Keep them cut so there are no green leaves to capture sunlight to convert into foods that feed the roots and eventually they will die. One year I picked up some bages of leaves that had lots of seeds from a Tree of Heaven and I still get seedlings of that germinating.

There is an answer to every type of weeds. Scuffle hoeing every day, or at least every other day. If the weeds are starting, try a Rogue hoe.

Google scuffle hoe and you will find images.

http://www.roguehoe.com/scufflehoes/scufflehoes.html

They cut the weeds off below the surface before they have a chance to grow. They also keep the surface loose which creates a "soil mulch" to retain moisture below.

It takes me less than 10 minutes to do 1,000 sq. feet of garden. It's pretty relaxing, too. If you have family members, buy two as it's something people like to do when they see someone else doing it.

Here is a video of me using the scuffle hoe.
http://www.youtube.com/user/bnaiorpueblo#p/c/64D03F492F3DBF8E

I have lot of old tops and I read someone used carpeting and rags for mulch. I was thinking of laying these old tops down and then putting mulch on top,would this work or would it take years or never for tops to disintergrate unless they were cotton.