The player will start jump (auto jump on) when walking to the bottom slab then walk to top slab with carpet above it, and when the auto jump option turned off you can not step up to the block. On PC Edition, or other platforms you can walk to it normally like step up from slab to slab.
Edit by @unknown
As of 1.0.0.0 Auto jump is still triggered when walking up onto a full block + carpet from a half slab (bug)
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Confirmed on 0.13.1 / iOS. The same thing happens when I replace the top-half slab with a solid opaque block and put a carpet on it. I think this is because carpet has a hitbox of 1/16 of a block, not zero. How does the PC version behave in this situation?

Oh I found the following description in the wiki:
A player can automatically step up from a lower to a higher height, if the difference is at most 0.6 of a block.
So this is actually a bug in MCPE because 1/2 + 1/16 ≦ 0.6

How did you know carpet are 1/16?
Carpet is 1/2 from snow layer so you are right it was 1/16. And yes, on PC Edition it just works normally without jumping.

How did you know carpet are 1/16?
I just read this page: http://minecraft.gamepedia.com/Carpet

Does this bug still affect 0.14.3? If so, please respond.

Well, this issue are mislabeled as "Fixed" instead of "Awaiting Response".
Agreed, this does affect 0.14.3
Redstone Comparators and Redstone Repeaters also have the same bug, not allowing the player to walk up onto the next slab.
[media]
Reopened.

Please respond and add the current version if you can reproduce this issue.

Confirmed for 0.15.4 (Tested on Windows 10.) Auto-jump should not activate when going from a slab to a slab with a carpet on top, and you should be able to walk from a slab to a slab and a carpet without jumping, as in the Java edition.

Reported for test and fix.

This is a bulk resolve of issues that haven't been updated to 0.16.0+ yet. Please comment below (To open the ticket) and update your affected versions. If you cannot reproduce the bug, please write in the comments so we can resolve it as "Cannot Reproduce". Thank You!

I believe this issue could be easily fixed by increasing either the player foot size, or just the default foot size for entities in general. Currently, the value is set to 0.5 when not specified in the entity's behavior JSON file, but if it was increased a bit, players and other mobs would be able to walk up heights that are slightly taller than 0.5 blocks tall.
For reference, the format of the foot size entity component looks like this:
“minecraft:foot_size”: {
value: 0.5
}

Hmmm, it would appear that, after testing, it looks like you can't change the foot_size of the player entity with a behavior pack. I made a test behavior pack that changed some player properties, and one of them, "minecraft:scale" was applied properly, but "minecraft:foot_size" was not applied. So due to the inability to change this property for players through behavior packs, I cannot tell if this would work or not. The inability to change certain player properties may actually be a bug itself... I'm not sure.

This doesn't affect carpets anymore but does affect repeaters and comparators
This has been partially fixed in 1.0.0.0:
Walking from a bottom half slab onto a full block height + carpet no longer requires jumping. (This has been fixed)
However, if you have autojump 'on', it will activate when walking up onto a block + carpet which is no longer needed. This is a bug and requires fixing.
N.B. Blocks with repeaters on will still require a 'jump' to get up onto, and this is confirmed as Working as Intended.

The remaining issue with this bug is working as intended and is on parity with Java.