# If an heuristic is not admissible, can it be consistent?

I am solving a problem in which, according to the given values, the heuristic is not admissible. According to my calculation from other similar problems, it should be consistent, as well as keeping in mind the values, but the solution says it's not consistent either. Can someone tell why?

• A consistent heuristic is always admissible, but an admissible heuristic does not have to be consistent. Someone may be able to give some examples for you in an answer. However, if you want to know why your heuristic is neither, you need to share details of the problem and your attempted solution Nov 9, 2019 at 13:00

For a heuristic to be admissible, it must never overestimate the distance from a state to the nearest goal state.

For a heuristic to be consistent, the heuristic's value must be less than or equal to the cost of moving from that state to the state nearest the goal that can be reached from it, plus the heurstic's estimate for that state.

What this means is that, as you move along the sequence of nodes from start to goal that the heuristic recommends, a consistent heuristic should monotonically decrease in value. A consistent heuristic is thus also always admissible.

Notice that this means that if a heuristic is not admissible (like yours), it is also not consistent (by the contrapositive).

Therefore, if you already know your heuristic is not admissible, you should not be surprised that it is not consistent.

It seems most likely that you may have confused the definition of consistent for monotone. A consistent heuristic is both monotone and admissible.

As Neil Says, if you want to know why your specific heuristic is inadmissible, you should post another question about it, or modify this one.

If a heuristic is not admissible, can it be consistent?

No. Consistency implies admissibility. In other words, if a heuristic is consistent, it is also admissible. However, admissibility does not imply consistency. In other words, an admissible heuristic is not necessarily consistent.

## Definitions

Given a graph $$G=(V, E)$$ representing the search space, where $$V$$ and $$E$$ are respectively the set of vertices and edges, and the function $$w: E \times E \rightarrow \mathbb{R}$$ that defines the weight (or cost) of each edge of $$G$$, an admissible heuristic $$h_{\text{a}}$$ is defined as

$$h_{\text{a}}(n) \leq h^*(n), \forall n \in V$$

where $$h^*(n)$$ is the optimal cost to reach a goal from $$n$$ (so $$h^*(n)$$ is the optimal heuristic).

On the other hand, a consistent heuristic $$h_{\text{c}}$$ is defined as

\begin{align} h_{\text{c}}(n) &\leq w(n, s) + h_{\text{c}}(s), \forall n \in V \setminus \mathcal{G}, \text{ and} \\ h_{\text{c}}(g) &= 0, \forall g \in \mathcal{G}, \end{align} where $$s$$ is a successor of $$n$$, $$g$$ is any goal node and $$\mathcal{G}$$ is the set of goal nodes of the graph $$G$$.

## Theorem

A consistent heuristic is an admissible heuristic.

### Proof

Let $$h$$ be a consistent heuristic. Given that $$h$$ is consistent, then $$h(g) = 0$$, for any goal node $$g$$, so it does not overestimate the cost of reaching the goal at any of the goal nodes (given that, if you already are at a goal node, the cost is $$0$$, and $$h(g) = 0$$ is not greater than $$0$$). Let $$g_{n}$$ be an arbitrary neighbour of an arbitrary goal node $$g$$. Given that $$h$$ is consistent, then $$h(g_{n}) \leq w(g_{n}, g) + h(g)$$. Given that $$h(g)$$ does not overestimate the cost to reach the goal from $$g$$, then $$w(g_{n}, g) + h(g)$$ also does not overestimate the cost of reaching the goal from $$g_n$$, given that $$w(g_{n}, g)$$ is the true cost of the edge $$(g_{n}, g) \in E$$ and the cost to reach the goal from $$g_n$$ must be at least $$w(g_{n}, g)$$. This reasoning can be applied inductively (or recursively) on $$g_n$$ (then on the neighbouring nodes of $$g_n$$, and so on), so $$h$$ must be admissible.

• A proof by induction usually requires you to prove 2 cases: 1) a base case (prove that a property holds for a base case of the statement) and 2) induction step (assume that a property holds for $n$ and prove it also holds for $n+1$ or $n-1$). This proof by induction does not have a proper induction step, but I hope it is intuitive enough. This is more like a recursive proof.
– nbro
Nov 10, 2019 at 16:10
• $h$ being consistent doesn't require that $h(g) = 0$. If $h(g) = 10$ you could still have a consistent heuristic everywhere that was not admissible. But, typically we assume that $h(g) = 0$. Feb 24, 2020 at 23:26
• @NathanS. I defined the consistent heuristic in a certain way (but I am not saying it can't be defined differently). The rest should follow unless there's a mistake in my reasoning (which I don't exclude).
– nbro
Feb 24, 2020 at 23:34
• Yes; I'm just being a bit pedantic. I would just change your proof to say that $h(g) = 0$ is by your definition of consistent. The literature on this isn't always consistent. : ) Feb 24, 2020 at 23:50
• @NathanS. Just to clarify. If you don't define $h(g) = 0$, then consistency does not necessarily imply admissibility, i.e. the set of consistent heuristics would not be a subset of the set of admissible heuristics, unless $h(g) \leq h^*(g)$. Do you agree? By the way, could you please provide an example of a paper or book where a consistent heuristic is not defined such that $h(g) = 0$, and so what are the implications in terms of the relationship between consistency and admissibility? It's been a while since I had to deal with these topics, but my knowledge is a bit rusty now.
– nbro
May 7, 2021 at 12:01