A few deductions I make in Kris Pixton's LOOPical game, that the program does not make (and I don't expect it to).

(all Case names were made up on the spurs of the moments, and the Cases are in no particular order)

Case 1 - Hugging Paths

Can you see how to force a closed loop?
So do the opposite, connect the two middle arms.
There are also more complicated examples, such as this one.
Can you see what would happen if you connect the yellow path to the pink one?
As a matter of fact, there is always another "slightly more complicated" example,
but I'll try to stay with simpler examples here and let you discover harder ones on your own.

Case 2 - Blues Carrying Red

If the path through the Red Clue goes Left-Right it will cause a loop.
So it (the path through the Red Clue) has to go Up-Down.

Case 3 - "-_" (aka Minus Underscore)

I'm sure you are acquainted with this situation.
And the program can tell you to do this.
But for the same reason (to prevent a premature loop), you should also do this.

Case 4 - Zig-Zag

And by the same logic, the situation in the first image here leads to the second one,
and so on, to the third image.
If you aren't convinced, try to put a Wall instead of one of those paths, and see the closed loop that ensues.

Case 5 - 4-In-Hall

Some people don't like this kind of deduction, but ...
- with four paths coming into a narrow "hallway" (in pairs)
- with some empty squares in-between
- with no additional Clues in-between
- with no other "exits" or paths in-between
the paths go straight down the length of the hallway.

Why? Because otherwise the solution would not be unique. (Do you see why?)
If you do not do this now, you will eventually find that the top two paths
(or the bottom two) get joined outside of the hallway (and become the one color),
and you will draw these paths (or Walls) to a prevent premature loop.

Sometimes this hallway goes around a corner (with or without the auto-corner path).
The logic is the same.

Case 6 - Need A 4th

A little harder to explain, but in short:
if you put a Wall (where I put the yellow path) you will have a dead end (the purple path probably).

And now the details:
- a two-by-two block, with three "ends" and one empty space
- the empty space has one Wall and one opening (away from the other "ends")
- the "end" which is diagonal from the empty space (the purple path)
  has no other exit available (here the white block blocks a would-be exit)
then you must put in a path segment from the empty space "out" through the opening.
Here is the simpler case for this type of deducction.  

Case 7 - Bottleneck

This is simply "don't divide the puzzle into two areas."

In the upper circled area, put a Wall between the two pink paths, (ok, "or the blue ones")
in the lower one, put a Wall between the two blue ones.

You will also be able to complete this "bottle" by doing so.

Here's another example ...

Case 8 - "=" (aka Equals)

This is somewhat of a special case (or a number of special cases), but it happens often enough to be worth mentioning.
Basically it is when you have two short path segments, like an Equals sign,
and you can force one to have nowhere to go except to the other one.

Usually this happens when, in the third row, there is one path end, and either a Wall, a Red Clue or a nearby Green Clue.

Here are two examples:

In this seemingly idyllic situation, at first glance there are no deductions to be made.

Or are there ?
At the marked spot we must either make a Wall or a path.
Let's try a Wall first.
That Wall forced the blue path, and forces a Wall in the other marked spot.
That will force both ends of the green path to connect to the purple path,
thus forming a closed loop.
So, instead, at the first location, we must make a path.
Like this.

In this case, if we put in a Wall at the marked location
it will force a path to the blue segment,
and the yellow will have nowhere to go, but up.
(Usually a good thing, perhaps, but not in this case).

Case 9 - Extenders

I'm sure you are acquainted with this situation.
A hint will tell you "... you can't make another straight on the far side of the Green Clue."
So you know which way the Red Clue goes.
Here you won't get a hint, but you can use the same logic to come to the same conclusion.

Case 10 - Parity

Because we have a closed loop (or even if we had a number of closed loops), any group of squares (an "area") will have an even number of paths going in (and/or out).

(Why is that true? There's a whole course in Number Theory hiding in that question.)

Anyway ... here is an example of how we can use that bit of info.
The "area" with the yellow border has three (green) paths going into it, and one spot where we want to know if there is a path or a wall (by the purple path).

So the question is, do you feel lucky? Er, I mean, is three even?

And in conclusion, the purple path goes left.

(Case 6 is a special case of this.)

You don't HAVE TO find or use any of these deductions, the program doesn't and IT manages to solve the puzzle,
but each extra little bit helps, and any one of these may be just the boost you need to many other deductions.