All about Lift and Project with unary coefficients

Proof Systems

• Lift and Project with unary coefficients simulates Resolution
• Source: [citation needed]
• Lift and Project with unary coefficients stronger than Truth table
• Source: uL&P → Res → regRes → tlRes → ttp
• Source: uL&P → Res → regRes → ordering → tlRes → ttp
• Lift and Project with unary coefficients stronger than Tree-like resolution
• Source: uL&P → Res → regRes → tlRes
• Source: uL&P → Res → regRes → ordering → tlRes
• Lift and Project with unary coefficients stronger than Regular resolution
• Source: uL&P → Res → regRes
• Source: uL&P → Res → poolRes → stone → regRes
• Lift and Project with unary coefficients stronger than Ordered resolution
• Source: uL&P → Res → regRes → ordRes
• Source: uL&P → Res → regRes → pearl → ordRes
• Lift and Project with unary coefficients simulates Pool resolution
• Source: uL&P → Res → poolRes
• Lift and Project with unary coefficients simulates Linear resolution
• Source: uL&P → Res → linRes
• Lift and Project with unary coefficients stronger than Reversible resolution
• Source: uL&P → Res → revRes
• Source: uL&P → Res → sod+xor → uSA → revRes
• Lift and Project with unary coefficients simulated by Cutting Planes
• Source: CP → uCP → uL&P
• Lift and Project with unary coefficients not simulated by Tree-like Cutting Planes
• Source: uL&P → Res → regRes → ordRes → peb+ind → tlCP
• Lift and Project with unary coefficients simulated by Cutting Planes with Unary Coefficients
• Source: HirschK06 Several notes on the power of Gomory-Chvátal cuts.
• Lift and Project with unary coefficients weaker than Semantic Cutting Planes
• Source: semanticCP → CP → uCP → uL&P
• Source: semanticCP → CliqueColouringeq → CP → uCP → uL&P
• Lift and Project with unary coefficients simulates Cutting Planes with Saturation
• Source: uL&P → Res → saturationCP
• Lift and Project with unary coefficients simulated by Stabbing Planes
• Source: SP → uSP → uCP → uL&P
• Lift and Project with unary coefficients simulated by Stabbing Planes with Unary Coefficients
• Source: uSP → uCP → uL&P
• Lift and Project with unary coefficients weaker than Res(CP)
• Source: Res(CP) → CP → uCP → uL&P
• Source: Res(CP) → Res(LP) → uRes(LP) → Res-k → CliqueColouringmlogm → CP → uCP → uL&P
• Lift and Project with unary coefficients weaker than Res(LP)
• Source: Res(LP) → uRes(LP) → uRes(L&P) → uL&P
• Source: Res(LP) → uRes(LP) → Res-k → CliqueColouringmlogm → CP → uCP → uL&P
• Lift and Project with unary coefficients weaker than Res(CP) with unary coefficients
• Source: uRes(CP) → uCP → uL&P
• Source: uRes(CP) → uRes(LP) → Res-k → CliqueColouringmlogm → CP → uCP → uL&P
• Lift and Project with unary coefficients weaker than Res(LP) with unary coefficients
• Source: uRes(LP) → uRes(L&P) → uL&P
• Source: uRes(LP) → Res-k → CliqueColouringmlogm → CP → uCP → uL&P
• Lift and Project with unary coefficients weaker than Res(L\(\&\)P)
• Source: Res(L&P) → L&P → uL&P
• Source: Res(L&P) → Res(LP) → uRes(LP) → Res-k → CliqueColouringmlogm → CP → uCP → uL&P
• Lift and Project with unary coefficients weaker than Res(L\(\&\)P) with unary coefficients
• Source: [subsystem]
• Source: uRes(L&P) → uRes(LP) → Res-k → CliqueColouringmlogm → CP → uCP → uL&P
• Lift and Project with unary coefficients [missing?] Semantic degree-k threshold system, treelike version
• Lift and Project with unary coefficients does not simulate Polynomial Calculus over \(\mathbb{F}_2\)
• Source: PC_F2 → NS_F2 → ts+ind → CP → uCP → uL&P
• Lift and Project with unary coefficients incomparable wrt Nullstellensatz over \(\mathbb{F}_2\)
• Source: uL&P → Res → regRes → ordRes → peb+ind → NS_F2
• Source: NS_F2 → ts+ind → CP → uCP → uL&P
• Lift and Project with unary coefficients weaker than ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Source: ResLin_Z → uResLin_Z → uRes(CP) → uCP → uL&P
• Source: ResLin_Z → uResLin_Z → uRes(CP) → uRes(LP) → Res-k → CliqueColouringmlogm → CP → uCP → uL&P
• Lift and Project with unary coefficients weaker than unary ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Source: uResLin_Z → uRes(CP) → uCP → uL&P
• Source: uResLin_Z → uRes(CP) → uRes(LP) → Res-k → CliqueColouringmlogm → CP → uCP → uL&P
• Lift and Project with unary coefficients [missing?] ResLin over \(\mathbb{F}_2\), Res(\(\oplus\))
• Lift and Project with unary coefficients [missing?] Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\))
• Lift and Project with unary coefficients does not simulate Polynomial Calculus over \(\mathbb{Q}\)
• Source: PC_Q → NS_Q → tsQ+ind → CP → uCP → uL&P
• Lift and Project with unary coefficients incomparable wrt Nullstellensatz over \(\mathbb{Q}\)
• Source: uL&P → Res → regRes → ordRes → peb+ind → NS_Q
• Source: NS_Q → tsQ+ind → CP → uCP → uL&P
• Lift and Project with unary coefficients stronger than Hitting
• Source: uL&P → Res → regRes → tlRes → hit
• Source: uL&P → Res → regRes → ordering → tlRes → hit
• Lift and Project with unary coefficients simulated by Lift and Project
• Source: [subsystem]
• Lift and Project with unary coefficients does not simulate Positivstellensatz Calculus
• Source: PSC → PS → SoS → sLSn+ → CliqueColouring → L&P → uL&P
• Lift and Project with unary coefficients does not simulate Positivstellensatz
• Source: PS → SoS → sLSn+ → CliqueColouring → L&P → uL&P
• Lift and Project with unary coefficients simulated by Lovász--Schrijver (LS)
• Source: LS → L&P → uL&P
• Lift and Project with unary coefficients simulated by Lovász--Schrijver with squares (LS\(_+\))
• Source: LS+ → LS → L&P → uL&P
• Lift and Project with unary coefficients weaker than Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree
• Source: LSd+ → LS+ → LS → L&P → uL&P
• Source: LSd+ → CliqueColouring → L&P → uL&P
• Lift and Project with unary coefficients weaker than Lovász--Schrijver with squares (LS\(_+^\infty\)), unbounded degree
• Source: LSn+ → LSd+ → LS+ → LS → L&P → uL&P
• Source: LSn+ → LSd+ → CliqueColouring → L&P → uL&P
• Lift and Project with unary coefficients weaker than Cone Proof System
• Source: CPS → LSn+ → LSd+ → LS+ → LS → L&P → uL&P
• Source: CPS → LSn+ → LSd+ → CliqueColouring → L&P → uL&P
• Lift and Project with unary coefficients [missing?] tl Lovász--Schrijver (LS)
• Lift and Project with unary coefficients [missing?] Lovász--Schrijver with squares (LS\(_+\))
• Lift and Project with unary coefficients [missing?] Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree, treelike
• Lift and Project with unary coefficients [missing?] static Lovász--Schrijver (static LS)
• Lift and Project with unary coefficients [missing?] static Lovász--Schrijver, with squares of linear functions (static LS\(_+\))
• Lift and Project with unary coefficients does not simulate static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\))
• Source: sLSn+ → CliqueColouring → L&P → uL&P
• Lift and Project with unary coefficients does not simulate Sum of Squares (Lasserre)
• Source: SoS → sLSn+ → CliqueColouring → L&P → uL&P
• Lift and Project with unary coefficients does not simulate Sherali--Adams
• Source: SA → NS_Q → tsQ+ind → CP → uCP → uL&P
• Lift and Project with unary coefficients does not simulate Circular resolution
• Source: circRes → SA → NS_Q → tsQ+ind → CP → uCP → uL&P
• Lift and Project with unary coefficients not simulated by Unary Sherali--Adams
• Source: uL&P → Res → sod+xor → uSA
• Lift and Project with unary coefficients weaker than Ideal Proof System
• Source: IPS → extFrege → Frege → CP → uCP → uL&P
• Source: IPS → extFrege → Frege → AC0Frege → CliqueColouring2mm → CP → uCP → uL&P
• Lift and Project with unary coefficients weaker than Extended Frege
• Source: extFrege → Frege → CP → uCP → uL&P
• Source: extFrege → Frege → AC0Frege → CliqueColouring2mm → CP → uCP → uL&P
• Lift and Project with unary coefficients weaker than Extended resolution
• Source: extRes → extFrege → Frege → CP → uCP → uL&P
• Source: extRes → extFrege → Frege → AC0Frege → CliqueColouring2mm → CP → uCP → uL&P
• Lift and Project with unary coefficients weaker than Frege
• Source: Frege → CP → uCP → uL&P
• Source: Frege → AC0Frege → CliqueColouring2mm → CP → uCP → uL&P
• Lift and Project with unary coefficients does not simulate \(\mathrm{AC}^0\)-Frege
• Source: AC0Frege → CliqueColouring2mm → CP → uCP → uL&P
• Lift and Project with unary coefficients does not simulate k-DNF Resolution
• Source: Res-k → CliqueColouringmlogm → CP → uCP → uL&P
• Lift and Project with unary coefficients weaker than \(\mathrm{TC}^0\)-Frege
• Source: TC0Frege → Res(CP) → CP → uCP → uL&P
• Source: TC0Frege → AC0Frege → CliqueColouring2mm → CP → uCP → uL&P
• Lift and Project with unary coefficients does not simulate \(\mathrm{AC}^0\)-Frege with mod 2 axioms
• Source: AC0Frege+Mod2axioms → AC0Frege → CliqueColouring2mm → CP → uCP → uL&P
• Lift and Project with unary coefficients does not simulate \(\mathrm{AC}^0\)-Frege with mod 2 gates
• Source: AC0(+)Frege → AC0Frege → CliqueColouring2mm → CP → uCP → uL&P
• Lift and Project with unary coefficients weaker than OBDD(join,weakening)
• Source: OBDDjoinweak → uCP → uL&P
• Source: OBDDjoinweak → CliqueColouring → L&P → uL&P
• Lift and Project with unary coefficients weaker than LK
• Source: LK → Frege → CP → uCP → uL&P
• Source: LK → Frege → AC0Frege → CliqueColouring2mm → CP → uCP → uL&P
• Lift and Project with unary coefficients weaker than Zermelo-Fraenkl Set Theory with the Axiom of Choice
• Source: ZFC → Res(CP) → CP → uCP → uL&P
• Source: ZFC → extFrege → Frege → AC0Frege → CliqueColouring2mm → CP → uCP → uL&P

Formulas

• The size complexity of Pigeonhole principle in Lift and Project with unary coefficients is [missing?]
• The size complexity of Pigeonhole principle with functionality axioms in Lift and Project with unary coefficients is [missing?]
• The size complexity of Pigeonhole principle with functionality and onto axioms in Lift and Project with unary coefficients is [missing?]
• The size complexity of Weak pigeonhole principle (2n pigeons, n holes) in Lift and Project with unary coefficients is [missing?]
• The size complexity of Weak pigeonhole principle (\(n^2\) pigeons, n holes) in Lift and Project with unary coefficients is [missing?]
• The size complexity of Bitwise pigeonhole principle in Lift and Project with unary coefficients is exponential
• Source: bPHP → CP → uCP → uL&P
• The size complexity of Relativized pigeonhole principle (n,\(n^2\),n-1) in Lift and Project with unary coefficients is [missing?]
• The size complexity of Ordering principle in Lift and Project with unary coefficients is polynomial
• Source: uL&P → Res → regRes → ordering
• The size complexity of Guarded ordering principle in Lift and Project with unary coefficients is polynomial
• Source: uL&P → Res → poolRes → ordering+guard
• The size complexity of Tseitin over \(\mathbb{F}_2\) in Lift and Project with unary coefficients is [missing?]
• The size complexity of Tseitin over \(\mathbb{F}_2\) with k-ary AND gadget in Lift and Project with unary coefficients is [missing?]
• The size complexity of Tseitin over \(\mathbb{Q}\) in Lift and Project with unary coefficients is [missing?]
• The size complexity of Flow in Lift and Project with unary coefficients is [missing?]
• The size complexity of Tseitin \(\mathbb{F}_2\) \(\circ\) IND in Lift and Project with unary coefficients is exponential
• Source: ts+ind → CP → uCP → uL&P
• The size complexity of Tseitin \(\mathbb{Q}\) \(\circ\) IND in Lift and Project with unary coefficients is exponential
• Source: tsQ+ind → CP → uCP → uL&P
• The size complexity of Random CNF in Lift and Project with unary coefficients is [missing?]
• The size complexity of Clique-Colouring in Lift and Project with unary coefficients is exponential
• Source: CliqueColouring → L&P → uL&P
• The size complexity of Clique-Colouring encoded as equalities in Lift and Project with unary coefficients is exponential
• Source: CliqueColouringeq → CP → uCP → uL&P
• The size complexity of Weak Clique-Colouring (2m clique, m colours) in Lift and Project with unary coefficients is exponential
• Source: CliqueColouring2mm → CP → uCP → uL&P
• The size complexity of Weak Clique-Colouring (m^1/2 clique, log^2 m colours) in Lift and Project with unary coefficients is superpolynomial
• Source: CliqueColouringmlogm → CP → uCP → uL&P
• The size complexity of Clique-Colouring composed with a permutation in Lift and Project with unary coefficients is exponential
• Source: CliqueColouringperm → OBDDjoinweak → uCP → uL&P
• The size complexity of Pebbling \(\circ\) IND in Lift and Project with unary coefficients is polynomial
• Source: uL&P → Res → regRes → ordRes → peb+ind
• The size complexity of Pebbling \(\circ\) XOR, then guarded in Lift and Project with unary coefficients is polynomial
• Source: uL&P → Res → poolRes → peb+xor+guard
• The size complexity of Stone formula in Lift and Project with unary coefficients is polynomial
• Source: uL&P → Res → poolRes → stone
• The size complexity of String of pearls in Lift and Project with unary coefficients is polynomial
• Source: uL&P → Res → regRes → pearl
• The size complexity of Sink of DAG \(\circ\) XOR in Lift and Project with unary coefficients is polynomial
• Source: uL&P → Res → sod+xor

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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