All about Semantic Cutting Planes

Proof Systems

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

Formulas

• The size complexity of Pigeonhole principle in Semantic Cutting Planes is polynomial
• Source: semanticCP → CP → uCP → PHP
• The size complexity of Pigeonhole principle with functionality axioms in Semantic Cutting Planes is polynomial
• Source: semanticCP → CP → uCP → PHP → fPHP
• The size complexity of Pigeonhole principle with functionality and onto axioms in Semantic Cutting Planes is polynomial
• Source: semanticCP → CP → uCP → PHP → fPHP → ofPHP
• The size complexity of Weak pigeonhole principle (2n pigeons, n holes) in Semantic Cutting Planes is polynomial
• Source: semanticCP → CP → uCP → PHP → PHP2nn
• The size complexity of Weak pigeonhole principle (\(n^2\) pigeons, n holes) in Semantic Cutting Planes is polynomial
• Source: semanticCP → CP → uCP → PHP → PHP2nn → PHPn2n
• The size complexity of Bitwise pigeonhole principle in Semantic Cutting Planes is [missing?]
• The size complexity of Relativized pigeonhole principle (n,\(n^2\),n-1) in Semantic Cutting Planes is [missing?]
• The size complexity of Ordering principle in Semantic Cutting Planes is polynomial
• Source: semanticCP → saturationCP → Res → regRes → ordering
• The size complexity of Guarded ordering principle in Semantic Cutting Planes is polynomial
• Source: semanticCP → saturationCP → Res → poolRes → ordering+guard
• The size complexity of Tseitin over \(\mathbb{F}_2\) in Semantic Cutting Planes is at most quasipolynomial
• Source: semanticCP → CP → tseitin
• The size complexity of Tseitin over \(\mathbb{F}_2\) with k-ary AND gadget in Semantic Cutting Planes is [missing?]
• The size complexity of Tseitin over \(\mathbb{Q}\) in Semantic Cutting Planes is [missing?]
• The size complexity of Flow in Semantic Cutting Planes is [missing?]
• The size complexity of Tseitin \(\mathbb{F}_2\) \(\circ\) IND in Semantic Cutting Planes is [missing?]
• The size complexity of Tseitin \(\mathbb{Q}\) \(\circ\) IND in Semantic Cutting Planes is [missing?]
• The size complexity of Random CNF in Semantic Cutting Planes is [missing?]
• The size complexity of Clique-Colouring in Semantic Cutting Planes is exponential
• Source: FHL16 Semantic Versus Syntactic Cutting Planes.
• The size complexity of Clique-Colouring encoded as equalities in Semantic Cutting Planes is polynomial
• Source: [citation needed]
• The size complexity of Weak Clique-Colouring (2m clique, m colours) in Semantic Cutting Planes is [missing?]
• The size complexity of Weak Clique-Colouring (m^1/2 clique, log^2 m colours) in Semantic Cutting Planes is [missing?]
• The size complexity of Clique-Colouring composed with a permutation in Semantic Cutting Planes is [missing?]
• The size complexity of Pebbling \(\circ\) IND in Semantic Cutting Planes is polynomial
• Source: semanticCP → saturationCP → Res → regRes → ordRes → peb+ind
• The size complexity of Pebbling \(\circ\) XOR, then guarded in Semantic Cutting Planes is polynomial
• Source: semanticCP → saturationCP → Res → poolRes → peb+xor+guard
• The size complexity of Stone formula in Semantic Cutting Planes is polynomial
• Source: semanticCP → saturationCP → Res → poolRes → stone
• The size complexity of String of pearls in Semantic Cutting Planes is polynomial
• Source: semanticCP → saturationCP → Res → regRes → pearl
• The size complexity of Sink of DAG \(\circ\) XOR in Semantic Cutting Planes is polynomial
• Source: semanticCP → saturationCP → 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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