All about Tree-like resolution

Proof Systems

• Tree-like resolution weaker than Resolution
• Source: Res → regRes → tlRes
• Source: Res → regRes → ordering → tlRes
• Tree-like resolution simulates Truth table
• Source: [citation needed]
• Tree-like resolution weaker than Regular resolution
• Source: [subsystem]
• Source: regRes → ordering → tlRes
• Tree-like resolution incomparable wrt Ordered resolution
• Source: tlRes → pearl → ordRes
• Source: ordRes → peb+ind → NS_F2 → tlRes
• Tree-like resolution weaker than Pool resolution
• Source: poolRes → regRes → tlRes
• Source: poolRes → regRes → ordering → tlRes
• Tree-like resolution weaker than Linear resolution
• Source: [citation needed]
• Source: linRes → ordering → tlRes
• Tree-like resolution simulated by Reversible resolution
• Source: [subsystem]
• Tree-like resolution weaker than Cutting Planes
• Source: CP → tlCP → tlRes
• Source: CP → uCP → PHP → tlResLin_F2 → tlRes
• Tree-like resolution simulated by Tree-like Cutting Planes
• Source: [citation needed]
• Tree-like resolution weaker than Cutting Planes with Unary Coefficients
• Source: uCP → Res → regRes → tlRes
• Source: uCP → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than Semantic Cutting Planes
• Source: semanticCP → CP → tlCP → tlRes
• Source: semanticCP → CliqueColouringeq → CP → tlCP → tlRes
• Tree-like resolution weaker than Cutting Planes with Saturation
• Source: saturationCP → Res → regRes → tlRes
• Source: saturationCP → Res → regRes → ordering → tlRes
• Tree-like resolution weaker than Stabbing Planes
• Source: SP → CP → tlCP → tlRes
• Source: SP → uSP → uCP → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than Stabbing Planes with Unary Coefficients
• Source: uSP → uCP → Res → regRes → tlRes
• Source: uSP → uCP → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than Res(CP)
• Source: Res(CP) → CP → tlCP → tlRes
• Source: Res(CP) → CP → uCP → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than Res(LP)
• Source: Res(LP) → uRes(LP) → Res → regRes → tlRes
• Source: Res(LP) → uRes(LP) → Res → regRes → ordering → tlRes
• Tree-like resolution weaker than Res(CP) with unary coefficients
• Source: uRes(CP) → uCP → Res → regRes → tlRes
• Source: uRes(CP) → uCP → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than Res(LP) with unary coefficients
• Source: uRes(LP) → Res → regRes → tlRes
• Source: uRes(LP) → Res → regRes → ordering → tlRes
• Tree-like resolution weaker than Res(L\(\&\)P)
• Source: Res(L&P) → Res(LP) → uRes(LP) → Res → regRes → tlRes
• Source: Res(L&P) → Res(LP) → uRes(LP) → Res → regRes → ordering → tlRes
• Tree-like resolution weaker than Res(L\(\&\)P) with unary coefficients
• Source: uRes(L&P) → uRes(LP) → Res → regRes → tlRes
• Source: uRes(L&P) → uRes(LP) → Res → regRes → ordering → tlRes
• Tree-like resolution simulated by Semantic degree-k threshold system, treelike version
• Source: tlTH(k) → tlCP → tlRes
• Tree-like resolution weaker than Polynomial Calculus over \(\mathbb{F}_2\)
• Source: PC_F2 → NS_F2 → tlRes
• Source: PC_F2 → Res → regRes → ordering → tlRes
• Tree-like resolution weaker than Nullstellensatz over \(\mathbb{F}_2\)
• Source: [citation needed]
• Source: NS_F2 → tseitin → Res → regRes → tlRes
• Tree-like resolution weaker than ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Source: ResLin_Z → uResLin_Z → ResLin_F2 → tlResLin_F2 → tlRes
• Source: ResLin_Z → uResLin_Z → uRes(CP) → uCP → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than unary ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Source: uResLin_Z → ResLin_F2 → tlResLin_F2 → tlRes
• Source: uResLin_Z → uRes(CP) → uCP → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than ResLin over \(\mathbb{F}_2\), Res(\(\oplus\))
• Source: ResLin_F2 → tlResLin_F2 → tlRes
• Source: ResLin_F2 → Res → regRes → ordering → tlRes
• Tree-like resolution simulated by Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\))
• Source: [subsystem]
• Tree-like resolution weaker than Polynomial Calculus over \(\mathbb{Q}\)
• Source: PC_Q → NS_Q → tlRes
• Source: PC_Q → Res → regRes → ordering → tlRes
• Tree-like resolution weaker than Nullstellensatz over \(\mathbb{Q}\)
• Source: [citation needed]
• Source: NS_Q → ofPHP → Res → regRes → tlRes
• Tree-like resolution equivalent Hitting
• Source: FHRSV23 Proving Unsatisfiability with Hitting Formulas.
• Tree-like resolution weaker than Lift and Project
• Source: L&P → uL&P → Res → regRes → tlRes
• Source: L&P → uL&P → Res → regRes → ordering → tlRes
• Tree-like resolution weaker than Lift and Project with unary coefficients
• Source: uL&P → Res → regRes → tlRes
• Source: uL&P → Res → regRes → ordering → tlRes
• Tree-like resolution weaker than Positivstellensatz Calculus
• Source: PSC → PS → SoS → SA → NS_Q → tlRes
• Source: PSC → PS → SoS → SA → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than Positivstellensatz
• Source: PS → SoS → SA → NS_Q → tlRes
• Source: PS → SoS → SA → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than Lovász--Schrijver (LS)
• Source: LS → tlLS → tlRes
• Source: LS → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than Lovász--Schrijver with squares (LS\(_+\))
• Source: LS+ → LS → tlLS → tlRes
• Source: LS+ → LS → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree
• Source: LSd+ → LS+ → LS → tlLS → tlRes
• Source: LSd+ → CliqueColouring → Res → regRes → tlRes
• Tree-like resolution weaker than Lovász--Schrijver with squares (LS\(_+^\infty\)), unbounded degree
• Source: LSn+ → LSd+ → LS+ → LS → tlLS → tlRes
• Source: LSn+ → LSd+ → CliqueColouring → Res → regRes → tlRes
• Tree-like resolution weaker than Cone Proof System
• Source: CPS → IPS → extFrege → hit → tlRes
• Source: CPS → IPS → extFrege → PHP → tlResLin_F2 → tlRes
• Tree-like resolution simulated by tl Lovász--Schrijver (LS)
• Source: [citation needed]
• Tree-like resolution simulated by Lovász--Schrijver with squares (LS\(_+\))
• Source: tlLS+ → tlLS → tlRes
• Tree-like resolution simulated by Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree, treelike
• Source: tlLSd+ → tlLS+ → tlLS → tlRes
• Tree-like resolution simulated by static Lovász--Schrijver (static LS)
• Source: sLS → tlLS → tlRes
• Tree-like resolution weaker than static Lovász--Schrijver, with squares of linear functions (static LS\(_+\))
• Source: sLS+ → tlLS+ → tlLS → tlRes
• Source: sLS+ → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\))
• Source: sLSn+ → sLS+ → tlLS+ → tlLS → tlRes
• Source: sLSn+ → sLS+ → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than Sum of Squares (Lasserre)
• Source: SoS → SA → NS_Q → tlRes
• Source: SoS → SA → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than Sherali--Adams
• Source: SA → NS_Q → tlRes
• Source: SA → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than Circular resolution
• Source: circRes → Res → regRes → tlRes
• Source: circRes → Res → regRes → ordering → tlRes
• Tree-like resolution simulated by Unary Sherali--Adams
• Source: uSA → revRes → tlRes
• Tree-like resolution weaker than Ideal Proof System
• Source: IPS → extFrege → hit → tlRes
• Source: IPS → extFrege → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than Extended Frege
• Source: extFrege → hit → tlRes
• Source: extFrege → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than Extended resolution
• Source: extRes → Res → regRes → tlRes
• Source: extRes → Res → regRes → ordering → tlRes
• Tree-like resolution weaker than Frege
• Source: Frege → CP → tlCP → tlRes
• Source: Frege → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than \(\mathrm{AC}^0\)-Frege
• Source: AC0Frege → Res-k → Res → regRes → tlRes
• Source: AC0Frege → Res-k → Res → regRes → ordering → tlRes
• Tree-like resolution weaker than k-DNF Resolution
• Source: Res-k → Res → regRes → tlRes
• Source: Res-k → Res → regRes → ordering → tlRes
• Tree-like resolution weaker than \(\mathrm{TC}^0\)-Frege
• Source: TC0Frege → Res(CP) → CP → tlCP → tlRes
• Source: TC0Frege → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than \(\mathrm{AC}^0\)-Frege with mod 2 axioms
• Source: AC0Frege+Mod2axioms → NS_F2 → tlRes
• Source: AC0Frege+Mod2axioms → NS_F2 → tseitin → Res → regRes → tlRes
• Tree-like resolution weaker than \(\mathrm{AC}^0\)-Frege with mod 2 gates
• Source: AC0(+)Frege → ResLin_F2 → tlResLin_F2 → tlRes
• Source: AC0(+)Frege → ResLin_F2 → Res → regRes → ordering → tlRes
• Tree-like resolution weaker than OBDD(join,weakening)
• Source: OBDDjoinweak → uCP → Res → regRes → tlRes
• Source: OBDDjoinweak → uCP → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than LK
• Source: LK → Frege → CP → tlCP → tlRes
• Source: LK → Frege → PHP → tlResLin_F2 → tlRes
• Tree-like resolution weaker than Zermelo-Fraenkl Set Theory with the Axiom of Choice
• Source: ZFC → extFrege → hit → tlRes
• Source: ZFC → extFrege → PHP → tlResLin_F2 → tlRes

Formulas

• The size complexity of Pigeonhole principle in Tree-like resolution is exponential
• Source: PHP → tlResLin_F2 → tlRes
• The size complexity of Pigeonhole principle with functionality axioms in Tree-like resolution is exponential
• Source: fPHP → ofPHP → Res → regRes → tlRes
• The size complexity of Pigeonhole principle with functionality and onto axioms in Tree-like resolution is exponential
• Source: ofPHP → Res → regRes → tlRes
• The size complexity of Weak pigeonhole principle (2n pigeons, n holes) in Tree-like resolution is [missing?]
• The size complexity of Weak pigeonhole principle (\(n^2\) pigeons, n holes) in Tree-like resolution is [missing?]
• The size complexity of Bitwise pigeonhole principle in Tree-like resolution is exponential
• Source: bPHP → CP → tlCP → tlRes
• The size complexity of Relativized pigeonhole principle (n,\(n^2\),n-1) in Tree-like resolution is exponential
• Source: rPHP → PC_F2 → NS_F2 → tlRes
• The size complexity of Ordering principle in Tree-like resolution is exponential
• Source: [citation needed]
• The size complexity of Guarded ordering principle in Tree-like resolution is exponential
• Source: ordering+guard → regRes → tlRes
• The size complexity of Tseitin over \(\mathbb{F}_2\) in Tree-like resolution is exponential
• Source: tseitin → Res → regRes → tlRes
• The size complexity of Tseitin over \(\mathbb{F}_2\) with k-ary AND gadget in Tree-like resolution is exponential
• Source: tseitin-and-k → tlTH(k) → tlCP → tlRes
• The size complexity of Tseitin over \(\mathbb{Q}\) in Tree-like resolution is [missing?]
• The size complexity of Flow in Tree-like resolution is exponential
• Source: flow → PC_F2 → NS_F2 → tlRes
• The size complexity of Tseitin \(\mathbb{F}_2\) \(\circ\) IND in Tree-like resolution is exponential
• Source: ts+ind → CP → tlCP → tlRes
• The size complexity of Tseitin \(\mathbb{Q}\) \(\circ\) IND in Tree-like resolution is exponential
• Source: tsQ+ind → CP → tlCP → tlRes
• The size complexity of Random CNF in Tree-like resolution is exponential
• Source: random → Res → regRes → tlRes
• The size complexity of Clique-Colouring in Tree-like resolution is exponential
• Source: CliqueColouring → Res → regRes → tlRes
• The size complexity of Clique-Colouring encoded as equalities in Tree-like resolution is exponential
• Source: CliqueColouringeq → CP → tlCP → tlRes
• The size complexity of Weak Clique-Colouring (2m clique, m colours) in Tree-like resolution is exponential
• Source: CliqueColouring2mm → CP → tlCP → tlRes
• The size complexity of Weak Clique-Colouring (m^1/2 clique, log^2 m colours) in Tree-like resolution is superpolynomial
• Source: CliqueColouringmlogm → CP → tlCP → tlRes
• The size complexity of Clique-Colouring composed with a permutation in Tree-like resolution is exponential
• Source: CliqueColouringperm → OBDDjoinweak → uCP → Res → regRes → tlRes
• The size complexity of Pebbling \(\circ\) IND in Tree-like resolution is exponential
• Source: peb+ind → NS_F2 → tlRes
• The size complexity of Pebbling \(\circ\) XOR, then guarded in Tree-like resolution is exponential
• Source: peb+xor+guard → regRes → tlRes
• The size complexity of Stone formula in Tree-like resolution is exponential
• Source: stone → regRes → tlRes
• The size complexity of String of pearls in Tree-like resolution is at most quasipolynomial
• Source: Johannsen01 Exponential Incomparability of Tree-like and Ordered Resolution
• The size complexity of Sink of DAG \(\circ\) XOR in Tree-like resolution is exponential
• Source: sod+xor → uSA → revRes → tlRes

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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