All about Lovász--Schrijver with squares (LS\(_+\))

Notes: tl, degree two, dynamic, with squares

Proof Systems

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

Formulas

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