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Set Constraints: Results, Applications and Future Directions
 In Second Workshop on the Principles and Practice of Constraint Programming
"... . Set constraints are a natural formalism for many problems that arise in program analysis. This paper provides a brief introduction to set constraints: what set constraints are, why they are interesting, the current state of the art, open problems, applications and implementations. 1 Introduction ..."
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Cited by 69 (3 self)
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. Set constraints are a natural formalism for many problems that arise in program analysis. This paper provides a brief introduction to set constraints: what set constraints are, why they are interesting, the current state of the art, open problems, applications and implementations. 1 Introduction Set constraints are a natural formalism for describing relationships between sets of terms of a free algebra. A set constraint has the form X ` Y , where X and Y are set expressions. Examples of set expressions are 0 (the empty set), ff (a setvalued variable), c(X; Y ) (a constructor application), and the union, intersection, or complement of set expressions. Recently, there has been a great deal of interest in program analysis algorithms based on solving systems of set constraints, including analyses for functional languages [AWL94, Hei94, AW93, AM91, JM79, MR85, Rey69], logic programming languages [AL94, HJ92, HJ90b, Mis84], and imperative languages [HJ91]. In these algorithms, sets of...
The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations
, 1991
"... ..."
Algorithmic Theories Of Everything
, 2000
"... The probability distribution P from which the history of our universe is sampled represents a theory of everything or TOE. We assume P is formally describable. Since most (uncountably many) distributions are not, this imposes a strong inductive bias. We show that P(x) is small for any universe x lac ..."
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Cited by 32 (15 self)
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The probability distribution P from which the history of our universe is sampled represents a theory of everything or TOE. We assume P is formally describable. Since most (uncountably many) distributions are not, this imposes a strong inductive bias. We show that P(x) is small for any universe x lacking a short description, and study the spectrum of TOEs spanned by two Ps, one reflecting the most compact constructive descriptions, the other the fastest way of computing everything. The former derives from generalizations of traditional computability, Solomonoff’s algorithmic probability, Kolmogorov complexity, and objects more random than Chaitin’s Omega, the latter from Levin’s universal search and a natural resourceoriented postulate: the cumulative prior probability of all x incomputable within time t by this optimal algorithm should be 1/t. Between both Ps we find a universal cumulatively enumerable measure that dominates traditional enumerable measures; any such CEM must assign low probability to any universe lacking a short enumerating program. We derive Pspecific consequences for evolving observers, inductive reasoning, quantum physics, philosophy, and the expected duration of our universe.
Gödel machines: Fully selfreferential optimal universal selfimprovers
 Goertzel and C. Pennachin, Artificial General Intelligence
, 2006
"... Summary. We present the first class of mathematically rigorous, general, fully selfreferential, selfimproving, optimally efficient problem solvers. Inspired by Kurt Gödel’s celebrated selfreferential formulas (1931), such a problem solver rewrites any part of its own code as soon as it has found ..."
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Cited by 25 (12 self)
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Summary. We present the first class of mathematically rigorous, general, fully selfreferential, selfimproving, optimally efficient problem solvers. Inspired by Kurt Gödel’s celebrated selfreferential formulas (1931), such a problem solver rewrites any part of its own code as soon as it has found a proof that the rewrite is useful, where the problemdependent utility function and the hardware and the entire initial code are described by axioms encoded in an initial proof searcher which is also part of the initial code. The searcher systematically and efficiently tests computable proof techniques (programs whose outputs are proofs) until it finds a provably useful, computable selfrewrite. We show that such a selfrewrite is globally optimal—no local maxima!—since the code first had to prove that it is not useful to continue the proof search for alternative selfrewrites. Unlike previous nonselfreferential methods based on hardwired proof searchers, ours not only boasts an optimal order of complexity but can optimally reduce any slowdowns hidden by the O()notation, provided the utility of such speedups is provable at all. 1
Ultimate Cognition à la Gödel
 COGN COMPUT
, 2009
"... "All life is problem solving," said Popper. To deal with arbitrary problems in arbitrary environments, an ultimate cognitive agent should use its limited hardware in the "best" and "most efficient" possible way. Can we formally nail down this informal statement, and der ..."
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Cited by 20 (11 self)
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"All life is problem solving," said Popper. To deal with arbitrary problems in arbitrary environments, an ultimate cognitive agent should use its limited hardware in the "best" and "most efficient" possible way. Can we formally nail down this informal statement, and derive a mathematically rigorous blueprint of ultimate cognition? Yes, we can, using Kurt Gödel’s celebrated selfreference trick of 1931 in a new way. Gödel exhibited the limits of mathematics and computation by creating a formula that speaks about itself, claiming to be unprovable by an algorithmic theorem prover: either the formula is true but unprovable, or math itself is flawed in an algorithmic sense. Here we describe an agentcontrolling program that speaks about itself, ready to rewrite itself in arbitrary fashion once it has found a proof that the rewrite is useful according to a userdefined utility function. Any such a rewrite is necessarily globally optimal—no local maxima!—since this proof necessarily must have demonstrated the uselessness of continuing the proof search for even better rewrites. Our selfreferential program will optimally speed up its proof searcher and other program parts, but only if the speed up’s utility is indeed provable—even ultimate cognition has limits of the Gödelian kind.
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 18 (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
The New AI: General & Sound & Relevant for Physics
 ARTIFICIAL GENERAL INTELLIGENCE (ACCEPTED 2002)
, 2003
"... Most traditional artificial intelligence (AI) systems of the past 50 years are either very limited, or based on heuristics, or both. The new millennium, however, has brought substantial progress in the field of theoretically optimal and practically feasible algorithms for prediction, search, induct ..."
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Cited by 15 (9 self)
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Most traditional artificial intelligence (AI) systems of the past 50 years are either very limited, or based on heuristics, or both. The new millennium, however, has brought substantial progress in the field of theoretically optimal and practically feasible algorithms for prediction, search, inductive inference based on Occam’s razor, problem solving, decision making, and reinforcement learning in environments of a very general type. Since inductive inference is at the heart of all inductive sciences, some of the results are relevant not only for AI and computer science but also for physics, provoking nontraditional predictions based on Zuse’s thesis of the computergenerated universe.
From Philosophical to Industrial Logics
"... One of the surprising developments in the area of program verification is how ideas introduced by logicians in the early part of the 20th Century ended up yielding by the 21 Century industrialstandard propertyspecification languages. This development was enabled by the equally unlikely transformat ..."
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Cited by 7 (0 self)
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One of the surprising developments in the area of program verification is how ideas introduced by logicians in the early part of the 20th Century ended up yielding by the 21 Century industrialstandard propertyspecification languages. This development was enabled by the equally unlikely transformation of the mathematical machinery of automata on infinite words, introduced in the early 1960s for secondorder logic, into effective algorithms for modelchecking tools. This paper attempts to trace the tangled threads of this development.
Randomness in physics
 Nature
, 2006
"... Summary. Most traditional artificial intelligence (AI) systems of the past 50 years are either very limited, or based on heuristics, or both. The new millennium, however, has brought substantial progress in the field of theoretically optimal and practically feasible algorithms for prediction, search ..."
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Cited by 5 (5 self)
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Summary. Most traditional artificial intelligence (AI) systems of the past 50 years are either very limited, or based on heuristics, or both. The new millennium, however, has brought substantial progress in the field of theoretically optimal and practically feasible algorithms for prediction, search, inductive inference based on Occam’s razor, problem solving, decision making, and reinforcement learning in environments of a very general type. Since inductive inference is at the heart of all inductive sciences, some of the results are relevant not only for AI and computer science but also for physics, provoking nontraditional predictions based on Zuse’s thesis of the computergenerated universe. 1