17 research outputs found

    From Nonstandard Analysis to various flavours of Computability Theory

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    As suggested by the title, it has recently become clear that theorems of Nonstandard Analysis (NSA) give rise to theorems in computability theory (no longer involving NSA). Now, the aforementioned discipline divides into classical and higher-order computability theory, where the former (resp. the latter) sub-discipline deals with objects of type zero and one (resp. of all types). The aforementioned results regarding NSA deal exclusively with the higher-order case; we show in this paper that theorems of NSA also give rise to theorems in classical computability theory by considering so-called textbook proofs.Comment: To appear in the proceedings of TAMC2017 (http://tamc2017.unibe.ch/

    On the Herbrand Functional Interpretation

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    We show that the types of the witnesses in the Herbrand functional interpretation can be simplified, avoiding the use of “sets of functionals” in the interpretation of implication and universal quantification. This is done by presenting an alternative formulation of the Herbrand functional interpretation, which we show to be equivalent to the original presentation. As a result of this investigation we also strengthen the monotonicity property of the original presentation, and prove a monotonicity property for our alternative definition

    Logical Dreams

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    We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic

    A Defence of Mathematical Pluralism

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    The problem of implementation and its relation to the philosophy of cognitive science

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    According to certain arguments, computation is observer-relative either in the sense that many physical systems implement many computations (Hilary Putnam), or in the sense that almost all physical systems implement all computations (John Searle). If sound, these arguments have a potentially devastating consequence for the computational theory of mind: if arbitrary physical systems can be seen to implement arbitrary computations, the notion of computation seems to lose all explanatory power as far as brains and minds are concerned. David Chalmers and B. Jack Copeland have attempted to counter these relativist arguments by placing certain constraints on the definition of implementation. In this thesis, I examine their proposals and find both wanting in some respects. During the course of this examination, I give a formal definition of the class of combinatorial-state automata , upon which Chalmers's account of implementation is based. I show that this definition implies two theorems (one an observation due to Curtis Brown) concerning the computational power of combinatorial-state automata, theorems which speak against founding the theory of implementation upon this formalism. Toward the end of the thesis, I sketch a definition of the implementation of Turing machines in dynamical systems, and offer this as an alternative to Chalmers's and Copeland's accounts of implementation. I demonstrate that the definition does not imply Searle's claim for the universal implementation of computations. However, the definition may support claims that are weaker than Searle s, yet still troubling to the computationalist. There remains a kernel of relativity in implementation at any rate, since the interpretation of physical systems seems itself to be an observer-relative matter, to some degree at least. This observation helps clarify the role the notion of computation can play in cognitive science. Specifically, I will argue that the notion should be conceived as an instrumental rather than as a fundamental or foundational one.ErÀiden argumenttien mukaan laskenta eli komputaatio on havaitsijarelatiivista siinÀ mielessÀ, ettÀ monien fysikaalisten systeemien voidaan nÀhdÀ implementoivan useita komputaatioita (Hilary Putnam), tai ettÀ miltei kaikkien fysikaalisten systeemien voidaan nÀhdÀ implementoivan kaikki komputaatiot (John Searle). SikÀli kuin nÀmÀ argumentit ovat pitÀviÀ, niillÀ voi olla kohtalokkaita seurauksia komputationaaliselle mielen teorialle. Jos mielivaltaiset fysikaaliset systeemit implementoivat mielivaltaisia komputaatioita, komputaation kÀsite nÀyttÀÀ kadottavan kaiken selittÀvÀn voimansa mielen ja aivojen toiminnan selittÀmisen yhteydessÀ. David Chalmers ja B. Jack Copeland ovat yrittÀneet vastata edellÀ mainittuihin relativistisiin argumentteihin asettamalla erÀitÀ rajoitteita implementaatiorelaation mÀÀritelmÀlle. TÀssÀ tutkielmassa tarkastelen heidÀn ehdotuksiaan ja esitÀn, ettÀ molempiin liittyy joitakin ongelmia. Tarkastelun myötÀ annan tÀsmÀllisen mÀÀritelmÀn kombinatoristen tilojen automaattien luokalle, jolle Chalmersin implementaation teoria perustuu. Osoitan, ettÀ mÀÀritelmÀstÀ seuraa kaksi tulosta (joista toinen on Curtis Brownin aiemmin esittÀmÀ huomautus) liittyen kombinatoristen tilojen automaattien laskennalliseen voimaan. Argumentoin, ettÀ nÀiden tulosten vuoksi implementaation teoriaa ei tulisi perustaa kombinatoristen tilojen automaattien formalismille. Tutkielman loppua kohden esitÀn vaihtoehtoisen implementaation analyysin, joka perustuu mÀÀritelmÀlle Turingin koneiden implementoitumisesta dynaamisissa systeemeissÀ. NÀytÀn, ettÀ ehdottamani mÀÀritelmÀ ei implikoi Searlen universaalin implementaation vÀitettÀ. TÀstÀ huolimatta on mahdollista, ettÀ mÀÀritelmÀstÀ seuraa tuloksia, jotka ovat Searlen vÀitettÀ heikompia mutta silti ongelmallisia komputationalismin kannalta. Implementaation teorian ytimessÀ nÀyttÀÀ joka tapauksessa sÀilyvÀn tietty relatiivisuus, sillÀ fysikaalisten systeemien tulkinta nÀyttÀisi itsessÀÀn olevan jossakin mÀÀrin havaitsijarelatiivista. TÀmÀ huomio auttaa selvittÀmÀÀn sitÀ roolia, joka komputaation kÀsitteellÀ on kognitiotieteessÀ. Erityisesti esitÀn, ettÀ kÀsite tulisi ymmÀrtÀÀ instrumentaalisena, ei fundamentaalisena tai perustan antavana kÀsitteenÀ

    First-Order Model Checking on Generalisations of Pushdown Graphs

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    We study the first-order model checking problem on two generalisations of pushdown graphs. The first class is the class of nested pushdown trees. The other is the class of collapsible pushdown graphs. Our main results are the following. First-order logic with reachability is uniformly decidable on nested pushdown trees. Considering first-order logic without reachability, we prove decidability in doubly exponential alternating time with linearly many alternations. First-order logic with regular reachability predicates is uniformly decidable on level 2 collapsible pushdown graphs. Moreover, nested pushdown trees are first-order interpretable in collapsible pushdown graphs of level 2. This interpretation can be extended to an interpretation of the class of higher-order nested pushdown trees in the collapsible pushdown graph hierarchy. We prove that the second level of this new hierarchy of nested trees has decidable first-order model checking. Our decidability result for collapsible pushdown graph relies on the fact that level 2 collapsible pushdown graphs are uniform tree-automatic. Our last result concerns tree-automatic structures in general. We prove that first-order logic extended by Ramsey quantifiers is decidable on all tree-automatic structures.Comment: phd thesis, 255 page

    On politics and social science – the subject-object problem in social science and Foucault’s engaged epistemology

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    The epistemological problem of the relationship between the subject of knowledge and the object being known has it’s form in social science as a problem of the relationship between a social scientist as a researcher and society and it’s phenomena as an object of this inquiry. As Berger and Kellner note in their book “Sociology Reinterpreted” a social scientist is necessarily a part of the object he studies, being embedded in a position in society from which he studies it. Hence social sciences as scientific endeavors face a problem of the inseperability of their researchers from object they study. Two main solutions two this problem have arisen: positivism and interpretivism. Positivism postulates that rigorous methods for research will insure that objective knowledge will be produced while interpretivism sees society only as an aggregate of individuals whose interactions should be interpreted. A third epistemological framework has arisen in the first half of the twentieth century usually called “critical theory”. Critical theory states that researchers should aim their research towards changing the object they are researching, therefore their scientific practice should have extra-scientific effects, namely political effects. This perspective violates Webers postulate of value neutrality which claims that social sciences can only study the state of affairs but can’t subscribe desirable ways of action. As we will see the main topic of our paper is the epistemological framework of the work of Michel Foucault and his contribution to the resolution of the problematic relation between a researcher and his research object in social science. We will claim that Foucault broadly falls into the critical theory paradigm but manages to solve it’s conflict with the value neutrality postulate. Foucault envisions society as an amalgam of discursive and non-discursive practices that interconnect in a way that gives them regularity and coherence through time. As Gayatri Spivak notices for Foucault discursive practices create meaning and in doing so chart a way for nondiscursive practices and therefore for action. This can be seen as an explanation for Foucault’s well known postulate of the relationship between power and knowledge, discursive practices create knowledge that makes visible certain paths for action. Both of these types of practices intertwine to create what Foucault calls “dispositifs” that can be seen as mechanisms that bind discursive and non-discursive practices in a coherent manner and enable their regular repetition through time. Foucault calls his methodology “genealogy” and sees it as a historical research of the emergence of dipositifs. Genealogy is a historical research of the contingent ways in which practices got interconnected in the past to create dispositifs we see today. As Foucault claims genealogy begins with a “question posed in the present” about a certain dispositive and then charts historical events and processes that led to its current form. The main aim of genealogy is to show that there is no transcendental necessity for a certain dispositif to exist in it’s current form by exposing the historical contingency that led to it’s current state. Foucault claimed that his intent was to show that there is no metaphysical necessity that grounds the existences of dispositifs and hence that their current form is arbitrary. As we can see Foucault follows his postulate on the relationship between knowledge and power and formulates his scientific practice as an opening of possibilities for different forms of action. This is way he calls his books “experiments” and claims that they are to be used for readers to re-examine their own links to the currently existing dispositifs and possibilities of their alternative arrangements. But as Foucault claims the genealogical method doesn’t include normative prescriptions and can be seen only as a form of an anti-metaphysical “unmasking” of current dispositifs. This unmasking doesn’t prescribe a desirable form to any dispositive but only shows that it can be arranged in different ways. Hence we can say that Foucault sees the relationship between a researcher and his object of study as a form of an intervention of the subject that aims at showing that the object is an arbitrary construction. In that regard Foucault falls into the critical theory paradigm. Where he differs from critical theory is his anti-normative stance that refuses to prescribe any desirable form of action unlike for example Horkheimer who in his essay on critical theory claims that “the task of the theorist is to push society towards justice”. Foucault claims that his research results should be used as “instruments” in political struggles but he himself doesn’t ever proclaim a desirable political goal. So we can conclude that Foucault solves the problem of the subject-object relation in social science by envisioning the research process as a practice of production of tools for social change. Therefore he connects social science to extra-scientific political goals but doesn’t violate the value neutrality postulate because his research doesn’t prescribe any concrete political goals but only shows the possibility for social change

    Acta Cybernetica : Volume 15. Number 4.

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    Programming Languages and Systems

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    This open access book constitutes the proceedings of the 28th European Symposium on Programming, ESOP 2019, which took place in Prague, Czech Republic, in April 2019, held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2019
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