What are the consequences of solving XOR 3-SAT in Logspace? XOR Formulas

Consider boolean formulas with connectives $\wedge$ (AND) and $\oplus$ (XOR).  Such a boolean formula is a valid instance for XOR SAT if it is a conjunction of $\oplus$-clauses.  An $\oplus$-clause contains literals that are connected with $\oplus$.

Example Formula:

$(v_1 \oplus \bar{v_2} \oplus v_3 \oplus v_4) \wedge (\bar{v_1} \oplus v_2) \wedge (\bar{v_2} \oplus v_3 \oplus \bar{v_4})$

Decision Problem


  Name: XOR 3-SAT
  
  Input: A boolean formula that is a conjunction of $\oplus$-clauses where each clause contains three literals.
  
  Question: Does there exist an assignment to the variables that satisfies the formulas?


More Background

XOR SAT can be reduced to solving a system of equations over $\mathbb{Z}_2$.  This is because every $\oplus$-clause can be thought of as an equation.  As a result, XOR SAT is solvable in polynomial time using gaussian elimination.

Further, XOR 2-SAT can be reduced to solving reachability in an undirected graph.  This is because every $\oplus$-clause with two literals defines an edge in an undirected graph.  As a result, XOR 2-SAT is solvable in logspace because reachability in an undirected graph is solvable in logspace (because SL = L).

Question


  (1) Is XOR 3-SAT solvable in logspace?
  
  (2) Are there any known consequences of solving XOR 3-SAT in logspace?
  
  (3) For example, would this imply that XOR $k$-SAT is solvable in
  logspace?
Take a look at https://www.sciencedirect.com/science/article/pii/S0022000008001141 "The complexity of satisfiability problems: Refining Schaefer's theorem" by Allender et al. which answer your questions:

(1) Open

(2) $\oplus L = L$

(3) Yes