At the core of a CIfly algorithm lies its ruletable. Here, we present an overview the ruletables that we use in our application examples. For personal use, just copy the ruletable into your project and modify it to your needs.

• Ancestors

  EDGES --> <--, <->
  SETS X
  START <-- AT X
  OUTPUT ...
  
  ... | <-- | true

  Graph type:

  ADMG

  Applications:

  Finding minimal d-separators, Finding conditional instruments

• Backdoor-connected

  EDGES --> <--, ---
  SETS X, C, W
  START <-- AT X
  OUTPUT ...
  
  --> | <-- | current in W
  --- | --- | current not in W
  ... | --> | current not in W and (current not in X or next not in C)
  <-- | ... | current not in W 

  Graph type:

  CPDAG

  Applications:

  Verifying adjustment sets

  References:

  Perković et al. (2018)

• Closure

  EDGES --> <--, <->
  SETS X, Z, A
  START <-- AT X
  OUTPUT ...
  
  -->, <-> | <--, <-> | next in A
  ...      | ...      | next in A and current not in Z

  Graph type:

  ADMG

  Applications:

  Finding minimal d-separators, Finding conditional instruments

  References:

  van der Zander, Liśkiewicz (2020)

• d-connected

  EDGES --> <--, <->
  SETS X, Z
  START <-- AT X
  OUTPUT ...
  
  -->, <-> | <--, <-> | current in Z
  ...      | ...      | current not in Z

  Graph type:

  ADMG

  Applications:

  Finding conditional instruments

  References:

  Shachter (1998)

• Descendants

  EDGES --> <--, <->
  SETS X
  START --> AT X
  OUTPUT ...
  
  ... | --> | true

  Graph type:

  ADMG

  Applications:

  Optimal conditional instruments, Finding conditional instruments

• Forbidden-path connected

  EDGES --> <--, ---
  SETS X, W
  COLORS pass, yield
  START --> [pass] AT X
  OUTPUT ... [yield]
  
  ... [pass]  | ---, --> [pass]  | next not in X and next not in W
  ... [pass]  | ---, --> [yield] | next not in X and next in W
  ... [yield] | ---, --> [yield] | next not in X

  Graph type:

  CPDAG

  Applications:

  Parent adjustment distance

  References:

  Henckel et al. (2024b)

• Non-causal connected

  EDGES --> <--, ---
  SETS X, W
  COLORS init, poss-causal, non-causal
  START ... [init] AT X
  OUTPUT ... [non-causal]
  
  ... [init]        | ---, --> [poss-causal] | next not in X
  ... [init]        | <--      [non-causal]  | next not in X
  --> [...]         | <--      [non-causal]  | next not in X and current in W
  --- [poss-causal] | ---      [poss-causal] | next not in X and current not in W
  ... [poss-causal] | -->      [poss-causal] | next not in X and current not in W
  --- [non-causal]  | ---      [non-causal]  | next not in X and current not in W
  <-- [non-causal]  | ...      [non-causal]  | next not in X and current not in W
  ... [non-causal]  | -->      [non-causal]  | next not in X and current not in W

  Graph type:

  CPDAG

  Applications:

  Parent adjustment distance

  References:

  Henckel et al. (2024b)

• Not-amenable

  EDGES --> <--, ---
  SETS X
  COLORS init, yield
  START ... [init] AT X
  OUTPUT ... [yield]
  
  ... [init]  | ---      [yield] | next not in X
  ... [yield] | ---, --> [yield] | next not in X

  Graph type:

  CPDAG

  Applications:

  Verifying adjustment sets, Parent adjustment distance

  References:

  Perković et al. (2018)

• Optimal IV

  EDGES  --> <--, <->
  SETS   S, D, F
  COLORS pass, yield
  START  <-- [pass] AT S
  OUTPUT ... [yield]
  
  ...         | -->              | false
  <-- [pass]  | <--      [pass]  | next in D     and next not in F
  <-- [pass]  | <--, <-> [yield] | next not in D and next not in F
  <-> [yield] | <--, <-> [yield] | next not in D and next not in F

  Graph type:

  ADMG

  Applications:

  Optimal conditional instruments

  References:

  Henckel et al. (2024a)

• Possible ancestors

  EDGES --> <--, ---
  SETS X, W
  START <-- AT X
  OUTPUT ...
  
  ... | <-- | next not in W
  ... | --- | next not in W

  Graph type:

  CPDAG

  Applications:

  Verifying adjustment sets

• Possible descendants

  EDGES --> <--, ---
  SETS X, W
  START --> AT X
  OUTPUT ...
  
  ... | --> | next not in W
  ... | --- | next not in W

  Graph type:

  CPDAG

  Applications:

  Verifying adjustment sets, Parent adjustment distance