What is this ground cover?

Our crushed-rock parking area has killed the various ground covers we tried to plant there, too dry or too much traffic or something else. But now a volunteer ground cover is spreading. It's low to the ground, seems not to want any water in our long, dry summer, puts up with foot traffic and car traffic, and is comfortable under bare feet.
So of course it's probably an invasive weed that I'm supposed to be eradicating furiously, right?
http://www.phred.org/~josh/PLANT_1.JPG is one whole plant.
http://www.phred.org/~josh/PLANT_2.JPG is a closer view.
For scale, it's on crushed rock, mostly 3/8" to 1/2".
Can you tell me what this is? Is it officially noxious, dangerous, undesirable? If I let it spread across the gravel, what will it do when it hits the lawn?
Thanks for any information!
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snipped-for-privacy@phred.org is Joshua Putnam
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Looks like one of the knotgrasses (Polygonum sensu strictu). I won't take a guess at the species; quite possibly you've got a different set of possibilities compare to my location.
--
Stewart Robert Hinsley

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says...

Oh, yes, location -- central Puget Sound, between Seattle and Tacoma.
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snipped-for-privacy@phred.org is Joshua Putnam
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Joshua Putnam wrote:

My guess is Purslane (portulaca oleracea)
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Bill
In Hamptonburgh, NY
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If that's it, it's tasty when the weather cools down. Put it in salads.
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