The prime number theorem, established by Hadamard and de la Vall'ee Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1 / ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erd"os in 1948. We describe a formally verified version of Selberg's proof, obtained using the Isabelle proof assistant.

The development of effective knowledge discovery techniques has become in the recent few years a very active research area due to the important impact it has in several relevant application areas. One interesting task thereof is that of singling out anomalous individuals from a given population, e.g., to detect rare events in time-series analysis settings, or to identify objects whose behavior is deviant w.r.t. a codified standard set of "social" rules. Such exceptional individuals are usually referred to as outliers in the literature. Recently, outlier detection has also emerged as a relevant KR&R problem. In this paper, we formally state the concept of outliers by generalizing in several respects an approach recently proposed in the context of default logic, for instance, by having outliers not being restricted to single individuals but, rather, in the more general case, to correspond to entire (sub)theories. We do that within the context of logic programming and, mainly through examples, we discuss its potential practical impact in applications. The formalization we propose is a novel one and helps in shedding some light on the real nature of outliers. Moreover, as a major contribution of this work, we illustrate the exploitation of minimality criteria in outlier detection. The computational complexity of outlier detection problems arising in this novel setting is thoroughly investigated and accounted for in the paper as well. Finally, we also propose a rewriting algorithm that transforms any outlier detection problem into an equivalent inference problem under the stable model semantics, thereby making outlier computation effective and realizable on top of any stable model solver.

We introduce a logic for reasoning about evidence that essentially views evidence as a function from prior beliefs (before making an observation) to posterior beliefs (after making the observation). We provide a sound and complete axiomatization for the logic, and consider the complexity of the decision problem. Although the reasoning in the logic is mainly propositional, we allow variables representing numbers and quantification over them. This expressive power seems necessary to capture important properties of evidence.

It is well known that the standard model of knowledge based on possible worlds is subject to the problem of logical omniscience, that is, the agents know all the logical consequences of the ir knowledge [Fagin, Halpern, Moses, and V ardi 1995, Chapter 9]. Thu s, possible-world definitions of knowledge make it difficult to reason about the knowledge tha t agents need to explicitly compute in order to make decisions and perform actions, or to capture si tuations where agents want to reason about the knowledge that other agents need to explicitly com pute in order to perform actions. This observation leads to a distinction between two forms of knowledge, implicit knowledge and explicit knowledge (or resource-bounded knowledge), a distinction long recog nized [Rosenschein 1985]. The classical AI approach known as the interpreted symbolic structures approach, where knowledge is based on information stored in data structures of the agent, can be seen as an instance of explicit knowledge. In contrast, the situated automata approach, which interprets knowledge based on information carried by the state of the machine, can be seen as an instance of implicit knowledge. Levesque [1984] makes a similar distinction bet ween implicit belief and explicit belief. While the possible-worlds approach is taken as the standard model for implicit knowledge, there is no standard model for explicit knowledge.

It is well known that, under certain conditions, it is possible to split logic programs under stable model semantics, i.e. to divide such a program into a number of different "levels", such that the models of the entire program can be constructed by incrementally constructing models for each level. Similar results exist for other non-monotonic formalisms, such as auto-epistemic logic and default logic. In this work, we present a general, algebraicsplitting theory for logics with a fixpoint semantics. Together with the framework of approximation theory, a general fixpoint theory for arbitrary operators, this gives us a uniform and powerful way of deriving splitting results for each logic with a fixpoint semantics. We demonstrate the usefulness of these results, by generalizing existing results for logic programming, auto-epistemic logic and default logic.

Sequential decision theory formally solves the problem of rational agents in uncertain worlds if the true environmental prior probability distribution is known. Solomonoff's theory of universal induction formally solves the problem of sequence prediction for unknown prior distribution. We combine both ideas and get a parameter-free theory of universal Artificial Intelligence. We give strong arguments that the resulting AIXI model is the most intelligent unbiased agent possible. We outline how the AIXI model can formally solve a number of problem classes, including sequence prediction, strategic games, function minimization, reinforcement and supervised learning. The major drawback of the AIXI model is that it is uncomputable. To overcome this problem, we construct a modified algorithm AIXItl that is still effectively more intelligent than any other time t and length l bounded agent. The computation time of AIXItl is of the order t x 2^l. The discussion includes formal definitions of intelligence order relations, the horizon problem and relations of the AIXI theory to other AI approaches.

Because query execution is the most crucial part of Inductive Logic Programming (ILP) algorithms, a lot of effort is invested in developing faster execution mechanisms. These execution mechanisms typically have a low-level implementation, making them hard to debug. Moreover, other factors such as the complexity of the problems handled by ILP algorithms and size of the code base of ILP data mining systems make debugging at this level a very difficult job. In this work, we present the trace-based debugging approach currently used in the development of new execution mechanisms in hipP, the engine underlying the ACE Data Mining system. This debugger uses the delta debugging algorithm to automatically reduce the total time needed to expose bugs in ILP execution, thus making manual debugging step much lighter.

This paper presents a property of propositional theories un der the answer sets semantics (called Equilibrium Logic for this general syntax): any theory can always be reexpress ed as a strongly equivalent disjunctive logic program, possib ly with negation in the head. We provide two different proofs for this result: one involvin g a syntactic transformation, and one that constructs a program starting from the counterm odels of the theory in the intermediate logic of here-and-there.

Open answer set programming (OASP) is an extension of answer set programming where one may ground a program with an arbitrary superset of the program's constants. We define a fixed point logic (FPL) extension of Clark's completion such that open answer sets correspond to models of FPL formulas and identify a syntactic subclass of programs, called (loosely) guarded programs. Whereas reasoning with general programs in OASP is undecidable, the FPL translation of (loosely) guarded programs falls in the decidable (loosely) guarded fixed point logic (mu(L)GF). Moreover, we reduce normal closed ASP to loosely guarded OASP, enabling for the first time, a characterization of an answer set semantics by muLGF formulas. We further extend the open answer set semantics for programs with generalized literals. Such generalized programs (gPs) have interesting properties, e.g., the ability to express infinity axioms. We restrict the syntax of gPs such that both rules and generalized literals are guarded. Via a translation to guarded fixed point logic, we deduce 2-exptime-completeness of satisfiability checking in such guarded gPs (GgPs). Bound GgPs are restricted GgPs with exptime-complete satisfiability checking, but still sufficiently expressive to optimally simulate computation tree logic (CTL). We translate Datalog lite programs to GgPs, establishing equivalence of GgPs under an open answer set semantics, alternation-free muGF, and Datalog lite.

Computability logic (CL) (see http://www.cis.upenn.edu/~giorgi/cl.html) is a recently launched program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth that logic has more traditionally been. Formulas in it represent computational problems, "truth" means existence of an algorithmic solution, and proofs encode such solutions. Within the line of research devoted to finding axiomatizations for ever more expressive fragments of CL, the present paper introduces a new deductive system CL12 and proves its soundness and completeness with respect to the semantics of CL. Conservatively extending classical predicate calculus and offering considerable additional expressive and deductive power, CL12 presents a reasonable, computationally meaningful, constructive alternative to classical logic as a basis for applied theories. To obtain a model example of such theories, this paper rebuilds the traditional, classical-logic-based Peano arithmetic into a computability-logic-based counterpart. Among the purposes of the present contribution is to provide a starting point for what, as the author wishes to hope, might become a new line of research with a potential of interesting findings -- an exploration of the presumably quite unusual metatheory of CL-based arithmetic and other CL-based applied systems.