I am trying to formalize the rules for application of different schedules in a reinforcement learning in special education. Children learn through trials. Each trial is successful if the child exhibits the correct behavior. Correct behavior is reinforced according to different schedules of reinforcement:
Continuous, (reinforce every correct answer),
Fixed Ratio (every n-th correct answer),
Variable Ratio (every, on average, n-th correct answer),
Fixed Interval (reinforce every n-th minute), and
Variable Interval (every n-th minute on average).
Here is what I have so far, and I would like to ask here if you can take a look and tell me if I've done it right. There is a separate challenge how to select what to average is, but for now I would like to ask you to take a look at my formulae and tell me if there is something to be fixed and how.
Continuous Reinforcement $CR$:
Formula: Let $s$ be a student under instruction, $R$ be a student's behavioral response, and $r$ be a suitable reinforcer.
We define Continuous Reinforcement schedule as:
$CR := \forall R$ Correct$(R) \to$ Reinforce$(s,r)$
Fixed Ratio $FR_n$
Formula: Let $s$ be a student under instruction, $n$ be a ratio of reinforcement (every n-th), $i_{lr}$ index of the trial when the behavior was last reinforced, $i_c$ index of the current trial, $R$ be a student's behavioral response, and $r$ be a suitable reinforcer.
We define Fixed Ratio schedule as:
$FR_n := ((\forall R,$ Correct$(R) \land ((i_c - i_{lr})=n)) \to$ Reinforce$(s,r))$
Fixed Interval $FI_t$
Let $s$ be a student under instruction, $R$ be a student's behavioral response, $t_l$ be the time since the last reinforcement, $t_c$ be the current time, $t$ be the interval of reinforcement, and $r$ be a suitable reinforcer.
We define Fixed Interval schedule as:
$FI_t := ((\forall R,$ Correct$(R) \land (t_c-t_l)\geq t) \to$(Reinforce$(s,r)) \land t_l = t_c $)