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    6W&MzpJi98)Sf9x-LcxC;uVRT zl%Uk9mq*(;B`Uhg$0^DG0IaK@K>eLOKh##D{mRPx$4X9A<<4I`k4~+6yA$pXs@H3^ zrVYMq)raG->xoK%^d4_0`6m_X^-?&UAtZG!m-{w&Y`E)Mw7V-%P6CxB@<$$-U(SwB z@Rv*4kM}q)$M|HYqtl-tY~$1ME7j`JlTvEMiO3!*_`R%bej(p%v2U^svLutxk0B_0 zH;nc!1s~^1YVng+0LS;0!&bJW9V$1yUS<@{S2AwZ%E) z{{Zsp*%u<^zB<(k*i_h)3yH=u&0UDUg(+f*A-M)ELiQM+V+^6BQo zr!03z*YSFK$o>+nZ?#80zv!=5siheDFsbAAh0{#*Ja?ut`a#Do776VMI3@ePG zBDpGcq8oV$j(`C?W9eS6R~r!1_e3?al@=4bn)!9K#EQdxCQ^*0B=UQj^?H+|5mN<% zwRt-Xj!hJ{S|U0(Ev%4u^DB=306O)0oW|DevCdA#WGevgR|ELfui?B_P)h-%bhedX zq#k~~>(%N>XvO*Ve2CO<7wsr%B?x^Ew&FsL6q3@EMn0e#>0cEiH)+r}X)ys|@RXo9 zl^lRnkbC@%dc8&)L&rzv587z~zZD_2oTWbkz~G$zWqQ(u#@`-2wes70Nj>)**Q?a; zVgTU{$RMd$N!kYE+K8n)P)I5SfIl&>SEvBSG6RU*p}4l9QctC4CDp0M&{RrCm=8?X ztJHj8N9gW~x51OAd^?tHmiA@cZN*b$MugKYyi>r+kdS=`W6!Q?#e)Jh>g%Xjp28)w zZHkpdL&@A3B?`$OTJ?IZ;P?UT;KmYU>qnq1ZU{-*Lk+m3Byc^5t8Y$P;xNs*w&S4) zQ^g7i9nvyDuUD#wSF~9=O34XO423hf;^FfZsNW1?*_RIXaj{v-bY8s8q>9oG}R1wW0793S34~anl?(b#6)I-S-??NKvC!U&_=CM zWW%0^TT;-`8v$Cl&wpC=dX_soyqXp+7C2Frxu$j~j1m^E-^QCXWLD%d${Ynr8OBt2 zi+h(>xL_0tK*~oA8i~s-t literal 0 HcmV?d00001 diff --git a/algorithms.html b/algorithms.html index a42dee9..638d093 100644 --- a/algorithms.html +++ b/algorithms.html @@ -54,6 +54,8 @@ ## What's all the fuss about? +![Alan Turing](alan-turing.jpg) + --- layout: true @@ -71,6 +73,7 @@ Back when computers were human... * As much notepaper as you wanted * Input → output +.float-right[![right-aligned Growth rates ](turing-machine.jpg)] ## Turing machines Simplified human computer. @@ -89,6 +92,14 @@ Algorithm is the instructions Repetition == iteration or recursion +What questions can we ask about algorithms? + +* Is it correct? +* Does it terminate? +* How long does it take? + +Different hardware works at different rates, so ask about how the run time changes with input size. + --- # Growth rates @@ -143,6 +154,9 @@ word1 = "cat", word2 = "catty" # Anagrams version 1a: checking off (fixed) + + + + + +
    ``` Given word1, word2 @@ -158,6 +172,26 @@ for pointer in range(len(word2)): anagram? = False return anagram? ``` + +``` +Given word1, word2 + +if len(word1) == len(word2): + anagram? = True + for character in word1: + for pointer in range(len(word2)): + if character == word2[pointer]: + word2[pointer] = None + else: + anagram? = False + return anagram? +else: + return False +``` +
    --- @@ -300,7 +334,8 @@ Don't forget bogosort! * Often leads to logarithmic growth "Guess the number" game - * What if I don't tell you the upper limit? + +* What if I don't tell you the upper limit? ## Dynamic programming @@ -314,6 +349,13 @@ Build a soluion bottom-up from subparts * Greedy algorithms * Backtracking +# Parallelism vs concurrency + +* Parallelism is implementation + * spread the load +* Concurrency is domain + * new task arrives before you've finished this one + --- # Making change @@ -421,6 +463,14 @@ The computer would either answer "Yes," "No," or never stop working. This is the Halting problem. +```python +while i > 0: + print(i) + i -= 1 +``` + +Will this halt? When will it not? + --- # Universal Turing machine @@ -445,7 +495,141 @@ A *universal* Turing machine can take the description of any Turing machine and * Virtual machines * ... +--- + +# Back to halting + +* Universal Turing machine can do any possible computation +* Has the same halting behaviour of the emulated machine + +Can we build another machine that detects if a machine halts when given some input? + +Assume we can... + +```python +def does_it_stop(program, input): + if very_clever_decision: + return True + else: + return False +``` + +But I can use the program listing as an input: + +```python +def stops_on_self(program): + return does_it_stop(program, program) +``` + +--- + +# Halting: the clever bit + + + + + + +
    +```python +def does_it_stop(program, input): + if very_clever_decision: + return True + else: + return False +``` + +```python +def stops_on_self(program): + return does_it_stop(program, program) +``` +
    + +Let's put an infinite loop in a program that detects infinite loops! + +```python +def bobs_yer_uncle(program): + if stops_on_self(program): + while True: + pass + else: + return True +``` + +If a `program` halts on on itself, `bobs_yer_uncle` doesn't halt. If `program` doesn't halt, `bobs_yer_uncle` returns `True`. + +-- + +What does this do? +```python +bobs_yer_uncle(bobs_yer_uncle) +``` + +--- + + + + + + + +
    +```python +def does_it_stop(program, input): + if very_clever_decision: + return True + else: + return False +``` + +```python +def stops_on_self(program): + return does_it_stop(program, program) +``` + +```python +def bobs_yer_uncle(program): + if stops_on_self(program): + while True: + pass + else: + return True +``` +
    + +What does this do? +```python +bobs_yer_uncle(bobs_yer_uncle) +``` + + + + + + + + + +
    If it halts...If doesn't halt...
    +
      +
    • `stops_on_self(bobs_yer_uncle)` returns `False`
    • +
    • `does_it_stop(bobs_yer_uncle, bobs_yer_uncle)` returns `False`
    • +
    • `bobs_yer_uncle(bobs_yer_uncle)` must loop forever
    • +
    +Contradiction! +
    +
      +
    • `stops_on_self(bobs_yer_uncle)` returns `True`
    • +
    • `does_it_stop(bobs_yer_uncle, bobs_yer_uncle)` returns `True`
    • +
    • `bobs_yer_uncle(bobs_yer_uncle)` must halt
    • +
    +Contradiction! +
    + +No flaw in the logic. The only place there could be a mistake is our assumption that `does_it_stop` is possible. + +(M269 has a different proof based on different sizes of infinity, and showing that there are infinitely more problems than there are Turing machines to solve them.) diff --git a/turing-machine.jpg b/turing-machine.jpg new file mode 100644 index 0000000000000000000000000000000000000000..6b512a17a12b04c538e82bcafb280fb34a6164b2 GIT binary patch literal 8211 zcmeHscTiN_vhR?iXoDUcFHD>c0EmTdS*f*IK{b-Me>p*WRmd7H-x7G)+5kK}008gL z1l%kGo&v~-iD_tPfDCl>j0}ISKt>=vGa~~d4J{Wv8#6Z(2auVGQ<8_39mGP<$aRMs zEGQ%@dYgqyLPlIzhF?Tf7>|scoQR0%)*ljLrehZRzaBSz09sN&0Um?^j~9SXi$_3< zcQXKB`(q$J!M~gOuR%nNPe6!A@@G?<27r%8fKNa`LP~~DNb(1Phfe??q$Q#wrsrYg zm0*xmH#Kw0zHbrutYrp^BLRY>WHkI@vT|NXoBO{5JIB3jnm+j>hv9#a{~wC}eU^k+I{?D z<#*1M7duf^Wc2r)pR>n~t;d;M>~F)X1|!rRzAzS_5^Uui-*2=H8KzEmKwFZVSpED5 z7WdCxi`Zwt>vx_Xxm6V&Unu=kl4C>RHB!F*05<^Bq=imDcmR}VZ+~p7z1Fxr{N(&h z-Q%CPTblW*mJH;Og1=vG|D3(;6rMAXJruxRU7t9|j$ieQ)udUXT98mlXmJLq(SU^Bq$1=bGwEysrl5tOHI2ep-EyZV>8>JG`(j{yN5}Pk@4UvpJy%f5(yqts@7Jg^M9ab+LquSIjAt{PjcQM^JIEE zj};MeZGG|q|5=i#DT9{MJ3`qIDCA0D9?KyPe0X=7Dt+3^TV##dHH}rIWm1^AY1M~c ziryV3t{laD*QUJ)lCt#W2J|g6d-e}%I59qcq+0OS$AE2kw3)ggv@5(R(7yZe5BHt zN=6rBcWvw-MoJ;m$obPm{ckARud zUm_)S^Rx~Mudv$?%@iAU%-*qG%_rjo#)h& zDmj76V9K)Qn2*E5Dmtd~DvYwM2-3`C?gIJsbP4IA4c`Z9RHsxg-kmIW8rPwvIey9c zgPS}2gZB9%|TpwrjJ+IEaL*UeI+sYQsgf<2Zf@&=$WTkz?^ zx*56A;Bsn-lT*QP-<#gOFFCQ|<{wdu2fiTboBeRV9gSl8)}K3^O_aWx0*O`iR~%;P zN|rr`?-5k9a3-DfbqEpVyc6~FVh!}u93N|yGv&tHdgjPcm)8GcDU!-a?cruuckL( z^)*J4wJ3S;WebA9R%TW4vzg7y0bQ}u5W=}^5`V?qTGPM0!3ucct?gBOF9ICG1(~0c3P{;&oF$Q_320@wt#7OV)m(BL3C9~a1vO)0sFf$=O)RATap#7omocH zhTd`^#4HlUiRjWjG~4uSSlOz+ySI;MDvwpi=?K}~Hf!HW2ABFPT82*?b3Xg(p!aiU z7+3riB^AtVgW84L=mhJ)3p%w+JX7aWr+QLEy^}2U9nmdEc(~x7Sm}6W^kl7~Eln@9 zstqzDn;yo(HzuC1W~Pq7izQ}m0tW)9UX}06)IYJmJ)CBUYLneQ1LWKm?u4Pv)a*k) zt*Y}3o;F+CcS(T;2`3R136^Or&(1GnuzhpAWT>#fWnHlIslE?G`ndo247c0e?UIs| zj5+?EPS*IB7~1;)h_wlG$+ELIB(j4Eo)CuGDk{R6D^4qi*> z7CU{RHjjFN1fkm&V4KZPnG)>X3wk5jewSi+v$FqlXL9zPDOPfos)IN-1 z`6Wx*3im-Y!D0auk782xAMzme=fr?|OXP(c2y zKXA_vhj>1D$L4q%$ri@ArH5`ur7>gL_q2`8d75G?ht15-9Vj=uc^Z*X_s7EgMP=I) zn5UnCTYAzuzcQbjyfCNl5H+@^oHFDTYTLnZ`;REQd3@6m>E?~+`^JkTmuoO1u_PG4 z!CwSbuLvPW?L1)LK=;nHk9Y8oHXSslpG(=xa z8TzSgXWH27)8a#%vbBa+DuybfLAc<}rj6Sg`h^spmE0rB5~B-j<}3`dfD6j4pXOF{ zUJcTU@ioh1r*;rH_i%Ejf#t?? 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