-type paradoxes.

quant-ph↗

arXiv2021

LQP: The Dynamic Logic of Quantum Information

The main contribution of this paper is the introduction of a dynamic logic formalism for reasoning about information flow in composite quantum systems. This builds on our previous work on a complete quantum dynamic logic for single systems. Here we extend that work to a sound (but not necessarily complete) logic for composite systems, which brings together ideas from the quantum logic tradition with concepts from (dynamic) modal logic and from quantum computation. This Logic of Quantum Programs (LQP) is capable of expressing important features of quantum measurements and unitary evolutions of multi-partite states, as well as giving logical characterisations to various forms of entanglement (for example, the Bell states, the GHZ states etc.). We present a finitary syntax, a relational semantics and a sound proof system for this logic. As applications, we use our system to give formal correctness proofs for the Teleportation protocol and for a standard Quantum Secret Sharing protocol; a whole range of other quantum circuits and programs, including other well-known protocols (for example, superdense coding, entanglement swapping, logic-gate teleportation etc.), can be similarly verified using our logic.

quant-ph↗

arXiv2021

Learning What Others Know

We propose a number of powerful dynamic-epistemic logics for multi-agent information sharing and acts of publicly or privately accessing other agents' information databases. The static base of our logics is obtained by adding to standard epistemic logic comparative epistemic assertions, that can express epistemic superiority between groups or individuals, as well as a common distributed knowledge operator (that combines features of both common knowledge and distributed knowledge). On the dynamic side, we introduce actions by which epistemic superiority can be acquired: "sharing all one knows" (by e.g. giving access to one's information database to all or some of the other agents), as well as more complex informational events, such as hacking. We completely axiomatize several such logics and prove their decidability.

cs.LO↗

arXiv2021

The Logic of Quantum Programs

We present a logical calculus for reasoning about information flow in quantum programs. In particular we introduce a dynamic logic that is capable of dealing with quantum measurements, unitary evolutions and entanglements in compound quantum systems. We give a syntax and a relational semantics in which we abstract away from phases and probabilities. We present a sound proof system for this logic, and we show how to characterize by logical means various forms of entanglement (e.g. the Bell states) and various linear operators. As an example we sketch an analysis of the teleportation protocol.

quant-ph↗

arXiv2020

Thinking About Causation: A Causal Language with Epistemic Operators

This paper proposes a formal framework for modeling the interaction of causal and (qualitative) epistemic reasoning. To this purpose, we extend the notion of a causal model with a representation of the epistemic state of an agent. On the side of the object language, we add operators to express knowledge and the act of observing new information. We provide a sound and complete axiomatization of the logic, and discuss the relation of this framework to causal team semantics.

cs.AI↗

arXiv2019

Learning Probabilities: Towards a Logic of Statistical Learning

We propose a new model for forming beliefs and learning about unknown probabilities (such as the probability of picking a red marble from a bag with an unknown distribution of coloured marbles). The most widespread model for such situations of 'radical uncertainty' is in terms of imprecise probabilities, i.e. representing the agent's knowledge as a set of probability measures. We add to this model a plausibility map, associating to each measure a plausibility number, as a way to go beyond what is known with certainty and represent the agent's beliefs about probability. There are a number of standard examples: Shannon Entropy, Centre of Mass etc. We then consider learning of two types of information: (1) learning by repeated sampling from the unknown distribution (e.g. picking marbles from the bag); and (2) learning higher-order information about the distribution (in the shape of linear inequalities, e.g. we are told there are more red marbles than green marbles). The first changes only the plausibility map (via a 'plausibilistic' version of Bayes' Rule), but leaves the given set of measures unchanged; the second shrinks the set of measures, without changing their plausibility. Beliefs are defined as in Belief Revision Theory, in terms of truth in the most plausible worlds. But our belief change does not comply with standard AGM axioms, since the revision induced by (1) is of a non-AGM type. This is essential, as it allows our agents to learn the true probability: we prove that the beliefs obtained by repeated sampling converge almost surely to the correct belief (in the true probability). We end by sketching the contours of a dynamic doxastic logic for statistical learning.

cs.AI↗

arXiv2019

A Complete Axiomatisation for the Logic of Lattice Effect Algebras

In a recent work Foulis and Pulmannov\' a \cite{Foulis2012} studied the logical connectives in lattice effect algebras. In this paper we extend their study and investigate further the logical calculus for which the lattice effect algebras can serve as semantic models. We shall first focus on some properties of lattice effect algebras and will then give a complete axiomatisation of this logic.

math.LO↗

arXiv2017

Argument-based Belief in Topological Structures

This paper combines two studies: a topological semantics for epistemic notions and abstract argumentation theory. In our combined setting, we use a topological semantics to represent the structure of an agent's collection of evidence, and we use argumentation theory to single out the relevant sets of evidence through which a notion of beliefs grounded on arguments is defined. We discuss the formal properties of this newly defined notion, providing also a formal language with a matching modality together with a sound and complete axiom system for it. Despite the fact that our agent can combine her evidence in a 'rational' way (captured via the topological structure), argument-based beliefs are not closed under conjunction. This illustrates the difference between an agent's reasoning abilities (i.e. the way she is able to combine her available evidence) and the closure properties of her beliefs. We use this point to argue for why the failure of closure under conjunction of belief should not bear the burden of the failure of rationality.

cs.AI↗

arXiv2016

On the Solvability of Inductive Problems: A Study in Epistemic Topology

We investigate the issues of inductive problem-solving and learning by doxastic agents. We provide topological characterizations of solvability and learnability, and we use them to prove that AGM-style belief revision is "universal", i.e., that every solvable problem is solvable by AGM conditioning.

cs.LO↗

arXiv2015

Revisable Justified Belief: Preliminary Report

The theory $\mathsf{CDL}$ of Conditional Doxastic Logic is the single-agent version of Board's multi-agent theory $\mathsf{BRSIC}$ of conditional belief. $\mathsf{CDL}$ may be viewed as a version of AGM belief revision theory in which Boolean combinations of revisions are expressible in the language. We introduce a theory $\mathsf{JCDL}$ of Justified Conditional Doxastic Logic that replaces conditional belief formulas $B^ψφ$ by expressions $t{\,:^ψ}φ$ made up of a term $t$ whose syntactic structure suggests a derivation of the belief $φ$ after revision by $ψ$. This allows us to think of terms $t$ as reasons justifying a belief in various formulas after a revision takes place. We show that $\mathsf{JCDL}$-theorems are the exact analogs of $\mathsf{CDL}$-theorems, and that this result holds the other way around as well. This allows us to think of $\mathsf{JCDL}$ as a theory of revisable justified belief.

cs.LO↗

arXiv2002

Logic of Dynamics & Dynamics of Logic; Some Paradigm Examples

This paper surveys some recent developments towards a dynamic quantum logic and outlines its explicite construction -- some analogies and contrasts with other logics of dynamics are indicated. Abstract: The development of ``(static) operational quantum logic" points out that classical boolean structures are too rigid to describe the actual and potential properties of quantum systems. On the other hand, an intuitionistic perspective on operational quantum logic, guides us in the direction of incorporating dynamics logically by reconsidering the primitive propositions required to describe the behavior of a quantum system, in particular in view of the emergent disjunctivity due to the non-determinism of quantum measurements. A further elaboration on "intuitionistic quantum logic" emerges into a "dynamic operational quantum logic", which allows us to express dynamic reasoning in the sense that we can capture how actual properties propagate, including their temporal causal structure, and provides a unified logical description of systems which evolve or which are submitted to measurements. This setting reveals that even static operational quantum logic bears a hidden dynamic ingredient in terms of what is called "the orthomodularity" of the lattice-structure. Focusing on the quantale semantics for dynamic operational quantum logic, we delineate some points of difference with the existing quantale semantics for (non)-commutative linear logic.

math.LO↗

arXiv2001

The Sasaki Hook is not a [Static] Implicative Connective but Induces a Backward [in Time] Dynamic One that Assigns Causes

In this paper we argue that the Sasaki adjunction, which formally encodes the logicality that different authors tried to attach to the Sasaki hook as a `quantum implicative connective', has a fundamental dynamic nature and encodes the so-called `causal duality' (Coecke, Moore and Stubbe 2001; quant-ph/0009100) for the particular case of a quantum measurement with a projector as corresponding self-adjoint operator. In particular: The action of the Sasaki hook $(a\stackrel{S}{\to}-)$ for fixed antecedent $a$ assigns to some property ``the weakest cause before the measurement of actuality of that property after the measurement'', i.e. ${(a\stackrel{S}{\to}b)}$ is the weakest property that guarantees actuality of $b$ after performing the measurement represented by the projector that has the `subspace $a$' as eigenstates for eigenvalue 1, say, the measurement that `tests' $a$ . From this we conclude that the logicality attributable to quantum systems contains a fundamentally dynamic ingredient: Causal duality actually provides a new dynamic interpretation of orthomodularity. We also reconsider the status of the Sasaki hook within `dynamic (operational) quantum logic' (DOQL). We can derive two labeled dynamic hooks (forwardly and backwardly) that encode how quantum measurements act on properties. In an even more radical perspective one could say that the transition from either classical or constructive/intuitionistic logic to quantum logic entails besides the introduction of an additional unary connective `operational resolution' (Coecke 2001a; math.LO/0011208) the shift from a binary connective implication to a ternary connective where two of the arguments refer to qualities of the system and the third, the new one, to an obtained outcome (in a measurement).

quant-ph↗

arXiv2001

A Quantum Structure Description of the Liar Paradox

In this article we propose an approach that models the truth behavior of cognitive entities (i.e. sets of connected propositions) by taking into account in a very explicit way the possible influence of the cognitive person (the one that interacts with the considered cognitive entity). Hereby we specifically apply the mathematical formalism of quantum mechanics because of the fact that this formalism allows the description of real contextual influences, i.e. the influence of the measuring apparatus on the physical entity. We concentrated on the typical situation of the liar paradox and have shown that (1) the truth-false state of this liar paradox can be represented by a quantum vector of the non-product type in a finite dimensional complex Hilbert space and the different cognitive interactions by the actions of the corresponding quantum projections, (2) the typical oscillations between false and truth - the paradox -is now quantum dynamically described by a Schrodinger equation. We analyse possible philosophical implications of this result.

quant-ph↗

arXiv2000

A logical description for perfect measurements

We reconsider the description for property transitions due to perfect measurements, viewing them as a special case of general transitions that are due to an externally imposed change. We propose a corresponding syntax involving operational quantum logic and a fragment of non-commutative linear logic.

quant-ph↗

arXiv2000

The Liar-paradox in a Quantum Mechanical Perspective

In this paper we concentrate on the nature of the liar paradox as a cognitive entity; a consistently testable configuration of properties. We elaborate further on a quantum mechanical model [Aerts, Broekaert, Smets 1999] that has been proposed to analyze the dynamics involved, and we focus on the interpretation and concomitant philosophical picture. Some conclusions we draw from our model favor an effective realistic interpretation of cognitive reality.

quant-ph↗