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Let L be a language that can be decided in linear space and let ϵ>0 be any constant. Let A be the exponential hardness assumption that for every n, membership in L for inputs of length n cannot be decided by circuits of size smaller than 2ϵn. We prove that for every function f:{0,1}∗→{0,1}, computable by a randomized logspace algorithm R, there exists a deterministic logspace algorithm D (attempting to compute f), such that on every input x of length n, the algorithm D outputs one of the following:1)The correct value f(x).2)The string: “I am unable to compute f(x) because the hardness assumption A is false”, followed by a (provenly correct) circuit of size smaller than 2ϵn′ for membership in L for inputs of length n′, for some n′=Θ(logn); that is, a circuit that refutes A. Moreover, D is explicitly constructed, given R.We note that previous works on the hardness-versus-randomness paradigm give derandomized algorithms that rely blindly on the hardness assumption. If the hardness assumption is false, the algorithms may output incorrect values, and thus a user cannot trust that an output given by the algorithm is correct. Instead, our algorithm D verifies the computation so that it never outputs an incorrect value. Thus, if D outputs a value for f(x), that value is certified to be correct. Moreover, if D does not output a value for f(x), it alerts that the hardness assumption was found to be false, and refutes the assumption.Our next result is a universal derandomizer for BPL (the class of problems solvable by bounded-error randomized logspace algorithms) 1 : We give a deterministic algorithm U that takes as an input a randomized logspace algorithm R and an input x and simulates the computation of R on x, deteriministically. Under the widely believed assumption BPL=L, the space ...
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On the Direction of Innovation
How are resources allocated across different R\&D areas, i.e. problems to be solved? As a result of dynamic congestion externalities, the competitive market allocates excessive resources into those of high return, being those with higher private (and social) payoffs. Good problems are tackled too soon, and as a result the distribution of open research problems in the socially optimal solution stochastically dominates that of the competitive equilibrium. Asevere form of rent dissipation occurs in the latter, where the total value of R\&D activity equals the value of allocating all resources to the least valuable problem solved. Resulting losses can be substantial.
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- Award ID(s):
- 1757134
- PAR ID:
- 10591560
- Publisher / Repository:
- University of Chicago Press
- Date Published:
- Journal Name:
- Journal of political ecology
- ISSN:
- 1073-0451
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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