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Lordito algoritmico: alcuni problemi algoritmici che hanno favorito il progresso scientifico. Alberto Policriti Dipartimento di Matematica e Informatica,

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Presentazione sul tema: "Lordito algoritmico: alcuni problemi algoritmici che hanno favorito il progresso scientifico. Alberto Policriti Dipartimento di Matematica e Informatica,"— Transcript della presentazione:

1 Lordito algoritmico: alcuni problemi algoritmici che hanno favorito il progresso scientifico. Alberto Policriti Dipartimento di Matematica e Informatica, Universita di Udine.

2 Di cosa parleremo Classi di problemi (i problemi specifici richiedono un trattamento tecnico) Problemi significativi (che legano lalgoritmica ad altre discipline) Complessita (perche, alla fine, e il vero problema dellalgoritmica)

3 Quali problemi Il problema della decisione ( Entscheidungsproblem) Problemi algoritmici in biologia computazionale Una riflessione sulla nozione di complessita Passato Presente Futuro

4 Le fonti principali M. Davis The Universal Computer: the road from Leibniz to Turing S. Feferman On the light of Logic E. Green Strategies for the systematic sequencing of complex genomes D. Knuth Papers on the foundation of Computer Science

5 Il problema della decisione Trovare un algoritmo per decidere le formule se una formula della logica al primordine e soddisfacibile. In a sense it [il problema della decisione] is the most general probem of mathematics. J. Herbrand

6 La logica del primordine xyu x y xu v y v vu Se x e y sono donne e x e felice con u, allora esiste v tale che y e felice con ve u e v sono amici Se x e y sono punti e x e sulla retta u, allora esiste v tale che y e sulla retta v e u e v sono parallele Esempio: Un algoritmo per risolvere il problema della decisione ci potrebbe dire se lipotesi di Riemann (ottavo problema di Hilbert) e vera o falsa!

7 David Hilbert D. Hilbert nel 1937 Born: 23 Jan 1862 in Königsberg, Prussia (now Kaliningrad, Russia) Died: 14 Feb 1943 in Göttingen, Germany The Entscheidungsproblem is solved when we know a procedure that allows for any given logical expression to decide by finitely many operations its validity or satisfiability. [...] The Entscheidungsproblem must be considered the main problem of mathematical logic. Principles of Mathematical Logic D. Hilbert and W. Ackermann 1928

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9 Hilbert sapeva porre problemi! The mathematicians present at an international conference in Paris in August 1900 inevitably wondered what the new century would bring to their subject. [...] he presented, as a challenge to the mathematicians of the twentieth century, 23 problems that seemed utterly inaccessible by the methods available at the time. The Universal Computer M. Davis In his work, Hilbert demonstrated an unusual combination of direct intuition and concern for absolute rigor. With exceptional technical power at his command, he would tackle outstanding problems, usually with a great originality of approach. The title of Hilberts lecture in Paris was simply, Mathematical problems. Deciding the undecidable: Wrestling with Hilberts problems S. Feferman

10 The great importance of definite problems for the progress of mathematical science in general... is undeniable.... [for] as long as a branch of knowledge supplies a surplus of such problems, it maintains its vitality.... every mathematician certainly shares..the conviction that every mathematical problem is necessarily capable of strict resolution... we hear within ourselves the constant cry: There is the problem, seek the solution. You can find it through pure thought... D. Hilbert

11 1. Lipotesi del continuo 2. La consistenza dellaritmetica 10. Lesistenza di un algoritmo per risolvere le equazioni diofantee The solution of three of Hilberts problems were to involve mathematical logic and the foundation of mathematics in an essential way; they are the ones numbered 1,2, and 10 in his list Non parleremo di 1. e 2. e il legame con il problema della decisione Deciding the undecidable: Wrestling with Hilberts problems S. Feferman

12 Esempio: E possibile scrivere una equazione diofantea che ammette soluzioni intere se e solo se lipotesi di Riemann e falsa. Equazioni diofantee: P(x 1,..., x k ) = 0 con P polinomio a coefficienti interi Contrary to Hilberts expectations, Problem 10 was eventually solved in the negative. This was accomplished in 1970 by a young russian mathematician, Yuri Matiyasevich, who built on earlier work in 1950s and 1960s by the American logicians Martin Davis, Hilary Putnam, and Julia Robinson. [...] Il decimo problema di Hilbert Gia nel 1920 si sospettava che problemi come il precedente fossero indecidibili. Ma come dimostrare che non esiste un algoritmo?? Deciding the undecidable: Wrestling with Hilberts problems S. Feferman

13 La soluzione del secondo problema: il simposio di Könisberg del 1930 During the days immediately preceding Hilberts address, a symposium on the foundations of mathematics took place in Königsberg. [...] At the round table discussion that concluded the event, a shy young man named Kurt Gödel [...] made a quiet announcement that, to those who grasped its import, signalled a new era in foundational studies. Von Neumann got the point at once, and concluded that the jig was up, that Hilberts program could not succeed. The Universal Computer M. Davis

14 Il programma di Hilbert consistenza 1.La consistenza dellaritmetica (secondo problema di Hilbert) completezza 2.La completezza della logica e dellaritmetica (Gödel 1928) 3.Il problema della decisione (Entscheidungsproblem)

15 Kurt Gödel Born: 28 April 1906 in Brünn, Austria-Hungary (now Brno, Czech Republic) Died: 14 Jan 1978 in Princeton, New Jersey, USA

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18 The crucial step in Gödels proof was his demonstration that the property of a natural number of being the code of a proposition provable in PM is itself expressible in PM. [...] - U says that some particular proposition is not provable in PM. - That particular proposition is none other than U itself. - Therefore, U says: U is not provable in PM. Gödel aveva scritto il primo compilatore e... decretato la fine del programma di Hilbert! The Universal Computer M. Davis

19 Cosa rimane del programma di Hilbert? Hilbert had also sought explicit calculational procedures by means of which it would always be possible to determine, given some premises and a proposed conclusion, written in the notation of what has come to be called first-order logic, whether Freges rules would enable that conclusion to be derived from those premises. The task of finding such procedures came to be known as Hilberts Entscheidungsproblem (literally: decision problem), The Universal Computer M. Davis Cerano risultati parziali e i granndi giovani matematici erano tutti attivi: F. P. Ramsey, W. Ackermann, P. Bernays, M. Shönfinkel e lo stesso Gödel

20 Apparently intrigued by these developments, Newman gave a lecture course in the spring term of 1935 on the foundations of mathematics featuring Gödels incompleteness theorem as its climax. Attending this course, Turing learned about Hilberts Entscheidungsproblem. Quite apart from the incredulity of such as Hardy, after Gödels work it was hard to believe that there could be an algorithm such as Hilbert had wanted. Alan Turing began to think about how it could be possible to prove that no such algorithm exists. The Universal Computer M. Davis

21 Now, if someone comes along with a proposed algorithm to settle a given decision problem in a positive way, one can check to see that it does the required work (or at least try to do so), without inquiring into the general nature of what constitutes an algorithm. But if it is to be shown that the problem is undecidable, one has to have a precise explanation of what algorithms can compute in general. Deciding the undecidable: Wrestling with Hilberts problems S. Feferman

22 Alan Turing His high pitched voice already stood out above the general murmur of well-behaved junior executives grooming themselves for promotion within the Bell corporation. Then he was suddenly heard to say: "No, I'm not interested in developing a powerful brain. All I'm after is just a mediocre brain, something like the President of the American Telephone and Telegraph Company." Quoted in A Hodges, Alan Turing the Enigma of Intelligence, (London 1983) 251.

23 [...] on the basis of Turings analysis of the notion of computation, it is possible to conclude that anything computable by any algorithmic process can be computed by a Turing machine. So if we can prove that some particular task can not be accomplished by a Turing machine, we can conclude that no algorithmic process can accomplish that task. That is how Turing proved that there is no algorithm for the Entscheidungsproblem. In addition, Turing showed how to produce one individual Turing machine that, all by itself, can do anything that could be done by any Turing machine whatever – a mathematical model of an all-purpose computer. The Universal Computer M. Davis

24 Il metodo diagonale nel lavoro di Turing Now, if we think of the halting set of a Turing machine as constituting a package and of the code number of that machine as labeling that package, then we have exactly the typical setup for applying the diagonal method: labeled packages in which the labels are exactly the kind of thing in the packages – in this case, natural numbers. The Universal Computer M. Davis

25 La macchina universale di Turing The universal machine also provides a model of a stored program computer [...] in which the machine makes no fundamental distinction between program and data. Finally, the universal machine shows how hardware [...] thought of as a description of the functioning of a mechanism, canbe replaced by equivalent software [...] stored on the tape of a universal machine. On computable numbers with an application to the `Entscheidungsproblem A. Turing Proc. of the London Mathematical Society 1937 The Universal Computer M. Davis

26 Turings universal computer was a marvelous conceptual device that all by-itself could execute any algorithmic task. But could one actually build such a thing? And aside from what such a machine could accomplish in principle, could it be designed and constructed so as to be able to solve real world problems in an acceptable time frame, and using reasonable available resources? By the end of 1945, Turing had produced his remarkable ACE (Automatic Computing Engine) Report. One detailed comparison of the ACE Report with von Neumann's EDVAC Report, notes that whereas the latter ``is a draft and is unfinished … more important … is incomplete …'' the ACE Report ``is a complete description of a computer, right down to the logical circuit diagrams'' and even including ``a cost estimate of £11,200.'' The Universal Computer M. Davis

27 ACE: la risposta (inglese) di Turing ad Edvac [It] is … very contrary to the line of development here, and much more in the American tradition of solving one's difficulties by means of much equipment rather than by thought. … Furthermore certain operations which we regard as more fundamental than addition and multiplication have been omitted Alan Turing

28 Problemi algoritmici in biologia computazionale Astronomy began when the Babylonians mapped the heavens. Our descendants will certainly not say that biology began with todays genome projects, but they may well recognize that a great acceleration in the accumulation of biological knowledge began in our era. To make sense of this knowledge is a challenge, and will require increased understanding of the biology of cells and organisms. But part of the challenge is simply to organise, classify and parse the immense richness of sequence data. Biological sequence analysis R. Durbin, S. Eddy, A. Krogh and G. Mitchinson

29 Un po di storia 1953: F. Crick e J. Watson scoprono la struttura a doppia elica del DNA anni 70: si sviluppano le tecniche per il sequenziamento di spezzoni di DNA (F. Sanger) anni 80: viene lanciato il progetto genoma e partono le prime sperimentazioni pilota (insieme alle prime compagnie per lo sfruttamento commerciale di queste ricerche) anni 90: vengono sequenziati i primi organismi (qualche M di paia di basi)

30 1990: viene pubblicato BLAST 1998: C. Venter annuncia la costituzione della compagnia privata Celera e sfida il consorzio pubblico per il sequenziaemnto del genoma umano: Celera otterra il risultato in 3 anni (e 300 M di $)

31 Human Genome Working Draft Sequence published February 15 & 16, 2001 Science and Nature

32 Clone-by-clone shotgun sequencing Dietro la sfida: Two main shotgun-sequencing strategies. Whole-genome shotgun sequencing

33 Programmi e algoritmi in bioinformatica [...] Yet other programs provide user-friendly viewers for inspection and editing of the resulting sequence assemblies. A particularly popular suite of programs for these various steps is Phred, Phrap and Consed,which are designed for base calling, sequence assembly and the viewing of sequence assemblies, respectively. [...] (21 occorrenze della parola programs 2 della parola algorithms) Strategies for the systematic sequencing of complex genomes Eric D. Green

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37 Programmi e algoritmi nella sfida Finally, perhaps the most essential element of any whole-genome shotgun-sequencing strategy is the availability of a robust assembly program that can accommodate the inevitably large collection of sequence reads. [...] include algorithms that account for the anticipated spatial relationship of read pairs emanating from individual subclones, which help to avoid misassemblies due to repetitive sequences. Strategies for the systematic sequencing of complex genomes Eric D. Green

38 Come finita la sfida?

39 Among the most useful computer-based tools in modern biology are those that involve sequence alignments of proteins, since these alignements oftem provide insights into gene and protein function. There are several types of alignments: global alignments of pairs of proteins, multiple alignments of members of protein families, and alignments made diring data base searches to detect homologies. S. Henikoff and J.G.Henikoff PNAS 1992 Lallineamento di sequenze

40 GTTGAT_TAGCTTATCCCAAAGCAAGGCACTGAAAATG_CTAGAT GT_GATGTAGCTTAACCCAA_GCAAGGCACTAAAAATGCCTAGAT Input: GTTGATTAGCTTATCCCAAAGCAAGGCACTGAAAATGCTAGAT GTGATGTAGCTTAACCCAAGCAAGGCACTAAAAATGCCTAGAT Output: Cose un allineamento?

41 Algoritmi Needelman-Wunsh 1970 Smith –Waterman 1981 Landau-Vishkin 1986 Wu-Manber 1992 Myers 1994 Chang-Lawler

42 GTTGATTAGCTTA G T G A T 432 G 543 T 654 A 765 GTTGATTAGCTTATCCCAAAGCAAGGCACTGAAAATGCTAGAT GTGATGTAGCTTAACCCAAGCAAGGCACTAAAAATGCCTAGAT GTTGAT_TAGCTTATCCCAAAGCAAGGCACTGAAAATG_CTAGAT GT_GATGTAGCTTAACCCAA_GCAAGGCACTAAAAATGCCTAGAT

43 Altri problemi algoritmici correlati exact-matching (un problema piu vecchio e forse meno applicativo, gli algoritmi per la cui soluzione si sono rivelati fondamentali) strutture dati (non conviene rappresentare in memoria sequenze come stringhe ma come sistemi di indici per tutti i possibili suffissi della sequenza) protein folding (un bel problema NP- completo che ci hanno regalato i biologi)...

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45 Riflessioni conclusive Il problema della decisione poteva essere difficile ma era enunciato in modo chiaro e preciso. Matematicamente pulito. I problemi algoritmici in biologia computazionale non sono sempre altrettanto puliti (forse, piu sono interessanti e piu sono sporchi). In cosa consiste veramente la complessita di un problema algoritmico?

46 Complessita: le risorse che abbiamo sono finite Advances in our ability to compute are bringing us substantially closer to ultimate limitations D. Knuth Mathematics and Computer Science: Coping with Finiteness

47 Che risorse (computazionali) abbiamo? 40 miliardi di anni luce cm Universo protone

48 (maggiore o uguale al) numero di protoni nelluniverso Se assumiamo una unita di tempo pari al tempo necessario alla luce a viaggiare per cm e assumiamo che luniverso sia nato 10 milioni di anni fa, il numero di unita di tempo trascorse e minore o uguale a 10 42

49 Che speranze abbiamo snail miles/h man 4 miles/h US auto 55 miles/h Jet 600 miles/h Supersonic jet 1200 miles/h man (pencil) 0.2/sec man (abacus) 1/sec calculator 4/sec computer /sec fast computer 2M/sec

50 start finish Grid problem: calcolare il numero di cammini da start a finish

51 Il problema e difficile non ci sono metodi noti per calcolare il numero di cammini (in a reasonable amount of time) possiamo comunque generare dei cammini random e usare un teorema di statistica che ci dice che la stima migliore e data dalla media dei reciproci delle probabilita osservate otteniamo una stima enorme: (1.6 ± 0.3) 10 24

52 il problema di stabilire una (qualunque) proprieta dei cammini sulla griglia e algoritmicamente trattabile? non possiamo contare nemmeno su una procedura esaustiva per enumerare i cammini! Forse abbiamo bisogno di una teoria della complessita algoritmica che ci permetta di classificare questo come un problema difficile Un problema semplice (da enunciare) e pulito, ma...

53 Conclusioni I problemi algoritmici costituiscono lossatura dellinformatica e le loro soluzioni richiedono uno sforzo (matematico) genuino e particolare I problemi algoritmici si sono rivelati essere dietro la scena in momenti cruciali dellavanzamento scientifico La complessita ed una teoria adeguata per il suo studio e probabilmente la piu interessante delle attuali sfide algoritmiche

54 My favorite way to describe computer science is to say that it is the study of algorithms. D.Knuth


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