All about Nullstellensatz over \(\mathbb{Q}\)

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

Proof Systems

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

Formulas

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