All about Positivstellensatz

Notes: Grigoriev's version, but with Boolean axioms

Proof Systems

• Positivstellensatz stronger than Resolution
• Source: PS → SoS → PC_Q → Res
• Source: PS → SoS → sLSn+ → CliqueColouring → Res
• Positivstellensatz stronger than Truth table
• Source: PS → SoS → SA → NS_Q → tlRes → ttp
• Source: PS → SoS → SA → PHP → tlResLin_F2 → tlRes → ttp
• Positivstellensatz stronger than Tree-like resolution
• Source: PS → SoS → SA → NS_Q → tlRes
• Source: PS → SoS → SA → PHP → tlResLin_F2 → tlRes
• Positivstellensatz stronger than Regular resolution
• Source: PS → SoS → PC_Q → Res → regRes
• Source: PS → SoS → sLSn+ → CliqueColouring → Res → regRes
• Positivstellensatz stronger than Ordered resolution
• Source: PS → SoS → PC_Q → Res → regRes → ordRes
• Source: PS → SoS → sLSn+ → CliqueColouring → Res → regRes → ordRes
• Positivstellensatz stronger than Pool resolution
• Source: PS → SoS → PC_Q → Res → poolRes
• Source: PS → SoS → sLSn+ → CliqueColouring → Res → poolRes
• Positivstellensatz stronger than Linear resolution
• Source: PS → SoS → PC_Q → Res → linRes
• Source: PS → SoS → sLSn+ → CliqueColouring → Res → linRes
• Positivstellensatz stronger than Reversible resolution
• Source: PS → SoS → SA → uSA → revRes
• Source: PS → SoS → sLSn+ → CliqueColouring → Res → revRes
• Positivstellensatz not simulated by Cutting Planes
• Source: PS → SoS → sLSn+ → CliqueColouring → CP
• Positivstellensatz not simulated by Tree-like Cutting Planes
• Source: PS → SoS → sLSn+ → CliqueColouring → CP → tlCP
• Positivstellensatz not simulated by Cutting Planes with Unary Coefficients
• Source: PS → SoS → sLSn+ → CliqueColouring → CP → uCP
• Positivstellensatz not simulated by Semantic Cutting Planes
• Source: PS → SoS → sLSn+ → CliqueColouring → semanticCP
• Positivstellensatz stronger than Cutting Planes with Saturation
• Source: PS → SoS → PC_Q → Res → saturationCP
• Source: PS → SoS → sLSn+ → CliqueColouring → Res → saturationCP
• Positivstellensatz not simulated by Stabbing Planes
• Source: PS → SoS → sLSn+ → CliqueColouring → SP
• Positivstellensatz not simulated by Stabbing Planes with Unary Coefficients
• Source: PS → SoS → sLSn+ → CliqueColouring → SP → uSP
• Positivstellensatz [missing?] Res(CP)
• Positivstellensatz [missing?] Res(LP)
• Positivstellensatz [missing?] Res(CP) with unary coefficients
• Positivstellensatz [missing?] Res(LP) with unary coefficients
• Positivstellensatz [missing?] Res(L\(\&\)P)
• Positivstellensatz [missing?] Res(L\(\&\)P) with unary coefficients
• Positivstellensatz [missing?] Semantic degree-k threshold system, treelike version
• Positivstellensatz not simulated by Polynomial Calculus over \(\mathbb{F}_2\)
• Source: PS → SoS → SA → PHP → PC_F2
• Positivstellensatz not simulated by Nullstellensatz over \(\mathbb{F}_2\)
• Source: PS → SoS → SA → PHP → PC_F2 → NS_F2
• Positivstellensatz [missing?] ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Positivstellensatz [missing?] unary ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Positivstellensatz [missing?] ResLin over \(\mathbb{F}_2\), Res(\(\oplus\))
• Positivstellensatz not simulated by Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\))
• Source: PS → SoS → SA → PHP → tlResLin_F2
• Positivstellensatz stronger than Polynomial Calculus over \(\mathbb{Q}\)
• Source: PS → SoS → PC_Q
• Source: PS → SoS → SA → PHP → PC_Q
• Positivstellensatz stronger than Nullstellensatz over \(\mathbb{Q}\)
• Source: PS → SoS → SA → NS_Q
• Source: PS → SoS → SA → PHP → PC_Q → NS_Q
• Positivstellensatz stronger than Hitting
• Source: PS → SoS → SA → NS_Q → tlRes → hit
• Source: PS → SoS → SA → PHP → tlResLin_F2 → tlRes → hit
• Positivstellensatz not simulated by Lift and Project
• Source: PS → SoS → sLSn+ → CliqueColouring → L&P
• Positivstellensatz not simulated by Lift and Project with unary coefficients
• Source: PS → SoS → sLSn+ → CliqueColouring → L&P → uL&P
• Positivstellensatz equivalent Positivstellensatz Calculus
• Source: Berkholz18 The Relation between Polynomial Calculus, Sherali-Adams, and Sum-of-Squares Proofs.
• Positivstellensatz [missing?] Lovász--Schrijver (LS)
• Positivstellensatz [missing?] Lovász--Schrijver with squares (LS\(_+\))
• Positivstellensatz [missing?] Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree
• Positivstellensatz simulated by Lovász--Schrijver with squares (LS\(_+^\infty\)), unbounded degree
• Source: LSn+ → PSC → PS
• Positivstellensatz simulated by Cone Proof System
• Source: CPS → LSn+ → PSC → PS
• Positivstellensatz simulates tl Lovász--Schrijver (LS)
• Source: PS → SoS → sLSn+ → sLS+ → tlLS+ → tlLS
• Positivstellensatz simulates Lovász--Schrijver with squares (LS\(_+\))
• Source: PS → SoS → sLSn+ → sLS+ → tlLS+
• Positivstellensatz [missing?] Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree, treelike
• Positivstellensatz simulates static Lovász--Schrijver (static LS)
• Source: PS → SoS → sLSn+ → sLS+ → sLS
• Positivstellensatz simulates static Lovász--Schrijver, with squares of linear functions (static LS\(_+\))
• Source: PS → SoS → sLSn+ → sLS+
• Positivstellensatz simulates static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\))
• Source: PS → SoS → sLSn+
• Positivstellensatz simulates Sum of Squares (Lasserre)
• Source: [subsystem]
• Positivstellensatz simulates Sherali--Adams
• Source: PS → SoS → SA
• Positivstellensatz simulates Circular resolution
• Source: PS → SoS → SA → circRes
• Positivstellensatz stronger than Unary Sherali--Adams
• Source: PS → SoS → SA → uSA
• Source: PS → SoS → PC_Q → Res → sod+xor → uSA
• Positivstellensatz [missing?] Ideal Proof System
• Positivstellensatz [missing?] Extended Frege
• Positivstellensatz [missing?] Extended resolution
• Positivstellensatz [missing?] Frege
• Positivstellensatz not simulated by \(\mathrm{AC}^0\)-Frege
• Source: PS → SoS → SA → NS_Q → ofPHP → AC0Frege
• Positivstellensatz not simulated by k-DNF Resolution
• Source: PS → SoS → SA → NS_Q → ofPHP → AC0Frege → Res-k
• Positivstellensatz [missing?] \(\mathrm{TC}^0\)-Frege
• Positivstellensatz [missing?] \(\mathrm{AC}^0\)-Frege with mod 2 axioms
• Positivstellensatz [missing?] \(\mathrm{AC}^0\)-Frege with mod 2 gates
• Positivstellensatz [missing?] OBDD(join,weakening)
• Positivstellensatz [missing?] LK
• Positivstellensatz [missing?] Zermelo-Fraenkl Set Theory with the Axiom of Choice

Formulas

• The size complexity of Pigeonhole principle in Positivstellensatz is polynomial
• Source: PS → SoS → SA → PHP
• The size complexity of Pigeonhole principle with functionality axioms in Positivstellensatz is polynomial
• Source: PS → SoS → SA → PHP → fPHP
• The size complexity of Pigeonhole principle with functionality and onto axioms in Positivstellensatz is polynomial
• Source: PS → SoS → SA → NS_Q → ofPHP
• The size complexity of Weak pigeonhole principle (2n pigeons, n holes) in Positivstellensatz is polynomial
• Source: PS → SoS → SA → PHP → PHP2nn
• The size complexity of Weak pigeonhole principle (\(n^2\) pigeons, n holes) in Positivstellensatz is polynomial
• Source: PS → SoS → SA → PHP → PHP2nn → PHPn2n
• The size complexity of Bitwise pigeonhole principle in Positivstellensatz is [missing?]
• The size complexity of Relativized pigeonhole principle (n,\(n^2\),n-1) in Positivstellensatz is [missing?]
• The size complexity of Ordering principle in Positivstellensatz is polynomial
• Source: PS → SoS → PC_Q → Res → regRes → ordering
• The size complexity of Guarded ordering principle in Positivstellensatz is polynomial
• Source: PS → SoS → PC_Q → Res → poolRes → ordering+guard
• The size complexity of Tseitin over \(\mathbb{F}_2\) in Positivstellensatz is [missing?]
• The size complexity of Tseitin over \(\mathbb{F}_2\) with k-ary AND gadget in Positivstellensatz is [missing?]
• The size complexity of Tseitin over \(\mathbb{Q}\) in Positivstellensatz is polynomial
• Source: PS → SoS → SA → NS_Q → tseitinQ
• The size complexity of Flow in Positivstellensatz is [missing?]
• The size complexity of Tseitin \(\mathbb{F}_2\) \(\circ\) IND in Positivstellensatz is [missing?]
• The size complexity of Tseitin \(\mathbb{Q}\) \(\circ\) IND in Positivstellensatz is at most quasipolynomial
• Source: PS → SoS → SA → NS_Q → tsQ+ind
• The size complexity of Random CNF in Positivstellensatz is [missing?]
• The size complexity of Clique-Colouring in Positivstellensatz is polynomial
• Source: PS → SoS → sLSn+ → CliqueColouring
• The size complexity of Clique-Colouring encoded as equalities in Positivstellensatz is [missing?]
• The size complexity of Weak Clique-Colouring (2m clique, m colours) in Positivstellensatz is polynomial
• Source: PS → SoS → sLSn+ → CliqueColouring → CliqueColouring2mm
• The size complexity of Weak Clique-Colouring (m^1/2 clique, log^2 m colours) in Positivstellensatz is polynomial
• Source: PS → SoS → sLSn+ → CliqueColouring → CliqueColouringmlogm
• The size complexity of Clique-Colouring composed with a permutation in Positivstellensatz is [missing?]
• The size complexity of Pebbling \(\circ\) IND in Positivstellensatz is polynomial
• Source: PS → SoS → PC_Q → Res → regRes → ordRes → peb+ind
• The size complexity of Pebbling \(\circ\) XOR, then guarded in Positivstellensatz is polynomial
• Source: PS → SoS → PC_Q → Res → poolRes → peb+xor+guard
• The size complexity of Stone formula in Positivstellensatz is polynomial
• Source: PS → SoS → PC_Q → Res → poolRes → stone
• The size complexity of String of pearls in Positivstellensatz is polynomial
• Source: PS → SoS → PC_Q → Res → regRes → pearl
• The size complexity of Sink of DAG \(\circ\) XOR in Positivstellensatz is polynomial
• Source: PS → SoS → PC_Q → 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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