Can TI CC2540 chip process simple logic by its own?
It is possible and Texas Instruments has the CC2540 Mini Development Kit available that might be a good starting point because it includes a debugger / programmer. The CC2540 product page contains all the information you should need along with some software and example downloads if you scroll down the page so you can take a look at what's involved in advance.

I haven't used that particular chipset but one thing to note is that there's likely to be quite a bit of effort getting up to speed with it. If it's for something low-volume the quicker path might be using a microcontroller considering suitable parts can be found for a few dollars or less, but it really depends on how confortable you are with hardware versus software development.

Something that Chris Stratton has pointed out is that the toolchain (development software) supplied with that kit is only a 30-day evaluation so also check into the licensensing costs before proceeding. From that point of view an AVR, PIC or similar microcontroller is likely to be cheaper because they have free compiler options available and hardware programmers tend to be around the $50 mark.