All about \(\mathrm{TC}^0\)-Frege

Proof Systems

• \(\mathrm{TC}^0\)-Frege stronger than Resolution
• Source: TC0Frege → AC0Frege → Res-k → Res
• Source: TC0Frege → PHP → PC_F2 → Res
• \(\mathrm{TC}^0\)-Frege stronger than Truth table
• Source: TC0Frege → Res(CP) → CP → tlCP → tlRes → ttp
• Source: TC0Frege → PHP → tlResLin_F2 → tlRes → ttp
• \(\mathrm{TC}^0\)-Frege stronger than Tree-like resolution
• Source: TC0Frege → Res(CP) → CP → tlCP → tlRes
• Source: TC0Frege → PHP → tlResLin_F2 → tlRes
• \(\mathrm{TC}^0\)-Frege stronger than Regular resolution
• Source: TC0Frege → AC0Frege → Res-k → Res → regRes
• Source: TC0Frege → PHP → PC_F2 → Res → regRes
• \(\mathrm{TC}^0\)-Frege stronger than Ordered resolution
• Source: TC0Frege → AC0Frege → Res-k → Res → regRes → ordRes
• Source: TC0Frege → PHP → PC_F2 → Res → regRes → ordRes
• \(\mathrm{TC}^0\)-Frege stronger than Pool resolution
• Source: TC0Frege → AC0Frege → Res-k → Res → poolRes
• Source: TC0Frege → PHP → PC_F2 → Res → poolRes
• \(\mathrm{TC}^0\)-Frege stronger than Linear resolution
• Source: TC0Frege → AC0Frege → Res-k → Res → linRes
• Source: TC0Frege → PHP → PC_F2 → Res → linRes
• \(\mathrm{TC}^0\)-Frege stronger than Reversible resolution
• Source: TC0Frege → AC0Frege → Res-k → Res → revRes
• Source: TC0Frege → PHP → PC_F2 → Res → revRes
• \(\mathrm{TC}^0\)-Frege stronger than Cutting Planes
• Source: TC0Frege → Res(CP) → CP
• Source: TC0Frege → AC0Frege → CliqueColouring2mm → CP
• \(\mathrm{TC}^0\)-Frege stronger than Tree-like Cutting Planes
• Source: TC0Frege → Res(CP) → CP → tlCP
• Source: TC0Frege → AC0Frege → Res-k → Res → regRes → ordRes → peb+ind → tlCP
• \(\mathrm{TC}^0\)-Frege stronger than Cutting Planes with Unary Coefficients
• Source: TC0Frege → Res(CP) → CP → uCP
• Source: TC0Frege → AC0Frege → CliqueColouring2mm → CP → uCP
• \(\mathrm{TC}^0\)-Frege [missing?] Semantic Cutting Planes
• \(\mathrm{TC}^0\)-Frege stronger than Cutting Planes with Saturation
• Source: TC0Frege → AC0Frege → Res-k → Res → saturationCP
• Source: TC0Frege → PHP → PC_F2 → Res → saturationCP
• \(\mathrm{TC}^0\)-Frege simulates Stabbing Planes
• Source: TC0Frege → Res(CP) → SP
• \(\mathrm{TC}^0\)-Frege stronger than Stabbing Planes with Unary Coefficients
• Source: TC0Frege → Res(CP) → uRes(CP) → uSP
• Source: TC0Frege → AC0Frege → CliqueColouring2mm → CP → uSP
• \(\mathrm{TC}^0\)-Frege simulates Res(CP)
• Source: [subsystem]
• \(\mathrm{TC}^0\)-Frege simulates Res(LP)
• Source: TC0Frege → Res(CP) → Res(LP)
• \(\mathrm{TC}^0\)-Frege simulates Res(CP) with unary coefficients
• Source: TC0Frege → Res(CP) → uRes(CP)
• \(\mathrm{TC}^0\)-Frege simulates Res(LP) with unary coefficients
• Source: TC0Frege → Res(CP) → Res(LP) → uRes(LP)
• \(\mathrm{TC}^0\)-Frege simulates Res(L\(\&\)P)
• Source: TC0Frege → Res(CP) → Res(LP) → Res(L&P)
• \(\mathrm{TC}^0\)-Frege simulates Res(L\(\&\)P) with unary coefficients
• Source: TC0Frege → Res(CP) → Res(LP) → uRes(LP) → uRes(L&P)
• \(\mathrm{TC}^0\)-Frege [missing?] Semantic degree-k threshold system, treelike version
• \(\mathrm{TC}^0\)-Frege stronger than Polynomial Calculus over \(\mathbb{F}_2\)
• Source: TC0Frege → AC0(+)Frege → PC_F2
• Source: TC0Frege → PHP → PC_F2
• \(\mathrm{TC}^0\)-Frege stronger than Nullstellensatz over \(\mathbb{F}_2\)
• Source: TC0Frege → AC0(+)Frege → AC0Frege+Mod2axioms → NS_F2
• Source: TC0Frege → PHP → PC_F2 → NS_F2
• \(\mathrm{TC}^0\)-Frege [missing?] ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• \(\mathrm{TC}^0\)-Frege [missing?] unary ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• \(\mathrm{TC}^0\)-Frege simulates ResLin over \(\mathbb{F}_2\), Res(\(\oplus\))
• Source: TC0Frege → AC0(+)Frege → ResLin_F2
• \(\mathrm{TC}^0\)-Frege stronger than Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\))
• Source: TC0Frege → AC0(+)Frege → ResLin_F2 → tlResLin_F2
• Source: TC0Frege → PHP → tlResLin_F2
• \(\mathrm{TC}^0\)-Frege not simulated by Polynomial Calculus over \(\mathbb{Q}\)
• Source: TC0Frege → PHP → PC_Q
• \(\mathrm{TC}^0\)-Frege not simulated by Nullstellensatz over \(\mathbb{Q}\)
• Source: TC0Frege → PHP → PC_Q → NS_Q
• \(\mathrm{TC}^0\)-Frege stronger than Hitting
• Source: TC0Frege → Res(CP) → CP → tlCP → tlRes → hit
• Source: TC0Frege → PHP → tlResLin_F2 → tlRes → hit
• \(\mathrm{TC}^0\)-Frege simulates Lift and Project
• Source: TC0Frege → Res(CP) → Res(LP) → Res(L&P) → L&P
• \(\mathrm{TC}^0\)-Frege stronger than Lift and Project with unary coefficients
• Source: TC0Frege → Res(CP) → CP → uCP → uL&P
• Source: TC0Frege → AC0Frege → CliqueColouring2mm → CP → uCP → uL&P
• \(\mathrm{TC}^0\)-Frege [missing?] Positivstellensatz Calculus
• \(\mathrm{TC}^0\)-Frege [missing?] Positivstellensatz
• \(\mathrm{TC}^0\)-Frege [missing?] Lovász--Schrijver (LS)
• \(\mathrm{TC}^0\)-Frege [missing?] Lovász--Schrijver with squares (LS\(_+\))
• \(\mathrm{TC}^0\)-Frege [missing?] Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree
• \(\mathrm{TC}^0\)-Frege [missing?] Lovász--Schrijver with squares (LS\(_+^\infty\)), unbounded degree
• \(\mathrm{TC}^0\)-Frege simulated by Cone Proof System
• Source: CPS → IPS → extFrege → Frege → TC0Frege
• \(\mathrm{TC}^0\)-Frege not simulated by tl Lovász--Schrijver (LS)
• Source: TC0Frege → Res(CP) → Res(LP) → uRes(LP) → tseitin → sLS+ → tlLS+ → tlLS
• \(\mathrm{TC}^0\)-Frege not simulated by Lovász--Schrijver with squares (LS\(_+\))
• Source: TC0Frege → Res(CP) → Res(LP) → uRes(LP) → tseitin → sLS+ → tlLS+
• \(\mathrm{TC}^0\)-Frege [missing?] Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree, treelike
• \(\mathrm{TC}^0\)-Frege not simulated by static Lovász--Schrijver (static LS)
• Source: TC0Frege → Res(CP) → Res(LP) → uRes(LP) → tseitin → sLS+ → sLS
• \(\mathrm{TC}^0\)-Frege not simulated by static Lovász--Schrijver, with squares of linear functions (static LS\(_+\))
• Source: TC0Frege → Res(CP) → Res(LP) → uRes(LP) → tseitin → sLS+
• \(\mathrm{TC}^0\)-Frege not simulated by static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\))
• Source: TC0Frege → Res(CP) → Res(LP) → uRes(LP) → tseitin → SoS → sLSn+
• \(\mathrm{TC}^0\)-Frege not simulated by Sum of Squares (Lasserre)
• Source: TC0Frege → Res(CP) → Res(LP) → uRes(LP) → tseitin → SoS
• \(\mathrm{TC}^0\)-Frege not simulated by Sherali--Adams
• Source: TC0Frege → Res(CP) → Res(LP) → uRes(LP) → tseitin → SoS → SA
• \(\mathrm{TC}^0\)-Frege not simulated by Circular resolution
• Source: TC0Frege → Res(CP) → Res(LP) → uRes(LP) → tseitin → SoS → SA → circRes
• \(\mathrm{TC}^0\)-Frege not simulated by Unary Sherali--Adams
• Source: TC0Frege → AC0Frege → Res-k → Res → sod+xor → uSA
• \(\mathrm{TC}^0\)-Frege simulated by Ideal Proof System
• Source: IPS → extFrege → Frege → TC0Frege
• \(\mathrm{TC}^0\)-Frege simulated by Extended Frege
• Source: extFrege → Frege → TC0Frege
• \(\mathrm{TC}^0\)-Frege simulated by Extended resolution
• Source: extRes → extFrege → Frege → TC0Frege
• \(\mathrm{TC}^0\)-Frege simulated by Frege
• Source: [citation needed]
• \(\mathrm{TC}^0\)-Frege stronger than \(\mathrm{AC}^0\)-Frege
• Source: [subsystem]
• Source: TC0Frege → PHP → fPHP → ofPHP → AC0Frege
• \(\mathrm{TC}^0\)-Frege stronger than k-DNF Resolution
• Source: TC0Frege → AC0Frege → Res-k
• Source: TC0Frege → PHP → fPHP → ofPHP → AC0Frege → Res-k
• \(\mathrm{TC}^0\)-Frege simulates \(\mathrm{AC}^0\)-Frege with mod 2 axioms
• Source: TC0Frege → AC0(+)Frege → AC0Frege+Mod2axioms
• \(\mathrm{TC}^0\)-Frege simulates \(\mathrm{AC}^0\)-Frege with mod 2 gates
• Source: MP97 None
• \(\mathrm{TC}^0\)-Frege [missing?] OBDD(join,weakening)
• \(\mathrm{TC}^0\)-Frege simulated by LK
• Source: LK → Frege → TC0Frege
• \(\mathrm{TC}^0\)-Frege simulated by Zermelo-Fraenkl Set Theory with the Axiom of Choice
• Source: ZFC → extFrege → Frege → TC0Frege

Formulas

• The size complexity of Pigeonhole principle in \(\mathrm{TC}^0\)-Frege is polynomial
• Source: Kra19 Proof complexity
• The size complexity of Pigeonhole principle with functionality axioms in \(\mathrm{TC}^0\)-Frege is polynomial
• Source: TC0Frege → PHP → fPHP
• The size complexity of Pigeonhole principle with functionality and onto axioms in \(\mathrm{TC}^0\)-Frege is polynomial
• Source: TC0Frege → PHP → fPHP → ofPHP
• The size complexity of Weak pigeonhole principle (2n pigeons, n holes) in \(\mathrm{TC}^0\)-Frege is polynomial
• Source: TC0Frege → PHP → PHP2nn
• The size complexity of Weak pigeonhole principle (\(n^2\) pigeons, n holes) in \(\mathrm{TC}^0\)-Frege is polynomial
• Source: TC0Frege → PHP → PHP2nn → PHPn2n
• The size complexity of Bitwise pigeonhole principle in \(\mathrm{TC}^0\)-Frege is [missing?]
• The size complexity of Relativized pigeonhole principle (n,\(n^2\),n-1) in \(\mathrm{TC}^0\)-Frege is [missing?]
• The size complexity of Ordering principle in \(\mathrm{TC}^0\)-Frege is polynomial
• Source: TC0Frege → AC0Frege → Res-k → Res → regRes → ordering
• The size complexity of Guarded ordering principle in \(\mathrm{TC}^0\)-Frege is polynomial
• Source: TC0Frege → AC0Frege → Res-k → Res → poolRes → ordering+guard
• The size complexity of Tseitin over \(\mathbb{F}_2\) in \(\mathrm{TC}^0\)-Frege is polynomial
• Source: TC0Frege → Res(CP) → Res(LP) → uRes(LP) → tseitin
• The size complexity of Tseitin over \(\mathbb{F}_2\) with k-ary AND gadget in \(\mathrm{TC}^0\)-Frege is [missing?]
• The size complexity of Tseitin over \(\mathbb{Q}\) in \(\mathrm{TC}^0\)-Frege is [missing?]
• The size complexity of Flow in \(\mathrm{TC}^0\)-Frege is [missing?]
• The size complexity of Tseitin \(\mathbb{F}_2\) \(\circ\) IND in \(\mathrm{TC}^0\)-Frege is at most quasipolynomial
• Source: TC0Frege → AC0(+)Frege → AC0Frege+Mod2axioms → NS_F2 → ts+ind
• The size complexity of Tseitin \(\mathbb{Q}\) \(\circ\) IND in \(\mathrm{TC}^0\)-Frege is [missing?]
• The size complexity of Random CNF in \(\mathrm{TC}^0\)-Frege is [missing?]
• The size complexity of Clique-Colouring in \(\mathrm{TC}^0\)-Frege is [missing?]
• The size complexity of Clique-Colouring encoded as equalities in \(\mathrm{TC}^0\)-Frege is [missing?]
• The size complexity of Weak Clique-Colouring (2m clique, m colours) in \(\mathrm{TC}^0\)-Frege is at most quasipolynomial
• Source: TC0Frege → AC0Frege → CliqueColouring2mm
• The size complexity of Weak Clique-Colouring (m^1/2 clique, log^2 m colours) in \(\mathrm{TC}^0\)-Frege is polynomial
• Source: TC0Frege → AC0Frege → Res-k → CliqueColouringmlogm
• The size complexity of Clique-Colouring composed with a permutation in \(\mathrm{TC}^0\)-Frege is [missing?]
• The size complexity of Pebbling \(\circ\) IND in \(\mathrm{TC}^0\)-Frege is polynomial
• Source: TC0Frege → AC0Frege → Res-k → Res → regRes → ordRes → peb+ind
• The size complexity of Pebbling \(\circ\) XOR, then guarded in \(\mathrm{TC}^0\)-Frege is polynomial
• Source: TC0Frege → AC0Frege → Res-k → Res → poolRes → peb+xor+guard
• The size complexity of Stone formula in \(\mathrm{TC}^0\)-Frege is polynomial
• Source: TC0Frege → AC0Frege → Res-k → Res → poolRes → stone
• The size complexity of String of pearls in \(\mathrm{TC}^0\)-Frege is polynomial
• Source: TC0Frege → AC0Frege → Res-k → Res → regRes → pearl
• The size complexity of Sink of DAG \(\circ\) XOR in \(\mathrm{TC}^0\)-Frege is polynomial
• Source: TC0Frege → AC0Frege → Res-k → 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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