Trace Algebra Sample Clauses

Trace Algebra. ‌ T In this section, we describe the trace algebra that underpins our generalised theory of reactive processes, the hybrid relational calculus, and ultimately the Modelica semantics. We define traces as an abstract set equipped with two operators: trace concatenation ^ : T → T → T , and the empty trace s : T , which obey the following axioms. T 3.1 (Trace algebra). A trace algebra ( , ^, s) is a cancellative monoid satis- fying the following axioms: x ^ (y ^ z) = (x ^ y) ^ z (TA1) s ^ x = x ^ s = x (TA2) x ^ y = x ^ z ⇒ y = z (TA3) x ^ z = y ^ z ⇒ x = y (TA4) x ^ y = s ⇒ x = s (TA5) As expected, ^ is associative and has left and right units. Axioms TA3 and TA4 show that ^ is injective in both arguments. As an aside, TA3 holds only in models without infinitely long traces, as such a trace x would usually annihilate y in x ^ y. Axiom TA5 ⇒ states that there are no “negative traces”, and so if x and y concatenate to s then x is s. We can also prove the reciprocal law: x ^ y = s y = s. From this algebraic basis, we derive a prefix relation and subtraction operator.
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