All about Res(LP)

Notes: aka R(LP); resolution over linear inequalities

Proof Systems

• Res(LP) stronger than Resolution
• Source: Res(LP) → uRes(LP) → Res
• Source: Res(LP) → uRes(LP) → tseitin → Res
• Res(LP) stronger than Truth table
• Source: Res(LP) → uRes(LP) → Res → regRes → tlRes → ttp
• Source: Res(LP) → uRes(LP) → Res → regRes → ordering → tlRes → ttp
• Res(LP) stronger than Tree-like resolution
• Source: Res(LP) → uRes(LP) → Res → regRes → tlRes
• Source: Res(LP) → uRes(LP) → Res → regRes → ordering → tlRes
• Res(LP) stronger than Regular resolution
• Source: Res(LP) → uRes(LP) → Res → regRes
• Source: Res(LP) → uRes(LP) → tseitin → Res → regRes
• Res(LP) stronger than Ordered resolution
• Source: Res(LP) → uRes(LP) → Res → regRes → ordRes
• Source: Res(LP) → uRes(LP) → Res → regRes → pearl → ordRes
• Res(LP) stronger than Pool resolution
• Source: Res(LP) → uRes(LP) → Res → poolRes
• Source: Res(LP) → uRes(LP) → tseitin → Res → poolRes
• Res(LP) stronger than Linear resolution
• Source: Res(LP) → uRes(LP) → Res → linRes
• Source: Res(LP) → uRes(LP) → tseitin → Res → linRes
• Res(LP) stronger than Reversible resolution
• Source: Res(LP) → uRes(LP) → Res → revRes
• Source: Res(LP) → uRes(LP) → tseitin → Res → revRes
• Res(LP) not simulated by Cutting Planes
• Source: Res(LP) → uRes(LP) → Res-k → CliqueColouringmlogm → CP
• Res(LP) not simulated by Tree-like Cutting Planes
• Source: Res(LP) → uRes(LP) → Res → regRes → ordRes → peb+ind → tlCP
• Res(LP) stronger than Cutting Planes with Unary Coefficients
• Source: Res(LP) → uRes(LP) → uRes(CP) → uCP
• Source: Res(LP) → uRes(LP) → Res-k → CliqueColouringmlogm → CP → uCP
• Res(LP) [missing?] Semantic Cutting Planes
• Res(LP) stronger than Cutting Planes with Saturation
• Source: Res(LP) → uRes(LP) → Res → saturationCP
• Source: Res(LP) → uRes(LP) → tseitin → Res → saturationCP
• Res(LP) [missing?] Stabbing Planes
• Res(LP) simulates Stabbing Planes with Unary Coefficients
• Source: Res(LP) → uRes(LP) → uRes(CP) → uSP
• Source: Res(LP) → uRes(LP) → Res-k → CliqueColouringmlogm → CP → uSP
• Res(LP) simulated by Res(CP)
• Source: [subsystem]
• Res(LP) simulates Res(CP) with unary coefficients
• Source: Res(LP) → uRes(LP) → uRes(CP)
• Res(LP) simulates Res(LP) with unary coefficients
• Source: [subsystem]
• Res(LP) equivalent Res(L\(\&\)P)
• Source: HirschK06 Several notes on the power of Gomory-Chvátal cuts.
• Res(LP) simulates Res(L\(\&\)P) with unary coefficients
• Source: Res(LP) → uRes(LP) → uRes(L&P)
• Res(LP) [missing?] Semantic degree-k threshold system, treelike version
• Res(LP) not simulated by Polynomial Calculus over \(\mathbb{F}_2\)
• Source: Res(LP) → uRes(LP) → uRes(CP) → uCP → PHP → PC_F2
• Res(LP) not simulated by Nullstellensatz over \(\mathbb{F}_2\)
• Source: Res(LP) → uRes(LP) → uRes(CP) → uCP → PHP → PC_F2 → NS_F2
• Res(LP) [missing?] ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Res(LP) [missing?] unary ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Res(LP) [missing?] ResLin over \(\mathbb{F}_2\), Res(\(\oplus\))
• Res(LP) not simulated by Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\))
• Source: Res(LP) → uRes(LP) → uRes(CP) → uCP → PHP → tlResLin_F2
• Res(LP) not simulated by Polynomial Calculus over \(\mathbb{Q}\)
• Source: Res(LP) → uRes(LP) → tseitin → PC_Q
• Res(LP) not simulated by Nullstellensatz over \(\mathbb{Q}\)
• Source: Res(LP) → uRes(LP) → tseitin → PC_Q → NS_Q
• Res(LP) stronger than Hitting
• Source: Res(LP) → uRes(LP) → Res → regRes → tlRes → hit
• Source: Res(LP) → uRes(LP) → Res → regRes → ordering → tlRes → hit
• Res(LP) simulates Lift and Project
• Source: Res(LP) → Res(L&P) → L&P
• Res(LP) stronger than Lift and Project with unary coefficients
• Source: Res(LP) → uRes(LP) → uRes(L&P) → uL&P
• Source: Res(LP) → uRes(LP) → Res-k → CliqueColouringmlogm → CP → uCP → uL&P
• Res(LP) [missing?] Positivstellensatz Calculus
• Res(LP) [missing?] Positivstellensatz
• Res(LP) [missing?] Lovász--Schrijver (LS)
• Res(LP) [missing?] Lovász--Schrijver with squares (LS\(_+\))
• Res(LP) [missing?] Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree
• Res(LP) [missing?] Lovász--Schrijver with squares (LS\(_+^\infty\)), unbounded degree
• Res(LP) simulated by Cone Proof System
• Source: CPS → IPS → extFrege → Frege → TC0Frege → Res(CP) → Res(LP)
• Res(LP) not simulated by tl Lovász--Schrijver (LS)
• Source: Res(LP) → uRes(LP) → tseitin → sLS+ → tlLS+ → tlLS
• Res(LP) not simulated by Lovász--Schrijver with squares (LS\(_+\))
• Source: Res(LP) → uRes(LP) → tseitin → sLS+ → tlLS+
• Res(LP) [missing?] Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree, treelike
• Res(LP) not simulated by static Lovász--Schrijver (static LS)
• Source: Res(LP) → uRes(LP) → tseitin → sLS+ → sLS
• Res(LP) not simulated by static Lovász--Schrijver, with squares of linear functions (static LS\(_+\))
• Source: Res(LP) → uRes(LP) → tseitin → sLS+
• Res(LP) not simulated by static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\))
• Source: Res(LP) → uRes(LP) → tseitin → SoS → sLSn+
• Res(LP) not simulated by Sum of Squares (Lasserre)
• Source: Res(LP) → uRes(LP) → tseitin → SoS
• Res(LP) not simulated by Sherali--Adams
• Source: Res(LP) → uRes(LP) → tseitin → SoS → SA
• Res(LP) not simulated by Circular resolution
• Source: Res(LP) → uRes(LP) → tseitin → SoS → SA → circRes
• Res(LP) not simulated by Unary Sherali--Adams
• Source: Res(LP) → uRes(LP) → Res → sod+xor → uSA
• Res(LP) simulated by Ideal Proof System
• Source: IPS → extFrege → Frege → TC0Frege → Res(CP) → Res(LP)
• Res(LP) simulated by Extended Frege
• Source: extFrege → Frege → TC0Frege → Res(CP) → Res(LP)
• Res(LP) simulated by Extended resolution
• Source: extRes → extFrege → Frege → TC0Frege → Res(CP) → Res(LP)
• Res(LP) simulated by Frege
• Source: Frege → TC0Frege → Res(CP) → Res(LP)
• Res(LP) not simulated by \(\mathrm{AC}^0\)-Frege
• Source: Res(LP) → uRes(LP) → tseitin → AC0Frege
• Res(LP) stronger than k-DNF Resolution
• Source: Res(LP) → uRes(LP) → Res-k
• Source: Res(LP) → uRes(LP) → tseitin → AC0Frege → Res-k
• Res(LP) simulated by \(\mathrm{TC}^0\)-Frege
• Source: TC0Frege → Res(CP) → Res(LP)
• Res(LP) [missing?] \(\mathrm{AC}^0\)-Frege with mod 2 axioms
• Res(LP) [missing?] \(\mathrm{AC}^0\)-Frege with mod 2 gates
• Res(LP) [missing?] OBDD(join,weakening)
• Res(LP) simulated by LK
• Source: LK → Frege → TC0Frege → Res(CP) → Res(LP)
• Res(LP) simulated by Zermelo-Fraenkl Set Theory with the Axiom of Choice
• Source: ZFC → Res(CP) → Res(LP)

Formulas

• 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 with functionality axioms in Res(LP) is polynomial
• Source: Res(LP) → uRes(LP) → uRes(CP) → uCP → PHP → fPHP
• The size complexity of Pigeonhole principle with functionality and onto axioms in Res(LP) is polynomial
• Source: Res(LP) → uRes(LP) → uRes(CP) → uCP → PHP → fPHP → ofPHP
• The size complexity of Weak pigeonhole principle (2n pigeons, n holes) in Res(LP) is polynomial
• Source: Res(LP) → uRes(LP) → uRes(CP) → uCP → PHP → PHP2nn
• The size complexity of Weak pigeonhole principle (\(n^2\) pigeons, n holes) in Res(LP) is polynomial
• Source: Res(LP) → uRes(LP) → uRes(CP) → uCP → PHP → PHP2nn → PHPn2n
• The size complexity of Bitwise pigeonhole principle in Res(LP) is [missing?]
• The size complexity of Relativized pigeonhole principle (n,\(n^2\),n-1) in Res(LP) is [missing?]
• The size complexity of Ordering principle in Res(LP) is polynomial
• Source: Res(LP) → uRes(LP) → Res → regRes → ordering
• The size complexity of Guarded ordering principle in Res(LP) is polynomial
• Source: Res(LP) → uRes(LP) → Res → poolRes → ordering+guard
• The size complexity of Tseitin over \(\mathbb{F}_2\) in Res(LP) is polynomial
• Source: Res(LP) → uRes(LP) → tseitin
• The size complexity of Tseitin over \(\mathbb{F}_2\) with k-ary AND gadget in Res(LP) is [missing?]
• The size complexity of Tseitin over \(\mathbb{Q}\) in Res(LP) is [missing?]
• The size complexity of Flow in Res(LP) is [missing?]
• The size complexity of Tseitin \(\mathbb{F}_2\) \(\circ\) IND in Res(LP) is [missing?]
• The size complexity of Tseitin \(\mathbb{Q}\) \(\circ\) IND in Res(LP) is [missing?]
• The size complexity of Random CNF in Res(LP) is [missing?]
• The size complexity of Clique-Colouring in Res(LP) is [missing?]
• The size complexity of Clique-Colouring encoded as equalities in Res(LP) is [missing?]
• The size complexity of Weak Clique-Colouring (2m clique, m colours) in Res(LP) is [missing?]
• The size complexity of Weak Clique-Colouring (m^1/2 clique, log^2 m colours) in Res(LP) is polynomial
• Source: Res(LP) → uRes(LP) → Res-k → CliqueColouringmlogm
• The size complexity of Clique-Colouring composed with a permutation in Res(LP) is [missing?]
• The size complexity of Pebbling \(\circ\) IND in Res(LP) is polynomial
• Source: Res(LP) → uRes(LP) → Res → regRes → ordRes → peb+ind
• The size complexity of Pebbling \(\circ\) XOR, then guarded in Res(LP) is polynomial
• Source: Res(LP) → uRes(LP) → Res → poolRes → peb+xor+guard
• The size complexity of Stone formula in Res(LP) is polynomial
• Source: Res(LP) → uRes(LP) → Res → poolRes → stone
• The size complexity of String of pearls in Res(LP) is polynomial
• Source: Res(LP) → uRes(LP) → Res → regRes → pearl
• The size complexity of Sink of DAG \(\circ\) XOR in Res(LP) is polynomial
• Source: Res(LP) → uRes(LP) → 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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