wmc8 – thessaloniki, greece some applications of spiking neural p systems mihai ionescu 1 &...
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WMC8 – Thessaloniki, Greece
Some Applications of Spiking Neural P Systems
Mihai Ionescu1 & Dragoş Sburlan2
1 URV, Research Group on Mathematical Linguistics, Spain 2 Ovidius University, Faculty of Mathematics and Informatics, Romania
WMC8 – Thessaloniki, Greece
Outline
1. On Spiking Neural P Systems Definition. Example. Exhaustive use of the rules. Example.
2. Simulating Logical Gates and Circuits NOT gate Example of a circuit
3. A Sorting Algorithm Example
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Definition1:
Π = (O, σ1, …, σm, syn, i0)
where:
1. O = { a } (the alphabet of objects contains only one object);
1. On Spiking Neural P Systems
2. σ1, …, σm are neurons, identified by tuples σi = (ni,Ri), 1 ≤ i ≤ m, where:
a) ni ≥ 0
a2 a
1 M. Ionescu, Gh. Paun, T. Yokomori, Spiking Neural P Systems, Fundamenta Informaticae, 71, 2-3(2006), 279-308.
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a2 ab) Ri is a finite set of rules:
(1) E/ar → a; t, where E
is a regular expresion over O, r ≥ 1,
t ≥ 0;
(2) as → λ, for some
s ≥ 1, as ∉ L(E) for any rule of
type (1) from Ri
a2->a;0 (aa)*/a3->a;1
a->a;0
a2->λ
1. On Spiking Neural P Systems
Definition (continued):Π = (O, σ1, …, σm, syn, i0)
...
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a2 a
1. On Spiking Neural P Systems
a2->a; 0 (aa)*/a3->a;1
a->a;0
a2->λ
Definition (continued):Π = (O, σ1, …, σm, syn, i0)
...
c) syn ⊆ {1, 2, … m} x {1, 2, … m}, with (i,i) ∉ syn, for 1≤ i ≤ m;
d) i0 € {1, 2, … m} indicates the output neuron
Spik2Pm(ruled, consp, forgq)
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Example – Initial Configuration
a2
a2 → a;0 a → λ
a2
a2 → a;0 a2 → a;1
a2
a2 → a;0 a → λ
a2
a2 → a;0 a3 → λ
1
2
3
4
1. On Spiking Neural P Systems
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Example – Step 1 – used rules
SPIKE
a2
a2 → a;0 a → λ
a2
a2 → a;0 a2 → a;1
a2
a2 → a;0 a → λ
a2
a2 → a;0 a3 → λ
1
2
3
4
1. On Spiking Neural P Systems
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Example – Step 1 – result
a2a3
a2 → a;0 a → λ
a1a2
a2 → a;0 a2 → a;1
a1a3
a2 → a;0 a → λ
a1a2a3
a2 → a;0 a3 → λ
1
2
3
4
1. On Spiking Neural P Systems
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Example – Step 2 – used rules
a2
a2 → a;0 a → λ
a2
a2 → a;0 a2 → a;1
a2
a2 → a;0 a → λ
a3
a2 → a;0 a3 → λ
1
2
3
4
1. On Spiking Neural P Systems
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Example – Step 2 – result
a2
a2 → a;0 a → λ
a2 → a;0 a2 → a;1
a1
a2 → a;0 a → λ
a1a2
a2 → a;0 a3 → λ
1
2
3
4
1. On Spiking Neural P Systems
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Example – Step 3 – used rules
SPIKE
aa2 → a;0
a → λ
a2 → a;0 a2 → a;1
aa2 → a;0
a → λ
a2
a2 → a;0 a3 → λ
1
2
3
4
1. On Spiking Neural P Systems
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Example – Step 3 – result
a3
a2 → a;0 a → λ
a2 → a;0 a2 → a;1
a3
a2 → a;0 a → λ
a3
a2 → a;0 a3 → λ
1
2
3
4
3-1 = 2
1. On Spiking Neural P Systems
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Results:
NFIN = Spik2P1(rule*, cons1, forg0) = Spik2P1(rule*, cons*, forg*) = Spik2P2(rule*, cons*, forg*)
Spik2P*(rulek, consp, forgq) = NRE, for all k ≥ 2, p ≥ 3, q ≥ 3.
SLIN1 = Spik2P*(rulek, consp, forgq, bounds), for all k ≥ 3, p ≥ 3, q ≥ 3, and s ≥ 3
1. Spiking Neural P Systems
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Exhaustive use of the rules. Example
a5
a(aa)*/a → a;0a(aa)*/a2 → a;0
1. On Spiking Neural P Systems
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Exhaustive use of the rules. Example
a5
a(aa)*/a → a;0a(aa)*/a2 → a;0
1. On Spiking Neural P Systems
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Exhaustive use of the rules. Example
a(aa)*/a → a;0a(aa)*/a2 → a;0
1. On Spiking Neural P Systems
a5
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Exhaustive use of the rules. Example
a5
a(aa)*/a → a;0a(aa)*/a2 → a;0
1. On Spiking Neural P Systems
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Exhaustive use of the rules. Example
a
a(aa)*/a → a;0a(aa)*/a2 → a;0
1. On Spiking Neural P Systems
a2
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2. Simulating Logical Gates and Circuits
Codification:
– Boolean value 1 : aa– Boolean value 0 : a
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2. Simulating Logical Gates and Circuits
NOT Gate:
aa2/a → a;0
a3 → a;0
1
a/a → a;0 a2/a2 → a;0
2
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2. Simulating Logical Gates and Circuits
NOT Gate: 1→0
aaaa2/a → a;0
a3 → a;0
1
a/a → a;0 a2/a2 → a;0
2
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2. Simulating Logical Gates and Circuits
NOT Gate: 1→0
a2/a → a;0 a3 → a;0
1a
a/a → a;0 a2/a2 → a;0
2
a
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2. Simulating Logical Gates and Circuits
NOT Gate: 0→1
aaa2/a → a;0
a3 → a;0
1
a/a → a;0 a2/a2 → a;0
2
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2. Simulating Logical Gates and Circuits
NOT Gate: 0→1
a2/a → a;0 a3 → a;0
1aa
a/a → a;0 a2/a2 → a;0
2
aa
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2. Simulating Logical Gates and Circuits
Lemma 1:– Boolean AND gate can be simulated by SN P systems
using one neuron and no delay on the rules, in one step.
Lemma 2:– Boolean OR gate can be simulated by SN P systems using
one neuron and no delay on the rules, in one step.
Lemma 3:– Boolean NOT gate can be simulated by SNP systems using
two neurons, no delay on the rules, in two steps.
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2. Simulating Logical Gates and Circuits
Circuits.Example:
f:{0,1}4 → {0,1}
f(x1,x2,x3,x4)=(x1 Λ x2) V ¬(x3 Λ x4)
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2. Simulating Logical Gates and Circuits
Circuits.Example:f:{0,1}4 → {0,1}
f(x1,x2,x3,x4)=(x1 Λ x2) V ¬(x3 Λ x4)
AND AND
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2. Simulating Logical Gates and Circuits
Circuits.Example:f:{0,1}4 → {0,1}
f(x1,x2,x3,x4)=(x1 Λ x2) V ¬(x3 Λ x4)
AND AND
NOT
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2. Simulating Logical Gates and Circuits
Circuits.Example:f:{0,1}4 → {0,1}
f(x1,x2,x3,x4)=(x1 Λ x2) V ¬(x3 Λ x4)
AND AND
NOTSYNC
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2. Simulating Logical Gates and Circuits
Circuits.Example:f:{0,1}4 → {0,1}
f(x1,x2,x3,x4)=(x1 Λ x2) V ¬(x3 Λ x4)
AND AND
NOTSYNC
OR
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2. Simulating Logical Gates and Circuits
Circuits.Example:
AND AND
NOTSYNC
OR
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2. Simulating Logical Gates and Circuits
Theorem:
Every Boolean circuit α, whose underlying graph structure is a rooted tree, can be simulated by a SN P system, Πα, in linear time. Πα is constructed from SN P systems of type ΠAND, ΠOR and ΠNOT, by reproducing in the architecture of the neural structure, the structure of the tree associated to the circuit.
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2. Simulating Logical Gates and Circuits – Further Ideas
Arbitrary circuits, hence not necessary rooted tree.
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3. A (Simple) Sorting Algorithm
Example. 1,3,2 Initial configuration
a*/a→a;0 a*/a→a;0
a3→a;0a2→λa→λ
a*/a→a;0
a2→a;0a3→λa→λ
a→a;0a2→λa3→λ
a a3 a2
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3. A (Simple) Sorting Algorithm
Example. 1,3,2 Step 1
a*/a→a;0 a*/a→a;0
a3→a;0a2→λa→λ
a*/a→a;0
a2→a;0a3→λa→λ
a→a;0a2→λa3→λ
a2 a
a3 a3 a3
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3. A (Simple) Sorting Algorithm
Example. 1,3,2 Step 2
a*/a→a;0 a*/a→a;0
a3→a;0a2→λa→λ
a*/a→a;0
a2→a;0a3→λa→λ
a→a;0a2→λa3→λ
a
a2 a2 a2
a a a
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3. A (Simple) Sorting Algorithm
Example. 1,3,2 Step 3
a*/a→a;0 a*/a→a;0
a3→a;0a2→λa→λ
a*/a→a;0
a2→a;0a3→λa→λ
a→a;0a2→λa3→λ
a a a
a a2 a2
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3. A (Simple) Sorting Algorithm
Example. 1,3,2 Step 4
a*/a→a;0 a*/a→a;0
a3→a;0a2→λa→λ
a*/a→a;0
a2→a;0a3→λa→λ
a→a;0a2→λa3→λ
a a2 a3
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3. A (Simple) Sorting Algorithm
Example. 1,3,2 Step 4
a*/a→a;0 a*/a→a;0
a3→a;0a2→λa→λ
a*/a→a;0
a2→a;0a3→λa→λ
a→a;0a2→λa3→λ
a a2 a3
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2. A Sorting Algorithm
Theorem:SN P systems can sort a vector of natural numbers where each number is given as number of spikes introduced in the neural structure.
Remarks:
- time complexity: O(T), T is the magnitude of the numbers to be sorted - Further research: magnitude, improvements of time complexity,
number of neurons