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B\"uchi automaton of records (BAR) has been proposed as a basic operationalsemantics for Reo coordination language. It is an extension of B\"uchiautomaton by using a set of records as its alphabet or transition labels.Records are used to express the synchrony between the externally visibleactions of coordinated components modeled by BARs. The main compositionoperator on the set of BARs is called as join which is the semantics of itscounterpart in Reo. In this paper, we define the notion of labeled transitionsystems of records as a generalization of the notion of BAR, abstracting awayfrom acceptance or rejection of strings. Then, we consider four equivalencerelations (semantics) over the set of labeled transition systems of records andinvestigate their congruency with respect to the join composition operator. Infact, we prove that the finite-traces-based, infinite-traces-based, andnondeterministic finite automata (NFA)-based equivalence relations all arecongruence relations over the set of all labeled transition systems of recordswith respect to the join operation. However, the equivalence relation usingB\"uchi acceptance condition is not so. In addition, using these results, weintroduce the language-theoretic definitions of the join operation consideringboth finite and infinite strings notions. Also, we show that there is nolanguage-based and structure-independent definition of the join operation onB\"uchi automata of records.

Joining Transition Systems of Records: Some Congruency and Language-Theoretic Results