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Containment, Equivalence and Coreness from CSP to QCSP and beyond

The constraint satisfaction problem (CSP) and its quantified extensions, whether without (QCSP) or with disjunction (QCSP_or), correspond naturally to the model checking problem for three increasingly stronger fragments of positive first-order logic. Their complexity is often studied when parameterised by a fixed model, the so-called template. It is a natural question to ask when two templates are equivalent, or more generally when one "contain" another, in the sense that a satisfied instance of the first will be necessarily satisfied in the second. One can also ask for a smallest possible equivalent template: this is known as the core for CSP. We recall and extend previous results on containment, equivalence and "coreness" for QCSP_or before initiating a preliminary study of cores for QCSP which we characterise for certain structures and which turns out to be more elusive.