We are building a software to verify the correctness and the safety of computer programs,
I would like to know the name of this software should be a "program verifier" or a "program verificator".
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I work in the software industry. Verificator is not a word I've ever come across. While I can understand why it might be a candidate on your list (because it's close to the noun "verification"), the word you're looking for is "verifier".
One side observation too - one would not normally say "a software", except in a context such as "a software program" or "a software suite". Just "software".