# Anul I, Semestrul I 2017/2018

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1 Logică Matematică şi Computaţională Anul I, Semestrul I 2017/2018 Laurenţiu Leuştean Pagina web: În prezentarea acestui curs sunt folosite parţial slideurile Ioanei Leuştean din Semestrul I 2014/

2 Ce este logica? logiké tékhné = ştiinţa raţionamentelor; logos = cuvânt, raţionament Aristotel (IV î.e.n.) Premiză Premiză Concluzie entries/aristotle-logic/ primul studiu formal al logicii a studiat silogismele, deducţii formate din două premize şi o concluzie. Barbara Toţi oamenii sunt muritori. Grecii sunt oameni. Deci grecii sunt muritori. 2

3 Logică şi Informatică... a computing machine is really a logic machine. Its circuits embody the distilled insights of a remarkable collection of logicians, developed over century. Nowadays, as computer technology advances with such breathtaking rapidity, as we admire the truly accomplishments of the engineers, it is all too easy to overlook the logicians whose ideas made it all possible. This book tells their story. 3

4 Gottfried Wilhelm Leibniz ( ) Visul lui Leibniz un limbaj matematic universal (lingua characteristica universalis) în care toată cunoaşterea umană poate fi exprimată şi reguli de calcul (calculus ratiocinator) pentru a deriva, cu ajutorul maşinilor, toate relaţiile logice: If controversies were to arise, there would be no more need of disputation between two philosophers than between two accountants. For it would suffice to take their pencils in their hands, and say to each other: Calculemus - Let us calculate. 4

5 George Boole ( ) The Mathematical Analysis of Logic (1847), The Laws of Thought (1854): a iniţiat analiza raţionamentelor logice prin metode asemănătoare calculului algebric. Silogismele lui Aristotel sunt despre clase de obiecte, care pot fi studiate algebric. The design of the following treatise is to investigate the fundamental laws of the operations of the mind by which reasoning is performed; to give expressions to them in the symbolic language of calculus, and upon this foundation to establish the science of logic and constructs its methods. 5

6 Gottlob Frege ( ) Begriffschrift (1879) A introdus sintaxa formală: obiecte, predicate, funcţii; conectori propoziţionali; cuantificatori. A inventat logica de ordinul întâi. van Heijenoort, From Frege to Godel, 1967: perhaps the most important single work ever written in logic. Exemplu: Toţi oamenii sunt muritori. Pentru orice x, dacă x este om, atunci x este muritor. x(om(x) Muritor(x)). 6

7 Georg Cantor ( ) A inventat teoria mulţimilor. A definit numerele cardinale, ordinale. A dezvoltat o teorie matematică a infinitului. Hilbert: No one shall be able to expel us from the paradise that Cantor created for us. 7

8 Georg Cantor ( ) Aristotel: Infinitum Actu Non Datur - nu există infinit actual. Leibniz: I am so in favor of the actual infinite that instead of admitting that Nature abhors it, I hold that Nature makes frequent use of it everywhere. Gauss: I protest above all the use of an infinite quantity as a completed one, which in mathematics is never allowed. Frege: For the infinite will eventually refuse to be excluded from arithmetics... Thus we can foresee that this issue will provide for a momentous and decisive battle. Poincaré: grave disease infecting mathematics. Kronecker despre Cantor: scientific charlatan, corrupter of youth Wittgenstein: utter nonsense Mittag-Leffler despre lucrările lui Cantor: about one hundred years too soon. 8

9 Criza fundamentelor matematicii Scrisoarea lui Bertrand Russell către Frege (16 iunie, 1902): I find myself in agreement with you in all essentials... I find in your work discussions, distinctions, and definitions that one seeks in vain in the work of other logicians... There is just one point where I have encountered a difficulty. Frege, apendix la The Fundamental Laws of Arithmetic, Vol. 2: There is nothing worse that can happen to a scientist than to have the foundation collapse just as the work is finished. I have been placed in this position by a letter from Mr. Bertrand Russell. 9

10 Criza fundamentelor matematicii Conform teoriei naive a mulţimilor, orice colecţie definibilă este mulţime. Fie U mulţimea tuturor mulţimilor. Paradoxul lui Russel (1902) Fie R = {A U A / A}. Atunci R este mulţime, deci R U. Obţinem că R / R R R. Criza fundamentelor matematicii Paradoxul lui Russel Sistemul logic al lui Frege inconsistent a declanşat criza fundamentelor matematicii ( foundations of mathematics ) s-a dezvoltat teoria axiomatică a mulţimilor: Zermelo-Fraenkel (ZF), ZFC: ZF+ Axioma alegerii (Axiom of Choice) 10

11 David Hilbert ( ) unul dintre matematicienii de vârf ai generaţiei sale unul dintre fondatorii teoriei demonstraţiei şi logicii matematice lista sa de 23 probleme deschise (1902) a influenţat foarte mult matematica secolului XX 11

12 Programul lui Hilbert Programul lui Hilbert (1921) Să se formalizeze matematica şi să se stabilească următoarele: Matematica este consistentă: un enunţ matematic şi negaţia sa nu pot fi demonstrate simultan. Matematica este completă: toate enunţurile matematice adevărate pot fi demonstrate. Matematica este decidabilă: există o regulă mecanică pentru a determina dacă un enunţ matematic dat este adevărat sau fals 12

13 Programul lui Hilbert Hilbert a fost convins că aceste obiective pot fi atinse: Every mathematical problem must necessarily be susceptible to an exact statement either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution. Once a logical formalism is established one can expect that a systematic, so-to-say computational, treatment of logic formulas is possible, which would somewhat correspond to the theory of equations in algebra. 13

14 Kurt Gödel ( ) Teoremele de incompletitudine ale lui Gödel ( ) Incompletitudinea aritmeticii obişnuite. Imposibilitatea de a demonstra consistenţa teoriei mulţimilor. Au marcat eşecul programului lui Hilbert. Este considerat cel mai mare logician al secolului XX. A introdus funcţiile calculabile. A demonstrat teorema de completitudine a logicii de ordinul l. A demonstrat că Axioma Alegerii şi Ipoteza Continuumului sunt consistente cu axiomele teoriei mulţimilor. 14

15 Kurt Gödel ( ) John von Neumann: Kurt Gödel s achievement in modern logic is singular and monumental - indeed it is more than a monument, it is a landmark which will remain visible far in space and time... The subject of logic has certainly completely changed its nature and possibilities with Gödel s achievement. Revista TIME (19 martie 1999) Gödel a fost inclus in lista cu cei mai importanţi 20 oameni de ştiinţă şi gânditori ai secolului XX. 15

16 Problema de decizie (Entscheidungsproblem) Hilbert şi Ackermann (1928): Există un algoritm pentru a verifica dacă o anumită formulă din logica de ordinul întâi este adevărată? Cu alte cuvinte: Este logica de ordinul întâi decidabilă? 16

17 Alan Turing( ) Turing, On computable numbers, with an application to the Entscheidungsproblem, Proc. London Math. Soc. 42 (1936). a demonstrat că logica de ordinul întâi este nedecidabilă (rezultat obţinut independent de Church (1936)). a introdus maşina Turing (universală) pentru a formaliza noţiunea de algoritm. părintele informaticii şi inteligenţei artificiale maşina Turing universală este model al calculatoarelor actuale 17

18 Alan Turing( ) Revista TIME (19 martie 1999) Turing a fost inclus in lista cu cei mai importanţi 20 oameni de ştiinţă şi gânditori ai secolului XX: Virtually all computers today from 10 million supercomputers to the tiny chips that power cell phones and Furbies, have one thing in common: they are all von Neumann machines, variations on the basic computer architecture that John von Neumann, building on the work of Alan Turing, laid out in the 1940 s. Premiul Turing decernat anual de către Association for Computing Machinery (ACM) pentru contribuţii în informatică este considerat un Premiu Nobel pentru Informatică 18

19 Logică şi Informatică E. W. Dijkstra, The next fifty years (EWD1243a). E.W. Dijkstra Archive. Center for American History, University of Texas at Austin: Computing and Computing Science unavoidably emerge as an exercise in formal mathematics or, if you wish an acronym, as exercise in VLSAL (Very Large Scale Application of Logic). Aaron R. Bradley, Zohar Manna, The Calculus of Computation Decision Procedures with Applications to Verification, Springer, 2007: Logic is the calculus of computation. Georg Gottlob, Logic and Artificial Intelligence, VSL 2014: Computer science is the continuation of logic by other means. 19

20 Logică şi Informatică Aplicatii ale logicii în informatică: calculabilitate şi complexitate arhitectura calculatoarelor (circuite logice) software engineering (verificare, model checking) limbaje de programare (semantică, programare logică, programare funcţională) baze de date (algebre de relaţii, teoria modelelor finite) inteligenţă artificială criptografie şi securitate J. Y. Halpern, R. Harper, N. Immerman, P.G.Kolaitis, M.Y. Vardi, V.Vianu, On the Unusual Effectiveness of Logic in Computer Science, Bulletin of Symbolic Logic 7(2001) 20

21 Logică şi Informatică în România Grigore C. Moisil ( ) Computer Pioneer Award of IEEE Computer Society S. Marcus, Grigore C. Moisil: A life becoming a myth, As a professor of the Bucharest University, he was the first to teach there mathematical logic. Articulating logic and automata, Moisil was well prepared to organize the Romanian development in the emergent field of Computer Science...we can say that 1957 is the date of birth of Romanian Computer Science, under the guidance of Professor Moisil and with the collaboration of engineers and mathematicians. 21

22 PRELIMINARII 22

23 Operaţii cu mulţimi Fie A, B, T mulţimi a.î. A, B T. A B = {x T x A sau x B} A B = {x T x A şi x B} A \ B = {x T x A şi x / B} C T A = T \ A = {x T x A} C T A se mai notează şi A când T este clar din context. Notaţii: N = {0, 1, 2,...} este mulţimea numerelor naturale; N = N \ {0}; Z este mulţimea numerelor întregi; R este mulţimea numerelor reale; Q este mulţimea numerelor raţionale. Mulţimea părţilor lui T este P(T ) = {A A T }. Se mai notează şi 2 T. Exemplu. P( ) = { }, P({ }) = {, { }}, P({, { }}) = {, { }, {{ }}, {, { }}}. 23

24 Produsul cartezian Notăm cu (a, b) perechea ordonată formată din a şi b (care sunt componentele lui (a, b)). Observaţii: dacă a b, atunci (a, b) (b, a); (a, b) {a, b}; (7, 7) este o pereche ordonată validă; două perechi ordonate (a, b) şi (c, d) sunt egale ddacă a = c şi b = d. În teoria mulţimilor, (a, b) se defineşte ca fiind mulţimea {{a}, {a, b}}. Definiţie Produsul cartezian a două mulţimi A şi B este definit astfel: A B = {(a, b) a A şi b B} Exerciţiu. A (B C) = (A B) (A C) A (B C) = (A B) (A C) 24

25 Relaţii binare Definiţie O relaţie binară între A şi B este o submulţime a produsului cartezian A B. O relaţie binară pe A este o submulţime a lui A A. Exemple N N = {(k, n) există m N a.î. mk = n} < N N < = {(k, n) există m N a.î. m 0 şi m + k = n} 25

26 Operaţii cu relaţii Fie A, B, C mulţimi. Dacă R A B, atunci relaţia inversă R 1 B A este definită astfel: R 1 = {(b, a) (a, b) R}. Dacă R A B şi Q B C, atunci compunerea lor Q R A C este definită astfel: Q R = {(a, c) există b B a.î. (a, b) R şi (b, c) Q}. Diagonala lui A este A = {(a, a) a A}. Exerciţiu Compunerea relaţiilor este asociativă. Dacă R A B atunci R A = R şi B R = R. 26

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