Let's say I'm writing a GA to find an optimal path to travel from point $A$ to point $B$. Genotypes are a list of directions (north, south, east, west) to follow. So a genotype NENWEE will move north once, east once, then north again, west once, and finally east twice.The directions are encoded as follows:N : 00E : 01W : 10S : 11Our first genotype, NENWEE (let's call it $P$), will thus be encoded as follows:$00\,01\,00\,10\,01\,01$Let $Q$ be a second genotype, say EENEWW, which is encoded as follows:$01\,01\,00\,01\,10\,10$Now let's do a one-point crossover operation on genotype $Q$, from $P$. The randomly-chosen crossover point is between the $9$-th and $10$-th bit, so bits $10$, $11$ and $12$ from genotype $P$ will replace those same bits from genotype $Q$. Let's call the resulting genotype $Q'$.P  :  00 01 00 10 01 01Q  :  01 01 00 01 10 10Q' :  01 01 00 01 11 01After decoding $Q'$ we find that the result is EENESE. However neither $P$ nor $Q$ contained direction south.My question is, do crossover operators imply a certain degree of mutation by definition?
Mutations as a crossover by product
genetic algorithms
I have contacted Inman Harvey from the University of Sussex and here is his answer to my question:In your example you have given each genotype $6 \,\,\,\, 2$-bit 'genes'. Let's look at how different your example parents $P$ and $Q$ are:In genes $2 (01)$ and $3 (00)$, the genes are identical in $P$ and $Q$  In genes $1, 4, 5$ and $6$ the $2$-bit genes differIe $2/3$ of the genes differ between parents.If you have $1$-point crossover at any of $12$ positions on the genotype (including here the non-functional case where the crossover-point is 'off-the-end', then:$6$ of the possible crossover points are between-genes and hence do not count as sort-of mutations.  $2$ of the possible crossover points are within the $2$-bit genes that are identical in $P$ and $Q$ – hence do not result in any 'crossover-style-mutation' in $Q'$$4$ of the possible crossover points are within non-identical genes, and hence DO result in a crossover-style-mutation in $Q'$.So -- with your specific example, $4$ out of $12$, ie $1/3$ of the time, crossover will produce a crossover-style-mutation. Sounds a lot. Sounds like maybe you should be worried about this in terms of impacting on effective mutation rates...BUT WRONG!! Your example was atypical. You will find in practice, after only a few generations of evolution, your parents $P$ and $Q$ will be genetically very similar to each other. This is the basic SAGA insight, see any of my papers with SAGA in the title. If you doubt this, then run any binary-style GA for as few as $10$ generations and check on how similar any $2$ randomly chosen parents are. It won't be (as in your example above) $67\%$ of genes differing between parents, it will be more like order of $1\%$ (if your genotypes are say $100$ bits long) differ between parents. Don't believe me, run the tests yourself, it is very easy to do!... and hence your concern about these crossover-style-mutations should, I suggest, diminish to near vanishing point when you take account of the reality that most parents have mostly identical genotypes. Anecdotally and simplistically, if human genotypes are $98\%$ similar to chimpanzee genotypes, then you should expect your average human-human parental couple to be $99.99\%$ genetically identical, and only $0.01\%$ open to the possibilities that worry you above (that $\sim 0.01\%$ is of course important and interesting for different reasons). Elephant mating with a flea, then you can expect problems -- in fact what you are discussing above relates closely to inter-species breeding problems. Your standard $GA$ does not run into such problems.I hadn't thought of it this way. So in fact crossover operators may introduce unwanted mutations in early generations (depending on how they are implemented) but these mutations will have little, if any, effect on the actual optimal mutation rate.