All about Nullstellensatz over \(\mathbb{F}_2\)

Notes: With twin variables \(\overline{x}=1-x\)

Proof Systems

• Nullstellensatz over \(\mathbb{F}_2\) incomparable wrt Resolution
• Source: NS_F2 → tseitin → Res
• Source: Res → regRes → ordRes → peb+ind → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) stronger than Truth table
• Source: NS_F2 → tlRes → ttp
• Source: NS_F2 → tseitin → Res → regRes → tlRes → ttp
• Nullstellensatz over \(\mathbb{F}_2\) stronger than Tree-like resolution
• Source: [citation needed]
• Source: NS_F2 → tseitin → Res → regRes → tlRes
• Nullstellensatz over \(\mathbb{F}_2\) incomparable wrt Regular resolution
• Source: NS_F2 → tseitin → Res → regRes
• Source: regRes → ordRes → peb+ind → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) incomparable wrt Ordered resolution
• Source: NS_F2 → tseitin → Res → regRes → ordRes
• Source: ordRes → peb+ind → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) incomparable wrt Pool resolution
• Source: NS_F2 → tseitin → Res → poolRes
• Source: poolRes → regRes → ordRes → peb+ind → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) not simulated by Linear resolution
• Source: NS_F2 → tseitin → Res → linRes
• Nullstellensatz over \(\mathbb{F}_2\) not simulated by Reversible resolution
• Source: NS_F2 → tseitin → Res → revRes
• Nullstellensatz over \(\mathbb{F}_2\) incomparable wrt Cutting Planes
• Source: NS_F2 → ts+ind → CP
• Source: CP → uCP → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) not simulated by Tree-like Cutting Planes
• Source: NS_F2 → ts+ind → CP → tlCP
• Nullstellensatz over \(\mathbb{F}_2\) incomparable wrt Cutting Planes with Unary Coefficients
• Source: NS_F2 → ts+ind → CP → uCP
• Source: uCP → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate Semantic Cutting Planes
• Source: semanticCP → CP → uCP → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) incomparable wrt Cutting Planes with Saturation
• Source: NS_F2 → tseitin → Res → saturationCP
• Source: saturationCP → Res → regRes → ordRes → peb+ind → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate Stabbing Planes
• Source: SP → uSP → uCP → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) incomparable wrt Stabbing Planes with Unary Coefficients
• Source: NS_F2 → ts+ind → CP → uSP
• Source: uSP → uCP → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate Res(CP)
• Source: Res(CP) → CP → uCP → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate Res(LP)
• Source: Res(LP) → uRes(LP) → uRes(CP) → uCP → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate Res(CP) with unary coefficients
• Source: uRes(CP) → uCP → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate Res(LP) with unary coefficients
• Source: uRes(LP) → uRes(CP) → uCP → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate Res(L\(\&\)P)
• Source: Res(L&P) → Res(LP) → uRes(LP) → uRes(CP) → uCP → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate Res(L\(\&\)P) with unary coefficients
• Source: uRes(L&P) → uRes(LP) → uRes(CP) → uCP → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) [missing?] Semantic degree-k threshold system, treelike version
• Nullstellensatz over \(\mathbb{F}_2\) weaker than Polynomial Calculus over \(\mathbb{F}_2\)
• Source: [subsystem]
• Source: PC_F2 → Res → regRes → ordRes → peb+ind → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Source: ResLin_Z → uResLin_Z → uRes(CP) → uCP → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate unary ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Source: uResLin_Z → uRes(CP) → uCP → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate ResLin over \(\mathbb{F}_2\), Res(\(\oplus\))
• Source: ResLin_F2 → Res → regRes → ordRes → peb+ind → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) [missing?] Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\))
• Nullstellensatz over \(\mathbb{F}_2\) incomparable wrt Polynomial Calculus over \(\mathbb{Q}\)
• Source: NS_F2 → tseitin → PC_Q
• Source: PC_Q → Res → regRes → ordRes → peb+ind → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) not simulated by Nullstellensatz over \(\mathbb{Q}\)
• Source: NS_F2 → tseitin → PC_Q → NS_Q
• Nullstellensatz over \(\mathbb{F}_2\) stronger than Hitting
• Source: NS_F2 → tlRes → hit
• Source: NS_F2 → tseitin → Res → regRes → tlRes → hit
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate Lift and Project
• Source: L&P → uL&P → Res → regRes → ordRes → peb+ind → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) incomparable wrt Lift and Project with unary coefficients
• Source: NS_F2 → ts+ind → CP → uCP → uL&P
• Source: uL&P → Res → regRes → ordRes → peb+ind → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate Positivstellensatz Calculus
• Source: PSC → PS → SoS → SA → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate Positivstellensatz
• Source: PS → SoS → SA → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate Lovász--Schrijver (LS)
• Source: LS → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate Lovász--Schrijver with squares (LS\(_+\))
• Source: LS+ → LS → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree
• Source: LSd+ → LS+ → LS → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate Lovász--Schrijver with squares (LS\(_+^\infty\)), unbounded degree
• Source: LSn+ → sLSn+ → sLS+ → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) weaker than Cone Proof System
• Source: CPS → IPS → extFrege → Frege → AC0(+)Frege → AC0Frege+Mod2axioms → NS_F2
• Source: CPS → IPS → extFrege → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) not simulated by tl Lovász--Schrijver (LS)
• Source: NS_F2 → tseitin → sLS+ → tlLS+ → tlLS
• Nullstellensatz over \(\mathbb{F}_2\) not simulated by Lovász--Schrijver with squares (LS\(_+\))
• Source: NS_F2 → tseitin → sLS+ → tlLS+
• Nullstellensatz over \(\mathbb{F}_2\) [missing?] Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree, treelike
• Nullstellensatz over \(\mathbb{F}_2\) not simulated by static Lovász--Schrijver (static LS)
• Source: NS_F2 → tseitin → sLS+ → sLS
• Nullstellensatz over \(\mathbb{F}_2\) incomparable wrt static Lovász--Schrijver, with squares of linear functions (static LS\(_+\))
• Source: NS_F2 → tseitin → sLS+
• Source: sLS+ → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) incomparable wrt static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\))
• Source: NS_F2 → tseitin → SoS → sLSn+
• Source: sLSn+ → sLS+ → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) incomparable wrt Sum of Squares (Lasserre)
• Source: NS_F2 → tseitin → SoS
• Source: SoS → SA → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) incomparable wrt Sherali--Adams
• Source: NS_F2 → tseitin → SoS → SA
• Source: SA → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) incomparable wrt Circular resolution
• Source: NS_F2 → tseitin → SoS → SA → circRes
• Source: circRes → SA → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) not simulated by Unary Sherali--Adams
• Source: NS_F2 → tseitin → SoS → SA → uSA
• Nullstellensatz over \(\mathbb{F}_2\) weaker than Ideal Proof System
• Source: IPS → extFrege → Frege → AC0(+)Frege → AC0Frege+Mod2axioms → NS_F2
• Source: IPS → extFrege → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) weaker than Extended Frege
• Source: extFrege → Frege → AC0(+)Frege → AC0Frege+Mod2axioms → NS_F2
• Source: extFrege → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) weaker than Extended resolution
• Source: extRes → extFrege → Frege → AC0(+)Frege → AC0Frege+Mod2axioms → NS_F2
• Source: extRes → extFrege → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) weaker than Frege
• Source: Frege → AC0(+)Frege → AC0Frege+Mod2axioms → NS_F2
• Source: Frege → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) incomparable wrt \(\mathrm{AC}^0\)-Frege
• Source: NS_F2 → tseitin → AC0Frege
• Source: AC0Frege → Res-k → Res → regRes → ordRes → peb+ind → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) incomparable wrt k-DNF Resolution
• Source: NS_F2 → tseitin → AC0Frege → Res-k
• Source: Res-k → Res → regRes → ordRes → peb+ind → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) weaker than \(\mathrm{TC}^0\)-Frege
• Source: TC0Frege → AC0(+)Frege → AC0Frege+Mod2axioms → NS_F2
• Source: TC0Frege → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) weaker than \(\mathrm{AC}^0\)-Frege with mod 2 axioms
• Source: IS06 Constant-depth Frege systems with counting axioms polynomially simulate Nullstellensatz refutations.
• Source: AC0Frege+Mod2axioms → AC0Frege → Res-k → Res → regRes → ordRes → peb+ind → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) weaker than \(\mathrm{AC}^0\)-Frege with mod 2 gates
• Source: AC0(+)Frege → AC0Frege+Mod2axioms → NS_F2
• Source: AC0(+)Frege → ResLin_F2 → Res → regRes → ordRes → peb+ind → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) does not simulate OBDD(join,weakening)
• Source: OBDDjoinweak → uCP → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) weaker than LK
• Source: LK → Frege → AC0(+)Frege → AC0Frege+Mod2axioms → NS_F2
• Source: LK → Frege → PHP → PC_F2 → NS_F2
• Nullstellensatz over \(\mathbb{F}_2\) weaker than Zermelo-Fraenkl Set Theory with the Axiom of Choice
• Source: ZFC → extFrege → Frege → AC0(+)Frege → AC0Frege+Mod2axioms → NS_F2
• Source: ZFC → extFrege → PHP → PC_F2 → NS_F2

Formulas

• The size complexity of Pigeonhole principle in Nullstellensatz over \(\mathbb{F}_2\) is exponential
• Source: PHP → PC_F2 → NS_F2
• The size complexity of Pigeonhole principle with functionality axioms in Nullstellensatz over \(\mathbb{F}_2\) is [missing?]
• The size complexity of Pigeonhole principle with functionality and onto axioms in Nullstellensatz over \(\mathbb{F}_2\) is polynomial
• Source: BR96 More on the relative strength of counting principles.
• The size complexity of Weak pigeonhole principle (2n pigeons, n holes) in Nullstellensatz over \(\mathbb{F}_2\) is [missing?]
• The size complexity of Weak pigeonhole principle (\(n^2\) pigeons, n holes) in Nullstellensatz over \(\mathbb{F}_2\) is [missing?]
• The size complexity of Bitwise pigeonhole principle in Nullstellensatz over \(\mathbb{F}_2\) is [missing?]
• The size complexity of Relativized pigeonhole principle (n,\(n^2\),n-1) in Nullstellensatz over \(\mathbb{F}_2\) is exponential
• Source: rPHP → PC_F2 → NS_F2
• The size complexity of Ordering principle in Nullstellensatz over \(\mathbb{F}_2\) is [missing?]
• The size complexity of Guarded ordering principle in Nullstellensatz over \(\mathbb{F}_2\) is [missing?]
• The size complexity of Tseitin over \(\mathbb{F}_2\) in Nullstellensatz over \(\mathbb{F}_2\) is polynomial
• Source: [citation needed]
• The size complexity of Tseitin over \(\mathbb{F}_2\) with k-ary AND gadget in Nullstellensatz over \(\mathbb{F}_2\) is [missing?]
• The size complexity of Tseitin over \(\mathbb{Q}\) in Nullstellensatz over \(\mathbb{F}_2\) is [missing?]
• The size complexity of Flow in Nullstellensatz over \(\mathbb{F}_2\) is exponential
• Source: flow → PC_F2 → NS_F2
• The size complexity of Tseitin \(\mathbb{F}_2\) \(\circ\) IND in Nullstellensatz over \(\mathbb{F}_2\) is at most quasipolynomial
• Source: GKRS19 Adventures in Monotone Complexity and TFNP.
• The size complexity of Tseitin \(\mathbb{Q}\) \(\circ\) IND in Nullstellensatz over \(\mathbb{F}_2\) is [missing?]
• The size complexity of Random CNF in Nullstellensatz over \(\mathbb{F}_2\) is [missing?]
• The size complexity of Clique-Colouring in Nullstellensatz over \(\mathbb{F}_2\) is [missing?]
• The size complexity of Clique-Colouring encoded as equalities in Nullstellensatz over \(\mathbb{F}_2\) is [missing?]
• The size complexity of Weak Clique-Colouring (2m clique, m colours) in Nullstellensatz over \(\mathbb{F}_2\) is [missing?]
• The size complexity of Weak Clique-Colouring (m^1/2 clique, log^2 m colours) in Nullstellensatz over \(\mathbb{F}_2\) is [missing?]
• The size complexity of Clique-Colouring composed with a permutation in Nullstellensatz over \(\mathbb{F}_2\) is [missing?]
• The size complexity of Pebbling \(\circ\) IND in Nullstellensatz over \(\mathbb{F}_2\) is exponential
• Source: BOCIP02 Homogenization and the polynomial calculus.
• The size complexity of Pebbling \(\circ\) XOR, then guarded in Nullstellensatz over \(\mathbb{F}_2\) is [missing?]
• The size complexity of Stone formula in Nullstellensatz over \(\mathbb{F}_2\) is [missing?]
• The size complexity of String of pearls in Nullstellensatz over \(\mathbb{F}_2\) is at most quasipolynomial
• Source: NS_F2 → tlRes → pearl
• The size complexity of Sink of DAG \(\circ\) XOR in Nullstellensatz over \(\mathbb{F}_2\) 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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