I'm looking for the complexity of satisfiability of a formula $\forall y_1, \dots,y_n, \exists x_1,\dots,x_m, \phi$ or of a formula $ \exists x_1,\dots,x_m \forall y_1, \dots,y_n,\phi$ where $\phi$ is formula of the form : $$\phi:= \phi \wedge\phi ~| ~\neg \phi ~| ~ \phi\to \phi~| ~\psi$$$$\psi := t>t~| ~t=t$$$$ t:= t+t~| ~x_i~|~y_i ~| ~c$$Where $c$ are the constant in $\mathbb{N}$, and the domain of variables $x_i,y_i$ is also $\mathbb{N}$.In fact the $y_i$ are either $0$ or $1$. Does that simplify the complexity?All answer with references would be gladly accepted.Thanks 
Complexity one-alternation SMT
cc.complexity theory;reference request;lo.logic