All about Lift and Project

Proof Systems

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

Formulas

• The size complexity of Pigeonhole principle in Lift and Project is [missing?]
• The size complexity of Pigeonhole principle with functionality axioms in Lift and Project is [missing?]
• The size complexity of Pigeonhole principle with functionality and onto axioms in Lift and Project is [missing?]
• The size complexity of Weak pigeonhole principle (2n pigeons, n holes) in Lift and Project is [missing?]
• The size complexity of Weak pigeonhole principle (\(n^2\) pigeons, n holes) in Lift and Project is [missing?]
• The size complexity of Bitwise pigeonhole principle in Lift and Project is [missing?]
• The size complexity of Relativized pigeonhole principle (n,\(n^2\),n-1) in Lift and Project is [missing?]
• The size complexity of Ordering principle in Lift and Project is polynomial
• Source: L&P → uL&P → Res → regRes → ordering
• The size complexity of Guarded ordering principle in Lift and Project is polynomial
• Source: L&P → uL&P → Res → poolRes → ordering+guard
• The size complexity of Tseitin over \(\mathbb{F}_2\) in Lift and Project is [missing?]
• The size complexity of Tseitin over \(\mathbb{F}_2\) with k-ary AND gadget in Lift and Project is [missing?]
• The size complexity of Tseitin over \(\mathbb{Q}\) in Lift and Project is [missing?]
• The size complexity of Flow in Lift and Project is [missing?]
• The size complexity of Tseitin \(\mathbb{F}_2\) \(\circ\) IND in Lift and Project is [missing?]
• The size complexity of Tseitin \(\mathbb{Q}\) \(\circ\) IND in Lift and Project is [missing?]
• The size complexity of Random CNF in Lift and Project is [missing?]
• The size complexity of Clique-Colouring in Lift and Project is exponential
• Source: Dash02 An Exponential Lower Bound on the Length of Some Classes of Branch-and-Cut Proofs.
• The size complexity of Clique-Colouring encoded as equalities in Lift and Project is [missing?]
• The size complexity of Weak Clique-Colouring (2m clique, m colours) in Lift and Project is [missing?]
• The size complexity of Weak Clique-Colouring (m^1/2 clique, log^2 m colours) in Lift and Project is [missing?]
• The size complexity of Clique-Colouring composed with a permutation in Lift and Project is [missing?]
• The size complexity of Pebbling \(\circ\) IND in Lift and Project is polynomial
• Source: L&P → uL&P → Res → regRes → ordRes → peb+ind
• The size complexity of Pebbling \(\circ\) XOR, then guarded in Lift and Project is polynomial
• Source: L&P → uL&P → Res → poolRes → peb+xor+guard
• The size complexity of Stone formula in Lift and Project is polynomial
• Source: L&P → uL&P → Res → poolRes → stone
• The size complexity of String of pearls in Lift and Project is polynomial
• Source: L&P → uL&P → Res → regRes → pearl
• The size complexity of Sink of DAG \(\circ\) XOR in Lift and Project is polynomial
• Source: L&P → 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.

Licensed under CC BY 4.0