All about OBDD(join,weakening)

Proof Systems

• OBDD(join,weakening) stronger than Resolution
• Source: OBDDjoinweak → uCP → Res
• Source: OBDDjoinweak → tseitin → Res
• OBDD(join,weakening) stronger than Truth table
• Source: OBDDjoinweak → uCP → Res → regRes → tlRes → ttp
• Source: OBDDjoinweak → uCP → PHP → tlResLin_F2 → tlRes → ttp
• OBDD(join,weakening) stronger than Tree-like resolution
• Source: OBDDjoinweak → uCP → Res → regRes → tlRes
• Source: OBDDjoinweak → uCP → PHP → tlResLin_F2 → tlRes
• OBDD(join,weakening) stronger than Regular resolution
• Source: OBDDjoinweak → uCP → Res → regRes
• Source: OBDDjoinweak → tseitin → Res → regRes
• OBDD(join,weakening) stronger than Ordered resolution
• Source: OBDDjoinweak → uCP → Res → regRes → ordRes
• Source: OBDDjoinweak → tseitin → Res → regRes → ordRes
• OBDD(join,weakening) stronger than Pool resolution
• Source: OBDDjoinweak → uCP → Res → poolRes
• Source: OBDDjoinweak → tseitin → Res → poolRes
• OBDD(join,weakening) stronger than Linear resolution
• Source: OBDDjoinweak → uCP → Res → linRes
• Source: OBDDjoinweak → tseitin → Res → linRes
• OBDD(join,weakening) stronger than Reversible resolution
• Source: OBDDjoinweak → uCP → Res → revRes
• Source: OBDDjoinweak → tseitin → Res → revRes
• OBDD(join,weakening) not simulated by Cutting Planes
• Source: OBDDjoinweak → CliqueColouring → CP
• OBDD(join,weakening) not simulated by Tree-like Cutting Planes
• Source: OBDDjoinweak → CliqueColouring → CP → tlCP
• OBDD(join,weakening) stronger than Cutting Planes with Unary Coefficients
• Source: AKV04 Constraint Propagation as a Proof System.
• Source: OBDDjoinweak → CliqueColouring → CP → uCP
• OBDD(join,weakening) not simulated by Semantic Cutting Planes
• Source: OBDDjoinweak → CliqueColouring → semanticCP
• OBDD(join,weakening) stronger than Cutting Planes with Saturation
• Source: OBDDjoinweak → uCP → Res → saturationCP
• Source: OBDDjoinweak → tseitin → Res → saturationCP
• OBDD(join,weakening) not simulated by Stabbing Planes
• Source: OBDDjoinweak → CliqueColouring → SP
• OBDD(join,weakening) not simulated by Stabbing Planes with Unary Coefficients
• Source: OBDDjoinweak → CliqueColouring → SP → uSP
• OBDD(join,weakening) [missing?] Res(CP)
• OBDD(join,weakening) [missing?] Res(LP)
• OBDD(join,weakening) [missing?] Res(CP) with unary coefficients
• OBDD(join,weakening) [missing?] Res(LP) with unary coefficients
• OBDD(join,weakening) [missing?] Res(L\(\&\)P)
• OBDD(join,weakening) [missing?] Res(L\(\&\)P) with unary coefficients
• OBDD(join,weakening) [missing?] Semantic degree-k threshold system, treelike version
• OBDD(join,weakening) not simulated by Polynomial Calculus over \(\mathbb{F}_2\)
• Source: OBDDjoinweak → uCP → PHP → PC_F2
• OBDD(join,weakening) not simulated by Nullstellensatz over \(\mathbb{F}_2\)
• Source: OBDDjoinweak → uCP → PHP → PC_F2 → NS_F2
• OBDD(join,weakening) [missing?] ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• OBDD(join,weakening) [missing?] unary ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• OBDD(join,weakening) [missing?] ResLin over \(\mathbb{F}_2\), Res(\(\oplus\))
• OBDD(join,weakening) not simulated by Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\))
• Source: OBDDjoinweak → uCP → PHP → tlResLin_F2
• OBDD(join,weakening) not simulated by Polynomial Calculus over \(\mathbb{Q}\)
• Source: OBDDjoinweak → tseitin → PC_Q
• OBDD(join,weakening) not simulated by Nullstellensatz over \(\mathbb{Q}\)
• Source: OBDDjoinweak → tseitin → PC_Q → NS_Q
• OBDD(join,weakening) stronger than Hitting
• Source: OBDDjoinweak → uCP → Res → regRes → tlRes → hit
• Source: OBDDjoinweak → uCP → PHP → tlResLin_F2 → tlRes → hit
• OBDD(join,weakening) not simulated by Lift and Project
• Source: OBDDjoinweak → CliqueColouring → L&P
• OBDD(join,weakening) stronger than Lift and Project with unary coefficients
• Source: OBDDjoinweak → uCP → uL&P
• Source: OBDDjoinweak → CliqueColouring → L&P → uL&P
• OBDD(join,weakening) [missing?] Positivstellensatz Calculus
• OBDD(join,weakening) [missing?] Positivstellensatz
• OBDD(join,weakening) [missing?] Lovász--Schrijver (LS)
• OBDD(join,weakening) [missing?] Lovász--Schrijver with squares (LS\(_+\))
• OBDD(join,weakening) [missing?] Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree
• OBDD(join,weakening) [missing?] Lovász--Schrijver with squares (LS\(_+^\infty\)), unbounded degree
• OBDD(join,weakening) [missing?] Cone Proof System
• OBDD(join,weakening) not simulated by tl Lovász--Schrijver (LS)
• Source: OBDDjoinweak → tseitin → sLS+ → tlLS+ → tlLS
• OBDD(join,weakening) not simulated by Lovász--Schrijver with squares (LS\(_+\))
• Source: OBDDjoinweak → tseitin → sLS+ → tlLS+
• OBDD(join,weakening) [missing?] Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree, treelike
• OBDD(join,weakening) not simulated by static Lovász--Schrijver (static LS)
• Source: OBDDjoinweak → tseitin → sLS+ → sLS
• OBDD(join,weakening) not simulated by static Lovász--Schrijver, with squares of linear functions (static LS\(_+\))
• Source: OBDDjoinweak → tseitin → sLS+
• OBDD(join,weakening) not simulated by static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\))
• Source: OBDDjoinweak → tseitin → SoS → sLSn+
• OBDD(join,weakening) not simulated by Sum of Squares (Lasserre)
• Source: OBDDjoinweak → tseitin → SoS
• OBDD(join,weakening) not simulated by Sherali--Adams
• Source: OBDDjoinweak → tseitin → SoS → SA
• OBDD(join,weakening) not simulated by Circular resolution
• Source: OBDDjoinweak → tseitin → SoS → SA → circRes
• OBDD(join,weakening) not simulated by Unary Sherali--Adams
• Source: OBDDjoinweak → uCP → Res → sod+xor → uSA
• OBDD(join,weakening) [missing?] Ideal Proof System
• OBDD(join,weakening) [missing?] Extended Frege
• OBDD(join,weakening) [missing?] Extended resolution
• OBDD(join,weakening) [missing?] Frege
• OBDD(join,weakening) not simulated by \(\mathrm{AC}^0\)-Frege
• Source: OBDDjoinweak → tseitin → AC0Frege
• OBDD(join,weakening) not simulated by k-DNF Resolution
• Source: OBDDjoinweak → tseitin → AC0Frege → Res-k
• OBDD(join,weakening) [missing?] \(\mathrm{TC}^0\)-Frege
• OBDD(join,weakening) [missing?] \(\mathrm{AC}^0\)-Frege with mod 2 axioms
• OBDD(join,weakening) [missing?] \(\mathrm{AC}^0\)-Frege with mod 2 gates
• OBDD(join,weakening) [missing?] LK
• OBDD(join,weakening) [missing?] Zermelo-Fraenkl Set Theory with the Axiom of Choice

Formulas

• The size complexity of Pigeonhole principle in OBDD(join,weakening) is polynomial
• Source: OBDDjoinweak → uCP → PHP
• The size complexity of Pigeonhole principle with functionality axioms in OBDD(join,weakening) is polynomial
• Source: OBDDjoinweak → uCP → PHP → fPHP
• The size complexity of Pigeonhole principle with functionality and onto axioms in OBDD(join,weakening) is polynomial
• Source: OBDDjoinweak → uCP → PHP → fPHP → ofPHP
• The size complexity of Weak pigeonhole principle (2n pigeons, n holes) in OBDD(join,weakening) is polynomial
• Source: OBDDjoinweak → uCP → PHP → PHP2nn
• The size complexity of Weak pigeonhole principle (\(n^2\) pigeons, n holes) in OBDD(join,weakening) is polynomial
• Source: OBDDjoinweak → uCP → PHP → PHP2nn → PHPn2n
• The size complexity of Bitwise pigeonhole principle in OBDD(join,weakening) is [missing?]
• The size complexity of Relativized pigeonhole principle (n,\(n^2\),n-1) in OBDD(join,weakening) is [missing?]
• The size complexity of Ordering principle in OBDD(join,weakening) is polynomial
• Source: OBDDjoinweak → uCP → Res → regRes → ordering
• The size complexity of Guarded ordering principle in OBDD(join,weakening) is polynomial
• Source: OBDDjoinweak → uCP → Res → poolRes → ordering+guard
• The size complexity of Tseitin over \(\mathbb{F}_2\) in OBDD(join,weakening) is polynomial
• Source: AKV04 Constraint Propagation as a Proof System.
• The size complexity of Tseitin over \(\mathbb{F}_2\) with k-ary AND gadget in OBDD(join,weakening) is [missing?]
• The size complexity of Tseitin over \(\mathbb{Q}\) in OBDD(join,weakening) is [missing?]
• The size complexity of Flow in OBDD(join,weakening) is [missing?]
• The size complexity of Tseitin \(\mathbb{F}_2\) \(\circ\) IND in OBDD(join,weakening) is [missing?]
• The size complexity of Tseitin \(\mathbb{Q}\) \(\circ\) IND in OBDD(join,weakening) is [missing?]
• The size complexity of Random CNF in OBDD(join,weakening) is [missing?]
• The size complexity of Clique-Colouring in OBDD(join,weakening) is polynomial
• Source: BIKS18 Reordering Rule Makes OBDD Proof Systems Stronger.
• The size complexity of Clique-Colouring encoded as equalities in OBDD(join,weakening) is [missing?]
• The size complexity of Weak Clique-Colouring (2m clique, m colours) in OBDD(join,weakening) is polynomial
• Source: OBDDjoinweak → CliqueColouring → CliqueColouring2mm
• The size complexity of Weak Clique-Colouring (m^1/2 clique, log^2 m colours) in OBDD(join,weakening) is polynomial
• Source: OBDDjoinweak → CliqueColouring → CliqueColouringmlogm
• The size complexity of Clique-Colouring composed with a permutation in OBDD(join,weakening) is exponential
• Source: Krajicek08 An exponential lower bound for a constraint propagation proof system based on ordered binary decision diagrams.
• The size complexity of Pebbling \(\circ\) IND in OBDD(join,weakening) is polynomial
• Source: OBDDjoinweak → uCP → Res → regRes → ordRes → peb+ind
• The size complexity of Pebbling \(\circ\) XOR, then guarded in OBDD(join,weakening) is polynomial
• Source: OBDDjoinweak → uCP → Res → poolRes → peb+xor+guard
• The size complexity of Stone formula in OBDD(join,weakening) is polynomial
• Source: OBDDjoinweak → uCP → Res → poolRes → stone
• The size complexity of String of pearls in OBDD(join,weakening) is polynomial
• Source: OBDDjoinweak → uCP → Res → regRes → pearl
• The size complexity of Sink of DAG \(\circ\) XOR in OBDD(join,weakening) is polynomial
• Source: OBDDjoinweak → uCP → 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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