All about Pigeonhole principle

Proof Systems

• The size complexity of Pigeonhole principle in Resolution is exponential
  • Source: PHP → PC_F2 → Res
• The size complexity of Pigeonhole principle in Truth table is exponential
  • Source: PHP → tlResLin_F2 → tlRes → ttp
• The size complexity of Pigeonhole principle in Tree-like resolution is exponential
  • Source: PHP → tlResLin_F2 → tlRes
• The size complexity of Pigeonhole principle in Regular resolution is exponential
  • Source: PHP → PC_F2 → Res → regRes
• The size complexity of Pigeonhole principle in Ordered resolution is exponential
  • Source: PHP → PC_F2 → Res → regRes → ordRes
• The size complexity of Pigeonhole principle in Pool resolution is exponential
  • Source: PHP → PC_F2 → Res → poolRes
• The size complexity of Pigeonhole principle in Linear resolution is exponential
  • Source: PHP → PC_F2 → Res → linRes
• The size complexity of Pigeonhole principle in Reversible resolution is exponential
  • Source: PHP → PC_F2 → Res → revRes
• The size complexity of Pigeonhole principle in Cutting Planes is polynomial
  • Source: CP → uCP → PHP
• The size complexity of Pigeonhole principle in Tree-like Cutting Planes is [missing?]
• The size complexity of Pigeonhole principle in Cutting Planes with Unary Coefficients is polynomial
  • Source: [citation needed]
• The size complexity of Pigeonhole principle in Semantic Cutting Planes is polynomial
  • Source: semanticCP → CP → uCP → PHP
• The size complexity of Pigeonhole principle in Cutting Planes with Saturation is exponential
  • Source: PHP → PC_F2 → Res → saturationCP
• The size complexity of Pigeonhole principle in Stabbing Planes is polynomial
  • Source: SP → uSP → uCP → PHP
• The size complexity of Pigeonhole principle in Stabbing Planes with Unary Coefficients is polynomial
  • Source: uSP → uCP → PHP
• The size complexity of Pigeonhole principle in Res(CP) is polynomial
  • Source: Res(CP) → CP → uCP → PHP
• The size complexity of Pigeonhole principle in Res(LP) is polynomial
  • Source: Res(LP) → uRes(LP) → uRes(CP) → uCP → PHP
• The size complexity of Pigeonhole principle in Res(CP) with unary coefficients is polynomial
  • Source: uRes(CP) → uCP → PHP
• The size complexity of Pigeonhole principle in Res(LP) with unary coefficients is polynomial
  • Source: uRes(LP) → uRes(CP) → uCP → PHP
• The size complexity of Pigeonhole principle in Res(L\(\&\)P) is polynomial
  • Source: Res(L&P) → Res(LP) → uRes(LP) → uRes(CP) → uCP → PHP
• The size complexity of Pigeonhole principle in Res(L\(\&\)P) with unary coefficients is polynomial
  • Source: uRes(L&P) → uRes(LP) → uRes(CP) → uCP → PHP
• The size complexity of Pigeonhole principle in Semantic degree-k threshold system, treelike version is [missing?]
• The size complexity of Pigeonhole principle in Polynomial Calculus over \(\mathbb{F}_2\) is exponential
  • Source: [citation needed]
• The size complexity of Pigeonhole principle in Nullstellensatz over \(\mathbb{F}_2\) is exponential
  • Source: PHP → PC_F2 → NS_F2
• The size complexity of Pigeonhole principle in ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals is polynomial
  • Source: ResLin_Z → uResLin_Z → uRes(CP) → uCP → PHP
• The size complexity of Pigeonhole principle in unary ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals is polynomial
  • Source: uResLin_Z → uRes(CP) → uCP → PHP
• The size complexity of Pigeonhole principle in ResLin over \(\mathbb{F}_2\), Res(\(\oplus\)) is [missing?]
• The size complexity of Pigeonhole principle in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is exponential
  • Source: IS20 Resolution over linear equations modulo two.
• The size complexity of Pigeonhole principle in Polynomial Calculus over \(\mathbb{Q}\) is exponential
  • Source: [citation needed]
• The size complexity of Pigeonhole principle in Nullstellensatz over \(\mathbb{Q}\) is exponential
  • Source: PHP → PC_Q → NS_Q
• The size complexity of Pigeonhole principle in Hitting is exponential
  • Source: PHP → tlResLin_F2 → tlRes → hit
• The size complexity of Pigeonhole principle in Lift and Project is [missing?]
• The size complexity of Pigeonhole principle in Lift and Project with unary coefficients is [missing?]
• The size complexity of Pigeonhole principle in Positivstellensatz Calculus is polynomial
  • Source: PSC → PS → SoS → SA → PHP
• The size complexity of Pigeonhole principle in Positivstellensatz is polynomial
  • Source: PS → SoS → SA → PHP
• The size complexity of Pigeonhole principle in Lovász--Schrijver (LS) is polynomial
  • Source: Pud99 On the complexity of the propositional calculus
• The size complexity of Pigeonhole principle in Lovász--Schrijver with squares (LS\(_+\)) is polynomial
  • Source: LS+ → LS → PHP
• The size complexity of Pigeonhole principle in Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree is polynomial
  • Source: LSd+ → LS+ → LS → PHP
• The size complexity of Pigeonhole principle in Lovász--Schrijver with squares (LS\(_+^\infty\)), unbounded degree is polynomial
  • Source: LSn+ → sLSn+ → sLS+ → PHP
• The size complexity of Pigeonhole principle in Cone Proof System is polynomial
  • Source: CPS → IPS → extFrege → PHP
• The size complexity of Pigeonhole principle in tl Lovász--Schrijver (LS) is [missing?]
• The size complexity of Pigeonhole principle in Lovász--Schrijver with squares (LS\(_+\)) is [missing?]
• The size complexity of Pigeonhole principle in Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree, treelike is [missing?]
• The size complexity of Pigeonhole principle in static Lovász--Schrijver (static LS) is [missing?]
• The size complexity of Pigeonhole principle in static Lovász--Schrijver, with squares of linear functions (static LS\(_+\)) is polynomial
  • Source: GHP02 Complexity of semialgebraic proofs
• The size complexity of Pigeonhole principle in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is polynomial
  • Source: sLSn+ → sLS+ → PHP
• The size complexity of Pigeonhole principle in Sum of Squares (Lasserre) is polynomial
  • Source: SoS → SA → PHP
• The size complexity of Pigeonhole principle in Sherali--Adams is polynomial
  • Source: Rod07 Rank Lower Bounds for the Sherali-Adams Operator.
• The size complexity of Pigeonhole principle in Circular resolution is polynomial
  • Source: circRes → SA → PHP
• The size complexity of Pigeonhole principle in Unary Sherali--Adams is [missing?]
• The size complexity of Pigeonhole principle in Ideal Proof System is polynomial
  • Source: IPS → extFrege → PHP
• The size complexity of Pigeonhole principle in Extended Frege is polynomial
  • Source: CR79 The Relative Efficiency of Propositional Proof Systems.
• The size complexity of Pigeonhole principle in Extended resolution is polynomial
  • Source: extRes → extFrege → PHP
• The size complexity of Pigeonhole principle in Frege is polynomial
  • Source: Buss87 Polynomial Size Proofs of the Propositional Pigeonhole Principle.
• The size complexity of Pigeonhole principle in \(\mathrm{AC}^0\)-Frege is exponential
  • Source: PHP → fPHP → ofPHP → AC0Frege
• The size complexity of Pigeonhole principle in k-DNF Resolution is exponential
  • Source: PHP → fPHP → ofPHP → AC0Frege → Res-k
• The size complexity of Pigeonhole principle in \(\mathrm{TC}^0\)-Frege is polynomial
  • Source: Kra19 Proof complexity
• The size complexity of Pigeonhole principle in \(\mathrm{AC}^0\)-Frege with mod 2 axioms is [missing?]
• The size complexity of Pigeonhole principle in \(\mathrm{AC}^0\)-Frege with mod 2 gates is [missing?]
• The size complexity of Pigeonhole principle in OBDD(join,weakening) is polynomial
  • Source: OBDDjoinweak → uCP → PHP
• The size complexity of Pigeonhole principle in LK is polynomial
  • Source: LK → Frege → PHP
• The size complexity of Pigeonhole principle in Zermelo-Fraenkl Set Theory with the Axiom of Choice is polynomial
  • Source: ZFC → extFrege → PHP

This database is still incomplete; missing data may indicate either the information was not yet recorded or an open problem. Users are encouraged to contribute missing proof systems and/or relations at https://gitlab.com/proofcomplexityzoo/zoo.

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