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Proof Planning as Understanding as Cortical Functions

Explore the correlation between proof planning and cortical functions, aiming to identify the algorithm for proof clustering. Discover the framework of memory prediction and the role of understanding in this context. Discuss how proof plans mimic human theorem proving and contribute to future work on automating learning of proof plans and refining definitions for CONSCS.

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Proof Planning as Understanding as Cortical Functions

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

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

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

  9. Understanding in the Memory-Prediction Framework

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

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

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