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Cycling in proofs and feasibility
 Transactions of the American Mathematical Society
, 1998
"... Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual b ..."
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Cited by 8 (4 self)
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Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual but for which some very large number is defined to be not feasible. Parikh shows that sufficiently short proofs in this theory can only prove true statements of arithmetic. We pursue these topics in light of logical flow graphs of proofs (Buss, 1991) and show that Parikh’s lower bound for concrete consistency reflects the presence of cycles in the logical graphs of short proofs of feasibility of large numbers. We discuss two concrete constructions which show the bound to be optimal and bring out the dynamical aspect of formal proofs. For this paper the concept of feasible numbers has two roles, as an idea with its own life and as a vehicle for exploring general principles on the dynamics and geometry of proofs. Cycles can be seen as a measure of how complicated a proof can be. We prove that short proofs must have cycles. 1.
A Logic for Perception and Belief
, 1991
"... We present a modal logic for reasoning about perception and belief, captured respectively by the operators P and B. The B operator is the standard belief operator used in recent years, and the P operator is similarly defined. The contribution of the paper is twofold. First, in terms of P we provide ..."
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Cited by 7 (2 self)
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We present a modal logic for reasoning about perception and belief, captured respectively by the operators P and B. The B operator is the standard belief operator used in recent years, and the P operator is similarly defined. The contribution of the paper is twofold. First, in terms of P we provide a definition of perceptual indistinguishability, such as arises out of limited visual acuity. The definition is concise, intuitive (we find), and avoids traditional paradoxes. Second, we explore the bimodal B \Gamma P system. We argue that the relationship between the two modalities varies among settings: The agent may or may not have confidence in its perception, may or may not be accurate in it, and so on. We therefore define a number of agent types corresponding to these various assumptions, and for each such agent type we provide a sound and complete axiomatization of the B \Gamma P system. 1 Introduction There is a longstanding interest in AI in defining the mental state of agents. ...
Qualitative Reasoning about Perception and Belief
 In M. E. Pollack (Ed.), Proceedings of the 15th International Joint Conference on Arti Intelligence (IJCAI '97
, 1997
"... We present a qualitative model for reasoning about perceptions, sensors, and belief, and a logic to reason about this model. Basic to our model is a distinction between precision and accuracy, for both of which we provide qualitative definitions. In our logic this distinction gives rise to two modal ..."
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We present a qualitative model for reasoning about perceptions, sensors, and belief, and a logic to reason about this model. Basic to our model is a distinction between precision and accuracy, for both of which we provide qualitative definitions. In our logic this distinction gives rise to two modal operatorsP for actual perception, and C p for perceptual capability, which is captured as a set of possible percepts. Adding to these operators the standard B operator to model belief, we end up with a logic combining standard Kripkestyle semantics with the almoststandard `neighborhood semantics.' We define various agent types in the logic, from agents who believe all and only what the sensors tell them, to much more skeptical agents. We define each agent both axiomatically and modeltheoretically, and provide soundness and completeness results relating the two types of definitions. 1 Introduction A great deal of attention in AI has been devoted to modeling states of information, whe...
In Proceedings of the 15th International Joint Conference on Arti
 In M. E. Pollack (Ed.), Proceedings of the 15th International Joint Conference on Arti Intelligence (IJCAI '97
, 1997
"... We present a qualitative model for reasoning about perceptions, sensors, and belief, and a logic to reason about this model. Basic to our model is a distinction between precision and accuracy, for both of which we provide qualitative de nitions. In our logic this distinction gives rise to two ..."
Abstract
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We present a qualitative model for reasoning about perceptions, sensors, and belief, and a logic to reason about this model. Basic to our model is a distinction between precision and accuracy, for both of which we provide qualitative de nitions. In our logic this distinction gives rise to two modal operatorsP for actual perception, and C p for perceptual capability, which is captured as a set of possible percepts. Adding to these operators the standard B operator to model belief, we end up with a logic combining standard Kripkestyle semantics with the almoststandard `neighborhood semantics.' We de ne various agent types in the logic, from agents who believe all and only what the sensors tell them, to much more skeptical agents. We de ne each agent both axiomatically and modeltheoretically, and provide soundness and completeness results relating the two types of de nitions.
A Logic for Approximate Reasoning
 Proc. Third International Conference on Principles of Knowledge Representation and Reasoning (KR '92
, 1992
"... We investigate the problem of reasoning with imprecise quantitative information. We give formal semantics to a notion of approximate observations, and define two types of entailment for a knowledge base with imprecise information: a cautious notion, which allows only completely justified conclusions ..."
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We investigate the problem of reasoning with imprecise quantitative information. We give formal semantics to a notion of approximate observations, and define two types of entailment for a knowledge base with imprecise information: a cautious notion, which allows only completely justified conclusions, and a bold one, which allows jumping to conclusions. Both versions of the entailment relation are shown to be decidable. We investigate the behavior of the two alternatives on various examples, and show that the answers obtained are intuitively desirable. The behavior of these two entailment relations is completely characterized for a certain sublanguage, in terms of the logic of true equality. We demonstrate various properties of the full logic, and show how it applies to many situations of interest.
Pedigreed Belief Change
, 2001
"... Revising beliefs given new information has long been a problem in artificial intelligence (AI). Much of this work has assumed little about the source of the information. However, such pedigree information is often readily accessible and useful in determining how to incorporate the new input. We shou ..."
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Revising beliefs given new information has long been a problem in artificial intelligence (AI). Much of this work has assumed little about the source of the information. However, such pedigree information is often readily accessible and useful in determining how to incorporate the new input. We should treat very dierently the information "It is raining" when we receive it from the weather reporter, from a notorious liar, or from our own eyes. The goal of this work is to enable the use of such pedigree information in semanticallyjustified ways. One source