# consistent heuristic - does it always exist?

I am studying for an artificial inteligence class, and i have to classify the following sentence

"For any search space, there is always an admissable and consistent A* heuristic".

Well, i know that there is always an admissable heuristic, for example zero, since its an underestimation of the real cost (althoug this would lead to uniform cost instead of a*).

What I would like to know is if there is always a consistent heuristic, and why.

Thanks for your help

## 1 Answer

The zero heuristic is trivially admissible, as you've noted. It's also trivially consistent as long as all step-costs are non-negative: if $h(x)=0$ for all $x$ and $c(x,y)\geq 0$ for all $x$ and $y$, then $$h(x)\leq c(x,y) + h(y)$$ for all $x$, $y$.

• you are right, i'm actualy ashamed i did not notice that haha. please just correct the sentence to consistent, and i will consider it as correct "It's also trivially admissible as long as all step-costs are non-negative" – Rafael Marques Nov 15 '17 at 19:35
• @RafaelMarques Yeah, I guess "it's also trivially admissible" was a bit of a trivial statement! :-D – David Richerby Nov 15 '17 at 19:37