Reachibility between first and last layer in grid graph in logspace I am trying to prove that there exists logspace deterministic Turing machine that check if exists path between first row and last row in grid graph. Grid graph is matrix of $0s$ and $1s$, the undirected edge exists between two $0s$  or $1s$ - one the left, right, up, down.   Path can contains only $0s$ or only $1s$.   

For example in following graph there exists such path:

0 0 1 0
0 0 1 1
0 0 0 1
0 0 0 1


For following there is no such path  

0 0 0 0
1 1 1 1
0 0 0 0
1 1 1 1 


How to prove it ?
I can't post a comment, which would be more appropriate:

Reingold's result gives a logspace algorithm, but reachability on undirected grid graphs was show to be in $L$ by Blum and Kozen about 30 years prior:

M. Blum and D. Kozen. On the power of the compass (or,
why mazes are easier to search than graphs). In IEEE Symposium
on Foundations of Computer Science (FOCS), pages
132–142, 1978.