All about Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\))

Proof Systems

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

Formulas

• The size complexity of Pigeonhole principle in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is exponential
• Source: IS20 Resolution over linear equations modulo two.
• The size complexity of Pigeonhole principle with functionality axioms in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Pigeonhole principle with functionality and onto axioms in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Weak pigeonhole principle (2n pigeons, n holes) in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Weak pigeonhole principle (\(n^2\) pigeons, n holes) in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Bitwise pigeonhole principle in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Relativized pigeonhole principle (n,\(n^2\),n-1) in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Ordering principle in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Guarded ordering principle in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Tseitin over \(\mathbb{F}_2\) in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Tseitin over \(\mathbb{F}_2\) with k-ary AND gadget in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Tseitin over \(\mathbb{Q}\) in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Flow in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Tseitin \(\mathbb{F}_2\) \(\circ\) IND in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Tseitin \(\mathbb{Q}\) \(\circ\) IND in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Random CNF in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Clique-Colouring in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Clique-Colouring encoded as equalities in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Weak Clique-Colouring (2m clique, m colours) in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Weak Clique-Colouring (m^1/2 clique, log^2 m colours) in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Clique-Colouring composed with a permutation in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Pebbling \(\circ\) IND in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Pebbling \(\circ\) XOR, then guarded in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Stone formula in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of String of pearls in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is at most quasipolynomial
• Source: tlResLin_F2 → tlRes → pearl
• The size complexity of Sink of DAG \(\circ\) XOR in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) 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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