From November 11-18, 2012, we held (what we now call) the 1st MIRI Workshop on Logic, Probability, and Reflection. This workshop had four participants:
- Eliezer Yudkowsky (MIRI)
- Paul Christiano (UC Berkeley)
- Marcello Herreshoff (Google)
- Mihály Bárász (Google)
The participants worked on the foundations of probabilistic reflective reasoning. In particular, they showed that a careful formalization of probabilistic logic can circumvent many classical paradoxes of self-reference. Applied to metamathematics, this framework provides (what seems to be) the first definition of truth which is expressive enough for use in reflective reasoning. Applied to set theory, this framework provides an implementation of probabilistic set theory based on unrestricted comprehension which is nevertheless powerful enough to formalize ordinary mathematical reasoning (in contrast with similar fuzzy set theories, which were originally proposed for this purpose but later discovered to be incompatible with mathematical induction).
These results suggest a similar approach may be used to work around Löb’s theorem, but this has not yet been explored. This work will be written up over the coming months.
In the meantime, MIRI is preparing for the 2nd MIRI Workshop on Logic, Probability, and Reflection, to take place from April 3-24, 2013. This workshop will be broken into two sections. The first section (Apr 3-11) will bring together the 1st workshop’s participants and 8 additional participants:
- Stuart Armstrong (Oxford University)
- Daniel Dewey (Oxford University)
- Benja Fallenstein (University of Vienna)
- Patrick LaVictoire (UW Madison)
- Jacob Steinhardt (Stanford University)
- Qiaochu Yuan (UC Berkeley)
- Andrew Critch (UC Berkeley)
- Jacob Taylor (Stanford)
The second section (Apr 12-24) will consist solely of the 4 participants from the 1st workshop.
Participants of this 2nd workshop will continue to work on the foundations of reflective reasoning, for example Gödelian obstacles to reflection, and decision algorithms for reflective agents (e.g. TDT).
Additional MIRI research workshops are also tentatively planned for the summer and fall of 2013.
Update: An early draft of the paper describing the first result from the 1st workshop is now available here.