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<p> Propositional proof complexity is a field of theoretical computer science which concerns itself with the lengths of formal proofs in various propositional proof systems. Frege systems are an important class of propositional proof systems. Extended Frege augments them by allowing the introduction of new variables to abbreviate formulas. Perhaps the largest open question in propositional proof complexity is whether or not Extended Frege is significantly more powerful that Frege. Several proof systems, each introducing new rules or syntax to Frege, have been developed in an attempt to shed some light on this problem.</p> <p> We introduce one such system, which we call H, which allows for the quantification of transpositions of propositional variables. We show that H is sound and complete, and that H's transposition quantifiers efficiently represent any permutation.</p> <p> The most important contribution is showing that a fragment of this proof system, H*1, is equivalent in power to Extended Frege. This is a complicated and rather technical result, and is achieved by showing that H*1 can efficiently prove translations of the first-order logical theory ∀PLA, a logical theory well suited for reasoning about linear algebra and properties of graphs.</p> / Thesis / Master of Science (MSc)

<p> Mechanized mathematics systems, especially Theorem Provers (TP) and Computer Algebra Systems (CAS), can play a very helpful role in handling relational calculus. Computer Algebra Systems help to automate tedious symbolic computations. However, they lack the ability to make sophisticated derivations of logical formulas. Correspondingly, a Theorem Prover is powerful in deriving the truth-value of a logical formula. Nevertheless, it is not suitable for dealing with symbolic expressions.</p> <p> The main goal for our research is to investigate the automation of relational calculus using existing mechanized mathematics technologies. Particularly, we elaborated a heuristic that enables the assignment of tasks to PVS and Maxima to help perform relational calculus. As well we built a proof-of-concept tool that supports this calculus.</p> <p> To fulfill our objective, we adopted the following steps: 1. Investigated and evaluated the characteristics and capabilities of TPs and CASs. This step led us to select PVS and Maxima as the tools to be used by our system. 2. Explored a strategy that governs setting tasks to PVS and Maxima in order to perform relational calculus. Then, we propose a task assignment heuristic based on this strategy. 3. Designed and built a proof-of-concept tool that makes use of PVS and Maxima to help perform relational calculus. 4. Assessed our tool by using it to handle some illustrative examples of operations on concrete relations.</p> <p> In our work, relations are given by their characteristic predicates. We assume as well that predicates that are provided to our proof-of-concept tool are in a Disjunctive Normal Form. We adopt a linear notation for the representation of propositions, quantifications, and expressions. We fall short of providing a user interface, which makes the use of the tool that we built slightly difficult.</p> / Thesis / Master of Science (MSc)

Since the creation of Bitcoin, the virtual currency has attracted the attention of many people and is now a household name synonymous with cryptocurrencies. Today, many thousands of different variants of cryptocurrencies exist, and more are being launched each day. The increase in popularity over the recent years has made them grow exponentially in value but at the same time also created a significant increase in energy consumption. Many of the cryptocurrencies we know today are based on Proof of Work, which is very energy intensive. There are also new and upcoming currencies based on alternative algorithms, such as Proof of Stake, which can considerably reduce the energy consumption of cryptocurrencies. However, Proof of Stake has not been proven to be as resilient and secure as Proof of Work. This study explores the future energy consumption and carbon emission of cryptocurrencies and reflects on their sustainability via exploratory scenario analysis. It includes three scenarios. Scenario 1 – Business as usual is the reference scenario. Scenario 2 – Change in the market, is based on the possibility of the market naturally switching over to PoS. Scenario 3 – under regulation is based on the possibility of a ban of PoW within the EU.  The results of this study indicate that the current emissions might be much lower than previously considered and that they might only be 30 percent of what had previously been reported. The fact that the emission is lower today does not mean that they will be sustainable in the future. Suppose Bitcoin and Ethereum energy consumption continues to grow as it has been doing for the last two years. In that case, the combined electricity consumption of the two currencies will have the possibility to surpass 650 TWh, which is an increase of over 300 percent from today's estimates. Banning Proof of Work within the EU would not yield the desired outcome of reducing carbon emissions but would instead increase carbon emissions. A continually growing Proof of Work based network, as it is used today, cannot be seen as sustainable. The recommendations to both industry and policymakers are to find and facilitate areas where Proof of Work would have the possibility to provide added value to society. / Sedan Bitcoin först lanserades har den virtuella valutan väckt mycket intresse. Många känner till Bitcoin men det finns flera tusen olika kryptovalutor och flera skapas varje dag. Det växande intresset har fått flertalet kryptovalutor att öka enormt i värde, men också i dess energiåtgång. Många av dagens kryptovalutor drivs av en algoritm som kallas Proof of Work, vilket är väldigt energikrävande. Det finns även nya och växande kryptovalutor baserade på alternativa algoritmer så som Proof of Stake, vilket har stora möjligheter att minska energiåtgången avsevärt. Dock har inte Proof of Stake bevisats vara lika motståndskraftigt vid attacker så som Proof of Work.  Denna studie utforskar den framtida energikonsumtionen och koldioxidutsläppen från kryptovalutor och har som avsikt att reflektera på hållbarheten via en utforskande scenarioanalys. Där tre scenarios utforskats. Scenario 1 – Business as usual är referensscenariot. Scenario 2 – Change in the market, är baserat på att marknaden själv glider över till PoS och Scenario 3 – under regulation vilket är baserat på ett förbud av PoW inom EU. Resultatet från studien visar att de nuvarande utsläppen kanske är mycket lägre än vad som tidigare trotts och kanske bara är 30 procent av det som tidigare rapporterats. Faktumet att kryptovalutor kanske släpper ut mindre koldioxid idag betyder inte att de kan anses hållbara i framtiden. Anta att Bicoin och Ethereum fortsätter växa som de har gjort de senaste två åren, då kommer de två valutorna ha en möjlighet att förbruka mer än 650 TWh per år vid 2025. Detta är en ökning med mer än 300 procent från dagens energikonsumtion. Att införa ett förbud på Ptoof of Work inom EU kommer dock inte ge de önskade förhoppningarna om att minska koldioxidutsläppen, utan skulle snarare kunna öka dem. Med det sagt så kan ett ständigt växande Proof of Work nätverk inte anses vara hållbart. Därför är rekommendationerna till industrin och beslutsfattarna att identifiera och främja områden där Proof of Work kan implementeras för att skapa ett mervärde till samhället.

The model-theoretic analysis of the concept of logical consequence has come under heavy criticism in the last couple of decades. The present work looks at an alternative approach to logical consequence where the notion of inference takes center stage. Formally, the model-theoretic framework is exchanged for a proof-theoretic framework. It is argued that contrary to the traditional view, proof-theoretic semantics is not revisionary, and should rather be seen as a formal semantics that can supplement model-theory. Specifically, there are formal resources to provide a proof-theoretic semantics for both intuitionistic and classical logic. We develop a new perspective on proof-theoretic harmony for logical constants which incorporates elements from the substructural era of proof-theory. We show that there is a semantic lacuna in the traditional accounts of harmony. A new theory of how inference rules determine the semantic content of logical constants is developed. The theory weds proof-theoretic and model-theoretic semantics by showing how proof-theoretic rules can induce truth-conditional clauses in Boolean and many-valued settings. It is argued that such a new approach to how rules determine meaning will ultimately assist our understanding of the apriori nature of logic.

In this thesis we examine the relationship between hypersequent and some types of labelled sequent calculi for a subset of intermediate logics—logics between intuitionistic (Int), and classical logics—that have geometric Kripke semantics, which we call Int∗/Geo. We introduce a novel calculus for a fragment of first-order classical logic, which we call partially-shielded formulae (or PSF for short), that is adequate for expressing the semantic validity of formulae in Int∗/Geo, and apply techniques from correspondence theory to provide translations of hypersequents, simply labelled sequents and relational sequents (simply labelled sequents with relational formulae) into PSF. Using these translations, we show that hypersequents and simply labelled sequents for calculi in Int∗/Geo share the same models. We also use these translations to justify various techniques that we introduce for translating simply labelled sequents into relational sequents and vice versa. In particular, we introduce a technique called "transitive unfolding" for translating relational sequents into simply labelled sequents (and by extension, hypersequents) which preserves linear models in Int∗/Geo. We introduce syntactic translations between hypersequent calculi and simply labelled sequent calculi. We apply these translations to a novel hypersequent framework HG3ipm∗ for some logics in Int∗/Geo to obtain a corresponding simply labelled sequent framework LG3ipm∗, and to an existing simply labelled calculus for Int from the literature to obtain a novel hypersequent calculus for Int. We introduce methods for translating a simply labelled sequent calculus into a cor- responding relational calculus, and apply these methods to LG3ipm∗ to obtain a novel relational framework RG3ipm∗ that bears similarities to existing calculi from the literature. We use transitive unfolding to translate proofs in RG3ipm∗ into proofs in LG3ipm∗ and HG3ipm∗ with the communication rule, which corresponds to the semantic restriction to linear models.

Law of evidence is a legal-based procedure for collecting and presenting proof. The aim of the process is to follow the basic principle of legal administrative proceedings, the principle of material truth. My thesis is based on the Administrative Code regulations, although I was also pointing to the fact that there are specific regulations that restrict or exclude the use of the Administrative Code. Despite the above-mentioned, the general and theoretical ideas of the law of evidence are applicable for all administrative proceedings. Except the principle of material truth there are other basic principles of vital importance to administrative proceedings, particularly the principle of legality and the principle of working out the case without undue delay. The core of law of evidence, however, is contained in provisions of the Administrative Code dealing with the bases for a decision in the administrative proceedings at first instance. For the law of evidence in administrative proceedings it is also often applicable doctrinal and judicial knowledge of civil and criminal law, as a connecting element between these different areas of law is the principle of material truth. The decision-making practice of the administrative courts respects the similarities between criminal and administrative punishment,...

Dans cette thèse nous présentons la formalisation de trois résultats principaux que sont la forme normale de Jordan d’une matrice, le théorème de Bolzano-Weierstraß et le théorème de Perron-Frobenius. Pour la formalisation de la forme normale de Jordan nous introduisons différents concepts d’algèbre linéaire tel que les matrices diagonales par blocs, les matrices compagnes, les facteurs invariants, ... Ensuite nous définissons et développons une théorie sur les espaces topologiques et métriques pour la formalisation du théorème de Bolzano-Weierstraß. La formalisation du théorème de Perron-Frobenius n’est pas terminée. La preuve de ce théorème utilise des résultats d’algèbre linéaire, mais aussi de topologie. Nous montrerons comment les précédents résultats seront réutilisés. / In this thesis we present the formalization of three principal results that are the Jordan normal form of a matrix, the Bolzano-Weierstraß theorem, and the Perron-Frobenius theorem. To formalize the Jordan normal form, we introduce many concepts of linear algebra like block diagonal matrices, companion matrices, invariant factors, ... The formalization of Bolzano-Weierstraß theorem needs to develop some theory about topological space and metric space. The Perron-Frobenius theorem is not completly formalized. The proof of this theorem uses both algebraic and topological results. We will show how we reuse the previous results.

GATA is a web-application used by truckdrivers transporting timber from assigned pick-up places in the forest to specific receiving locations. GATA is used to facilitate and ease the navigation, communication and planning behind each delivery. This create situations where GATA needs to be used while driving, this is both dangerous and illegal. To solve this problem a voice-controlled proof-of-concept application for the most common features used during driving was developed and given the name GATA-Assistenten. The application was designed with a concept of having a few basic features built upon a stable platform. Dialogflow was chosen as tool to create the AI-voice control and to get it integrated with Google Assistant. Two tracks were created, one for drivers to change the estimated time of arrival and one to get the status on the receiving sites. According to statics at least 0,5% of all traffic related accidents can be directly contributed to interacting with communication devices, this is without taking the unrecorded cases into account. The conclusion based on these statistics is that GATA-Assistenten can not only help avoid accidents but also save lives. If a company wants to invest in safety, it is paramount to use voice-control, however it is also important to do research and development on new technologies to find out and fix the causes behind accidents. / GATA är en webbapplikation som används av lastbilschaufförer vid frakt av timmer från skogarna till mottagningsplatser för att lättare navigera, kommunicera och planera körningar på uppdrag av SCA. Det innebär att chaufförerna ibland behöver använda GATA under färd, något som inte bara kan vara en trafikfara utan även är olagligt. För att lösa detta problem skall ett proof-of-concept på röststyrning utvecklas för några av de vanligaste funktionerna som används under färd i GATA. Röststyrningsapplikationen som fick namnet GATA-Assistenten designades utifrån ett koncept om att ha få grundläggande funktioner byggt på en stabil plattform. Dialogflow valdes som verktyg för att skapa en AI-baserad röststyrning med integration via Google Assistant. GATA- Assistenten bestod slutligen av två huvudspår, det ena för att ändra den av chauffören angivna ankomsttiden till mottagningsplatsen och det andra för att ta reda på aviseringsläge på mottagningsplatserna. Enligt statistik är minst 0,5% av alla olyckor direkt orsakade av en förares interaktion med någon form av kommunikationsutrustning, detta utan att ta mörkertalet i beaktande. Slutsatsen som kan tas av detta är att GATA-Assistenten kan hjälpa till att undvika olyckor och rädda liv. Vill ett företag satsa på säkerhet är röststyrning ett måste, om det finns en nollvision kring olyckor i trafiken måste dock mer forskning och utveckling läggas på ny teknik för att ta reda på samt åtgärda orsakerna bakom alla olyckor.

1 The burden of proof in the disputes for unjust enrichment Abstract This diploma thesis deals with the institute of the burden of proof. Its aim is to analyse the distribution of the burden of proof in disputes for unjust enrichment. The thesis consists of an introduction, four main chapters, which are further internally divided and the conclusion. The introduction is focused mainly on the reasons of author's choice of the topic. The first chapter contains the general characteristics and interpretation of key concepts as well as related institutes, whose basic knowledge will be a key to other parts of the text. It explains the differences between the concepts of proof, counter-proof and objection. It then analyses the standard of proof needed to prove certain facts. It also describes the state of non liquet and analyses the individual procedural obligations and burdens of the parties. The subject of the second chapter of the thesis is the term of the burden of proof, which can be divided into the objective and subjective burden of proof. The subjective burden of proof can be further subdivided into the abstract and concrete burden of proof. The chapter goes on to analyse selected theories of distribution of the burden of proof, while not neglecting the negative theory of proof. In the partial conclusion,...