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Invariants are key to formal loop verification as they capture loopproperties that are valid before and after each loop iteration. Yet, generatinginvariants is a notorious task already for syntactically restricted classes ofloops. Rather than generating invariants for given loops, in this paper wesynthesise loops that exhibit a predefined behaviour given by an invariant.From the perspective of formal loop verification, the synthesised loops arethus correct by design and no longer need to be verified.  To overcome the hardness of reasoning with arbitrarily strong invariants, inthis paper we construct simple (non-nested) while loops with linear updatesthat exhibit polynomial equality invariants. Rather than solving arbitrarypolynomial equations, we consider loop properties defined by a single quadraticinvariant in any number of variables. We present a procedure that, given aquadratic equation, decides whether a loop with affine updates satisfying thisequation exists. Furthermore, if the answer is positive, the proceduresynthesises a loop and ensures its variables achieve infinitely many differentvalues.

Linear Loop Synthesis for Quadratic Invariants