The quantum measurement problem and physical reality: a computation theoretic perspective

Is the universe computable? If yes, is it computationally a polynomial place? In standard quantum mechanics, which permits infinite parallelism and the infinitely precise specification of states, a negative answer to both questions is not ruled out. On the other hand, empirical evidence suggests that NP-complete problems are intractable in the physical world. Likewise, computational problems known to be algorithmically uncomputable do not seem to be computable by any physical means. We suggest that this close correspondence between the efficiency and power of abstract algorithms on the one hand, and physical computers on the other, finds a natural explanation if the universe is assumed to be algorithmic; that is, that physical reality is the product of discrete sub-physical information processing equivalent to the actions of a probabilistic Turing machine. This assumption can be reconciled with the observed exponentiality of quantum systems at microscopic scales, and the consequent possibility of implementing Shor's quantum polynomial time algorithm at that scale, provided the degree of superposition is intrinsically, finitely upper-bounded. If this bound is associated with the quantum-classical divide (the Heisenberg cut), a natural resolution to the quantum measurement problem arises. From this viewpoint, macroscopic classicality is an evidence that the universe is in BPP, and both questions raised above receive affirmative answers. A recently proposed computational model of quantum measurement, which relates the Heisenberg cut to the discreteness of Hilbert space, is briefly discussed. A connection to quantum gravity is noted. Our results are compatible with the philosophy that mathematical truths are independent of the laws of physics.Comment: Talk presented at "Quantum Computing: Back Action 2006", IIT Kanpur, India, March 200

Computability and Algorithmic Complexity in Economics

This is an outline of the origins and development of the way computability theory and algorithmic complexity theory were incorporated into economic and finance theories. We try to place, in the context of the development of computable economics, some of the classics of the subject as well as those that have, from time to time, been credited with having contributed to the advancement of the field. Speculative thoughts on where the frontiers of computable economics are, and how to move towards them, conclude the paper. In a precise sense - both historically and analytically - it would not be an exaggeration to claim that both the origins of computable economics and its frontiers are defined by two classics, both by Banach and Mazur: that one page masterpiece by Banach and Mazur ([5]), built on the foundations of Turing’s own classic, and the unpublished Mazur conjecture of 1928, and its unpublished proof by Banach ([38], ch. 6 & [68], ch. 1, #6). For the undisputed original classic of computable economics is Rabinís effectivization of the Gale-Stewart game ([42];[16]); the frontiers, as I see them, are defined by recursive analysis and constructive mathematics, underpinning computability over the computable and constructive reals and providing computable foundations for the economist’s Marshallian penchant for curve-sketching ([9]; [19]; and, in general, the contents of Theoretical Computer Science, Vol. 219, Issue 1-2). The former work has its roots in the Banach-Mazur game (cf. [38], especially p.30), at least in one reading of it; the latter in ([5]), as well as other, earlier, contributions, not least by Brouwer.

The Mathematical Universe

I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters, randomness and initial conditions to broader issues like consciousness, parallel universes and Godel incompleteness. I hypothesize that only computable and decidable (in Godel's sense) structures exist, which alleviates the cosmological measure problem and help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems.Comment: Replaced to match accepted Found. Phys. version, 31 pages, 5 figs; more details at http://space.mit.edu/home/tegmark/toe.htm

The Fundamental Theorems of Welfare Economics, DSGE and the Theory of Policy - Computable & Constructive Foundations

The genesis and the path towards what has come to be called the DSGE model is traced, from its origins in the Arrow-Debreu General Equilibrium model (ADGE), via Scarf's Computable General Equilibrium model (CGE) and its applied version as Applied Computable General Equilibrium model (ACGE), to its ostensible dynamization as a Recursive Competitive Equilibrium (RCE). It is shown that these transformations of the ADGE - including the fountainhead - are computably and constructively untenable. The policy implications of these (negative) results, via the Fundamental Theorems of Welfare Economics in particular, and against the backdrop of the mathematical theory of economic policy in general, are also discussed (again from computable and constructive points of view). Suggestions for going 'beyond DSGE' are, then, outlined on the basis of a framework that is underpinned - from the outset - by computability and constructivity considerationsComputable General Equilibrium, Dynamic Stochastic General Equilibrium, Computability, Constructivity, Fundamental Theorems of Welfare Economics, Theory of Policy, Coupled Nonlinear Dynamic

Agent-Based Modeling: The Right Mathematics for the Social Sciences?

This study provides a basic introduction to agent-based modeling (ABM) as a powerful blend of classical and constructive mathematics, with a primary focus on its applicability for social science research.� The typical goals of ABM social science researchers are discussed along with the culture-dish nature of their computer experiments. The applicability of ABM for science more generally is also considered, with special attention to physics. Finally, two distinct types of ABM applications are summarized in order to illustrate concretely the duality of ABM: Real-world systems can not only be simulated with verisimilitude using ABM; they can also be efficiently and robustly designed and constructed on the basis of ABM principles. �