All about Cone Proof System

Definition from: Semi-algebraic proofs, IPS lower bounds, and the τ-conjecture: can a natural number be negative?

Proof Systems

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

Formulas

• The size complexity of Pigeonhole principle in Cone Proof System is polynomial
• Source: CPS → IPS → extFrege → PHP
• The size complexity of Pigeonhole principle with functionality axioms in Cone Proof System is polynomial
• Source: CPS → IPS → extFrege → PHP → fPHP
• The size complexity of Pigeonhole principle with functionality and onto axioms in Cone Proof System is polynomial
• Source: CPS → IPS → extFrege → PHP → fPHP → ofPHP
• The size complexity of Weak pigeonhole principle (2n pigeons, n holes) in Cone Proof System is polynomial
• Source: CPS → IPS → extFrege → PHP → PHP2nn
• The size complexity of Weak pigeonhole principle (\(n^2\) pigeons, n holes) in Cone Proof System is polynomial
• Source: CPS → IPS → extFrege → PHP → PHP2nn → PHPn2n
• The size complexity of Bitwise pigeonhole principle in Cone Proof System is [missing?]
• The size complexity of Relativized pigeonhole principle (n,\(n^2\),n-1) in Cone Proof System is [missing?]
• The size complexity of Ordering principle in Cone Proof System is polynomial
• Source: CPS → IPS → extFrege → extRes → Res → regRes → ordering
• The size complexity of Guarded ordering principle in Cone Proof System is polynomial
• Source: CPS → IPS → extFrege → extRes → Res → poolRes → ordering+guard
• The size complexity of Tseitin over \(\mathbb{F}_2\) in Cone Proof System is polynomial
• Source: CPS → LSn+ → LSd+ → tseitin
• The size complexity of Tseitin over \(\mathbb{F}_2\) with k-ary AND gadget in Cone Proof System is [missing?]
• The size complexity of Tseitin over \(\mathbb{Q}\) in Cone Proof System is polynomial
• Source: CPS → LSn+ → PSC → PS → SoS → SA → NS_Q → tseitinQ
• The size complexity of Flow in Cone Proof System is [missing?]
• The size complexity of Tseitin \(\mathbb{F}_2\) \(\circ\) IND in Cone Proof System is at most quasipolynomial
• Source: CPS → IPS → extFrege → Frege → AC0(+)Frege → AC0Frege+Mod2axioms → NS_F2 → ts+ind
• The size complexity of Tseitin \(\mathbb{Q}\) \(\circ\) IND in Cone Proof System is at most quasipolynomial
• Source: CPS → LSn+ → PSC → PS → SoS → SA → NS_Q → tsQ+ind
• The size complexity of Random CNF in Cone Proof System is [missing?]
• The size complexity of Clique-Colouring in Cone Proof System is polynomial
• Source: CPS → LSn+ → LSd+ → CliqueColouring
• The size complexity of Clique-Colouring encoded as equalities in Cone Proof System is [missing?]
• The size complexity of Weak Clique-Colouring (2m clique, m colours) in Cone Proof System is polynomial
• Source: CPS → LSn+ → LSd+ → CliqueColouring → CliqueColouring2mm
• The size complexity of Weak Clique-Colouring (m^1/2 clique, log^2 m colours) in Cone Proof System is polynomial
• Source: CPS → LSn+ → LSd+ → CliqueColouring → CliqueColouringmlogm
• The size complexity of Clique-Colouring composed with a permutation in Cone Proof System is [missing?]
• The size complexity of Pebbling \(\circ\) IND in Cone Proof System is polynomial
• Source: CPS → IPS → extFrege → extRes → Res → regRes → ordRes → peb+ind
• The size complexity of Pebbling \(\circ\) XOR, then guarded in Cone Proof System is polynomial
• Source: CPS → IPS → extFrege → extRes → Res → poolRes → peb+xor+guard
• The size complexity of Stone formula in Cone Proof System is polynomial
• Source: CPS → IPS → extFrege → extRes → Res → poolRes → stone
• The size complexity of String of pearls in Cone Proof System is polynomial
• Source: CPS → IPS → extFrege → extRes → Res → regRes → pearl
• The size complexity of Sink of DAG \(\circ\) XOR in Cone Proof System is polynomial
• Source: CPS → IPS → extFrege → extRes → 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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