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- Matteo Brunelli, Andrew Critch, Michele Fedrizzi
- Applied Mathematics and Computation
- 2013

Analyzing the consistency of preferences is an important step in decision making with pairwise comparison matrices, and several indices have been proposed in order to estimate it. In this paper we prove the proportionality between some consistency indices in the framework of the Analytic Hierarchy Process. Knowing such equivalences eliminates redundancy in… (More)

We quantify the representational power of matrix product states (MPS) for entangled qubit systems by giving polynomial expressions in a pure quantum state’s amplitudes which hold if and only if the state is a translation invariant matrix product state or a limit of such states. For systems with few qubits, we give these equations explicitly, considering… (More)

- Andrew Critch
- 2013

Algebraic Geometry of Hidden Markov and Related Models

We survey eight research areas organized around one question: As learning systems become increasingly intelligent and autonomous, what design principles can best ensure that their behavior is aligned with the interests of the operators? We focus on two major technical obstacles to AI alignment: the challenge of specifying the right kind of objective… (More)

- Andrew Critch
- ArXiv
- 2016

Löb’s theorem and Gödel’s theorems make predictions about the behavior of systems capable of self-reference with unbounded computational resources with which to write and evaluate proofs. However, in the real world, systems capable of self-reference will have limited memory and processing speed, so in this paper we introduce an effective version of Löb’s… (More)

- Scott Garrabrant, Tsvi Benson-Tilsen, Andrew Critch, Nate Soares, Jessica Taylor
- Electronic Colloquium on Computational Complexity
- 2016

We present a computable algorithm that assigns probabilities to every logical statement in a given formal language, and refines those probabilities over time. For instance, if the language is Peano arithmetic, it assigns probabilities to all arithmetical statements, including claims about the twin prime conjecture, the outputs of long-running computations,… (More)

- Andrew Critch
- ArXiv
- 2017

Existing multi-objective reinforcement learning (MORL) algorithms do not account for objectives that arise from players with differing beliefs. Concretely, consider two players with different beliefs and utility functions who may cooperate to build a machine that takes actions on their behalf. A representation is needed for how much the machine’s policy… (More)

We present the logical induction criterion for computable algorithms that assign probabilities to every logical statement in a given formal language, and refine those probabilities over time. The criterion is motivated by a series of stock trading analogies. Roughly speaking, each logical sentence φ is associated with a stock that is worth $1 per share if φ… (More)

We present a computable algorithm that assigns probabilities to every logical statement in a given formal language, and refines those probabilities over time. For instance, if the language is Peano arithmetic, it assigns probabilities to all arithmetical statements, including claims about the twin prime conjecture, the outputs of long-running computations,… (More)

It is often argued that an agent making decisions on behalf of two or more principals who have different utility functions should adopt a Pareto-optimal policy, i.e., a policy that cannot be improved upon for one agent without making sacrifices for another. A famous theorem of Harsanyi shows that, when the principals have a common prior on the outcome… (More)

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