From 7f0392be0b1f801366171b58f565ecffddea5920 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Sun, 9 Nov 2025 07:12:11 +0100 Subject: [PATCH] [TI] Compact: More notes --- .../ti-compact/parts/02_finite-automata.tex | 3 ++- semester3/ti-compact/ti-compact.pdf | Bin 331209 -> 331334 bytes 2 files changed, 2 insertions(+), 1 deletion(-) diff --git a/semester3/ti-compact/parts/02_finite-automata.tex b/semester3/ti-compact/parts/02_finite-automata.tex index a62b96f..24d5c45 100644 --- a/semester3/ti-compact/parts/02_finite-automata.tex +++ b/semester3/ti-compact/parts/02_finite-automata.tex @@ -119,7 +119,8 @@ That is a contradiction, which concludes our proof To show that a language needs \textit{at least} $n$ states, use Lemma 3.3 and $n$ words. We thus again do a proof by contradiction: \begin{enumerate} \item Assume that there exists FA with $|Q| < n$. We now choose $n$ words (as short as possible), as we would for non-regularity proofs using Lemma 3.3 (i.e. find some prefixes). - It is usually beneficial to choose prefixes with $|w|$ small (consider just one letter, $\lambda$, then two and more letter words) + It is usually beneficial to choose prefixes with $|w|$ small (consider just one letter, $\lambda$, then two and more letter words). + An ``easy'' way to find the prefixes is to construct a finite automaton and then picking a prefix from each class \item Construct a table for the suffixes using the $n$ chosen words such that one of the words at entry $x_{ij}$ is in the language and the other is not. ($n \times n$ matrix, see below in example) \item Conclude that we have reached a contradiction as every field $x_{ij}$ contains a suffix such that one of the two words is in the language and the other one is not. \end{enumerate} diff --git a/semester3/ti-compact/ti-compact.pdf b/semester3/ti-compact/ti-compact.pdf index c1912f15e37b40aabd02ff575efd81484ad1de06..07bea4bba5a8456c78672fc1d1e13be9db14048d 100644 GIT binary patch delta 29697 zcmV(@K-Ry>n-s>J6tK$*0X?(K3JwAr)rQJgRTbQ&1pEOsZ-V*bkq5v$&g3_ZGxDsH z^$aQjGLvBhD6>cnp9dXNUWX{A?$*h3I3DN4{N2;Oe&)tiT#w}67^&Q&vjr8579Yy) zgfV*@XcYr8Ff@~41So&q8(ojww(B0WTwdNTQM!s1w9((ln@H z1AR#Ipxf2mD%MIHNgJR2>pR0CwUm}R=i)Xk3fQ;&9FoJC;e08H?xQ67^UW8pZ$5n? zbX2e+6It~7E>Z={k}S###fm(QUhkvt#MQemCa@=rHsGqj;Hfj^;myX@vB{3 zwY!SCw$z3r%H6oximb?Z{7u=`1NHpnJig!j@cNsZuU_B$bi-jANyIP9a>jKL?T$C! z|Bys`==&y0m@IVk){H#D$c!bKgmM$Tx_Nn%*rO1nBzhCM(qDsZ*K4Mdr4pAk%}Ou@ z2j_z6B8&Pex`RbRfD#{vFynuM!&I=eNC9xhQeF@a|5ms4aU(^1J^~)JjN6UmSb?Tn zXc4hNCvDxQ-ahi)tRM)4Q_C=RcR6rhi)Tu 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