Jeroen Goudsmit

On a problem by Friedman and its solution by Rybakov

Sun, 22 Aug 2021 20:21:00 +0200

In September 1984, Vladimir Rybakov published his algorithm to determine the admissible rules of intuitionistic propositional logic. Creating this algorithm was no easy matter: the conceptual machinery required to do so had to be created nearly from scratch. Numerous logicians have used and explored comparable concepts in the years since. At the end of my time as a PhD candidate, I connected Rybakov’s approach to these now well-understood concepts. The resulting paper Decidability of Admissibility has just been published.

Proofs and refutations

A formal logical system typically consists of a language to talk about (formulas) and a mechanism of axioms and rules to construct proofs of some of these formulas. These formulas can be interpreted in hypothetical situations (models). In a well-designed system, every formula that can be proven is true in all models (soundness) and every formula that cannot be proven can be refuted by a model (completeness). Many concepts that logicians work with exist either on the side of the models or on the side of the proofs, and ideally, have a mirror image on the other side.

Can you create an algorithm that determines whether a formula can be proven? That depends on the logic. A common approach to create such an algorithm is to step through the mirror. Instead of searching for a proof of the formula, you can search for a model that refutes it. After all, if no such model exists then there must a proof. You can therefore determine whether a proof exists for a formula by checking all models that could refute it one-by-one. There is a significant caveat: this only works if you don’t need to check too many models and checking them doesn’t take too much time.

Making sure that these caveats are met is no small feat, as in principle there may be infinitely many models — or infinitely large models – you would have to check. The challenge is in showing that it suffices to only check a bounded and finite amount of things. This approach works for intuitionistic propositional logic, as first shown in 1946 — although an algorithm had been already provided through other means in 1934.

A problem by Friedman

When you add new rules to a logical system you create new ways of making proofs. As a consequence, a formula that wasn’t provable before now very well may be. When adding a rule doesn’t make any new formula provable, the rule is said to be an admissible rule. In his 1975 paper of 102 problems, Harvey Friedman asked for an algorithm that determines whether a rule is admissible in intuitionistic propositional logic. Rybakov provided the answer in his groundbreaking paper in 1984.

Rybakov applied what amounts to an intricate variant of the trick we described above. His approach uses theoretical concepts that were still very new at the time. Hence these intricacies had to be solved from scratch. Metaphorically, he was painting a multi-storied mural without pre-made scaffolding. This works out well if you’re tall enough or can jump really well, but it definitely adds to the challenge.

Theoretical scaffolding

In the years since 1984, the concepts used in Rybakov’s approach have been explored in multiple settings. When working on my dissertation, I came across well-studied notions that could be applied as theoretical scaffolding to Rybakov’s approach. A key role is played by exact formulas (related to proofs of formulas) and their mirror-image extendible formulas (related to models that refute formulas).

In my paper, I introduced generalized versions of these concepts based on the 2013 preprint by Nick & Guram Bezhanishvili. These generalizations provided adequate scaffolding to re-paint Rybakov’s mural, making the overall approach more accessible. The concepts as described in my paper are used throughout my dissertation — especially chapter 4 on decidability.

Looking back

When writing all of this in 2014, I definitely had the benefit of hindsight. I am very grateful for having had the opportunity to learn from the logicians who work in this field. Dick de Jongh, who first published about exact formulas in 1982, introduced me to this notion and to his approach to universal models. I learned about the early origins of this field through Alex Citkin, who has been writing about admissible rules since 1977 and whose early insights shaped the field.

Vladimir Rybakov met with me in March 2014 to discuss my ideas, which helped me to better understand his way of thinking. Later that year, I submitted my paper to the Bulletin of Symbolic Logic. This paper has finally been published in March 2021. It took quite some time, but looking back, I’m grateful for having had the opportunity to make this journey.

Models and the Compliance Officer

Fri, 01 Jan 2021 21:23:00 +0100

Many organisations rely on technological systems to help mitigate compliance risk. Think for example of the post-event transaction monitoring systems used throughout the financial sector to mitigate risks of money laundering and terrorism financing. Increasingly sophisticated quantitative methods such as machine learning are finding their way into the field. How can the compliance professional stay on top of these developments?

You are not alone

Each of these technological system have a clear goal: the mitigation of a particular type of risk. They receive input data, and through some quantitative approach, output an estimate of the risk at hand. This fits well within the established definition of a model as provided by the Office of the Comptroller of the Currency and the Board of Governors of the Federal Reserve System in their Sound Practices for Model Risk Management (2011). Compliance professionals can take comfort in knowing that they’re not alone, and that the challenge they’re facing is not completely new.

Sound model risk management

These sound practices explain how an organisation can take good care of its models. These models are simplifications of reality that — unavoidably and by their very design — focus attention on key factors and ignore others. Any such simplification is based on numerous assumptions that don’t necessarily hold in general. Hence when you’re using a model, there’s a risk that it’s fundamentally incorrect or that you’re misusing it. This is called model risk and the sound practices provide guidance on its management.

As a compliance professional, you can play a crucial role in management of model risk. After all, you’re intimately familiar with the compliance risks your organisation faces. Your expertise is invaluable to effectively challenge the assumptions underlying any compliance model. Moreover, you have deep insights in the risks that the model is meant to mitigate. Any outcome analysis would greatly benefit from such knowledge. Even without a quantitative background, you can thus contribute towards ensuring the soundness of these models and uncovering their limitations.

Fruitful discussion

On December 12th, I was a guest together with Maurice Jongmans on the Dutch podcast Compliance Adviseert: Automated Learning & AI hosted by Erik Reissenweber. We spoke for about an hour and covered the role of the compliance officer in the management of model risk. Additionally, we addressed the integrity risks that can arise from using artificial intelligence as described in DNB’s General principles for the use of Artificial Intelligence in the financial sector (2019).

During the podcast, Maurice spoke about the model used within his organisation to mitigate the risks of fraud, money laundering and terrorism financing. Seemingly effortlessly, a fruitful discussion arose between Erik and Maurice on the soundness, efficacy and efficiency of the model. It’s precisely through these types of conversations that a compliance professional can start to effectively challenge a model — regardless of the modelling techniques used.

Machine learning-based classification of viewing behavior

Sun, 20 Sep 2020 23:00:00 +0200

Sharing scientific thinking with a lay audience can have interesting consequences — intended and otherwise. Back in 2016, I attended one such lecture on the topic of “attention”. I suppose it is fair to say that neither me nor the speaker could have foreseen this consequence: a machine-learning based classification model of viewing behaviour.

Origin story

Having just read Stefan van der Stigchel’s book on attention, I was excited to hear his public lecture on the topic. Stefan and I were affiliated to the same Bachelor’s program on Artificial Intelligence back when I was a PhD candidate. Hence, after the lecture was over, I walked over to have a chat.

Stefan spoke at length about eye tracking: an extremely fruitful paradigm used to investigate attention. Having recently read about continuous authentication through biometrics, this piqued my interest. More specifically, I had heard about machine learning models in the context of behavioural classification of mouse dynamics. I wondered whether such an approach might work here. Both excited by the possibility, we agreed to follow-up.

The journey was long but extremely satisfying. The result is a paper by a team of six authors with complementary expertise.

Abstract

Since the seminal work of Yarbus (1967), multiple studies have demonstrated the influence of task-set on oculomotor behavior and the current cognitive state. In more recent years, this field of research has expanded by evaluating the costs of abruptly switching between such different tasks. At the same time, the field of classifying oculomotor behavior has been moving toward more advanced, data-driven methods of decoding data.

For the current study, we used a large dataset compiled over multiple experiments and implemented separate state-of-the-art machine learning methods for decoding both cognitive state and task-switching. We found that, by extracting a wide range of oculomotor features, we were able to implement robust classifier models for decoding both cognitive state and task-switching. Our decoding performance highlights the feasibility of this approach, even invariant of image statistics. Additionally, we present a feature ranking for both models, indicating the relative magnitude of different oculomotor features for both classifiers. These rankings indicate a separate set of important predictors for decoding each task, respectively. Finally, we discuss the implications of the current approach related to interpreting the decoding results.

Bestuurlijke borging van integere algoritmes

Wed, 01 Jul 2020 00:00:00 +0200

Compliance heeft toe te zien op het integer gebruik van algoritmes. Algoritmes zijn zelden waardenvrij. In hun implementatie wordt al dan niet bewust stelling genomen in de spanningen tussen meerdere conflicterende waarden. De uitdaging is om te borgen dat deze stellingnames bewust plaatsvinden in de daarvoor gepaste gremia. De conflicten tussen relevante waarden rondom de implementatie van algoritmes kunnen gezien worden als integriteitsrisico’s. Door gestructureerd waardenconflicten rondom het ontwerp en de inzet van algoritmes te identificeren, analyseren en beoordelen kan een organisatie borgen dat deze maatschappelijk betamelijk zullen zijn. In dit proces kan gebruik gemaakt worden van de methode van Value Sensitive Design. Hiermee kan men niet alleen de waarden van belanghebbenden grondig verkennen maar ook zoeken naar beheersmaatregelen.

Zodoende wordt uiteindelijk bewust besloten of het nettointegriteitsrisico dat voortkomt uit de stellingname acceptabel is. Dit besluit hoort thuis in de gebruikelijke beslisstructuur van de organisatie. Hierin kan compliance op gepaste wijze tegenwicht bieden. Op deze manier wordt de betamelijkheid van algoritmes bestuurlijk geborgd.

Tussen data en theorie

Tue, 12 May 2020 00:00:00 +0200

Algoritmen helpen om op grote schaal beslissingen te nemen. Het is echter lastig om achteraf toe te zien op de kwaliteit van deze beslissingen. Toezicht zou zich met name moeten richten op het wordingsproces van algoritmes: de stappen die genomen worden om te komen van probleemomschrijving tot een geïmplementeerd algoritme. In dit proces worden immers de principiële keuzes gemaakt die bepalend zijn voor de manier waarop het algoritme zal handelen en de wijze waarop het uitwerking heeft op de maatschappij. Door deze keuzes expliciet en onder de juiste overwegingen te maken verkleint het risico op misdragingen. Toezichthouders en ontwikkelaars van algoritmen tezamen kunnen hier een handreiking voor opstellen.

Finite Frames Fail

Thu, 12 May 2016 00:00:00 +0200

Many intermediate logics, even extremely well-behaved ones such as IPC, lack the finite model property for admissible rules. We give conditions under which this failure holds. We show that frames which validate all admissible rules necessarily satisfy a certain closure condition, and we prove that this condition, in the finite case, ensures that the frame is of width 2. Finally, we indicate how this result is related to some classical results on finite, free Heyting algebras.

Intuitionistic Rules

Fri, 29 May 2015 00:00:00 +0200

The totality of admissible rules is an invariant associated to a logic. In my dissertation, I explain how one can describe the admissible rules of a logic and I hint at several ways in which admissible rules provide one with information about a given logic. To be a bit more specific, I describe methods by means of which one can discern between those rules that are and those rules that are not admissible in certain intermediate logics. Moreover, I show that some intermediate logics can be characterised by means of their admissible rules.

Admissibility and refutation

Fri, 01 Aug 2014 00:00:00 +0200

Refutation systems are formal systems for inferring the falsity of formulae. These systems can, in particular, be used to syntactically characterise logics. In this paper, we explore the close connection between refutation systems and admissible rules. We develop technical machinery to construct refutation systems, employing techniques from the study of admissible rules. Concretely, we provide a refutation system for the intermediate logics of bounded branching, known as the Gabbay–de Jongh logics. We show that this gives a characterisation of these logics in terms of their admissible rules. To illustrate the technique, we also provide a refutation system for Medvedev’s logic.

On unification and admissible rules in Gabbay–de Jongh logics

Sat, 01 Feb 2014 00:00:00 +0100

In this paper we study the admissible rules of intermediate logics. We establish some general results on extensions of models and sets of formulas. These general results are then employed to provide a basis for the admissible rules of the Gabbay–de Jongh logics and to show that these logics have finitary unification type.

The Admissible Rules of BD2 and GSc

Tue, 03 Sep 2013 00:00:00 +0200

The Visser rules form a basis of admissibility for the intuitionistic propositional calculus. We show how one can characterize the existence of covers in certain models by means of formulae. Through this characterization, we provide a new proof of the admissibility of a weak form of the Visser rules. Finally, we use this observation, coupled with a description of a generalization of the disjunction property, to provide a basis of admissibility for the intermediate logics 𝖡𝖣2 and GSc.