All about Linear resolution

Proof Systems

• Linear resolution simulated by Resolution
• Source: [subsystem]
• Linear resolution stronger than Truth table
• Source: linRes → tlRes → ttp
• Source: linRes → ordering → tlRes → ttp
• Linear resolution stronger than Tree-like resolution
• Source: [citation needed]
• Source: linRes → ordering → tlRes
• Linear resolution not simulated by Regular resolution
• Source: linRes → ordering+guard → regRes
• Linear resolution not simulated by Ordered resolution
• Source: linRes → ordering+guard → regRes → ordRes
• Linear resolution [missing?] Pool resolution
• Linear resolution [missing?] Reversible resolution
• Linear resolution weaker than Cutting Planes
• Source: CP → uCP → Res → linRes
• Source: CP → uCP → PHP → PC_Q → Res → linRes
• Linear resolution [missing?] Tree-like Cutting Planes
• Linear resolution weaker than Cutting Planes with Unary Coefficients
• Source: uCP → Res → linRes
• Source: uCP → PHP → PC_F2 → Res → linRes
• Linear resolution weaker than Semantic Cutting Planes
• Source: semanticCP → saturationCP → Res → linRes
• Source: semanticCP → CP → uCP → PHP → PC_Q → Res → linRes
• Linear resolution simulated by Cutting Planes with Saturation
• Source: saturationCP → Res → linRes
• Linear resolution weaker than Stabbing Planes
• Source: SP → uSP → uCP → Res → linRes
• Source: SP → uSP → uCP → PHP → PC_F2 → Res → linRes
• Linear resolution weaker than Stabbing Planes with Unary Coefficients
• Source: uSP → uCP → Res → linRes
• Source: uSP → uCP → PHP → PC_F2 → Res → linRes
• Linear resolution weaker than Res(CP)
• Source: Res(CP) → CP → uCP → Res → linRes
• Source: Res(CP) → CP → uCP → PHP → PC_Q → Res → linRes
• Linear resolution weaker than Res(LP)
• Source: Res(LP) → uRes(LP) → Res → linRes
• Source: Res(LP) → uRes(LP) → tseitin → Res → linRes
• Linear resolution weaker than Res(CP) with unary coefficients
• Source: uRes(CP) → uCP → Res → linRes
• Source: uRes(CP) → uRes(LP) → tseitin → Res → linRes
• Linear resolution weaker than Res(LP) with unary coefficients
• Source: uRes(LP) → Res → linRes
• Source: uRes(LP) → tseitin → Res → linRes
• Linear resolution weaker than Res(L\(\&\)P)
• Source: Res(L&P) → Res(LP) → uRes(LP) → Res → linRes
• Source: Res(L&P) → Res(LP) → uRes(LP) → tseitin → Res → linRes
• Linear resolution weaker than Res(L\(\&\)P) with unary coefficients
• Source: uRes(L&P) → uRes(LP) → Res → linRes
• Source: uRes(L&P) → uRes(LP) → tseitin → Res → linRes
• Linear resolution [missing?] Semantic degree-k threshold system, treelike version
• Linear resolution weaker than Polynomial Calculus over \(\mathbb{F}_2\)
• Source: PC_F2 → Res → linRes
• Source: PC_F2 → NS_F2 → tseitin → Res → linRes
• Linear resolution does not simulate Nullstellensatz over \(\mathbb{F}_2\)
• Source: NS_F2 → tseitin → Res → linRes
• Linear resolution weaker than ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Source: ResLin_Z → uResLin_Z → ResLin_F2 → Res → linRes
• Source: ResLin_Z → uResLin_Z → ResLin_F2 → tseitin → Res → linRes
• Linear resolution weaker than unary ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Source: uResLin_Z → ResLin_F2 → Res → linRes
• Source: uResLin_Z → ResLin_F2 → tseitin → Res → linRes
• Linear resolution weaker than ResLin over \(\mathbb{F}_2\), Res(\(\oplus\))
• Source: ResLin_F2 → Res → linRes
• Source: ResLin_F2 → tseitin → Res → linRes
• Linear resolution [missing?] Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\))
• Linear resolution weaker than Polynomial Calculus over \(\mathbb{Q}\)
• Source: PC_Q → Res → linRes
• Source: PC_Q → NS_Q → ofPHP → Res → linRes
• Linear resolution does not simulate Nullstellensatz over \(\mathbb{Q}\)
• Source: NS_Q → ofPHP → Res → linRes
• Linear resolution stronger than Hitting
• Source: linRes → tlRes → hit
• Source: linRes → ordering → tlRes → hit
• Linear resolution simulated by Lift and Project
• Source: L&P → uL&P → Res → linRes
• Linear resolution simulated by Lift and Project with unary coefficients
• Source: uL&P → Res → linRes
• Linear resolution weaker than Positivstellensatz Calculus
• Source: PSC → PS → SoS → PC_Q → Res → linRes
• Source: PSC → PS → SoS → sLSn+ → CliqueColouring → Res → linRes
• Linear resolution weaker than Positivstellensatz
• Source: PS → SoS → PC_Q → Res → linRes
• Source: PS → SoS → sLSn+ → CliqueColouring → Res → linRes
• Linear resolution weaker than Lovász--Schrijver (LS)
• Source: LS → L&P → uL&P → Res → linRes
• Source: LS → PHP → PC_F2 → Res → linRes
• Linear resolution weaker than Lovász--Schrijver with squares (LS\(_+\))
• Source: LS+ → LS → L&P → uL&P → Res → linRes
• Source: LS+ → LS → PHP → PC_F2 → Res → linRes
• Linear resolution weaker than Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree
• Source: LSd+ → LS+ → LS → L&P → uL&P → Res → linRes
• Source: LSd+ → CliqueColouring → Res → linRes
• Linear resolution weaker than Lovász--Schrijver with squares (LS\(_+^\infty\)), unbounded degree
• Source: LSn+ → PSC → PS → SoS → PC_Q → Res → linRes
• Source: LSn+ → LSd+ → CliqueColouring → Res → linRes
• Linear resolution weaker than Cone Proof System
• Source: CPS → IPS → extFrege → extRes → Res → linRes
• Source: CPS → LSn+ → LSd+ → CliqueColouring → Res → linRes
• Linear resolution [missing?] tl Lovász--Schrijver (LS)
• Linear resolution [missing?] Lovász--Schrijver with squares (LS\(_+\))
• Linear resolution [missing?] Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree, treelike
• Linear resolution [missing?] static Lovász--Schrijver (static LS)
• Linear resolution does not simulate static Lovász--Schrijver, with squares of linear functions (static LS\(_+\))
• Source: sLS+ → PHP → PC_F2 → Res → linRes
• Linear resolution does not simulate static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\))
• Source: sLSn+ → CliqueColouring → Res → linRes
• Linear resolution weaker than Sum of Squares (Lasserre)
• Source: SoS → PC_Q → Res → linRes
• Source: SoS → sLSn+ → CliqueColouring → Res → linRes
• Linear resolution weaker than Sherali--Adams
• Source: SA → circRes → Res → linRes
• Source: SA → NS_Q → ofPHP → Res → linRes
• Linear resolution weaker than Circular resolution
• Source: circRes → Res → linRes
• Source: circRes → SA → NS_Q → ofPHP → Res → linRes
• Linear resolution [missing?] Unary Sherali--Adams
• Linear resolution weaker than Ideal Proof System
• Source: IPS → extFrege → extRes → Res → linRes
• Source: IPS → extFrege → PHP → PC_F2 → Res → linRes
• Linear resolution weaker than Extended Frege
• Source: extFrege → extRes → Res → linRes
• Source: extFrege → PHP → PC_F2 → Res → linRes
• Linear resolution weaker than Extended resolution
• Source: extRes → Res → linRes
• Source: extRes → extFrege → PHP → PC_F2 → Res → linRes
• Linear resolution weaker than Frege
• Source: Frege → AC0Frege → Res-k → Res → linRes
• Source: Frege → AC0(+)Frege → ResLin_F2 → tseitin → Res → linRes
• Linear resolution weaker than \(\mathrm{AC}^0\)-Frege
• Source: AC0Frege → Res-k → Res → linRes
• Source: AC0Frege → CliqueColouring2mm → CP → uCP → Res → linRes
• Linear resolution weaker than k-DNF Resolution
• Source: Res-k → Res → linRes
• Source: Res-k → CliqueColouringmlogm → CP → uCP → Res → linRes
• Linear resolution weaker than \(\mathrm{TC}^0\)-Frege
• Source: TC0Frege → AC0Frege → Res-k → Res → linRes
• Source: TC0Frege → PHP → PC_F2 → Res → linRes
• Linear resolution weaker than \(\mathrm{AC}^0\)-Frege with mod 2 axioms
• Source: AC0Frege+Mod2axioms → AC0Frege → Res-k → Res → linRes
• Source: AC0Frege+Mod2axioms → NS_F2 → tseitin → Res → linRes
• Linear resolution weaker than \(\mathrm{AC}^0\)-Frege with mod 2 gates
• Source: AC0(+)Frege → ResLin_F2 → Res → linRes
• Source: AC0(+)Frege → ResLin_F2 → tseitin → Res → linRes
• Linear resolution weaker than OBDD(join,weakening)
• Source: OBDDjoinweak → uCP → Res → linRes
• Source: OBDDjoinweak → tseitin → Res → linRes
• Linear resolution weaker than LK
• Source: LK → Frege → AC0Frege → Res-k → Res → linRes
• Source: LK → Frege → AC0(+)Frege → ResLin_F2 → tseitin → Res → linRes
• Linear resolution weaker than Zermelo-Fraenkl Set Theory with the Axiom of Choice
• Source: ZFC → extFrege → extRes → Res → linRes
• Source: ZFC → extFrege → PHP → PC_F2 → Res → linRes

Formulas

• The size complexity of Pigeonhole principle in Linear resolution is exponential
• Source: PHP → PC_F2 → Res → linRes
• The size complexity of Pigeonhole principle with functionality axioms in Linear resolution is exponential
• Source: fPHP → ofPHP → Res → linRes
• The size complexity of Pigeonhole principle with functionality and onto axioms in Linear resolution is exponential
• Source: ofPHP → Res → linRes
• The size complexity of Weak pigeonhole principle (2n pigeons, n holes) in Linear resolution is [missing?]
• The size complexity of Weak pigeonhole principle (\(n^2\) pigeons, n holes) in Linear resolution is [missing?]
• The size complexity of Bitwise pigeonhole principle in Linear resolution is exponential
• Source: bPHP → CP → uCP → Res → linRes
• The size complexity of Relativized pigeonhole principle (n,\(n^2\),n-1) in Linear resolution is exponential
• Source: rPHP → PC_F2 → Res → linRes
• The size complexity of Ordering principle in Linear resolution is polynomial
• Source: BJ16 On Linear Resolution.
• The size complexity of Guarded ordering principle in Linear resolution is polynomial
• Source: BJ16 On Linear Resolution.
• The size complexity of Tseitin over \(\mathbb{F}_2\) in Linear resolution is exponential
• Source: tseitin → Res → linRes
• The size complexity of Tseitin over \(\mathbb{F}_2\) with k-ary AND gadget in Linear resolution is [missing?]
• The size complexity of Tseitin over \(\mathbb{Q}\) in Linear resolution is [missing?]
• The size complexity of Flow in Linear resolution is exponential
• Source: flow → PC_F2 → Res → linRes
• The size complexity of Tseitin \(\mathbb{F}_2\) \(\circ\) IND in Linear resolution is exponential
• Source: ts+ind → CP → uCP → Res → linRes
• The size complexity of Tseitin \(\mathbb{Q}\) \(\circ\) IND in Linear resolution is exponential
• Source: tsQ+ind → CP → uCP → Res → linRes
• The size complexity of Random CNF in Linear resolution is exponential
• Source: random → Res → linRes
• The size complexity of Clique-Colouring in Linear resolution is exponential
• Source: CliqueColouring → Res → linRes
• The size complexity of Clique-Colouring encoded as equalities in Linear resolution is exponential
• Source: CliqueColouringeq → CP → uCP → Res → linRes
• The size complexity of Weak Clique-Colouring (2m clique, m colours) in Linear resolution is exponential
• Source: CliqueColouring2mm → CP → uCP → Res → linRes
• The size complexity of Weak Clique-Colouring (m^1/2 clique, log^2 m colours) in Linear resolution is superpolynomial
• Source: CliqueColouringmlogm → CP → uCP → Res → linRes
• The size complexity of Clique-Colouring composed with a permutation in Linear resolution is exponential
• Source: CliqueColouringperm → OBDDjoinweak → uCP → Res → linRes
• The size complexity of Pebbling \(\circ\) IND in Linear resolution is [missing?]
• The size complexity of Pebbling \(\circ\) XOR, then guarded in Linear resolution is [missing?]
• The size complexity of Stone formula in Linear resolution is [missing?]
• The size complexity of String of pearls in Linear resolution is at most quasipolynomial
• Source: linRes → tlRes → pearl
• The size complexity of Sink of DAG \(\circ\) XOR in Linear 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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