# Decidability of $\{p|∃y : \operatorname{Dom}(φ p ) ⊇ \operatorname{Dom}(φ y )\}$

I need to classify the set $$\{p|∃y : \operatorname{Dom}(φ p ) ⊇ \operatorname{Dom}(φ y )\}$$ as decidable, semidecidable or not semidecidable.

I don't know how to start. Any ideas?

Dom (φp) it's all the inputs that are accepted by a turing machine "p", and the same for Dom (φy). So if My(some input) accepts then Mp(same input) accepts too, Mp can accept more inputs than My but not less.

But I don't know how work with that.

• What are your ideas on the topic? – Yuval Filmus Dec 4 '16 at 22:12
• I have added my ideas on the question. – Crider7 Dec 5 '16 at 12:21
• I am unfamiliar with your notation, but it seems that your set contains everything, since you cal always choose $y = p$. – Yuval Filmus Dec 5 '16 at 12:23