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A Model for Belief Revision
, 1988
"... It is generally recognized that the possibdity of detecting contradictions and identifying their sources is an important feature of an intelligent system. Systems that are able to detect contradictions, identify their causes, or readjust their knowledge bases to remove the contradiction, called Beli ..."
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Cited by 131 (31 self)
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It is generally recognized that the possibdity of detecting contradictions and identifying their sources is an important feature of an intelligent system. Systems that are able to detect contradictions, identify their causes, or readjust their knowledge bases to remove the contradiction, called Belief Revision Systems. Truth Maintenance Systems, or Reason Maintenance Systems. have been studied by several researchers in Artificial bttelligence ( AI). In this paper, we present a logic suitable for supporting belief revision systems, discuss the properties that a belief revision system based on this logic will exhibit, and present a particular intplementation of our model of a belief revision system. The system we present differs from most of the systems developed so far in three respects: First, it is baseti on a logic that was developed to support belief revision systems. Second, it uses the rules of inference of the logic to automatically compute the dependencies among propositions rather than having to force the user to do titis, as in many existing systems. Third, it was the first belief revision system whose implementation relies on the manipulation of sets of assumptions, not justifications.
The ProofTheory and Semantics of Intuitionistic Modal Logic
, 1994
"... Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpret ..."
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Cited by 130 (0 self)
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Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpreted in an intuitionistic metatheory then the induced modal logics no longer satisfy certain intuitionistically invalid principles. This thesis investigates the intuitionistic modal logics that arise in this way. Natural deduction systems for various intuitionistic modal logics are presented. From one point of view, these systems are selfjustifying in that a possible world interpretation of the modalities can be read off directly from the inference rules. A technical justification is given by the faithfulness of translations into intuitionistic firstorder logic. It is also established that, in many cases, the natural deduction systems induce wellknown intuitionistic modal logics, previously given by Hilbertstyle axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their
Formalized mathematics
 TURKU CENTRE FOR COMPUTER SCIENCE
, 1996
"... It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In c ..."
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Cited by 27 (0 self)
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It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.
Using Hypothetical Reasoning as a Method for Belief Ascription
, 1993
"... A key cognitive faculty that enables humans to communicate with each other is their ability to incrementally construct and use models describing the mental states of others. Every such model describing some other cognitive agent will realistically contain only a finite number of sentences in some la ..."
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Cited by 13 (3 self)
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A key cognitive faculty that enables humans to communicate with each other is their ability to incrementally construct and use models describing the mental states of others. Every such model describing some other cognitive agent will realistically contain only a finite number of sentences in some language of thought, hence, assuming sufficiently powerful inference rules, some of its consequences will remain implicit. To make them explicit, the person holding the model could employ a kind of reasoning that can be paraphrased as "what would I believe if I were the other person believing everything I believe that person believes", a strategy that can be viewed as a simulation of the other person's reasoning using the model of that person in conjunction with the reasoning abilities of the simulator. If we want to equip an artificial cognitive agent with such a simulative reasoning ability we have to cope with problems such as simulation at various levels of nesting, metareasoning to make ...
Mathematical Knowledge Management in MIZAR
 Proc. of MKM 2001
, 2001
"... We report on how mathematics is done in the Mizar system. Mizar oers a language for writing mathematics and provides software for proofchecking. Mizar is used to build Mizar Mathematical Library (MML). This is a long term project aiming at building a comprehensive library of mathematical knowled ..."
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We report on how mathematics is done in the Mizar system. Mizar oers a language for writing mathematics and provides software for proofchecking. Mizar is used to build Mizar Mathematical Library (MML). This is a long term project aiming at building a comprehensive library of mathematical knowledge. The language and the checking software evolve and the evolution is driven by the growing MML.
Linear Logic by Levels and Bounded Time Complexity
, 2009
"... This work deals with the characterization of elementary and deterministic polynomial time computation in linear logic through the proofsasprograms correspondence. Girard’s seminal results, concerning elementary and light linear logic, use a principle called stratification to ensure the complexity b ..."
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Cited by 10 (1 self)
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This work deals with the characterization of elementary and deterministic polynomial time computation in linear logic through the proofsasprograms correspondence. Girard’s seminal results, concerning elementary and light linear logic, use a principle called stratification to ensure the complexity bound on the cutelimination procedure. Here, we propose a more flexible control principle, that of indexing, which allows us to extend Girard’s systems while keeping the same complexity properties. A consequence of the higher flexibility of indexing with respect to stratification is the absence of boxes for handling the § modality. We finally propose a variant of our polytime system in which the § modality is only allowed on atoms, and which may thus serve as a basis for developing λcalculus type assignment systems with more efficient typing algorithms than existing ones.
SIMBA: Belief Ascription by Way of Simulative Reasoning
, 1996
"... A key cognitive faculty that enables humans to communicate with each other is their ability to incrementally construct and use models describing the mental states of others, in particular, models of their beliefs. Not only do humans have beliefs about the beliefs of others, they can also reason with ..."
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Cited by 8 (0 self)
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A key cognitive faculty that enables humans to communicate with each other is their ability to incrementally construct and use models describing the mental states of others, in particular, models of their beliefs. Not only do humans have beliefs about the beliefs of others, they can also reason with these beliefs even if they do not hold them themselves. If we want to build an artificial or computational cognitive agent that is similarly capable, we need a formalism that is fully adequate to represent the beliefs of other agents, and that also specifies how to reason with them. Standard formalizations of knowledge or belief, in particular the various epistemic and doxastic logics, seem to be not very well suited to serve as the formal device upon which to build an actual computational agent. They neglect either representation problems, or the reasoning aspect, or the defeasibility that is inherent in reasoning about somebody else's beliefs, or they use idealizations which are problema...
Diamonds are a Philosopher's Best Friends. The Knowability Paradox and Modal Epistemic Relevance Logic (Extended Abstract)
 Journal of Philosophical Logic
, 2002
"... Heinrich Wansing Dresden University of Technology The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the ..."
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Cited by 7 (0 self)
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Heinrich Wansing Dresden University of Technology The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the unwanted disappearance of the diamond may be escaped. The emphasis is not laid on a discussion of the contentious premise of the knowability paradox, namely that all truths are possibly known, but on how from this assumption the conclusion is derived that all truths are, in fact, known. Nevertheless, the solution o#ered is in the spirit of the constructivist attitude usually maintained by defenders of the antirealist premise. In order to avoid the paradoxical reasoning, a paraconsistent constructive relevant modal epistemic logic with strong negation is defined semantically. The system is axiomatized and shown to be complete.
Proof Assistants: history, ideas and future
"... In this paper we will discuss the fundamental ideas behind proof assistants: What are they and what is a proof anyway? We give a short history of the main ideas, emphasizing the way they ensure the correctness of the mathematics formalized. We will also briefly discuss the places where proof assista ..."
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In this paper we will discuss the fundamental ideas behind proof assistants: What are they and what is a proof anyway? We give a short history of the main ideas, emphasizing the way they ensure the correctness of the mathematics formalized. We will also briefly discuss the places where proof assistants are used and how we envision their extended use in the future. While being an introduction into the world of proof assistants and the main issues behind them, this paper is also a position paper that pushes the further use of proof assistants. We believe that these systems will become the future of mathematics, where definitions, statements, computations and proofs are all available in a computerized form. An important application is and will be in computer supported modelling and verification of systems. But their is still along road ahead and we will indicate what we believe is needed for the further proliferation of proof assistants.