All about Regular resolution

Proof Systems

• Regular resolution weaker than Resolution
• Source: [subsystem]
• Source: Res → poolRes → stone → regRes
• Regular resolution stronger than Truth table
• Source: regRes → tlRes → ttp
• Source: regRes → ordering → tlRes → ttp
• Regular resolution stronger than Tree-like resolution
• Source: [subsystem]
• Source: regRes → ordering → tlRes
• Regular resolution stronger than Ordered resolution
• Source: [subsystem]
• Source: regRes → pearl → ordRes
• Regular resolution weaker than Pool resolution
• Source: vGel05 Pool Resolution and Its Relation to Regular Resolution and DPLL with Clause Learning.
• Source: poolRes → stone → regRes
• Regular resolution does not simulate Linear resolution
• Source: linRes → ordering+guard → regRes
• Regular resolution [missing?] Reversible resolution
• Regular resolution weaker than Cutting Planes
• Source: CP → uCP → Res → regRes
• Source: CP → uCP → PHP → PC_Q → Res → regRes
• Regular resolution not simulated by Tree-like Cutting Planes
• Source: regRes → ordRes → peb+ind → tlCP
• Regular resolution weaker than Cutting Planes with Unary Coefficients
• Source: uCP → Res → regRes
• Source: uCP → Res → poolRes → stone → regRes
• Regular resolution weaker than Semantic Cutting Planes
• Source: semanticCP → saturationCP → Res → regRes
• Source: semanticCP → CP → uCP → PHP → PC_Q → Res → regRes
• Regular resolution weaker than Cutting Planes with Saturation
• Source: saturationCP → Res → regRes
• Source: saturationCP → Res → poolRes → stone → regRes
• Regular resolution weaker than Stabbing Planes
• Source: SP → uSP → uCP → Res → regRes
• Source: SP → uSP → uCP → Res → poolRes → stone → regRes
• Regular resolution weaker than Stabbing Planes with Unary Coefficients
• Source: uSP → uCP → Res → regRes
• Source: uSP → uCP → Res → poolRes → stone → regRes
• Regular resolution weaker than Res(CP)
• Source: Res(CP) → CP → uCP → Res → regRes
• Source: Res(CP) → CP → uCP → PHP → PC_Q → Res → regRes
• Regular resolution weaker than Res(LP)
• Source: Res(LP) → uRes(LP) → Res → regRes
• Source: Res(LP) → uRes(LP) → tseitin → Res → regRes
• Regular resolution weaker than Res(CP) with unary coefficients
• Source: uRes(CP) → uCP → Res → regRes
• Source: uRes(CP) → uRes(LP) → tseitin → Res → regRes
• Regular resolution weaker than Res(LP) with unary coefficients
• Source: uRes(LP) → Res → regRes
• Source: uRes(LP) → tseitin → Res → regRes
• Regular resolution weaker than Res(L\(\&\)P)
• Source: Res(L&P) → Res(LP) → uRes(LP) → Res → regRes
• Source: Res(L&P) → Res(LP) → uRes(LP) → tseitin → Res → regRes
• Regular resolution weaker than Res(L\(\&\)P) with unary coefficients
• Source: uRes(L&P) → uRes(LP) → Res → regRes
• Source: uRes(L&P) → uRes(LP) → tseitin → Res → regRes
• Regular resolution [missing?] Semantic degree-k threshold system, treelike version
• Regular resolution weaker than Polynomial Calculus over \(\mathbb{F}_2\)
• Source: PC_F2 → Res → regRes
• Source: PC_F2 → NS_F2 → tseitin → Res → regRes
• Regular resolution incomparable wrt Nullstellensatz over \(\mathbb{F}_2\)
• Source: regRes → ordRes → peb+ind → NS_F2
• Source: NS_F2 → tseitin → Res → regRes
• Regular resolution weaker than ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Source: ResLin_Z → uResLin_Z → ResLin_F2 → Res → regRes
• Source: ResLin_Z → uResLin_Z → ResLin_F2 → tseitin → Res → regRes
• Regular resolution weaker than unary ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Source: uResLin_Z → ResLin_F2 → Res → regRes
• Source: uResLin_Z → ResLin_F2 → tseitin → Res → regRes
• Regular resolution weaker than ResLin over \(\mathbb{F}_2\), Res(\(\oplus\))
• Source: ResLin_F2 → Res → regRes
• Source: ResLin_F2 → tseitin → Res → regRes
• Regular resolution [missing?] Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\))
• Regular resolution weaker than Polynomial Calculus over \(\mathbb{Q}\)
• Source: PC_Q → Res → regRes
• Source: PC_Q → NS_Q → ofPHP → Res → regRes
• Regular resolution incomparable wrt Nullstellensatz over \(\mathbb{Q}\)
• Source: regRes → ordRes → peb+ind → NS_Q
• Source: NS_Q → ofPHP → Res → regRes
• Regular resolution stronger than Hitting
• Source: regRes → tlRes → hit
• Source: regRes → ordering → tlRes → hit
• Regular resolution weaker than Lift and Project
• Source: L&P → uL&P → Res → regRes
• Source: L&P → uL&P → Res → poolRes → stone → regRes
• Regular resolution weaker than Lift and Project with unary coefficients
• Source: uL&P → Res → regRes
• Source: uL&P → Res → poolRes → stone → regRes
• Regular resolution weaker than Positivstellensatz Calculus
• Source: PSC → PS → SoS → PC_Q → Res → regRes
• Source: PSC → PS → SoS → sLSn+ → CliqueColouring → Res → regRes
• Regular resolution weaker than Positivstellensatz
• Source: PS → SoS → PC_Q → Res → regRes
• Source: PS → SoS → sLSn+ → CliqueColouring → Res → regRes
• Regular resolution weaker than Lovász--Schrijver (LS)
• Source: LS → L&P → uL&P → Res → regRes
• Source: LS → PHP → PC_F2 → Res → regRes
• Regular resolution weaker than Lovász--Schrijver with squares (LS\(_+\))
• Source: LS+ → LS → L&P → uL&P → Res → regRes
• Source: LS+ → LS → PHP → PC_F2 → Res → regRes
• Regular resolution weaker than Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree
• Source: LSd+ → LS+ → LS → L&P → uL&P → Res → regRes
• Source: LSd+ → CliqueColouring → Res → regRes
• Regular resolution weaker than Lovász--Schrijver with squares (LS\(_+^\infty\)), unbounded degree
• Source: LSn+ → PSC → PS → SoS → PC_Q → Res → regRes
• Source: LSn+ → LSd+ → CliqueColouring → Res → regRes
• Regular resolution weaker than Cone Proof System
• Source: CPS → IPS → extFrege → extRes → Res → regRes
• Source: CPS → LSn+ → LSd+ → CliqueColouring → Res → regRes
• Regular resolution [missing?] tl Lovász--Schrijver (LS)
• Regular resolution [missing?] Lovász--Schrijver with squares (LS\(_+\))
• Regular resolution [missing?] Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree, treelike
• Regular resolution [missing?] static Lovász--Schrijver (static LS)
• Regular resolution does not simulate static Lovász--Schrijver, with squares of linear functions (static LS\(_+\))
• Source: sLS+ → PHP → PC_F2 → Res → regRes
• Regular resolution does not simulate static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\))
• Source: sLSn+ → CliqueColouring → Res → regRes
• Regular resolution weaker than Sum of Squares (Lasserre)
• Source: SoS → PC_Q → Res → regRes
• Source: SoS → sLSn+ → CliqueColouring → Res → regRes
• Regular resolution weaker than Sherali--Adams
• Source: SA → circRes → Res → regRes
• Source: SA → NS_Q → ofPHP → Res → regRes
• Regular resolution weaker than Circular resolution
• Source: circRes → Res → regRes
• Source: circRes → Res → poolRes → stone → regRes
• Regular resolution [missing?] Unary Sherali--Adams
• Regular resolution weaker than Ideal Proof System
• Source: IPS → extFrege → extRes → Res → regRes
• Source: IPS → extFrege → PHP → PC_F2 → Res → regRes
• Regular resolution weaker than Extended Frege
• Source: extFrege → extRes → Res → regRes
• Source: extFrege → PHP → PC_F2 → Res → regRes
• Regular resolution weaker than Extended resolution
• Source: extRes → Res → regRes
• Source: extRes → Res → poolRes → stone → regRes
• Regular resolution weaker than Frege
• Source: Frege → AC0Frege → Res-k → Res → regRes
• Source: Frege → AC0(+)Frege → ResLin_F2 → tseitin → Res → regRes
• Regular resolution weaker than \(\mathrm{AC}^0\)-Frege
• Source: AC0Frege → Res-k → Res → regRes
• Source: AC0Frege → Res-k → Res → poolRes → stone → regRes
• Regular resolution weaker than k-DNF Resolution
• Source: Res-k → Res → regRes
• Source: Res-k → Res → poolRes → stone → regRes
• Regular resolution weaker than \(\mathrm{TC}^0\)-Frege
• Source: TC0Frege → AC0Frege → Res-k → Res → regRes
• Source: TC0Frege → PHP → PC_F2 → Res → regRes
• Regular resolution weaker than \(\mathrm{AC}^0\)-Frege with mod 2 axioms
• Source: AC0Frege+Mod2axioms → AC0Frege → Res-k → Res → regRes
• Source: AC0Frege+Mod2axioms → NS_F2 → tseitin → Res → regRes
• Regular resolution weaker than \(\mathrm{AC}^0\)-Frege with mod 2 gates
• Source: AC0(+)Frege → ResLin_F2 → Res → regRes
• Source: AC0(+)Frege → ResLin_F2 → tseitin → Res → regRes
• Regular resolution weaker than OBDD(join,weakening)
• Source: OBDDjoinweak → uCP → Res → regRes
• Source: OBDDjoinweak → tseitin → Res → regRes
• Regular resolution weaker than LK
• Source: LK → Frege → AC0Frege → Res-k → Res → regRes
• Source: LK → Frege → AC0(+)Frege → ResLin_F2 → tseitin → Res → regRes
• Regular resolution weaker than Zermelo-Fraenkl Set Theory with the Axiom of Choice
• Source: ZFC → extFrege → extRes → Res → regRes
• Source: ZFC → extFrege → PHP → PC_F2 → Res → regRes

Formulas

• The size complexity of Pigeonhole principle in Regular resolution is exponential
• Source: PHP → PC_F2 → Res → regRes
• The size complexity of Pigeonhole principle with functionality axioms in Regular resolution is exponential
• Source: fPHP → ofPHP → Res → regRes
• The size complexity of Pigeonhole principle with functionality and onto axioms in Regular resolution is exponential
• Source: ofPHP → Res → regRes
• The size complexity of Weak pigeonhole principle (2n pigeons, n holes) in Regular resolution is [missing?]
• The size complexity of Weak pigeonhole principle (\(n^2\) pigeons, n holes) in Regular resolution is [missing?]
• The size complexity of Bitwise pigeonhole principle in Regular resolution is exponential
• Source: bPHP → CP → uCP → Res → regRes
• The size complexity of Relativized pigeonhole principle (n,\(n^2\),n-1) in Regular resolution is exponential
• Source: rPHP → PC_F2 → Res → regRes
• The size complexity of Ordering principle in Regular resolution is polynomial
• Source: Stalmarck96 Short Resolution Proofs for a Sequence of Tricky Formulas.
• The size complexity of Guarded ordering principle in Regular resolution is exponential
• Source: AJPU07 An Exponential Separation between Regular and General Resolution.
• The size complexity of Tseitin over \(\mathbb{F}_2\) in Regular resolution is exponential
• Source: tseitin → Res → regRes
• The size complexity of Tseitin over \(\mathbb{F}_2\) with k-ary AND gadget in Regular resolution is [missing?]
• The size complexity of Tseitin over \(\mathbb{Q}\) in Regular resolution is [missing?]
• The size complexity of Flow in Regular resolution is exponential
• Source: flow → PC_F2 → Res → regRes
• The size complexity of Tseitin \(\mathbb{F}_2\) \(\circ\) IND in Regular resolution is exponential
• Source: ts+ind → CP → uCP → Res → regRes
• The size complexity of Tseitin \(\mathbb{Q}\) \(\circ\) IND in Regular resolution is exponential
• Source: tsQ+ind → CP → uCP → Res → regRes
• The size complexity of Random CNF in Regular resolution is exponential
• Source: random → Res → regRes
• The size complexity of Clique-Colouring in Regular resolution is exponential
• Source: CliqueColouring → Res → regRes
• The size complexity of Clique-Colouring encoded as equalities in Regular resolution is exponential
• Source: CliqueColouringeq → CP → uCP → Res → regRes
• The size complexity of Weak Clique-Colouring (2m clique, m colours) in Regular resolution is exponential
• Source: CliqueColouring2mm → CP → uCP → Res → regRes
• The size complexity of Weak Clique-Colouring (m^1/2 clique, log^2 m colours) in Regular resolution is superpolynomial
• Source: CliqueColouringmlogm → CP → uCP → Res → regRes
• The size complexity of Clique-Colouring composed with a permutation in Regular resolution is exponential
• Source: CliqueColouringperm → OBDDjoinweak → uCP → Res → regRes
• The size complexity of Pebbling \(\circ\) IND in Regular resolution is polynomial
• Source: regRes → ordRes → peb+ind
• The size complexity of Pebbling \(\circ\) XOR, then guarded in Regular resolution is exponential
• Source: Urquhart11 A Near-Optimal Separation of Regular and General Resolution.
• The size complexity of Stone formula in Regular resolution is exponential
• Source: AJPU07 An Exponential Separation between Regular and General Resolution.
• The size complexity of String of pearls in Regular resolution is polynomial
• Source: BEGJ00 On the Relative Complexity of Resolution Refinements and Cutting Planes Proof Systems.
• The size complexity of Sink of DAG \(\circ\) XOR in Regular resolution is [missing?]

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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