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We prove that the solvability of systems of linear equations and related linear algebraic properties are definable in a fragment of fixed-point logic with counting that only allows polylogarithmically many iterations of the fixed-point operators. This enables us to separate the descriptive complexity of solving linear equations from full fixed-point logic with counting by logical means. As an application...
A cyclic proof system, called CLKID-omega, gives us another way of representing inductive definitions and efficient proof search. The 2011 paper by Brotherston and Simpson showed that the provability of CLKID-omega includes the provability of the classical system of Martin-Lof's inductive definitions, called LKID, and conjectured the equivalence. By this year the equivalence has been left an open...
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