Alternative machine models

Transcription

1 Alternative machine models Computational complexity thesis: All reasonable computer models can simulate one another in polynomial time (i.e. P is robust or machine independent ). But the Turing machine is based on a classical physics model of the universe, whereas current physical theory asserts that the universe is quantum physical! Question: Can we build more powerful computing devices based on quantum physics? (Another interesting kind of computing device is the biological computer... ) Autumn of 9

2 Quantum Computers or outsmarting complexity According to quantum physics a particle (elctron, proton, etc) can be in several different quantum states at the same time. A quantum computer can follow several different path in the computation tree at the same time, and therefore somehow act as a NTM. Several quantum machine models have been proposed, e.g. a quantum Turing machine. In 1994 Peter W. Shor showed a polynomial time quantum algorithm for FACTORING and DISCRETE LOG, two problems that seem to be difficult on a classical TM, and whose intractability modern cryptography relies upon. Autumn of 9

3 E. Bernstein and U. Vazirani have recently showed that a certain problem the recursive Fourier sampling problem can be solved in polynomial time on a quantum Turing machine, but requires superpolynomial time on a classical TM unless P=NP. This was the first evidence ever contradicting the Computational Complexity Thesis! It has recently been proven that the class NPcannot be solved on a quantum Turing machine in time o(2 n/2 ) unless P=NP. To this date (1998) the largest quantum computer actually build has 2 bits, but there is much research going on. Many excellent articles on quantum computing and complexity can be found in SIAM Journal on Computing Vol. 26, No. 5, pp Autumn of 9

4 Cryptography or cultivating complexity Security & legal issues limit the use of computers. A foundation stone: Public Key Cryptosystem. Public key (function) Encoding e Dino y = E(e, x) Secret key (function) Decoding d Dino x = D(d, y) D(d, E(e, x)) = E(e, D(d, x)) = x The system depends upon the existence of one-way functions functions that are easy to compute, but difficult to invert. Autumn of 9

5 Example The RSA (Rivest, Shamir, Adleman) cryptosystem (1978) encoding: y = x e mod pq, p and q large primes decoding: x = y d mod pq Note: The scheme can be broken (and x computed from (y, pq, e)ifpq canbefactored (i.e. if p and q can be computed from their product). Autumn of 9

6 Cryptographic protocols Example: Secret letters with digital signatures. Two persons Alice and Bob with their public (e A, e B )andsecret(d A,d B )keys. Alice computes the letter consisting of message x (in plain text) and signature D(d A,x)(using her secret key), and encodes the whole thing using Bob s public key. Bob decodes the letter using his secret key (the message x is then readable to him) and then computes (encodes) the signature E (e A,D(d A,x)) using Alice s public key. If the result is equal to x,heknowsthat Alice is the sender. Autumn of 9

7 PKCs are based on one-way functions which are easy to compute, but difficult to invert. RSA uses essentially PRIMALITY as the easy function and FACTORING as the supposedly difficult function. PRIMALITY can be shown to belong to NP Co-NP. It is also proven that PRIMALITY belongs to ZPP, meaning that it can be solved by a Las Vegas algorithm. There exists no polynomial-time algorithm for FACTORING on a classical TM, but FACTORING can be solved efficiently on a quantum TM. Note: If P=NP then any public key cryptosystem can be broken. Co NP P NP PRIMALITY Autumn of 9

8 Expressive/computational power of machines & languages or expressing complexity Sample results Modeling (Mc Culloh, Pitts, ca. 1950): Neural networks are Turing equivalent. Neuron k x 1 x 2 x 3 y k t k... n 1 x i t? x n Logic (Expressive power of first-order logic): First-order graph properties are in P. First-order logic: x y( x F y) T problems algorithms properties, FLs TMs theories logic Autumn of 9

9 PL design (Expressive power of programming languages): Simula is Turing equivalent (applicative PL) Prolog? (declarative PL) Query language design (Expressive power of database query languages): Datalog queries are polynomial-time computabel Grammars, compiler design, etc. Relationship between logic & complexity, (query) language design Fagin (1976): NP = existential second-order logic graph ( Rφ(G, R)) P = graph first-order + while + successor = first-order + fixpoint + successor = Horn existential second-order + succ. Autumn of 9