It has been hypothesized that illusory agency detection is at the basis of belief in supernatural agents and paranormal beliefs. In the present study a biological motion perception task was used to study illusory agency detection in a group of skeptics and a group of paranormal believers. Participants were required to detect the presence or absence of a human agent in a point-light display. It was found that paranormal believers had a lower perceptual sensitivity than skeptics, which was due to (...) a response bias to ‘yes’ for stimuli in which no agent was present. The relation between paranormal beliefs and illusory agency detection held only for stimuli with low to intermediate ambiguity, but for stimuli with a high number of visual distractors responses of believers and skeptics were at the same level. Furthermore, it was found that illusory agency detection was unrelated to traditional religious belief and belief in witchcraft, whereas paranormal beliefs were strongly related to illusory agency detection. These findings qualify the relation between illusory pattern perception and supernatural and paranormal beliefs and suggest that paranormal beliefs are strongly related to agency detection biases. (shrink)

In recent years, the human ability to reasoning about mental states of others in order to explain and predict their behavior has come to be a highly active area of research. Researchers from a wide range of fields { from biology and psychology through linguistics to game theory and logic{ contribute new ideas and results. This interdisciplinary workshop, collocated with the Thirteenth International Conference on Theoretical Aspects of Rationality and Knowledge (TARK XIII), aims to shed light on models of social (...) reasoning that take into account realistic resource bounds. People reason about other people's mental states in order to understand and predict the others' behavior. This capability to reason about others' knowledge, beliefs and intentions is often referred to as theory of mind. Idealized rational agents are capable of recursion in their social reasoning, and can reason about phenomena like common knowledge. Such idealized social reasoning has been modeled by modal logics such as epistemic logic and BDI (belief, desire, intention) logics and by epistemic game theory. However, in real-world situations, many people seem to lose track of such recursive social reasoning after only a few levels. The workshop provides a forum for researchers that attempt to analyze, understand and model how resource-bounded agents reason about other minds. (shrink)

The effects of public announcements, private communications, deceptive messages to groups, and so on, can all be captured by a general mechanism of updating multi-agent models with update action models, now in widespread use. There is a natural extension of the definition of a bisimulation to action models. Surely enough, updating with bisimilar action models gives the same result (modulo bisimulation). But the converse turns out to be false: update models may have the same update effects without being bisimilar. We (...) propose action emulation as a notion of equivalence more appropriate for action models, and generalizing standard bisimulation. It is proved that action emulation provides a full characterization of update effect. We first concentrate on the general case, and next focus on the important case of action models with propositional preconditions. Our notion of action emulation yields a simplification procedure for action models, and it gives designers of multi-agent systems a useful tool for comparing different ways of representing a particular communicative action. (shrink)

Notice: This PDF version was distributed by request to members of the Friends of the SEP Society and by courtesy to SEP content contributors. It is solely for their fair use. Unauthorized distribution is prohibited. To learn how to join the Friends of the..

Discourse Representation Theory is a specific name for the work of Hans Kamp in the area of dynamic interpretation of natural language. Also, it has gradually become a generic term for proposals for dynamic interpretation of natural language in the same spirit. These proposals have in common that each new sentence is interpreted in terms of the contribution it makes to an existing piece of interpreted discourse. The interpretation conditions for sentences are given as instructions for updating the representation of (...) the discourse. (shrink)

Computer software is written in languages like C, Java or Haskell. In many cases social software is expressed in natural language. The paper explores connections between the areas of natural language analysis and analysis of social protocols, and proposes an extended program for natural language semantics, where the goals of natural language communication are derived from the demands of specific social protocols.

Guarded actions are changes with preconditions acting as a guard. Guarded action models are multimodal Kripke models with the valuations replaced by guarded actions. Call guarded action logic the result of adding product updates with guarded action models to PDL (propositional dynamic logic). We show that guarded action logic reduces to PDL.

Dynamic logic, broadly conceived, is the logic that analyses change by decomposing actions into their basic building blocks and by describing the results of performing actions in given states of the world. The actions studied by dynamic logic can be of various kinds: actions on the memory state of a computer, actions of a moving robot in a closed world, interactions between cognitive agents performing given communication protocols, actions that change the common ground between speaker and hearer in a conversation, (...) actions that change the contextually available referents in a conversation, and so on. (shrink)

Three key ways of updating one’s knowledge are (i) perception of states of affairs, e.g., seeing with one’s own eyes that something is the case, (ii) reception of messages, e.g., being told that something is the case, and (iii) drawing new conclusions from known facts. If one represents knowledge by means of Kripke models, the implicit assumption is that drawing conclusions is immediate. This assumption of logical omniscience is a useful abstraction. It leaves the distinction between (i) and (ii) to (...) be accounted for. In current versions of Update Logic (Dynamic Epistemic Logic, Logic of Communication and Change) perception and message reception are not distinguished. This paper proposes an extension of Update Logic that makes this distinction explicit. The logic deals with three kinds of updates: announcements, changes of the world, and observations about the world in the presence of witnesses. The resulting logic is shown to be complete by means of a reduction to epistemic propositional dynamic logic by a well known method. (shrink)

We present a direct reduction of dynamic epistemic logic in the spirit of [4] to propositional dynamic logic (PDL) [17, 18] by program transformation. The program transformation approach associates with every update action a transformation on PDL programs. These transformations are then employed in reduction axioms for the update actions. It follows that the logic of public announcement, the logic of group announcements, the logic of secret message passing, and so on, can all be viewed as subsystems of PDL. Moreover, (...) the program transformation approach can be used to generate the appropriate reduction axioms for these logics. Our direct reduction of dynamic epistemic logic to PDL was inspired by the reduction of dynamic epistemic logic to automata PDL of [13]. Our approach shows how the detour through automata can be avoided. (shrink)

In this paper we investigate Kripke models, used to model knowledge or belief in a static situation, and action models, used to model communicative actions that change this knowledge or belief. The appropriate notion for structural equivalence between modal structures such as Kripke models is bisimulation: Kripke models that are bisimilar are modally equivalent. We would like to find a structural relation that can play the same role for the action models that play a prominent role in information updating. Two (...) action models are equivalent if they yield the same results when updating Kripke models. More precisely, two action models are equivalent if it holds for all Kripke models that the result of updating with one action model is bisimilar to the result of updating with the other action model. We propose a new notion of action emulation that characterizes the structural equivalence of the important class of canonical action models. Since every action model has an equivalent canonical action model, this gives a method to decide the equivalence of any pair of action models. We also give a partial result that holds for the class of all action models. Our results extend the work in van Eijck et al. (Synthese 185(1):131–151, 2012). (shrink)

Non-determined game logic is the logic of two player board games where the game may end in a draw: unlike the case with determined games, a loss of one player does not necessarily constitute of a win of the other player. A calculus for non-determined game logic is given in [4] and shown to be complete. The calculus adds a new rule for the treatment of greatest fixpoints, and a new unfolding axiom for iterations of the universal player. The technique (...) of the completeness proof is inspired by the canonical model construction for propositional dynamic logic (PDL). In this paper, this is extended to the logic of determined games. It is proved that the calculus for nondetermined game logic, together with the axiom of determinacy, is complete for determined game logic. Next, it is shown that the axioms and rules of the new calculus can all be derived from the calculus proposed by Parikh in [5], for which the completeness was still open. This proves Parikh’s conjecture that his calculus is complete for determined games. (shrink)

Model checking techniques for communication protocols usually are phrased in terms of processes, basically labelled arcs in a labelled transition system. We propose to lift checking for such protocols to a more abstract level by analysing the protocols as composite communicative actions, with a communicative action viewed as a mapping on an appropriate class of epistemic models. As an example, we analyse an anonymous broadcast protocol (Chaum’s well-known dining cryptographers protocol) and an electronic voting protocol.

We will study game trees as representations of rational choice and as representations of player preferences, and promises as public announcements of genuine intentions. Promises in a game change what players know about the preferences of other players. They can be modelled as operations that change a given game into a different game where players know more about the effects of their strategies.

This paper defines the grammar class of sequentially indexed grammars. Sequentially indexed grammars are the result of a change in the index stack handling mechanism of indexed grammars [Aho68, Aho69]. Sequentially indexed grammars are different from linear indexed grammars [Gaz88]. Like indexed languages, sequentially indexed languages are a fully abstract language class. Unlike indexed languages, sequentially indexed languages allow polynomial parsing algorithms. We give a polynomial algorithm for parsing with sequentially indexed gramamrs that is an extension of the Earley algorithm (...) for parsing with context free grammars. (shrink)

• Een programma puzzle • Puzzelen met steentjes • Functies en functioneel programmeren • Functies maken met lambda abstractie • Eigenschappen van dingen en karakteristieke functies • De ‘filter’ functie • Oneindige lijsten • Priemgetallen herkennen en genereren • Opdrachten..

Verifying an epistemic protocol involves creating a formalized version of the protocol in a suitable logical language, and next showing (i) that the steps of the protocol are in one to one correspondence with the steps in its formalized version, (ii) that the formalized version satisfies certain correctness conditions, and (iii) hence, that the original version also satisfies these conditions. We will show that DEL is a suitable medium for carrying out this program for an interesting example protocol.

In this paper we forge a connection between dynamic epistemic logics of belief revision on one hand and studies of collective judgement and multi-agent preference change on the other. Belief revision in the spirit of dynamic epistemic logic uses updating with relational substitutions to change the beliefs of individual agents. Collective judgement in social choice theory studies the collective outcomes of individual belief changes. We start out from the logic of communication and change (LCC), which is basically epistemic propositional dynamic (...) logic (PDL) extended with update action modalities. We show how epistemic PDL can be made to handle belief operators. Our new version of epistemic/doxastic PDL does not impose any constraints on the basic relations. Because of this it does not suffer from the drawback of LCC that constraints on epistemic relations, such as transitivity and reflexivity, may get lost under updates that are admitted by the system. After introducing the base system without constraints, we study the effects of imposing a single constraint, namely the constraint that the agent’s preference relations are linked. Linkedness is a natural extension of local connectedness to the multi-agent case. It assures that if Alice prefers y to x and Bob prefers z to x, then both Alice and Bob can make up their minds when given a choice between y and z. Since general belief changes may not preserve linkedness, we propose a recipe for belief change that does preserve it. Finally, we show that the resulting framework can be used to model consensus seeking procedures. We focus on the case of plenary Dutch meetings. In Dutch meetings, a belief change (or rather: preference change) is performed for all agents in the meeting if a majority believes (or: is in favour of) the proposition that is under discussion. A special case of these meetings is judgement aggregation, and we apply our framework to the discursive dilemma in.. (shrink)

The communicative effect of a collective message from the Dutch former minister of finance Wouter Bos to inform all his contacts about his new email address is completely different from that of a set of individual messages to the same list. The talk will explain how differences of this kind can be modelled in epistemic logic (the logic of knowledge). A central notion here is common knowledge. We will explain the general framework for describing update effects of messages as mappings (...) on epistemic models (“knowledge models”), and we will give a sketch of some recent work in this area. (shrink)

The effects of public announcements, private communications, deceptive messages to groups, and so on, can all be captured by a general mechanism of updating multi-agent models with update action models, now in widespread use. There is a natural extension of the definition of a bisimulation to action models. Surely enough, updating with bisimilar action models gives the same result. But the converse turns out to be false: update models may have the same update effects without being bisimilar. We propose action (...) emulation as a notion of equivalence more appropriate for action models, and generalizing standard bisimulation. It is proved that action emulation provides a full characterization of update effect. We first concentrate on the general case, and next focus on the important case of action models with propositional preconditions. Our notion of action emulation yields a simplification procedure for action models, and it gives designers of multi-agent systems a useful tool for comparing different ways of representing a particular communicative action. (shrink)

Almost forty years ago Richard Montague proposed to analyse natural language with the same tools as formal languages. In particular, he gave formal semantic analyses of several interesting fragments of English in terms of typed logic. This led to the development of Montague grammar as a particular style of formal analysis of natural language.

In this editorial contribution, two issues relevant to the question, what should be at the top of the research agenda for ethics and technology, are identified and discussed. Firstly: can, and do, engineers make a difference to the degree to which technology leads to morally desirable outcomes? What role does professional autonomy play here, and what are its limits? And secondly, what should be the scope of engineers’ responsibility; that is to say, on which issues are they, as engineers, morally (...) obliged to reflect? The research agendas proposed by the authors contributing to this special section, implicitly, give different answers to these questions. We suggest that an explicit discussion of these issues would greatly help in constructing a common research agenda. (shrink)

Current dynamic epistemic logics for analyzing effects of informational events often become cumbersome and opaque when common knowledge is added for groups of agents. Still, postconditions involving common knowledge are essential to successful multi-agent communication. We propose new systems that extend the epistemic base language with a new notion of ‘relativized common knowledge’, in such a way that the resulting full dynamic logic of information flow allows for a compositional analysis of all epistemic postconditions via perspicuous ‘reduction axioms’. We also (...) show how such systems can deal with factual alteration, rather than just information change, making them cover a much wider range of realistic events. After a warm-up stage of analyzing logics for public announcements, our main technical results are expressivity and completeness theorems for a much richer logic that we call LCC. This is a dynamic epistemic logic whose static base is propositional dynamic logic (PDL), interpreted epistemically. This system is capable of expressing all model-shifting operations with finite action models, while providing a compositional analysis for a wide range of informational events. This makes LCC a serious candidate for a standard in dynamic epistemic logic, as we illustrate by analyzing some complex communication scenarios, including sending successive emails with both ‘cc’ and ‘bcc’ lines, and other private announcements to subgroups. Our proofs involve standard modal techniques, combined with a new application of Kleene’s theorem on finite automata, as well as new Ehrenfeucht games of model comparison. (shrink)

This paper introduces DEMO, a Dynamic Epistemic Modelling tool. DEMO allows modelling epistemic updates, graphical display of update results, graphical display of action models, formula evaluation in epistemic models, translation of dynamic epistemic formulas to PDL formulas, and so on. The paper implements the reduction of dynamic epistemic logic [16, 2, 3, 1] to PDL given in [12]. The reduction of dynamic epistemic logic to automata PDL from [24] is also discussed and implemented. Epistemic models are minimized under bisimulation, and (...) update action models are minimized under action emulation (the appropriate structural notion for having the same update effect, cf. [13]). The paper is an exemplar of tool building for epistemic update logic. It contains the full code of an implementation in Haskell [22], in ‘literate programming’ style [23], of DEMO. (shrink)

This paper contains the full code of an implementation in Haskell [2], in ‘literate programming’ style [3], of an approach to deductive parsing based on [4]. We focus on the case of the Earley [1] parsing algorithm for CF languages.

This talk shows how propositional dynamic logic (PDL) can be interpreted as a logic for multi-agent knowledge update and belief revision, or as a logic of preference change, if the basic relations are read as preferences instead of plausibilities. Our point of departure is the logic of communication and change (LCC) of [9]. Like LCC, our logic uses PDL as a base epistemic language. Unlike LCC, we start out from agent plausibilities, add their converses, and build knowledge and belief operators (...) from these with the PDL constructs. We extend the update mechanism of LCC to an update mechanism that handles belief change as relation substitution, and we show that the update part of this logic is more expressive than either that of LCC or that of epistemic/doxastic PDL with a belief change modality. Next, we show that the properties of knowledge and belief are preserved under any update, unlike in LCC. We prove completeness of the logic and give examples of its use. If there is time, we will also look at the preference interpretation of the system, and at preference change scenarios that can be modelled with it. (shrink)

A new system of dynamic logic is introduced and motivated, witha novel approach to variable binding for incremental interpretation. Thesystem is shown to be equivalent to first order logic and complete.The new logic combines the dynamic binding idea from DynamicPredicate Logic with De Bruijn style variable free indexing. Quantifiersbind the next available variable register; the indexing mechanismguarantees that active registers are never overwritten by newquantifiers actions. Apart from its interest in its own right, theresulting system has certain advantages over Dynamic (...) Predicate Logic orDiscourse Representation Theory. It comes with a more well behaved(i.e., transitive) consequence relation, it gives a more explicitaccount of how anaphoric context grows as text gets processed, and ityields new insight into the dynamics of anaphoric linking in reasoning.Incremental dynamics also points to a new way of handling contextdynamically in Montague grammar. (shrink)

We implement the extension of the logical consequence relation to a partial order ≤ on arbitary types built from e (entities) and t (Booleans) that was given in [1], and the definition of monotonicity preserving and monotonicity reversing functions in terms of ≤. Next, we present a new algorithm for polarity marking, and implement this for a particular fragment of syntax. Finally, we list the reseach agenda that these definitions and this algorithm suggest. The implementations use Haskell [8], and are (...) given in ‘literate programming’ style [9]. (shrink)

Long ago, when Alexander the Great asked the mathematician Menaechmus for a crash course in geometry, he got the famous reply ``There is no royal road to mathematics.'' Where there was no shortcut for Alexander, there is no shortcut for us. Still, the fact that we have access to computers and mature programming languages means that there are avenues for us that were denied to the kings and emperors of yore. The purpose of this book is to teach logic and (...) mathematical reasoning in practice, and to connect logical reasoning with computer programming in Haskell. Haskell emerged in the 1990s as a standard for lazy functional programming, a programming style where arguments are evaluated only when the value is actually needed. Haskell is a marvelous demonstration tool for logic and maths because its functional character allows implementations to remain very close to the concepts that get implemented, while the laziness permits smooth handling of infinite data structures. This book does not assume the reader to have previous experience with either programming or construction of formal proofs, but acquaintance with mathematical notation, at the level of secondary school mathematics is presumed. Everything one needs to know about mathematical reasoning or programming is explained as we go along. After proper digestion of the material in this book, the reader will be able to write interesting programs, reason about their correctness, and document them in a clear fashion. The reader will also have learned how to set up mathematical proofs in a structured way, and how to read and digest mathematical proofs written by others. This is the updated, expanded, and corrected second edition of a much-acclaimed textbook. Praise for the first edition: 'Doets and van Eijck's ``The Haskell Road to Logic, Maths and Programming'' is an astonishingly extensive and accessible textbook on logic, maths, and Haskell.' Ralf Laemmel, Professor of Computer Science, University of Koblenz-Landau. (shrink)

We will present relational tools for analysing software systems, based on the Haskell datatype for relations defined in Chapter 5 of Doets and Van Eijck, The Haskell Road to Logic, Maths and Programming, King’s College Publications, London 2004 [DvE04]. The main purpose is to demonstrate some very concrete applications of abstract relations, and to make the point that functional programming is highly relevant to software engineering.

We explore some logics of change, focusing on commands to change the world in such a way that certain elementary propositions become true or false. This investigation starts out from the following two simplifying assumptions: (1) the world is a collection of facts (Wittgenstein), and (2), the world can be changed by changing elementary facts (Marx). These assumptions allow us to study the logic of imperatives in the simplest possible setting.

In [11] it is shown how propositional dynamic logic (PDL) can be interpreted as a logic of belief revision that extends the logic of communication and change (LCC) given in [7]. This new version of epistemic/doxastic PDL does not impose any constraints on the basic relations and because of this it does not suffer from the drawback of LCC that these constraints may get lost under updates that are admitted by the system. Here, we will impose one constraint, namely that (...) the agent’s plausibility relations are linked. Linkedness is a natural extension of local connectedness to the multi-agent case and it assures that we know the agent’s preferences between all relevant alternatives. Since the belief updates that are used in [11] may not preserve linkedness, we limit ourselves to a particular kind of belief change that does preserve it. Our framework has obvious connections to coalition logic [17] and social choice theory [19]. We show how we can use it to model consensus seeking in plenary Dutch meetings. In Dutch meetings, a belief update is done for all agents in the meeting if a majority beliefs the proposition that is under discussion. A special case of these meetings is judgement aggregation, and we apply our framework to the discursive dilemma in this field [15]. (shrink)

Presuppositions of utterances are the pieces of information you convey with an utterance no matter whether your utterance is true or not We rst study presupposition in a very simple framework of updating propo sitional information with examples of how presuppositions of complex propositional updates can be calculated Next we move on to presupposi tions and quanti cation in the context of a dynamic version of predicate logic suitably modi ed to allow for presupposition failure In both the propositional and (...) the quanti cational case presupposition failure can be viewed as error abortion of procedures Thus a dynamic assertion logic which describes the preconditions for error abortion is the suitable tool for analysing presupposition.. (shrink)

The paper gives a formal analysis of public lies, explains how public lying is related to public announcement, and describes the process of recoveries from false beliefs engendered by public lying. The framework treats two kinds of public lies: simple lying update and two-step lying, which consists of suggesting that the lie may be true followed by announcing the lie. It turns out that agents’ convictions of what is true are immune to the first kind, but can be shattered by (...) the second kind. Next, recovery from public lying is analyzed. Public lies that are accepted by an audience cannot be undone simply by announcing their negation. The paper proposes a recovery process that works well for restoring beliefs about facts but cannot be extended to beliefs about beliefs. The formal machinery of the paper consists of KD45 models and conditional neighbourhood models, with various update procedures on them. Completeness proofs for a number of reasoning systems (converse belief logic, public lies logic, lying and recovery logic, conditional neighbourhood logic, plus its dynamic version) are included. (shrink)

Current dynamic epistemic logics often become cumbersome and opaque when common knowledge is added for groups of agents. Still, postconditions regarding common knowledge express the essence of what communication achieves. We present some methods that yield so-called reduction axioms for common knowledge. We investigate the expressive power of public announcement logic with relativized common knowledge, and present reduction axioms that give a detailed account of the dynamics of common knowledge in some major communication types.

• We study a new composition operation on (epistemic) multiagent models and update actions that takes vocabulary extensions into account.

The paper presents a simple format for typed logics with states by adding a function for register update to standard typed lambda calculus. It is shown that universal validity of equality for this extended language is decidable . This system is next extended to a full fledged typed dynamic logic, and it is illustrated how the resulting format allows for very simple and intuitive representations of dynamic semantics for natural language and denotational semantics for imperative programming. The proposal is compared (...) with some alternative approaches to formulating typed versions of dynamic logics. (shrink)

The European Workshop on Logics in Artificial Intelligence was held at the Centre for Mathematics and Computer Science in Amsterdam, September 10-14, 1990. This volume includes the 29 papers selected and presented at the workshop together with 7 invited papers. The main themes are: - Logic programming and automated theorem proving, - Computational semantics for natural language, - Applications of non-classical logics, - Partial and dynamic logics.

Dynamic First Order Logic results from interpreting quantification over a variable v as change of valuation over the v position, conjunction as sequential composition, disjunction as non-deterministic choice, and negation as test for continuation. We present a tableau style calculus for DFOL with explicit binding, prove its soundness and completeness, and point out its relevance for programming with DFOL, for automated program analysis including loop invariant detection, and for semantics of natural language. We also extend this to an infinitary calculus (...) for DFOL with iteration and connect up with other work in dynamic logic. (shrink)

This paper shows how propositional dynamic logic can be interpreted as a logic for multi-agent belief revision. For that we revise and extend the logic of communication and change of [9]. Like LCC, our logic uses PDL as a base epistemic language. Unlike LCC, we start out from agent plausibilities, add their converses, and build knowledge and belief operators from these with the PDL constructs. We extend the update mechanism of LCC to an update mechanism that handles belief change as (...) relation substitution, and we show that the update part of this logic is more expressive than either that of LCC or that of doxastic/epistemic PDL with a belief change modality. It is shown that the properties of knowledge and belief are preserved under any update, and that the logic is complete. © 2008 Springer-Verlag Berlin Heidelberg. (shrink)

• Een voorbeeld van direct inzicht • Puzzelen met steentjes en programma’s • Functies en functioneel programmeren • Functies maken met lambda abstractie • Eigenschappen van dingen en karakteristieke functies • De ‘filter’ functie • Oneindige lijsten • Priemgetallen herkennen en genereren • Opdrachten..

Time discounting is the phenomenon that a desired result in the future is perceived as less valuable than the same result now. Economic theories can take this psychological fact into account in several ways. In the economic literature the most widely used type of additive time discounting is exponential discounting. In exponential discounting, the fall of valuation depends by a constant factor on the length of the delay period. It is well known, however, that exponential time discounting often does not (...) describe well how people actually behave. Most people are averse to short delays in gratification now, while their future selves may not mind a bit of extra waiting. This behaviour can be described well by non-exponential discounting functions such as hyperbolic discounting. In hyperbolic discounting, valuations fall rapidly for small delay periods, but the fall gets slower for longer delay periods. Hyperbolic discounting captures phenomena such as procrastination, addiction and in general inconsistency over time. This chapter investigates whether forms of non-exponential discounting, in particular close to the so called Quasi-Hyperbolic model, could also be characterized in terms of dynamically consistent choices when individuals discount the welfare of future selves as well as their payoffs. (shrink)

Mijn wetenschappelijke bijdrage sluit aan bij het stuk van Jan Willem Klop in deze zelfde afscheidsbundel, dat ik van Jan Willem onder embargo te lezen heb gekregen. Je zult je herinneren dat Jan Willem in de CWI lezing ter gelegenheid van zijn eredoctoraat kort refereerde aan de Thue Morse reeks. Noem deze reeks M . Jan Willem gaf de versie die start met 1. Noem het resultaat van omwisselen van nullen en enen in de Thue Morse reeks M . De (...) reeks M is wat je krijgt als je het Thue Morse proces start met 0. (shrink)