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Proof Complexity of Systems of (Non-Deterministic) Decision Trees and Branching Programs

Authors Sam Buss, Anupam Das, Alexander Knop

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Author Details

Sam Buss
  • Dept. of Mathematics, UC San Diego, USA
Anupam Das
  • Dept. of Computer Science, University of Copenhagen, Denmark
Alexander Knop
  • Dept. of Mathematics, UC San Diego, USA

Funding

  • Buss, Sam: This work supported in part by Simons Foundation grant 578919.
  • Das, Anupam: This work was supported by a Marie Skłodowska-Curie fellowship, Monotonicity in Logic and Complexity, ERC project 753431.

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Sam Buss, Anupam Das, and Alexander Knop. Proof Complexity of Systems of (Non-Deterministic) Decision Trees and Branching Programs. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.CSL.2020.12

Abstract

This paper studies propositional proof systems in which lines are sequents of decision trees or branching programs, deterministic or non-deterministic. Decision trees (DTs) are represented by a natural term syntax, inducing the system LDT, and non-determinism is modelled by including disjunction, ∨, as primitive (system LNDT). Branching programs generalise DTs to dag-like structures and are duly handled by extension variables in our setting, as is common in proof complexity (systems eLDT and eLNDT). 
Deterministic and non-deterministic branching programs are natural nonuniform analogues of log-space (L) and nondeterministic log-space (NL), respectively. Thus eLDT and eLNDT serve as natural systems of reasoning corresponding to L and NL, respectively.
The main results of the paper are simulation and non-simulation results for tree-like and dag-like proofs in LDT, LNDT, eLDT and eLNDT. We also compare them with Frege systems, constant-depth Frege systems and extended Frege systems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • proof complexity
  • decision trees
  • branching programs
  • logspace
  • sequent calculus
  • non-determinism
  • low-depth complexity

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