can every LR(k) grammer be transformed to LR(1)
Remember that the point of parsing (in almost all practical applications) is not simply to recognize whether or not a sentence is in a language, but rather to find the parse tree corresponding to the sentence in order to do further processing (such as compilation). Or, to put it another way, few people would purchase a "compiler" which took some input and whose output was limited to one of two sentences:


That input is a syntactically valid C program
That input is not a syntactically valid C program.


By a large, the desired product is an executable, not just an assurance that the program could be compiled if you had a compiler.

So, it is true that if there is an LR(k) grammar for a language, then there is also an LR(1) grammar. But it is not the same grammar (unless k is 1) and it will therefore not produce the same parse tree.

It is possible to mechanically transform the LR(k) grammar into an LR(1) grammar in a way that the original parse tree can be recovered, but it is not a procedure you want to try by hand. (Trust me on this.)

So if you have an LR(k) grammar and you want to build a parser, you could:


Find an LR(k) parser generator
Find a program which mechanically tranforms an LR(k) grammar into the corresponding LR(1) grammar and the code necessary to transform a parse tree produced by the LR(1) grammar into the corresponding parse tree which would have been produced by an LR(k) parse. Then feed the LR(1) grammar into an LR(1) parser generator and add the parse tree recovery code to the semantic actions.


These two tasks are not particularly different. The mechanical transformation of an LR(k) grammar into an LR(1) grammar blows up the size of the grammar much in the way that the LR(k) parse tables are much larger than the LR(1) parse tables. From a practical viewpoint, it is roughly equally difficult to find an LR(k) parser generator as it is to find an automatic LR(k)&rarr;LR(1) transformer.