Algoritmi Evolutivi e Teoria della Computabilità

Algoritmi Evolutivi e Teoria della Computabilità Prof. Mario Pavone CdL Magistr
ale in Inform
a
Dip. Matem
atica ed Info tica rmatica mpavone@
dmi.unict.it
http://www
.dmi.unict.it
/mpavone/
COMPUTAZIONE NATURALE
—  Sono sistemi computazionali che prendono ispirazione
— 
— 
— 
— 
— 
dalla Natura
Nature Inspired Computation: SIMULANO E NON
COPIANO
Sottocampo di “Intelligenza Artificiale” che coinvolge
problemi di ottimizzazione continua e combinatoria
Sono considerati come metodi di ottimizzazione globali
Le tecniche di computazione naturale vengono utilizzati
per problemi di ottimizzazione e learning
Uno dei principi presi in prestito è la sopravvivenza del
migliore
selezione del migliore
—  mutazione
— 
Mario Pavone, AA. 2014/2015 – Computazione Naturale, CdL Magistrale Informatica, DMI, UniCT – [email protected] COMPUTAZIONE NATURALE
—  BlackBox Problems: tecniche di computazione
naturale non richiedono alcuna conoscenza del
dominio di applicazione
—  Adatti in problemi con imprecisione, incertezza, verità
parziale e approssimazione
—  Principio Guida: sfruttare imprecisione, incertezza e
approssimazione per ottenere trattabilità, robustezza e
soluzioni low cost
—  Processo iterativo: crescita e sviluppo di una
popolazione => (evoluzione)
Mario Pavone, AA. 2014/2015 – Computazione Naturale, CdL Magistrale Informatica, DMI, UniCT – [email protected] ALGORITMO EVOLUTIVO
Parents
Popolazione
di Soluzioni
Scambio delle
caratteristiche
dei parents
Cambiamenti
casuali
Offspring
Mario Pavone, AA. 2014/2015 – Computazione Naturale, CdL Magistrale Informatica, DMI, UniCT – [email protected] ALGORITMO EVOLUTIVO
Parents
Popolazione
di Soluzioni
Scambio delle
caratteristiche
dei parents
Cambiamenti
casuali
Offspring
Mario Pavone, AA. 2014/2015 – Computazione Naturale, CdL Magistrale Informatica, DMI, UniCT – [email protected] COMPUTAZIONE NATURALE
—  Many aspects of such an evolutionary process
are stochastic
—  Changed pieces of information due to
recombination and mutation are randomly
chosen
— 
—  s e l e c t i o n o p e r a t o r s c a n
deterministic, or stochastic
be either
—  individuals with a higher fitness have a
higher chance to be selected
—  typically even the weak individuals have a
chance to become a parent or to survive
Mario Pavone, AA. 2014/2015 – Computazione Naturale, CdL Magistrale Informatica, DMI, UniCT – [email protected] PERCHÉ LA COMPUTAZIONE
NATURALE
— Flessibilità: applicabile a differenti problemi
— Robustezza: in grado di affrontare incertezza
— Adattivi: in grado di gestire applicazioni in ambienti
dinamici attraverso l'auto-adattamento
— Autonomi: possono funzionare senza l’intervento
dell’utente
— Decentrata: senza un autorità centrale
Mario Pavone, AA. 2014/2015 – Computazione Naturale, CdL Magistrale Informatica, DMI, UniCT – [email protected] TEORIA DELLA
COMPLESSITÀ
Problem 0  A problem is a general question to be answered,
usually possessing several parameters, or free
variables, whose values are left unspecified.
0  A problem is described by giving:
0  a general description of all its parameters, and
0  a statement of what properties the answer, or solution,
is required to satisfy
0  An instance of a problem is obtained by
specifying particular values for all the problem
parameters
Mario Pavone, AA. 2014/2015 – Computazione Naturale, CdL Magistrale Informatica, DMI, UniCT – [email protected] A Classical Example: TSP problem •  The parameters of this problem consist of a finite set
C = c1, c2, . . . , cm
of "cities" and, for each pair of cities ci, cj in C, the "distance“
d(ci , cj) between them
•  A solution is an ordering
< cЛ(1), c Л(2) , . . . , c Л(m)>
•  of the given cities that minimizes
⎛ m−1
⎞
⎜ ∑ d (cπ (i ) , cπ (i +1) )⎟ + d (cπ ( m ) , cπ (1) )
⎝ i =1
⎠
this expression gives the length of the "tour" that starts at
cЛ(1), visits each city in sequence, and then returns directly to
cЛ(1) from the last city cЛ(m)
Mario Pavone, AA. 2014/2015 – Computazione Naturale, CdL Magistrale Informatica, DMI, UniCT – [email protected] Combinatorial Landscapes ¢  The notion of landscape is among the rare existing concepts which help to understand the behaviour of search algorithms and heuristics and to characterize the dif4iculty of a combinatorial problem. Mario Pavone, AA. 2014/2015 – Computazione Naturale, CdL Magistrale Informatica, DMI, UniCT – [email protected] Mario Pavone, AA. 2014/2015 – Computazione Naturale, CdL Magistrale Informatica, DMI, UniCT – [email protected] Search Space ¢ Given a combinatorial problem P, a search space associated to a mathematical formulation of P is deQined by a couple (S,f ) —  where S is a Qinite set of conQigurations (or nodes or points) and —  f a cost function which associates a real number to each conQigurations of S. ¢ For this structure two most common measures are the minimum and the maximum costs. In this case we have the combinatorial optimization problems. Mario Pavone, AA. 2014/2015 – Computazione Naturale, CdL Magistrale Informatica, DMI, UniCT – [email protected] Example: K-­‐SAT ¢ An instance of the K-­‐SAT problem consists of a set V of variables, a collection C of clauses over V such that each clause c ∈ C has |c|= K ¢ The problem is to Qind a satisfying truth assignment for C ¢ The search space for the 2-­‐SAT with |V|=2 is (S,f ) where —  S={ (T,T), (T,F), (F,T), (F,F) } and —  the cost function for 2-­‐SAT computes only the number of satisQied clauses fsat (s)= #Satis4iedClauses(F,s), s ∈ S Mario Pavone, AA. 2014/2015 – Computazione Naturale, CdL Magistrale Informatica, DMI, UniCT – [email protected] An Example of SEARCH SPACE Let we consider F = (A ∨ ¬B) ∧ (¬ A ∨ ¬B)
A
T
T
B
T
F
fsat(F,s)
1
2
F
F
T
F
1
2
Mario Pavone, AA. 2014/2015 – Computazione Naturale, CdL Magistrale Informatica, DMI, UniCT – [email protected] SEARCH LANDSCAPE ¢ Given a search space (S,f), a search landscape is deQined by a triplet (S,n,f) where n is a neighborhood function which veriQies n : S → 2S -­‐ {0} ¢ This landscape, also called energy landscape, can be considered as a neutral one since no search process is involved. ¢ It can be conveniently viewed as weighted graph G=(S,n,F) where the weights are deQined on the nodes, not on the edges. Mario Pavone, AA. 2014/2015 – Computazione Naturale, CdL Magistrale Informatica, DMI, UniCT – [email protected] Example and Relevance of Landscape ¢  The search Landscape for the K-­‐SAT problem is a N dimensional hypercube with N = number of variables = |V| ¢  Combinatorial optimization problems are often hard to solve since such problems may have huge and complex search landscape Mario Pavone, AA. 2014/2015 – Computazione Naturale, CdL Magistrale Informatica, DMI, UniCT – [email protected] Example: HYPERCUBES PROCESS LANDSCAPE ¢  Given a search landscape (S,n,f), a process landscape is deQined by a quadruplet (S,n,f,φ) where φ is a search process ¢  The process landscape represents a particular view of the neutral landscape (S,n,f) seen by a search algorithm. ¢  Examples of search algorithms: — 
— 
— 
Local Search Algorithms Complete Algorithms (e. g. Davis-­‐Putnam algorithm) Evolutionary Algorithms: Genetic Algorithms, Genetic Programming, Evolution Strategies, Evolution Programming, Immune Algorithms Mario Pavone, AA. 2014/2015 – Computazione Naturale, CdL Magistrale Informatica, DMI, UniCT – [email protected] ALGORITHMS ¢ Algorithms are general, step-­‐by-­‐step procedures for solving problems ¢ For concreteness, we can think of them simply as being computer programs, written in some precise computer language. ¢ An algorithm is said to solve a problem P if that algorithm can be applied to any instance I of P and is guaranteed always to produce a solution for that instance I ¢ For example: —  An algorithm does not "solve" the TSP unless it always constructs an ordering that gives a minimum length tour Mario Pavone, AA. 2014/2015 – Computazione Naturale, CdL Magistrale Informatica, DMI, UniCT – [email protected] BIBLIOGRAPHY
M. R. Garey and D. S. Johnson, Computers
and Intractability: A Guide to the Theory of
NP-Completness, W. H. Freeman, 1st ed.
(1979).
OR:
A compendium of NP optimization
problems at the link:
http://www.csc.kth.se/~viggo/problemlist/