All about static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\))

Notes: unbounded degree, with arbitrary squares

Proof Systems

• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) not simulated by Resolution
• Source: sLSn+ → CliqueColouring → Res
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) stronger than Truth table
• Source: sLSn+ → sLS+ → tlLS+ → tlLS → tlRes → ttp
• Source: sLSn+ → sLS+ → PHP → tlResLin_F2 → tlRes → ttp
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) stronger than Tree-like resolution
• Source: sLSn+ → sLS+ → tlLS+ → tlLS → tlRes
• Source: sLSn+ → sLS+ → PHP → tlResLin_F2 → tlRes
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) not simulated by Regular resolution
• Source: sLSn+ → CliqueColouring → Res → regRes
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) not simulated by Ordered resolution
• Source: sLSn+ → CliqueColouring → Res → regRes → ordRes
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) not simulated by Pool resolution
• Source: sLSn+ → CliqueColouring → Res → poolRes
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) not simulated by Linear resolution
• Source: sLSn+ → CliqueColouring → Res → linRes
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) not simulated by Reversible resolution
• Source: sLSn+ → CliqueColouring → Res → revRes
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) incomparable wrt Cutting Planes
• Source: sLSn+ → CliqueColouring → CP
• Source: CP → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) not simulated by Tree-like Cutting Planes
• Source: sLSn+ → CliqueColouring → CP → tlCP
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) not simulated by Cutting Planes with Unary Coefficients
• Source: sLSn+ → CliqueColouring → CP → uCP
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) incomparable wrt Semantic Cutting Planes
• Source: sLSn+ → CliqueColouring → semanticCP
• Source: semanticCP → CP → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) not simulated by Cutting Planes with Saturation
• Source: sLSn+ → CliqueColouring → Res → saturationCP
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) incomparable wrt Stabbing Planes
• Source: sLSn+ → CliqueColouring → SP
• Source: SP → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) not simulated by Stabbing Planes with Unary Coefficients
• Source: sLSn+ → CliqueColouring → SP → uSP
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate Res(CP)
• Source: Res(CP) → Res(LP) → uRes(LP) → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate Res(LP)
• Source: Res(LP) → uRes(LP) → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate Res(CP) with unary coefficients
• Source: uRes(CP) → uRes(LP) → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate Res(LP) with unary coefficients
• Source: uRes(LP) → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate Res(L\(\&\)P)
• Source: Res(L&P) → Res(LP) → uRes(LP) → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate Res(L\(\&\)P) with unary coefficients
• Source: uRes(L&P) → uRes(LP) → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) [missing?] Semantic degree-k threshold system, treelike version
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) incomparable wrt Polynomial Calculus over \(\mathbb{F}_2\)
• Source: sLSn+ → sLS+ → PHP → PC_F2
• Source: PC_F2 → NS_F2 → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) incomparable wrt Nullstellensatz over \(\mathbb{F}_2\)
• Source: sLSn+ → sLS+ → PHP → PC_F2 → NS_F2
• Source: NS_F2 → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Source: ResLin_Z → uResLin_Z → ResLin_F2 → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate unary ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Source: uResLin_Z → ResLin_F2 → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate ResLin over \(\mathbb{F}_2\), Res(\(\oplus\))
• Source: ResLin_F2 → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) not simulated by Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\))
• Source: sLSn+ → sLS+ → PHP → tlResLin_F2
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) not simulated by Polynomial Calculus over \(\mathbb{Q}\)
• Source: sLSn+ → sLS+ → PHP → PC_Q
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) not simulated by Nullstellensatz over \(\mathbb{Q}\)
• Source: sLSn+ → sLS+ → PHP → PC_Q → NS_Q
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) stronger than Hitting
• Source: sLSn+ → sLS+ → tlLS+ → tlLS → tlRes → hit
• Source: sLSn+ → sLS+ → PHP → tlResLin_F2 → tlRes → hit
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) not simulated by Lift and Project
• Source: sLSn+ → CliqueColouring → L&P
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) not simulated by Lift and Project with unary coefficients
• Source: sLSn+ → CliqueColouring → L&P → uL&P
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) simulated by Positivstellensatz Calculus
• Source: PSC → PS → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) simulated by Positivstellensatz
• Source: PS → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) [missing?] Lovász--Schrijver (LS)
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) [missing?] Lovász--Schrijver with squares (LS\(_+\))
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree
• Source: LSd+ → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) weaker than Lovász--Schrijver with squares (LS\(_+^\infty\)), unbounded degree
• Source: [subsystem]
• Source: LSn+ → LSd+ → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) weaker than Cone Proof System
• Source: CPS → LSn+ → sLSn+
• Source: CPS → LSn+ → LSd+ → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) simulates tl Lovász--Schrijver (LS)
• Source: sLSn+ → sLS+ → tlLS+ → tlLS
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) simulates Lovász--Schrijver with squares (LS\(_+\))
• Source: sLSn+ → sLS+ → tlLS+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) [missing?] Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree, treelike
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) simulates static Lovász--Schrijver (static LS)
• Source: sLSn+ → sLS+ → sLS
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) simulates static Lovász--Schrijver, with squares of linear functions (static LS\(_+\))
• Source: [subsystem]
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) simulated by Sum of Squares (Lasserre)
• Source: AtseriasH19 Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs.
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) [missing?] Sherali--Adams
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) [missing?] Circular resolution
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) [missing?] Unary Sherali--Adams
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate Ideal Proof System
• Source: IPS → extFrege → Frege → AC0(+)Frege → ResLin_F2 → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate Extended Frege
• Source: extFrege → Frege → AC0(+)Frege → ResLin_F2 → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate Extended resolution
• Source: extRes → extFrege → Frege → AC0(+)Frege → ResLin_F2 → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate Frege
• Source: Frege → AC0(+)Frege → ResLin_F2 → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) not simulated by \(\mathrm{AC}^0\)-Frege
• Source: sLSn+ → sLS+ → PHP → fPHP → ofPHP → AC0Frege
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) not simulated by k-DNF Resolution
• Source: sLSn+ → sLS+ → PHP → fPHP → ofPHP → AC0Frege → Res-k
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate \(\mathrm{TC}^0\)-Frege
• Source: TC0Frege → Res(CP) → Res(LP) → uRes(LP) → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate \(\mathrm{AC}^0\)-Frege with mod 2 axioms
• Source: AC0Frege+Mod2axioms → NS_F2 → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate \(\mathrm{AC}^0\)-Frege with mod 2 gates
• Source: AC0(+)Frege → ResLin_F2 → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate OBDD(join,weakening)
• Source: OBDDjoinweak → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate LK
• Source: LK → Frege → AC0(+)Frege → ResLin_F2 → tseitin → SoS → sLSn+
• static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) does not simulate Zermelo-Fraenkl Set Theory with the Axiom of Choice
• Source: ZFC → Res(CP) → Res(LP) → uRes(LP) → tseitin → SoS → sLSn+

Formulas

• The size complexity of Pigeonhole principle in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is polynomial
• Source: sLSn+ → sLS+ → PHP
• The size complexity of Pigeonhole principle with functionality axioms in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is polynomial
• Source: sLSn+ → sLS+ → PHP → fPHP
• The size complexity of Pigeonhole principle with functionality and onto axioms in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is polynomial
• Source: sLSn+ → sLS+ → PHP → fPHP → ofPHP
• The size complexity of Weak pigeonhole principle (2n pigeons, n holes) in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is polynomial
• Source: sLSn+ → sLS+ → PHP → PHP2nn
• The size complexity of Weak pigeonhole principle (\(n^2\) pigeons, n holes) in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is polynomial
• Source: sLSn+ → sLS+ → PHP → PHP2nn → PHPn2n
• The size complexity of Bitwise pigeonhole principle in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is [missing?]
• The size complexity of Relativized pigeonhole principle (n,\(n^2\),n-1) in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is [missing?]
• The size complexity of Ordering principle in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is [missing?]
• The size complexity of Guarded ordering principle in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is [missing?]
• The size complexity of Tseitin over \(\mathbb{F}_2\) in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is exponential
• Source: tseitin → SoS → sLSn+
• The size complexity of Tseitin over \(\mathbb{F}_2\) with k-ary AND gadget in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is [missing?]
• The size complexity of Tseitin over \(\mathbb{Q}\) in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is [missing?]
• The size complexity of Flow in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is [missing?]
• The size complexity of Tseitin \(\mathbb{F}_2\) \(\circ\) IND in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is [missing?]
• The size complexity of Tseitin \(\mathbb{Q}\) \(\circ\) IND in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is [missing?]
• The size complexity of Random CNF in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is [missing?]
• The size complexity of Clique-Colouring in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is polynomial
• Source: [citation needed]
• The size complexity of Clique-Colouring encoded as equalities in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is [missing?]
• The size complexity of Weak Clique-Colouring (2m clique, m colours) in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is polynomial
• Source: sLSn+ → CliqueColouring → CliqueColouring2mm
• The size complexity of Weak Clique-Colouring (m^1/2 clique, log^2 m colours) in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is polynomial
• Source: sLSn+ → CliqueColouring → CliqueColouringmlogm
• The size complexity of Clique-Colouring composed with a permutation in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is [missing?]
• The size complexity of Pebbling \(\circ\) IND in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is [missing?]
• The size complexity of Pebbling \(\circ\) XOR, then guarded in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is [missing?]
• The size complexity of Stone formula in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is [missing?]
• The size complexity of String of pearls in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is at most quasipolynomial
• Source: sLSn+ → sLS+ → tlLS+ → tlLS → tlRes → pearl
• The size complexity of Sink of DAG \(\circ\) XOR in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) 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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