I can't say it's impossible, but it sounds a little like Star Trek or something. Garbage has plenty of carbon in it, so you can easily syntheisize methane or other hydrocarbons. It would be a pretty slick trick if all you got was methane and a bunch of insoluble solids in a fully contained system. That doesn't make it cleaner than a natural gas generator (natural gas is almost 100% methane, and methane is the cleanest hydrocarbon fuel), unless you get a lot of heat off the synthesis process directly, or unless you generate hydrogen to burn, which seems unlikely.
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