We present a formal system to reason about and specify the behavior of multiple intelligent artificial agents. Essentially, each agent can perform certain actions, and it may possess a variety of information in order to reason about its and other agent's actions. Thus, our KARO-framework tries to deal formally with the notion of Knowledge, possessed by the agents, and their possible execution of actions. In particular, each agent may reason about its ---or, alternatively, other's--- Abilities to perform certain actions, the possible Results of such an execution and the availability of the Opportunities to take a particular action. Formally, we combine dynamic and epistemic logic into one modal system, and add the notion of ability to it. We demonstrate that there are several options to define the ability to perform a sequentially composed action, and we outline several properties under two alternative choices. Also, the agents' views on the correctness and feasibility of their plans a...

Abstract. Unlike computer algebra systems, automated theorem provers have not yet achieved considerable recognition and relevance in mathematical practice. A significant shortcoming of mathematical proof assistance systems is that they require the fully formal representation of mathematical content, whereas in mathematical practice an informal, natural-language-like representation where obvious parts are omitted is common. We aim to support mathematical paper writing by integrating a scientific text editor and mathematical assistance systems such that mathematical derivations authored by human beings in a mathematical document can be automatically checked. To this end, we first define a calculus-independent representation language for formal mathematics that allows for underspecified parts. Then we provide two systems of rules that check if a proof is correct and at an acceptable level of granularity. These checks are done by decomposing the proof into basic steps that are then passed on to proof assistance systems for formal verification. We illustrate our approach using an example textbook proof. 1

A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of `proof theoretic ordinals' to theories, gauging their `consistency strength' and `computational power'. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories. Though this area of mathematical logic has is roots in Hilbert's "Beweistheorie " - the aim of which was to lay to rest all worries about the foundations of mathematics once and for all by securing mathematics via an absolute proof of consistency - technical results in pro...

Abstract. The last section of “Lecture at Zilsel’s ” [9, §4] contains an interesting but quite condensed discussion of Gentzen’s first version of his consistency proof for P A [8], reformulating it as what has come to be called the no-counterexample interpretation. I will describe Gentzen’s result (in game-theoretic terms), fill in the details (with some corrections) of Gödel’s reformulation, and discuss the relation between the two proofs. 1. Let me begin with a description of Gentzen’s consistency proof. As had already been noted in [5], we may express it in terms of a game. 1 To simplify things, we can assume that the logical constants of the classical system of number theory, P A, are ∧, ∨, ∀ and ∃ and that negations are applied only to atomic formulas. ¬φ in general is represented by the complement φ of φ, obtained by interchanging ∧ with ∨, ∀ with ∃, and atomic sentences with their negations. The components of the sentences φ ∨ ψ and φ ∧ ψ are φ and ψ. The components of the sentences ∃xφ(x) and ∀xφ(x) are the sentences φ(¯n) for each numeral ¯n. A ∧- or ∀-sentence, called a �-sentence, is thus expressed by the conjunction of its components and a ∨- or ∃-sentence, called a �-sentence, is expressed by the disjunction of them; and so it follows that every sentence can be represented as an infinitary propositional formula built up from prime sentences— atomic or negated atomic sentences. Disjunctive and conjunctive sentences with the components φn (where the range of n is 1, 2 or ω) will be denoted respectively by

Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s

In recent years a number of criticisms have been raised against the formal systems of

The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1811–1832) duel; the deaths of consumption of Gotthold Eisenstein (1823–1852) (who sometimes lectured his few students from his bedside) and of Gustav Roch (1839–1866) in Venice; the drowning of the topologist Pavel Samuilovich Urysohn (1898–1924) on vacation; the burial of Raymond Paley (1907–1933) in an avalanche at Deception Pass in the Rocky Mountains; as well as the fatal imprisonment of Gerhard Gentzen (1909–1945) in Prague1 — these are tales most scholars of logic and mathematics have heard in their student days. Jacques Herbrand, a young prodigy admitted to the École Normale Supérieure as the best student of the year1925, when he was17, died only six years later in a mountaineering accident in La Bérarde (Isère) in France. He left a legacy in logic and mathematics that is outstanding.

The objective of this paper is to introduce an ordinal representation system which has been employed in the determination of the proof--theoretic strength of \Pi 1 2 comprehension and related systems. 1 Introduction The purpose of this paper is to provide an ordinal representation system for the system of \Pi 1 2 analysis, which is the subsystem of formal second order arithmetic, Z 2 , with comprehension confined to \Pi 1 2 -formulae. The ordinal representation can also be used to provide ordinal analyses for theories that are reducible to iterated \Pi 1 2 comprehension, e.g. \Delta 1 3 comprehension. The details will be laid out in the second part of this paper. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. Gentzen fostered hopes that with sufficiently large constructive ordinals one could establish the consistency of analysis, i.e., Z 2 . Considerable progress has been made...

this paper is, in general, to report the state of the art of ordinal analysis and, in particular, the recent success in obtaining an ordinal analysis for the system of #

Abstract. In previous work, the author gave a higher-order analysis of focusing proofs (in the sense of Andreoli’s search strategy), with a role for infinitary rules very similar in structure to Buchholz’s Ω-rule. Among other benefits, this “pattern-based ” description of focusing simplifies the cut-elimination procedure, allowing cuts to be eliminated in a connective-generic way. However, interpreted literally, it is problematic as a representation technique for proofs, because of the difficulty of inspecting and/or exhaustively searching over these infinite objects. In the spirit of infinitary proof theory, this paper explores a view of pattern-based focusing proofs as façons de parler, describing how to compile them down to first-order derivations through defunctionalization, Reynolds ’ program transformation. Our main result is a representation of pattern-based focusing in the Twelf logical framework, whose core type theory is too weak to directly encode infinitary rules—although this weakness directly enables so-called “higher-order abstract syntax ” encodings. By applying the systematic defunctionalization transform, not only do we retain the benefits of the higher-order focusing analysis, but we can also take advantage of HOAS within Twelf, ultimately arriving at a proof representation with surprisingly little bureaucracy. 1