1. Proof Planning as Understanding as Cortical Functions Brendan Juba With Manuel Blum Matt Humphrey Ryan Williams

2. 2 WHAT and WHY WANTED: proof-clustering algorithm –Characterizes high-level idea –Aid theorem-provers WANTED: define CONSCSness –Aid in answering fundamental questions –Basis for developing protocols –Directing development of robots, etc.

3. 3 CONTENTS NO algorithms –– INSTEAD: –Where to discover an algorithm Viewing neocortex as a proof planner –Why expect suitability Link to understanding

4. 4 Proof Planning in the Memory- Prediction Framework Suppose Alice is studying proofs… Under the Framework: –Regions of cortex representing proof steps switch on in sequence –Hierarchically higher regions form “names” “names” and “names of patterns of names” –Alice can recall the patterns later Patterns serve as proof plans (more…)

5. 5 Patterns serve as proof plans? Proof plan: –Generates sequence of proof steps –Features: Expectancy Generality Satisfied by named patterns in cortex –Proof steps encoded in lower regions

6. 6 Where’s the algorithm? Critical link between cortical regions –Cortical region forms name for an input pattern –Translated: forms proof plan from pattern of already-formed proof plans –Our algorithm! Presently: not understood.

7. 7 Proof Planning and the Cortical Algorithms Conservative learning algorithm  lower bounds – Proof Planning: restricted domain Decoded cortical algorithms  system for learning and utilizing proof plans – “But, is it any good?”

8. 8 YOU ARE HERE CONTENTS –Where to discover an algorithm Viewing neocortex as a proof planner –Why expect suitability  Link to understanding

9. 9 Understanding in the Memory- Prediction Framework

10. 10 Understanding as Proof Plans Share several characteristics Identifying a proof plan permits –Prediction –Correction of “minor” mistakes –Re-use of ideas and/or techniques –Generation of summaries

11. 11 Ideal Proof Plans Goals of proof-planning –Mimic human theorem-proving –Produce human-oriented output Goal for CONSCS –Characterize high-level ideas

12. 12 Directions for Future Work 1.Decipher cortical algorithms!! 2.Automate learning of proof plans 3.Analyze cortical functions 4.Refine definitions for CONSCS