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Size matters! Minimizing DFAs

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Minimising states = Minimising cost Minimising number of states = Minimising number of memory cells (hardware size) = Minimising cost (Computing time is not minimised by this process.) D CLK Q Q’ reset input pulse (“1”s) Use this first before inputting anything Bulb remains glowing if no. of 1s inputted so far is odd A simple hardware to realize the DFA shown 12 evenodd 1 1 DFA that checks if the number of 1’s is even or odd

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To glow or not to glow--- that is the question What do we mean by “minimising a DFA”? 0123 a abb Fed with a given string, a DFA could exhibit only TWO kinds of behaviour: {“bulb glows”, “bulb doesn’t glow”}. Details like which particular state the machine finally settles in do not count. “bulb glows”: The machine finds itself in one of the final states after consuming the entire string. “bulb doesn’t glow”: The machine finds itself in one of the non-final states after consuming the entire string. We eliminate (strike out) a few states from the DFA and yet are able to retain its (original) “behaviour”—the same set of strings are accepted/rejected as before.

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Minimizing states: The idea It’s simple: locate those states that are indistinguishable from (equivalent to) one another; they can all be replaced by one common “representative” (state).

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s1 Indistinguishable states s2 w M s2 : Start machine M in state s 2 M s1 : Start machine M in state s 1 Consider a machine M= ( Q, ∑, δ, DFA_start, F ) with states s 1 and s 2. s 1 and s 2 are indistinguishable if both M s1 and M s2 exhibit the same kind of behaviour for each input string w ε Σ*: given an input w, either both machines accept w (glow) or both of them don’t (no glow). I.e., either (i) delta_bar ( s 1, w ) є F, delta_bar ( s 2, w ) є F or (ii) delta_bar ( s 1, w ) є Q \ F, delta_bar ( s 2, w ) є Q \ F, for every w є Σ*. (s 1, s 2 are said to be k -indistinguishable, if the above holds for any w of length k.)

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Tracking down indistinguishable states We try to group together indistinguishable states, proceeding step-by-step: We first group together states that are “ indistinguishable ” when only very short strings (say, of length k ) are considered as input; then we gradually refine such a grouping to form newer groups that remain “ indistinguishable ” even when we allow slightly longer strings (say, of length k +1) as input. We continue till we arrive at groups of (literally) indistinguishable states. Non-final statesFinal states “indistinguishable” when only strings of length 0 (i.e., ε) are considered

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Tracking down indistinguishable states s1s1 s2s2 δ (s 1, a ) δ (s 2, a ) final / non-final pool of states strings of length k a a strings of length k+ 1 k- indistinguishable ↔ (k +1)-indistinguishable Lemma: s 1 and s 2 are ( k +1)-indistinguishable iff for each a ε Σ, δ ( s 1, a ) and δ ( s 2, a ) are k -indistinguishable.

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Tracking down indistinguishable states Stage I Indistinguishable by ε Stage II Indistinguishable not just by ε, but also by strings of length 1 a b a b “pool” of non- final states “pool” of final states States p and q will remain members of a group (i.e., p and q are upto-( k +1)-indistinguishabe*) iff: (i)p and q were members of the same group in the previous step (i.e., p and q are “upto- k -indistinguishabe”), AND (ii)δ ( p, c ) and δ ( p, c ), for each c ε ∑, both belong to the same group (of the previous step) (i.e., they are “upto- k -indistinguishable”) (see lemma on previous slide) p q * s 1 and s 2 are said to be upto- k -indistinguishable if they are x -indistinguishable for x = 0 to k.

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Tracking down indistinguishable states “pool” of non- final states “pool” of final states Stage I Stage II Indistinguishable by ε Indistinguishable not just by ε, but also by strings of length 1 p q a b a b We stop this “refinement” process when we can’t refine any further, i.e. the members of each and every group remain (stubbornly) unchanged. The members (states) in a group at this stage are literally indistinguishable (by any string). (We can now choose a common representative for states in the same group.)

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PROOF (as to why, after reaching the “final stage”, the states in each group are indistinguishable by ANY string): Tracking down indistinguishable states In the final stage, clearly, the states in each group are upto-k-indistinguishable (for some k); also, they (the states in each group) automatically qualify as upto-(k+1)-indistinguishable states, which in turn makes them upto-(k+1)+1-indistinguishable and so on (see lemma in previous slide). This forms an inductive proof for the fact that they are indistinguishable by strings of any length k.

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An example 1 a a b 3 a 2 b L = {a x | x є {a, b}*} 4 a b b

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{1, 2} non-finalfinal {3, 4}Stage I δ(1, a) = 3 (group 2) δ(2, a) = 2 (group 1) δ(3, a) = 3 (group 2) δ(4, a) = 3 (group 2) δ(3, b) = 4 (group 2) δ(4, b) = 4 (group 2) {1}{3, 4}Stage II δ(3, a) = 3 (group 3) δ(4, a) = 3 (group 3) δ(3, b) = 4 (group 3) δ(4, b) = 4 (group 3) {2} {1}{3, 4}Stage III δ(3, a) = 3 (group 3) δ(4, a) = 3 (group 3) δ(3, b) = 4 (group 3) δ(4, b) = 4 (group 3) {2} No further splitting is possible different groups choose a common representative 3 Eliminate state 4 from the original DFA; any transition going to state 4 should now go to state 3 in the original DFA. An example same group

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1 a a b 3 a 2 b b An example L = {a x | x є {a, b}*} Minimized DFA:

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