I think ASP's a bit of a merger of the two cultures. The implementations are heavily SAT-influenced, with all existing ones based on eliminating variables to ground to a finite program that's then solved. But its semantics (stable-model semantics) originated in the early-90s flurry of attempts to come up with a solid logical semantics for LP (alongside other attempts like the well-founded semantics), to replace Prolog's operationally defined "whatever SLDNF resolution returns" semantics, and especially its displeasing-to-logicians property of the semantics depending on statement order. And the most widely used surface syntax, sometimes called "AnsProlog", is an adaptation of Prolog syntax.

This research group in particular is now pushing it heavily in the programmability/expressivity direction, which is pretty interesting: http://potassco.sourceforge.net/ . There is even some other research floating around on more software-engineeringish things like ASP debugging techniques.