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Hence, we have the key definition. 1 (Well-structured transition system with strong compatibility) A well-structured transition system with strong compatibility is a transition system TS = (S, ⇒), equipped with a quasi-order ⇐ on S, such that the two following conditions hold: 1. ⇐ is a well-quasi-order; 2. ⇐ is strongly (upward) compatible with ⇒, that is, for all s1 ⇐ t1 and all transitions s1 ⇒ s2 , there exists a state t2 such that t1 ⇒ t2 and s2 ⇐ t2 holds. The next theorem is a special case of a result in  and will be used to obtain our decidability result.
Note that the previous result does not exclude the existence of weak encodings. , a variant of CCS, yet it is possible to provide a weak termination preserving encoding of Turing machines in CCS! (essentially by adding spurious nonterminating computations). We conjecture that such an encoding is not possible for CHRωa (C). Note also that previous results imply that range-restricted CHRωa (C) is strictly less expressive than CHRωa (C): in fact there exists a termination preserving encoding of Turing machines into CHRωa (C) [32, 116].
1 Notation As mentioned before, the computational power of CHR depends on several aspects, including the number of atoms allowed in the heads, the underlying signature Σ on which programs are defined, and the constraint theory CT , defining the built-ins. In particular, the language under consideration in this chapter is the CHR defined over a signature which contains no function symbol of arity >0 and interpreted using the ωa semantics. We will indicate this language as CHRωa (C). , Σ = ∧). Finally, CHRωa (F) indicates the CHR language which allows functor symbols and the = built-in.