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We report on the mechanization of (preference-based) conditional normative reasoning. Our focus is on Aqvist's system E for conditional obligation, and its extensions. Our mechanization is achieved via a shallow semantical embedding in…
When a decision, such as the approval or denial of a bank loan, is delegated to a computer, an explanation of that decision ought to be given with it. This ethical need to explain the decisions leads to the search for a formal definition of…
Training powerful AI systems to exhibit desired behaviors hinges on the ability to provide accurate human supervision on increasingly complex tasks. A promising approach to this problem is to amplify human judgement by leveraging the power…
The problem 2-Xor-Sat asks for the probability that a random expression, built as a conjunction of clauses $x \oplus y$, is satisfiable. We revisit this classical problem by giving an alternative, explicit expression of this probability. We…
Argumentation accommodates various rhetorical devices, such as questions, reported speech, and imperatives. These rhetorical tools usually assert argumentatively relevant propositions rather implicitly, so understanding their true meaning…
We present a case study in {\it experimental} yet {\it rigorous} mathematics by describing an algorithm, fully implemented in both Mathematica and Maple, that {\it automatically conjectures}, and then {\it automatically proves}, closed-form…
By considering probability distributions over the set of assignments the expected truth values assignment to propositional variables are extended through linear operators, and the expected truth values of the clauses at any given…
We introduce a notion of real-valued reward testing for probabilistic processes by extending the traditional nonnegative-reward testing with negative rewards. In this richer testing framework, the may and must preorders turn out to be…
Deciding termination is a fundamental problem in the analysis of probabilistic imperative programs. We consider the qualitative and quantitative probabilistic termination problems for an imperative programming model with discrete…
Using mechanised reasoning we prove that combinatorial Vickrey auctions are soundly specified in that they associate a unique outcome (allocation and transfers) to any valid input (bids). Having done so, we auto-generate verified executable…
Automated theorem provers and formal proof assistants are general reasoning systems that are in theory capable of proving arbitrarily hard theorems, thus solving arbitrary problems reducible to mathematics and logical reasoning. In…
Performance prediction or forecasting sporting outcomes involves a great deal of insight into the particular area one is dealing with, and a considerable amount of intuition about the factors that bear on such outcomes and performances. The…
We give a new proof of a_4\phi_3 summation due to G.E. Andrews and confirm another_4\phi_3 summation conjectured by him recently. Some variations of these two_4\phi_3 summations are also given.
We consider the problem of rational uncertainty about unproven mathematical statements, remarked on by G\"odel and others. Using Bayesian-inspired arguments we build a normative model of fair bets under deductive uncertainty which draws…
We consider a setting where in a known future time, a certain continuous random variable will be realized. There is a public prediction that gradually converges to its realized value, and an expert that has access to a more accurate…
In this paper, we present two methods to provide explanations for reasoning with belief functions in the valuation-based systems. One approach, inspired by Strat's method, is based on sensitivity analysis, but its computation is simpler…
In the quest for Explainable Artificial Intelligence (XAI) one of the questions that frequently arises given a decision made by an AI system is, ``why was the decision made in this way?'' Formal approaches to explainability build a formal…
Explainable Recommendation has been gaining attention over the last few years in industry and academia. Explanations provided along with recommendations in a recommender system framework have many uses: particularly reasoning why a…
In this paper we give a conditional improvement to the Elekes-Szab\'{o} problem over the rationals, assuming the Uniformity Conjecture. Our main result states that for $F\in \mathbb{Q}[x,y,z]$ belonging to a particular family of…
We present a benchmark of 29687 problems derived from the On-Line Encyclopedia of Integer Sequences (OEIS). Each problem expresses the equivalence of two syntactically different programs generating the same OEIS sequence. Such programs were…