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Formalising Abilities and Opportunities of Agents
 Fundamenta Informaticae
, 1998
"... 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 KAROframework tries ..."
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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 KAROframework 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...
Hilbert’s Program Then and Now
, 2005
"... 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 els ..."
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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
Textbook proofs meet formal logic  the problem of underspecification and granularity
 Proceedings of MKM’05, volume 3863 of LNAI, IUB
, 2006
"... 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, ..."
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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, naturallanguagelike 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 calculusindependent 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
The growth of mathematical knowledge: an open world view
 The growth of mathematical knowledge, Kluwer, Dordrecht 2000
"... mathematical knowledge: “The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but ..."
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Cited by 5 (5 self)
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mathematical knowledge: “The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past ” (Poincaré 1958, p. 14). The view criticized by Poincaré corresponds to Frege’s idea that the development of mathematics can be described as an activity of system building, where each system is supposed to provide a complete representation for a certain mathematical field and must be pitilessly torn down whenever it fails to achieve such an aim. All facts concerning any mathematical field must be fully organized in a given system because “in mathematics we must always strive after a system that is complete in itself ” (Frege 1979, p. 279). Frege is aware that systems introduce rigidity and are in conflict with the actual development of mathematics because “in history we have development; a system is static”, but he sticks
Gödels reformulation of Gentzen’s first consistency proof for arithmetic: the nocounterexample interpretation
 The. Bulletin of Symbolic Logic
, 2005
"... 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 nocounterexample interpretation. I will describe Gentzen’s result ( ..."
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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 nocounterexample interpretation. I will describe Gentzen’s result (in gametheoretic 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
Gödel’s incompleteness theorem and the philosophy of open systems
 7, Centre de Recherches Sémiologiques, Université de Neuchâtel (Neuchâtel
, 1992
"... In recent years a number of criticisms have been raised against the formal systems of ..."
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In recent years a number of criticisms have been raised against the formal systems of
Defunctionalizing Focusing Proofs (Or, How Twelf Learned To Stop Worrying And Love The Ωrule)
"... Abstract. In previous work, the author gave a higherorder 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 Ωrule. Among other benefits, this “patternbased ” description of focusing simplifies the cut ..."
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Abstract. In previous work, the author gave a higherorder 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 Ωrule. Among other benefits, this “patternbased ” description of focusing simplifies the cutelimination procedure, allowing cuts to be eliminated in a connectivegeneric 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 patternbased focusing proofs as façons de parler, describing how to compile them down to firstorder derivations through defunctionalization, Reynolds ’ program transformation. Our main result is a representation of patternbased focusing in the Twelf logical framework, whose core type theory is too weak to directly encode infinitary rules—although this weakness directly enables socalled “higherorder abstract syntax ” encodings. By applying the systematic defunctionalization transform, not only do we retain the benefits of the higherorder focusing analysis, but we can also take advantage of HOAS within Twelf, ultimately arriving at a proof representation with surprisingly little bureaucracy. 1
Gödel on Intuition and on Hilbert’s finitism
"... There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the con ..."
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There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, Gödel’s writings represent a smooth evolution, with just one rather small doublereversal, of his view of finitism. He used the term “finit ” (in German) or “finitary ” or “finitistic ” primarily to refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [Gödel, 1938a] and the lecture notes for a lecture at Yale [Gödel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of firstorder number theory, P A; but starting in the Dialectica paper