# Is it possible to make a grammar LL($1$) which recognizes palindroms?

Is it possible to make an algebraic grammar LL($$1$$) which recognizes palindroms for an alphabet $$\{a,b\}$$?

The language of even-length palindromes $$L_{epal}=\{ww^R \mid w\in\Sigmạ^*\}$$ is a classic example of a non-deterministic context-free language, and you can find the outline of a proof in Hopcroft & Ullman (and other standard texts). The restriction to even-length palindromes simplifies the proof, but it can be extended to the language of all palindromes $$L_{pal}=\{w \mid w=w^R, w\in\Sigmạ^* \}$$. Without doing more work, though, it's evident that $$L_{epal}=L_{pal}\cap (\Sigma\Sigma)^*$$, and deterministic CFLs are closed against intersection with regular languages.