Abstract: Motivated by the cohomology theory of loop spaces we consider a concept of a special class of higher order homotopy commutative differential graded algebras. For such algebras the so-called filtered Hirsch model is constructed. As an application we converse a theorem of Borel by proving that for a simply connected space $X$ with the polynomial cohomology algebra $H^{\ast}(X;\Bbbk)=S(U),$ the loop space cohomology $H^{\ast}(\Omega X;\Bbbk)= \Lambda(s^{_{-1}}U)$ is the exterior algebra if $\Bbbk=\Bbb{Z},$ or if and only if $\Bbbk=\mathbb{Z}_2$ and the Steenrod operation $Sq_1$ is multiplicatively decomposable on $H^{\ast}(X;\mathbb{Z}_2).$