12.3 THE Posr Con,n,pspoNt)ENcE Pnoer.nlr
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Exarnine Figurc 12.10 r:a,refully to see wlnt is htr,ppening. We want to construct an MPC-solution, so we must sta.rt with tu1, that is, f.S +. This string contains 5, so to rnatch it we have to use u1s or u11. Irr this instance, we u$e tr10;this brings irr'rr.'16, lending us to the secondstring in the partial derivatiorr. Looking at several rnore $teprJlwe see that the string u.,rur;ur1.., is always longer than the corresponding string a1uiaj...t and that the first is exactly one step ahead in the derivation. The only exception is the last step, where uo rnlmt he applied to let the u-string catch up. The complete MPC-solutiorr is shown in Figure 12.11. Tho c:onstruction, together witlr the example, indicate tho lines along which the rrext rcsult is established. I
Let G : (V,T,S,P) be any unrestricted gra,mmar, with u any string irr T+ . Let (A, B) be the correspondencepair c:t)rr$tructeclfiom G and trr be the process exhibited in Figure 12.8. Therr thc pir.ir ('{, B) permits an MPC-solution if arrd only if w e L (G). Proof: The proof involves a forrrrir,linductive argument based orr the outlined reirsonine. We will omit the details. I
Witlt this result, we ca,rtreduce the rrretrbership problem for recursively enurnerabkl lrr,ngua,gesto the rnodified Post correspondence problem and thereby dernonstrrrtethe undecidability of the lattcr.