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Review of Shadows of the Mind By Roger Penrose
, 1994
"... This book and its predecessor The Emperor's New Mind argue that natural minds cannot be understood and artificial minds cannot be constructed without new physics, about which the book gives some ideas. We have no objection to new physics but don't see it as necessary for artificial intel ..."
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This book and its predecessor The Emperor's New Mind argue that natural minds cannot be understood and artificial minds cannot be constructed without new physics, about which the book gives some ideas. We have no objection to new physics but don't see it as necessary for artificial intelligence. We see artificial intelligence research as making definite progress on difficult scientific problems. I take it that students of natural intelligence also see present physics as adequate for understanding mind. This review concerns only some problems with the first part of the book.
Making Robots Conscious of their Mental States
, 1995
"... In AI, consciousness of self consists in a program having certain kinds of facts about its own mental processes and state of mind. We discuss what consciousness of its own mental structures a robot will need in order to operate in the common sense world and accomplish the tasks humans will give it. ..."
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Cited by 90 (7 self)
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In AI, consciousness of self consists in a program having certain kinds of facts about its own mental processes and state of mind. We discuss what consciousness of its own mental structures a robot will need in order to operate in the common sense world and accomplish the tasks humans will give it. It's quite a lot. Many features of human consciousness will be wanted, some will not, and some abilities not possessed by humans have already been found feasible and useful in limited contexts. We give preliminary fragments of a logical language a robot can use to represent information about its own state of mind. A robot will often have to conclude that it cannot decide a question on the basis of the information in memory and therefore must seek information externally. Godel's idea of relative consistency is used to formalize nonknowledge. Programs with the kind of consciousness discussed in this article do not yet exist, although programs with some components of it exist. Thinking about c...
Set theory for verification: I. From foundations to functions
 J. Auto. Reas
, 1993
"... A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherord ..."
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Cited by 52 (20 self)
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A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherorder syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor’s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey’s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics. Key words. Isabelle, set theory, generic theorem proving, Ramsey’s Theorem,
Uniqueness theorems for certain triangulated categories possessing an Adams spectral sequence
, 139
"... 1.2. The axioms ..."
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The inverse conjecture for the Gowers norm over finite fields via the correspondence principle. Arxiv preprint arXiv:0810.5527
 The State University of New Jersey, Department of
, 2008
"... The inverse conjecture for the Gowers norms U d(V) for finitedimensional vector spaces V over a finite field F asserts, roughly speaking, that a bounded function f has large Gowers norm ‖ f ‖Ud (V) if and only if it correlates with a phase polynomial φ = eF(P) of degree at most d−1, thus P:V→F is a ..."
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The inverse conjecture for the Gowers norms U d(V) for finitedimensional vector spaces V over a finite field F asserts, roughly speaking, that a bounded function f has large Gowers norm ‖ f ‖Ud (V) if and only if it correlates with a phase polynomial φ = eF(P) of degree at most d−1, thus P:V→F is a polynomial of degree at most d − 1. In this paper, we develop a variant of the Furstenberg correspondence principle which allows us to establish this conjecture in the large characteristic case char F> d from an ergodic theory counterpart, which was recently established by Bergelson, Tao and Ziegler. In low characteristic we obtain a partial result, in which the phase polynomial φ is allowed to be of some larger degree C(d). The full inverse conjecture remains open in low characteristic; the counterexamples found so far in this setting can be avoided by a slight reformulation of the conjecture. 1.
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Cited by 24 (0 self)
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
Does Mathematics Need New Axioms?
 American Mathematical Monthly
, 1999
"... this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called f ..."
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Cited by 23 (2 self)
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this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called for the pursuit of new axioms to settle undecided arithmetical problems. And from 1947 on, with the publication of his unusual article, "What is Cantor's continuum problem?" [11], he called in addition for the pursuit of new axioms to settle Cantor's famous conjecture about the cardinal number of the continuum. In both cases, he pointed primarily to schemes of higher infinity in set theory as the direction in which to seek these new principles. Logicians have learned a great deal in recent years that is relevant to Godel's program, but there is considerable disagreement about what conclusions to draw from their results. I'm far from unbiased in this respect, and you'll see how I come out on these matters by the end of this essay, but I will try to give you a fair presentation of other positions along the way so you can decide for yourself which you favor.
Biform theories in Chiron
 Towards Mechanized Mathematical Assistants, volume 4573 of Lecture Notes in Computer Science
, 2007
"... Abstract. An axiomatic theory represents mathematical knowledge declaratively as a set of axioms. An algorithmic theory represents mathematical knowledge procedurally as a set of algorithms. A biform theory is simultaneously an axiomatic theory and an algorithmic theory. It represents mathematical k ..."
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Abstract. An axiomatic theory represents mathematical knowledge declaratively as a set of axioms. An algorithmic theory represents mathematical knowledge procedurally as a set of algorithms. A biform theory is simultaneously an axiomatic theory and an algorithmic theory. It represents mathematical knowledge both declaratively and procedurally. Since the algorithms of algorithmic theories manipulate the syntax of expressions, biform theories—as well as algorithmic theories—are difficult to formalize in a traditional logic without the means to reason about syntax. Chiron is a derivative of vonNeumannBernaysGödel (nbg) set theory that is intended to be a practical, generalpurpose logic for mechanizing mathematics. It includes elements of type theory, a scheme for handling undefinedness, and a facility for reasoning about the syntax of expressions. It is an exceptionally wellsuited logic for formalizing biform theories. This paper defines the notion of a biform theory, gives an overview of Chiron, and illustrates how biform theories can be formalized in Chiron. 1