IST 4 Informtion nd Logic
T = tody x= hw#x out x= hw#x due mon tue wed thr 28 M1 oh 1 4 oh M1 11 oh oh 1 2 M2 18 oh oh 2 fri oh oh = office hours oh 25 oh M2 2 3 oh midterms oh Mx= MQx out 9 oh 3 T 4 oh Mx= MQx due 16 oh oh 4 5 23 30 oh 5 oh oh oh
Lst time DNF is wy to express syntx ox using formul lger meets syntx oxes Ide: construct DNF y dding the norml terms tht correspond to the norml ssignments (f=1) m(,) 00 01 10 11 1 1 1 0 00 01 10
Lst time DNF is wy to express syntx ox using formul lger meets syntx oxes Ide: construct DNF y dding the norml terms tht correspond to the norml ssignments (f=1) 00 01 10 11 m(,) 1 1 1 0
Lst time DNF is wy to express syntx ox using formul 00 01 10 11 XOR(,) 0 1 1 0 00 01 10 11 XOR(,) 1 0 0 1
Lst time Boolen Alger lnguge for inry mgic oxes m 0 0 0 1 1 0 1 1 1 1 1 0 Cn relize the opertions of the lger cn compute ny DNF mgicl for Binry
HW#3, perspective:?? with 5 m-oxes?
Leiniz d1 d2 c 2 symol dder c s
When ws the first inry dder uilt? Shnnon Leiniz Boole d1 d2 c 2 symol dder c s sum crry
The First Digitl Adder George Stiitz, 1904-1995 Grduted with PhD from Cornel in 1930 He worked t Bell Ls in New York In the fll of 1937 Stiitz used surplus relys, tin cn strips, flshlight uls, nd other common items to construct his "Model K" K stnds for kitchen...
Bell Ls Model 1-1939 Bell Ls Model 1 Complex Clcultor Computed with complex numers 450 relys Remote opertion... vi telegrph Drtmouth college to New York Invented the term digitl s opposed to nlog
Logic to Physics Clude Shnnon s
Shnnon 1916-2001 Clude Elwood Shnnon ws orn in Petoskey, Michign, on April 30, 1916. The first sixteen yers of Shnnon's life were spent in Gylord, Michign
Shnnon 1916-2001 Shnnon s Bckground Clude Elwood Shnnon ws orn in Petoskey, Michign, on April 30, 1916 In 1932 (16) he entered the University of Michign, where he took course tht introduced him to the work of George Boole He grduted in 1936 (20) with two chelor's degrees, one in electricl engineering nd one in mthemtics Joined MIT in 1936, received the msters in electricl engineering nd doctorte in Mthemtics, t the 1940 (24) commencement
Shnnon s Inspirtion Joined MIT in 1936 Shnnon 1916-2001 Vnnevr Bush 1890 1974 Smuel Cldwell 1904-1960 The differentil nlyzer t MIT (1931) ws the first generl eqution solver It could hndle sixth-order differentil equtions
Shnnon 1916-2001 Connection Between Boolen Clculus nd Physicl Circuits Shnnon 1938 Hitchcock 1875-1957 Shnnon s dvisor oth MSc nd PhD mthemticin 78 yers go
Hitchcock 1875-1957 Bush 1890-1974 Sutherlnd ws fculty t Cltech from 1974 to 1978 (lso MS degree) Served s the founding chir of the CS Deprtment t Cltech Shnnon 1916-2001 Ivn Sutherlnd 1938-
Bush 1890-1974 Shnnon 1916-2001 1940, MIT 1924, MIT Posted on the clss wesite Termn 1900-1982 Buss 1913-2004 Cumming 1927- Goodmn 1989, Stnford 1940, Stnford 1955, Stnford 1963, Stnford Bruck
Logic to Physics The lnguge of lines s
Boolen Clculus nd Physicl Circuits Single Lines nd Composition The lnguge of lines: A line cn hve only two possile colors: lue or red Two lines cn e composed in two possile wys In prllel: In series:
Boolen Clculus nd Physicl Circuits Endpoints Lines hve endpoints Compositions hve endpoints Two lines cn e composed in two possile wys In prllel:
Boolen Clculus nd Physicl Circuits Endpoints Lines hve endpoints Compositions hve endpoints Two lines cn e composed in two possile wys In series:
Boolen Clculus nd Physicl Circuits Composition In prllel: Compositions of lines cn e composed in two different wys, using their endpoints
Boolen Clculus nd Physicl Circuits Composition In prllel: Compositions of lines cn e composed in two different wys, using their endpoints
Boolen Clculus nd Physicl Circuits Composition Compositions of lines cn e composed in two different wys, using their endpoints In series:
Boolen Clculus nd Physicl Circuits Composition In series: Compositions of lines cn e composed in two different wys, using their endpoints
Boolen Clculus nd Physicl Circuits Color of Composition Wht is the color of composition? color = lue color = red???
Boolen Clculus nd Physicl Circuits Color of Composition Wht is the color of composition? color = lue color = red The color of composition is red if there is red pth etween the endpoints Otherwise, the color is lue
The two-color line composition is 0-1 Boolen lger! How cn we prove it?
Boolen Alger Algeric system: set of elements B, two inry opertions + nd B hs t lest two elements (0 nd 1) If the following xioms re true then it is Boolen Alger: A1. identity A2. complement A3. commuttive A4. distriutive
Two-Colored Line Composition nd 0-1 Boolen Alger Algeric system: set of elements B, two inry opertions + nd B hs t lest two elements (0 nd 1) Elements: 0 1
Two-Colored Line Composition nd 0-1 Boolen Alger Algeric system: set of elements B, two inry opertions + nd B hs t lest two elements (0 nd 1) Elements: Opertions: Compose in prllel: The color of composition is red if there is red pth etween the endpoints 0 1 Otherwise, the color is lue + Compose in series:
Two-Colored Line Composition nd 0-1 Boolen Alger 0 The color of the composition equls the color of The color of the composition equls the color of 1 + Compose in prllel Compose in series The color of composition is red if there is red pth etween the endpoints Otherwise, the color is lue
Two-Colored Line Composition nd 0-1 Boolen Alger The color of the composition is red = 1 The color of the composition is lue = 0 + Compose in prllel Compose in series The color of composition is red if there is red pth etween the endpoints. Otherwise, the color is lue
Two-Colored Line Composition nd 0-1 Boolen Alger By the definition of the color of composition + Compose in prllel Compose in series The color of composition is red if there is red pth etween the endpoints. Otherwise, the color is lue
Two-Colored Line Composition nd 0-1 Boolen Alger Two pths: nd c c c
Is the two-color line composition 0-1 Boolen lger?
circuits = lger In Shnnon s words: The lger of logic originted y George Boole, is symolic method of investigting logicl reltionships. The symols of Boolen lger dmit of two logicl interprettions. If interpreted in terms of clsses, the vriles re not limited to the two possile vlues 0 nd 1. E. V. Huntington gives the following set of postultes for symolic logic: We re now in position to demonstrte the equivlence of this clculus with certin elementry prts of the clculus of propositions.
Shnnon 1916-2001 Shnnon used relys nd connected them in series-prllel circuits Rely on the edge controlled y 0-1 vrile 0 1 0 1
Shnnon 1916-2001 Connection Between Boolen Clculus nd Physicl Circuits Shnnon 1938 0 1 The vlue of circuit is 1 if there is connected pth etween the endpoints Otherwise, it is 0 0 1 The color of composition is red if there is red pth etween the endpoints Otherwise, the color is lue
Rely Circuits nlysis nd synthesis s
Shnnon 1916-2001 Connection Between Boolen Clculus nd Physicl Circuits Shnnon 1938 No mention of computers... they did not exist A concept tht is missing in the text?
In Shnnon s words: nlysis of circuits..ny circuit is represented y set of equtions, The terms of the equtions corresponding to the vrious relys nd switches in the circuit. A clculus is developed for mnipulting these equtions y simple mthemticl processes most of which re similr to ordinry lgeric lgorisms. This clculus is shown to e exctly nlogous to the clculus of propositions used in the symolic study of logic.
In Shnnon s words: synthesis of circuits For the synthesis prolem the desired chrcteristics re first written s system of equtions, nd the equtions re then mnipulted into the form representing the simplest circuit. The circuit my then e immeditely drwn from the equtions. desired chrcteristic system of equtions simplified set of equtions simple circuit
In Shnnon s words: synthesis of circuits For the synthesis prolem the desired chrcteristics re first written s system of equtions, nd the equtions re then mnipulted into the form representing the simplest circuit. The circuit my then e immeditely drwn from the equtions. desired chrcteristic The lnguge of system of equtions Logic Design is orn! simplified set of equtions simple circuit
Rely Circuits nlysis
A rely circuit corresponds to formul rely circuits Boolen functions Boolen sum of ll the pths etween endpoints
Anlysis of Rely Circuits Exmple 1: - series-prllel - independent pths etween endpoints d c e c d e
Exmple 1: - series-prllel - independent pths etween endpoints d e c e c d
Exmple 1: - series-prllel - independent pths etween endpoints d e c e c d
Exmple 1: - series-prllel - independent pths etween endpoints d e c e c d
Exmple 1: - series-prllel - independent pths etween endpoints d e c e c d
Anlysis of Rely Circuits Exmple 2: - non series-prllel - dependent pths etween endpoints c d e
Anlysis of Rely Circuits c d e
Anlysis of Rely Circuits c d e
Anlysis of Rely Circuits c d e
Anlysis of Rely Circuits c d e
Anlysis of Rely Circuits Exmple 3: multiple terminls (lso in HW#4) How mny functions? c d e
Anlysis of Rely Circuits Exmple 3: multiple terminls (lso in HW#4) c d e
Anlysis of Rely Circuits Exmple 3: multiple terminls (lso in HW#4) c d e
Anlysis of Rely Circuits Exmple 3: multiple terminls (lso in HW#4) c d e
Anlysis of Rely Circuits Exmple 3: multiple terminls (lso in HW#4) c d e
Anlysis of Rely Circuits Exmple 3: multiple terminls (lso in HW#4) c d e
Anlysis of Rely Circuits Exmple 3: multiple terminls (lso in HW#4) c d e
Anlysis of Rely Circuits Exmple 3: multiple terminls (lso in HW#4) c d e
Exmple 3: 6 functions multiple terminls c d e
Anlysis of Rely Circuits Exmple 4: - series-prllel - MANY dependent pths etween endpoints Q: how mny FORWARD pths?
Anlysis of Rely Circuits Exmple 4: - series-prllel - MANY dependent pths etween endpoints Q: how mny FORWARD pths? 3X3X3X3 =81
Anlysis of Rely Circuits Exmple 4: - series-prllel - MANY dependent pths etween endpoints Red = vrile Blue = complement of d c c d e
Anlysis of Rely Circuits Exmple 4: - series-prllel - MANY dependent pths etween endpoints Red = vrile Blue = complement of d c c d e Q: Is (=0, =1, c=1, d=1, e=1) stisfying ssignment? NO
Anlysis of Rely Circuits Exmple 4: - series-prllel - MANY dependent pths etween endpoints Red = vrile Blue = complement of d c c d e Q: Is (=1, =1, c=1, d=1, e=1) stisfying ssignment? YES
Anlysis of Rely Circuits Is there stisfying ssignment?
Anlysis of Rely Circuits Exmple 5: - series-prllel - MANY dependent pths etween endpoints Red = vrile Blue = complement of c c Q: Is there stisfying ssignment?? NO
Anlysis of Rely Circuits Exmple 5: - series-prllel - MANY dependent pths etween endpoints Red = vrile Blue = complement of c c Q: Is there stisfying ssignment? must e 1
Anlysis of Rely Circuits Exmple 5: - series-prllel - MANY dependent pths etween endpoints Red = vrile Blue = complement of c c Q: Is there stisfying ssignment? must e 0 Contrdiction!
Efficient lgorizms? Questions on stisfying ssignments? The SAT prolem Is given ssignment stisfying? Is there stisfying ssignment?
There is n efficient lgorizm for verifying given SAT solution for ny structure! d c c d e It is relted to lgorizms for solving connectivity prolems grphs
Algorizms for finding SAT ssignment? c c If the circuit hs width 2: There is n efficient lgorizm for finding stisfying ssignment...
However, if the circuit hs width 3: No efficient lgorizm is known! Likely, n efficient lgorizm does not exist d c c d e
Efficient lgorizms? Questions on stisfying (SAT) ssignments? Is given ssignment stisfying? Is there stisfying ssignment? Shnnon s connection etween computtion nd Boolen lger is t the core of Algorithms nd Complexity! P vs NP question
Wht is the function?
Wht Anlysis is the of Rely function? Circuits Red = vrile Blue = complement of c c d c d c
Wht Anlysis is the of Rely function? Circuits The key: 1 - switch etween low nd high 0 - sty t the sme level Red = vrile Blue = complement of c d odd prity c d c even prity
Wht Anlysis is the of Rely function? Circuits The key: 1 - switch etween low nd high 0 - sty t the sme level odd prity =1 d=1 =1 c=0 even prity
Wht Anlysis is the of Rely function? Circuits The key: 1 - switch etween low nd high 0 - sty t the sme level Red = vrile Blue = complement of odd prity even prity
Rely Circuits teching the next genertion
The First Book on Switching Circuits Keister, Ritchie nd Wshurn, 1951 1951 Willim Keister 1907-1997
SpinOut Keister Willim Keister 1907-1997 Willim Keister ws pioneer in switching theory nd design t Bell Ls Keister egn working in his spre time to prove tht puzzles could e solved using Boolen lger U.S. Ptent 3637215 (1972): SpinOut U.S. Ptent 3637216 (1972): The Hexdeciml Puzzle
The First Book on Switching Circuits Keister, Ritchie nd Wshurn, 1951 C nd Unix Dennis Ritchie 1941-2011 1951 Son of
The First Book on Switching Circuits Keister, Ritchie nd Wshurn, 1951 Being recognized y the president with co-inventor Ken Thompson C nd Unix Dennis Ritchie 1941-2011 1951 Son of
The First Book on Switching Circuits Keister, Ritchie nd Wshurn, 1951
Importnt note Shnnon Used the Dul Nottion In this HW set use the nottion from clss! 1 = closed rely/circuit 0 = open rely/circuit
One circuit with multiple terminls for mny functions
One circuit with multiple terminls for mny functions
Wit until fter the clss on Thursdy...
Wit until fter the clss on Thursdy...