Web Search API that permits automated queries

Are there any web search APIs or search engines that allow automated queries?

Engines that offer paid service terms:

• Gigablast

Engines that offer limited free service:

• Google (100 per day)
• Bingᴿ (5,000 per month)

Engines that offer free service:

• FAROOᴿ
• YaCy (P2P)

ᴿ Registration required

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Deprecated, obsolete, or no longer exist:

• DuckDuckGo
• Yahoo
• AltaVista
• DogPile
• Ask.com
• Ixquick
• ScrubTheWeb (uses CAPTCHA)
• Blekko
• Yahoo! BOSS