At 17:20 your program would get called and would read in a small file (lets call it /tmp/last). In that file would be stored
1) a file positon from tell
2) two boolean values for the time slot 16:50 to 16:55 (note the difference to above)
3) two boolean values for the time slot 16:55 to 17:00
...
6) two boolean values for the time slot 17:10 to 17:15

Now your program would open the logfile, seek to the file position from 1), read the logfile from there and calculate the two boolean values (is there a success line? is there a non-zero value ?). This would be the result for time slot 17:15 to 17:20

Now the program can calculate the endresult for the half hour by ANDing the values from 2) to 6) and the newly calculated values.

Finally the program overwrites /tmp/last and stores the following into it:
1) the file position from tell
2) two boolean values for the time slot 16:55 to 17:00
2) two boolean values for the time slot 17:00 to 17:05
...
6) the newly calculated boolean values for the time slot 17:15 to 17:20

After 5 minutes your program gets called and does all of the above again. The temporary file is like a pipeline storing only the last 5 results, shifting out an old one and getting a new one.