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Pratt's action logic is an equational theory ACT conservatively extending the equational theory REG of regular expressions with operations preimplication a→b (had a then b) and postimplication b←a (b if-ever a).
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The algebra formed by regular languages is known as Kleene algebra, and furthermore, there is something called action algebra, which includes residuals. This might be something close to what I've been searching for. https://t.co/KSyEPx19SO
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Action logic 初耳だ : http://t.co/tW3X8Soaeu