1. Solving Numerical NP-complete Problems with Spiking Neural P Systems Dipartimento di Informatica, Sistemistica e Comunicazione Università degli Studi di Milano – Bicocca WMC 8 Thessaloniki, June 25 – 28, 2007 Alberto Leporati  Claudio Zandron Claudio Ferretti  Giancarlo Mauri {leporati, zandron, ferretti, mauri}@disco.unimib.it

2. WMC 8 - Thessaloniki - June 25-28, 2007 2 Talk outline Spiking Neural (SN) P systems standard version an extended version Computational power of SN P systems universality (Turing completeness) nondeterminism, determinism, maximal parallelism Solving NP-complete problems with nondeterministic SN P systems 3-SAT vs. numerical problems (Subset Sum et al.) conversion from binary to unary, and back Uniform family of SN P systems that solves Subset Sum with instances encoded in binary form Directions for further research

3. WMC 8 - Thessaloniki - June 25-28, 2007 3 Standard SN P systems An SN P system of degree m  1 is a construct:   (O,  1,  2, …,  m, syn, in) where: O  {a} is the singleton alphabet cells (neurons)  i  (n i, R i ) are placed in the nodes of the synapse graph syn, where: n i  0 : initial contents R i : finite set of rules syn  {1,2,…,m}  {1,2,…,m} : directed graph of synapses, without self-loops in  {1,2,…,m} : input neuron

4. WMC 8 - Thessaloniki - June 25-28, 2007 4 Standard SN P systems The rules can be: firing rules: E/a c  a; d with c  1, d  0 integers (if E  a c we simply write a c  a; d ) forgetting rules: a s  with s  1 and a s  L(E) for any firing rule E/a c  a; d in the neuron initial configuration: n 1, n 2, …, n m spikes in the neurons all neurons are open configuration (during computations): for each neuron: number of spikes in the neuron number of steps to wait until the neuron becomes open

5. WMC 8 - Thessaloniki - June 25-28, 2007 5 Standard SN P systems Computation of f:    the system starts in the initial configuration input n enters in in, encoded as the time elapsed between two spikes (unary notation) computation proceeds sequentially in every neuron, and in parallel between the neurons if  2 rules can be applied in a neuron, a nondeterministic choice is made the system is deterministic if L(E i )  L(E j ) =  for all i  j if the system halts, then the output f(n) is read as the time elapsed between the first two spikes emitted by the system

6. WMC 8 - Thessaloniki - June 25-28, 2007 6 Standard SN P systems (Already) many variants/possibilities: different halting conditions different ways to encode input and output values (ex: number of spikes) different ways to compute f:      directly (spike trains in input and output) through bijections (     and     ) generative case: we ignore the input accepting case: we ignore the output

7. WMC 8 - Thessaloniki - June 25-28, 2007 7 Description size Description size of  : number of bits required to represent it: no bits for the alphabet  m 2 bits for the synapse graph in (and out ): lg m bits for every neuron  i : n i  N  lg N bits at most R rules; for each rule E/a c  a; d o type  {firing, accepting}  1 bit o regular expression E  size  S bits o two numbers  2 lg N bits m 2 + 2 lg m + m (lg N + R (1 + S + 2 lg N)) bits

8. WMC 8 - Thessaloniki - June 25-28, 2007 8 Description size Remark: we work with unary languages every string is bijectively associated with its length a compact representation of E works on the lengths regular expressions on natural numbers union and Kleene star are computed as usual L 1  L 2 : all numbers of L 1 are summed with those of L 2 in all possible ways Example: {2,3}  {2,5} = {4,5,7,8}

9. WMC 8 - Thessaloniki - June 25-28, 2007 9 Computational power SN P systems are universal (Turing complete) also if we remove forgetting rules or delays also if the outdegree of syn is 2 also with regular expressions of very simple forms nondeterministic systems are able to solve 3-SAT deterministic systems with regular expressions of very restricted forms can be simulated by DTM with a polynomial slowdown using general regular expressions, choosing the rule to apply is at least as difficult as solving Subset Sum

10. WMC 8 - Thessaloniki - June 25-28, 2007 10 The 3-SAT problem variables X = {x 1, x 2, …, x n } literal X i : directed ( x i ) or negated (  x i ) variable 3-clause: disjunction of exactly three literals assignment: function a : X  { false, true } The 3-SAT problem: instance: set C = {C 1,C 2,…,C m } of 3-clauses built on X (equiv.,  = C 1  C 2  …  C m ) question: is there an assignment which satisfies all the clauses? (equiv., which satisfies  ) Note: m  (2n) 3 = 8n 3

11. WMC 8 - Thessaloniki - June 25-28, 2007 11 Solving 3-SAT with SN P systems a “circuit” with four layers first two layers: one module for each variable This appears only once

12. WMC 8 - Thessaloniki - June 25-28, 2007 12 Solving 3-SAT with SN P systems next two layers: one module for each variable semi-uniform solution generative approach solution given in constant time

13. WMC 8 - Thessaloniki - June 25-28, 2007 13 Deterministic SN P systems explicit simulation by DTMs given in the Technical Report of the last Brainstorming t steps of any deterministic SN P system  can be simulated in poly(t, Description size) steps of a DTM crucial assumption: regular expressions are of very restricted form, for example : a i, with i  3 a(aa) + (the membership problem must be polynomial also the succint version)

14. WMC 8 - Thessaloniki - June 25-28, 2007 14 The Subset Sum problem instance: a (multi)set V = {v 1,v 2,…,v n } of positive integer values a positive integer value S question: is there a sub(multi)set A  V such that numerical, pseudo-polynomial NP-complete problem instance size: let K = max{v 1,v 2,…,v n, S} each number requires lg K bits to be represented total size:  (n lg K)

15. WMC 8 - Thessaloniki - June 25-28, 2007 15 Solving Subset Sum with SN P systems let ({v 1,v 2,…,v n }, S) be an instance of Subset Sum consider the languages (in succint form): L i = {0,v i }, for i  {1,2,…,n} let L = L 1  L 2  …  L n membership problem: is S  L ? the answer is yes if and only if the instance of Subset Sum is positive deciding whether the rule E/a c  a; d can be applied, when L = L(E) and the neuron contains S spikes, can be difficult (hidden power)

16. WMC 8 - Thessaloniki - June 25-28, 2007 16 Extending SN P systems Differences with traditional SN P systems: we allow rules of type E/a c  a p ; d with c  1, p  0, d  0 p = 1 : traditional rules p = 0 : forgetting rules no mutual exclusion between firing and forgetting rules output neuron ( out ); if out = 0, then the output is sent to the environment different interpretations of output: number of spikes in a given time interval number of spikes contained in out

17. WMC 8 - Thessaloniki - June 25-28, 2007 17 Solving Subset Sum with SN P systems

18. WMC 8 - Thessaloniki - June 25-28, 2007 18 Solving Subset Sum with SN P systems generative system semi-uniform approach (input encoded both in the initial configuration and in the structure of the system) solution given in constant time nondeterminism: at least one computation chooses the correct v i, if possible (again magic power) in particular, we don’t like: rules operate on large ( O(K) ) numbers of spikes the initial configuration contains O(nK) spikes

19. WMC 8 - Thessaloniki - June 25-28, 2007 19 Dynamic programming solution boolean matrix M[1..n, 0..S] M[i, j] = 1  there exists B  {v 1,…,v i } such that space and time complexity:  (nK) solution

20. WMC 8 - Thessaloniki - June 25-28, 2007 20 Dynamic programming solution pseudo-code:

21. WMC 8 - Thessaloniki - June 25-28, 2007 21 Encoding inputs in binary form : a k -bit non-negative integer number usual binary encoding using k consecutive time steps; at time k–1–j (with j  {0,1,…, k–1} ): presence of a spike if and only if N j = 1 conversion from binary to unary (next slide) N enters the system starting from msb one (undesired) spike is produced at the 2 nd step at the end of the k -th step, neuron j contains 2 j spikes if N j = 1, and 0 spikes otherwise at the ( k +1)-th step, neuron out becomes open and receives N spikes

22. WMC 8 - Thessaloniki - June 25-28, 2007 22 Encoding inputs in binary form maximal parallelism: one of the active rules is nondetermi- nistically chosen and is applied as many times as possible

23. WMC 8 - Thessaloniki - June 25-28, 2007 23 Solving Subset Sum (2 nd version) these rules remove spurious spikes instance (V, S) is positive if and only if one spike is emitted at the (2k+2) -nd step

24. WMC 8 - Thessaloniki - June 25-28, 2007 24 Solving Subset Sum (2 nd version) we can also use two neurons in the last layer nondeterministic and semi-uniform solution rules depend on the value of S idea: read also S in binary convert it compare it with spike if equal A better idea:  keep S in binary form  convert sum of B from unary to binary  perform a bit-by-bit comparison

25. WMC 8 - Thessaloniki - June 25-28, 2007 25 Conversion from unary to binary uses maximal parallelism at each step, an integer division by 2 is performed the result is propagated the remainder stays in the current neuron

26. WMC 8 - Thessaloniki - June 25-28, 2007 26 Conversion from unary to binary The conversion cannot be performed in polynomial time t(k) without using maximal parallelism regular expressions of restricted forms Q = max number of spikes produced by any rule the number of spikes is incremented at most by mQ units at each computation step M spikes in the initial configuration Hence, at most M + mQt(k) spikes after t(k) steps a polynomial quantity !

27. WMC 8 - Thessaloniki - June 25-28, 2007 27 Uniform solution of Subset Sum the output has to be observed exactly after 3k+6 steps length of delay subsystem: 3k+2 steps

28. WMC 8 - Thessaloniki - June 25-28, 2007 28 family {  (  n,k  )} n,k   such that for all n,k  ,  (  n,k  ) solves all possible instances ({v 1,…,v n }, S) where v 1,…,v n, S are all k -bit natural numbers the number of neurons and of rules is polynomial in n and k some spurious spikes may be emitted before the result the delay subsystem: Uniform solution of Subset Sum

29. WMC 8 - Thessaloniki - June 25-28, 2007 29 The Comparison subsystem emits one spike if and only if the two numbers given in input (expressed in binary form) are equal the subsystem computes the following boolean function:

30. WMC 8 - Thessaloniki - June 25-28, 2007 30 Conclusions and future research nondeterministic SN P systems solve NP-complete problems such as 3-SAT deterministic SN P systems with simple regexp can be simulated by Turing machines with a polynomial slowdown deterministic SN P systems with general regexp solve NP-complete problems such as Subset Sum Hence it should be interesting to investigate: influence of regular unary languages on the computational power of SN P systems the role of maximal parallelism (and different flavors of it)

31. WMC 8 - Thessaloniki - June 25-28, 2007 31 Last slide ! Thanks for your attention !