Eric Bonnetier
email: eric.bonnetier@univ-grenoble-alpes.fr

Faouzi Triki
email: faouzi.triki@univ-grenoble-alpes.fr

Chun-Hsiang Tsou
email: chun-hsiang.tsou@univ-grenoble-alpes.fr

On the electro-sensing of weakly electric fish

Keywords: 1991 Mathematics Subject Classification. Primary: 35R30 Inverse problems, uniqueness, multifrequency, unique continuation, electrolocalization, stability estimates, weakly electric fish

In this paper we are interested in electro-sensing inverse problems. Our objective is to understand the electro-perception mechanism of weakly electric fish. These species of fish have the ability to recognize the environment around them in complete darkness by generating a weak electrical field at different frequencies, and perceiving the transdermal potential perturbation. Assuming that the target has a known conductivity profile, the electro-sensing inverse problem consists in recovering the shape and location of the target from measurements of the electric potential over the skin. Using an original spectral decomposition of the solution to the direct problem in terms of Poincaré variational eigenfunctions, we retrieve the Cauchy data of the electric potential over the fish skin corresponding to the case where we substitute the target by a perfect conductor with the same shape and position. We then identify the target from the recovered Cauchy data. We derive uniqueness and stability estimates to the considered electro-sensing inverse problem. The numerical validation of our theoretical approach is realized by reconstructing different targets using synthetic data in dimension two. The numerical experiments are conducted using gradient descent optimization algorithms.

1. Introduction 2. The Mathematical Model 3. Well-posedness of the direct problem 4. The Poincaré Variational Problem 5. Spectral decomposition of the solution u(x, ω) 6. Uniqueness and stability estimates 7. Numerical identification of the target fish 8. Conclusion 9. Acknowledgments References

Introduction

Some species of fish have the ability to recognize the environment around them by generating a weak electrical field at different frequencies. Their sensorial organs can generate electric fields, and their skins are sensitive to nearby electric fields. Their central nervous systems can identify objects by the response perceived on their skin to the electric impulses they generated. From the mathematical point of view these electric waves can be described by the Maxwell equations in the quasi-statistic regime, and understanding the behavior of the electrical fields in the neighborhood of the fish leads to an inverse conductivity problem with a finite number of excitations at different frequencies, and a finite number of boundary measurements.

To identify its targets the weak electric fish solves an inverse problem that has many similarities with the multifrequency electric impedance tomography. This latter method is a recent imaging technique of biological tissues where one tries to take advantage of the fact that most biological tissues exhibit frequency dependent conductivities, when excited by electric waves with frequencies ranging roughly around 10kHz [START_REF] Ammari | Identification of an inclusion in multifrequency electric impedance tomography[END_REF][START_REF] Habib Ammari | Spectroscopic imaging of a dilute cell suspension[END_REF][START_REF] Moller | Electric fishes: history and behavior[END_REF]. Experiments indicate that electric fish produce time harmonic electric waves at different frequencies, to gather information about their environment. Assuming that the target is a different fish, our objective here is to study how a weakly electric fish might identify it. The proposed inverse problem has many potential bio-inspired applications in underwater robotics [START_REF] Lebastard | Underwater robot navigation around a sphere using electrolocation sense and kalman filter[END_REF][START_REF] Curet | Aquatic manoeuvering with counter-propagating waves: a novel locomotive strategy[END_REF].

In this paper we are interested in the case where the electric fish and the target fish occupy respectively the bounded domains Ω ⊂ R d , d = 2, 3, and D ⊂ R d \ Ω. We assume that the conductivity distribution around the electric fish, is given by γ(x, ω) = 1 + (k(ω) -1)χ D (x), where χ D denotes the characteristic function of D. The conductivity of the background is normalized to be equal to one, k(ω) : R + → C \ R -denotes the conductivity of the target fish, and ω the frequency of the electric wave u produced by the electric fish. Recently, shape recognition and classification methods have been applied on small volume target fish [START_REF] Habib Ammari | Target identification using dictionary matching of generalized polarization tensors[END_REF][START_REF] Habib Ammari | Shape recognition and classification in electro-sensing[END_REF]. In the present work we adapt the method developed in [START_REF] Ammari | Identification of an inclusion in multifrequency electric impedance tomography[END_REF] to the weak electric fish inverse problems. We take advantage of multifrequency measurements and combine unique continuation techniques for meromorphic complex functions with a particular spectral representation of u that involves the eigenfunctions of the Poincaré variational operator, to reconstruct the conductivity map γ. The spectral decomposition can be written in the form u(x, ω) = u 0 (x) + u f (x, k(ω)), where the function u 0 is independent of the frequency ω, and represents the limiting solution when the contrast k(ω) tends to ∞. In fact u 0 is exactly the electric potential when we substitute the target fish by a perfect conductor with the same shape and position. Thus, it is not surprising that one can uniquely identify D from only one boundary measurement u 0 | ∂Ω . The problem of determining D from u 0 | ∂Ω has been studied by several authors (see e.g. [START_REF] Al Bukhgeim | Conditional stability in an inverse problem of determining a non-smooth boundary[END_REF][START_REF] Alessandrini | Optimal stability for inverse elliptic boundary value problems with unknown boundaries[END_REF]).

In this paper, we are concerned with the multifrequency inverse inclusion problem in unbounded domains. We first describe the mathematical model of electroreception. After introducing the weighted Sobolev space W 1,-1 (R d ), which is the natural space for solving the Laplace equation using variational techniques in unbounded domains, we prove the existence and uniqueness of solutions to the direct problem. We then set up a spectral decomposition using a modified Poincaré variational operator on W 1,-1 (R d ). Using the same techniques of reconstruction as in [START_REF] Ammari | Identification of an inclusion in multifrequency electric impedance tomography[END_REF], we derive the uniqueness of solutions to the inverse problem, and obtain stability estimates. To validate our theoretical approach we reconstruct different targets using synthetic data in dimension two.

This paper is organized as follows. In the next section, we present the mathematical model of the weakly electric fish. In section 3, we study the well-posedness of the partial differential system associated to the forward problem. In section 4, we introduce the Poincaré variational operator, and study its spectrum. Next, we derive the spectral decomposition of the unique solution to the forward problem in section 5. We derive uniqueness and stability estimates to the inverse problem in section 6. Section 7 is devoted to numerical illustrations of our theoretical results. A similar spectral decomposition is used to derive the frequency independent part u 0 in a bounded truncated numerical domain. We introduce an optimization scheme to reconstruct the position and shape of the target fish and present some numerical experiments.

The Mathematical Model

In this section we adapt the mathematical model of the weakly electric fish introduced by Ammari and al. in [START_REF] Habib Ammari | Modeling active electrolocation in weakly electric fish[END_REF]. Let Ω ⊂ R d , d = 2, 3, be an open connected and bounded region, which represents the electric fish. We suppose that ∂Ω is of class C 1,α for some α ∈ (0, 1]. We denote by Ω e the exterior of Ω, that is

Ω e := R d \ Ω.
The target fish D is also assumed to be an open connected region in Ω e . We assume that there exists δ > 0 such that dist(∂Ω, ∂D) > δ. We define a class of inclusions on which we study the uniqueness and stability of the inverse problem. We assume that the inclusion D contains the origin. Let b 1 = dist(0, ∂Ω) and let b 0 < b 1 . For m > 2 and ς ∈ (0, 1], we define the class of inclusions

D := D := {x ∈ R d : |x| < Υ(x), x = x |x| }; b 0 < Υ(x) < b 1 -δ; Υ C 2,ς ≤ m .
We assume that the conductivity is equal to 1 everywhere in Ω e except in the target D where the conductivity is equal to k(ω). We denote by γ(x, ω) := 1 + (k(ω) -1)χ D , the conductivity distribution in Ω e . Let u be the electric voltage produced by the electric fish, which satisfies the following system

           u = J s in Ω, div[γ(x, ω)∇u] = 0 in Ω e , ∂ ν u| -= 0 on ∂Ω, u| + -u| -= ξ∂ ν u| + on ∂Ω, |u| = O( 1 |x| d-1 ) as |x| → ∞, (2.1)
where the constant ξ > 0 is the effective fish skin thickness, where J s represents the electric current source generated by the fish organs, and where ∂ ν denotes the derivative with respect to the outward normal vector ν. We assume that J s can be written as the sum of Dirac masses

J s = M j=1 α j δ x (j) s , (2.2)
where x (j) s ∈ Ω and the electric charges α j ∈ R satisfy the neutrality condition

(2.3) M j=1 α j = 0.
Note that the equation (2.1) can also be rewritten as

(2.4)                    u = J s in Ω, u = 0 in Ω e \ ∂D, ∂ ν u| -= 0 on ∂Ω, u| + -u| -= ξ∂ ν u| + on ∂Ω, u| + = u| - on ∂D, ∂ ν u| + = k(ω)∂ ν u| - on ∂D, |u| = O( 1 |x| d-1 ) as |x| → ∞.
Assuming that k(ω) is a given continuous function, the inverse problem of weakly electric fish consists in recovering the shape and the position of the inclusion D from measurements of the voltages u(x, ω) on the boundary ∂Ω for ω ∈ (ω, ω), where ω, ω are fixed constants.

Well-posedness of the direct problem

In this section, we study the well-posedness of the direct problem (2.1). Firstly, we introduce the Sobolev space W 1,-1 (Ω e ). Secondly, we establish the existence and uniqueness of the solution to (2.1) in W 1,-1 (Ω e ).

3.1. Sobolev space W 1,-1 (Ω e ). In this subsection, we establish using variational techniques the existence and uniqueness for the Laplace equations with a Robin boundary condition in Ω e . To overcome the difficulties of integrating by parts in the unbounded exterior domain Ω e , we introduce the following weighted Sobolev spaces [START_REF] Nédélec | Acoustic and electromagnetic equations: integral representations for harmonic problems[END_REF].

Definition 3.1. (3.1) W 1,-1 (Ω e ) := u; u(x) (1 + |x| 2 ) 1/2 (log(2 + |x| 2 )) 3-d ∈ L 2 (Ω e ), ∇u ∈ L 2 (Ω e ) .
Remark 3.1. We make the following observations.

-From the boundedness of the weighted functions, W (Ω e ) := {u ∈ W 1,-1 (Ω e ); u| ∂Ω = 0}. -The functions v ∈ W 1,-1 (Ω e ) satisfy the following decay behavior far from

Ω v(x) = O( 1 |x| d-2 ), |x| → +∞. (3.2)
In particular, the constant functions belong to W 1,-1 (Ω e ) when d = 2.

We introduce the space

L 2 -1 (Ω e ) := u; u(x) (1 + |x| 2 ) 1/2 (log(2 + |x| 2 )) 3-d ∈ L 2 (Ω e ) .
We state now a useful compact embedding result involving these weighted Sobolev spaces.

Lemma 3.1. The embedding W 1,-1 (Ω e ) → L 2 -1
(Ω e ) is compact. The proof follows the same approach used in [START_REF] Robert | Compact sobolev imbeddings for unbounded domains with discrete boundaries[END_REF].

Proof. If the embedding W 1,-1 (R d ) → L 2 -1 (R d
) is compact, then it is also compact for any connected domain O ⊂ R d . So, without loss of generality, we only consider the case R d , d = 2.

Let u ∈ D(R 2 ), we have

(3.3) u(ρ, θ) = - ˆ∞ ρ ∂ r u(r, θ)dr,
where (r, θ) are the polar coordinates. Then,

(3.4) |u(ρ, θ)| 2 (1 + ρ 2 )(log(2 + ρ 2 ) 2 ) ≤ C 1 ρ 2 (log(ρ)) 2 ˆ∞ ρ |∂ r u(r, θ)u(r, θ)|dr, for ρ ≥ ρ 0 > 0.
Denoting by B c R the exterior of the ball B R = B(0, R), we have, ˆBc

R |u(x)| 2 (1 + |x| 2 ) log(2 + |x| 2 ) 2 dx ≤ C ˆ2π 0 dθ ˆ∞ R 1 ρ(log(ρ)) 2 ˆ∞ ρ |∂ r u(r, θ)u(r, θ)|drdρ ≤ C 1 log(R) u 2 W 1,-1 (R 2 ) (3.5)
By density, this inequality holds for any

u ∈ W 1,-1 (R 2 ). Let (u i ) i∈N be a bounded sequence in W 1,-1 (R 2 ). To prove that this sequence is a precompact in L 2 -1 (R 2 ), it is sufficient to show i) for ε > 0, there exists R such that u i L 2 -1 (B c R ) < ε for all i. ii) for any bounded part O ⊂ R 2 , (u i | O ) i∈N is a precompact.
The first point is a direct consequence of the previous inequality. The second point can be obtained by applying the Rellich-Kondrachov Theorem. Thus, the result of the lemma follows.

Well-posedness.

In this subsection we establish the existence and uniqueness to (2.1)

in W 1,-1 (Ω e ). Lemma 3.2. If d = 2, let α > 0. Then there exists β > 0 such that ∀u ∈ W 1,-1 (Ω e ), (3.6) ˆΩe |∇u| 2 dx + α ˆ∂Ω u 2 dσ ≥ β u 2 L 2 -1 (Ωe) .
Proof. We suppose by a contradiction argument that (3.6) is not true. The opposite of the statement of the lemma implies that there exists a sequence (u m ) m∈N with

u m ∈ W 1,-1 (Ω e ), u m W 1,-1 (Ωe) = 1, such that (3.7) ˆΩe |∇u m | 2 dx + α ˆ∂Ω u 2 m dσ < 1 m u m 2 L 2 -1 (Ωe) .
From the previous compact imbedding result, there is a subsequence, still denoted by (u m ) m∈N , which converges in L 2 -1 (Ω e ). Moreover, (3.7) implies that (∇u m ) m∈N converges to 0 in L 2 (Ω e ). So, the sequence (u m ) m∈N converges to a constant in W 1,-1 (Ω e ). We deduce again from relation (3.7) that the trace of u m converge to 0, thus the sequence (u m ) m∈N converges to 0, which is a contradiction with the normalization assumption.

Theorem 3.1. Let α > 0, let f be in the dual space (W 1,-1 (Ω e )) * , and let g ∈ H -1/2 (∂Ω). Then, the following Laplace equation with Robin boundary condition

(3.8) -u = f in Ω e , u -α∂ ν u = g on ∂Ω.
admits a unique weak solution in W 1,-1 (Ω e ).

Proof. A variational formulation to (3.8) is given by

∀v ∈ W 1,-1 (Ω e ), ˆΩe ∇u∇vdx + 1 α ˆ∂Ω uvdσ = ˆΩe f vdx + 1 α ˆ∂Ω gvdσ (3.9)
It follows from the trace theorem that the left-hand side is a bounded bilinear form and the right -hand side is a bounded linear form. We study separately the 2d and 3d cases for the coercivity of the bilinear form. i) If d = 2, the coercivity is a direct consequence of lemma 3.2. ii) If d = 3, we deduce from Theorem 2.5.13 in [START_REF] Nédélec | Acoustic and electromagnetic equations: integral representations for harmonic problems[END_REF], that there exists a constant c > 0 such that ∀u ∈ W 1,-1 (Ω e ), (3.10)

u W 1,-1 (Ωe) ≤ c ∇u L 2 (Ωe)
Then, the left-hand side bilinear form in (3.9) is coercive.Thus, the result follows from the Lax-Milgram Theorem.

From the previous theorem, and without considering the asymptotic behavior

u(x) = O( 1 |x| d-1
) as |x| → ∞, we next construct solutions to (2.1) up to a constant. In a second step in order to establish the uniqueness of the solution, we will give necessary conditions on the trace of the interior and exterior solutions u| ∂Ω-and u| ∂Ω+ respectively, such that the exterior solution satisfies the desired asymptotic behavior.

In the first step we split the equation (2.1) into two parts: the interior part

(3.11)    u i = J s in Ω, ∂ui ∂ν = 0 on ∂Ω, ´∂Ω u i dσ = 0,
and the exterior part

(3.12)    div[(1 + (k(ω) -1)χ D )∇u] = 0 in Ω e , u = ξ∂ ν u + u i + c u on ∂Ω, |u| = O( 1 |x| d-1 ) as |x| → ∞
where c u is a constant depending on u that has to be fixed. It is clear that there exists a unique solution u i to (3.11). For a given constant c u theorem (3.1) guarantee the existence and uniqueness of solution to (3.12) in W

= u i +c u . ∀v ∈ W 1,-1 (Ω e ), ˆΩe ∇u∇vdx + 1 α ˆ∂Ω (u -c u )vdσ = 1 α ˆ∂Ω u i vdσ.
We distinguish two different cases:

i) d = 2.
Since the constants belong to W 1,-1 (Ω e ), by taking v = 1 in the previous variational formulation we obtain the desired result. ii) d = 3.

Let B R be a ball centered at 0 with radius R > 0, and assume that R is large enough such that Ω ⊂ B R . Multiplying (3.12) by 1 and integrating by parts lead to

1 α ˆ∂Ω (u -c u )dσ = R 2 ˆS2 ∂ ν u(Rt)dt,
where S 2 is the unit sphere in R 3 . Since u harmonic in B c R , and decays as O( 1|x| 2 ) when |x| → +∞, by expanding it in the spherical harmonic basis [START_REF] Nédélec | Acoustic and electromagnetic equations: integral representations for harmonic problems[END_REF], we can easily obtain that

∂ ν u(Rt) = O( 1 R 3 ) as R → +∞ uniformly in t ∈ S 2 .
Consequently, the right hand term in the previous inequality tends to zero as R → +∞, which achieves the proof of the lemma.

Regarding the results of lemma 3.3, we can rewrite the problem (2.1) as follows

(3.14)    div[(1 + (k(ω) -1)χ D )∇u] = 0 in Ω e , u -1 |∂Ω| ´∂Ω udσ -ξ∂ ν u = u i on ∂Ω, |u| = O( 1 |x| d-1 ) as |x| → +∞,
where u i is the unique solution to the system (3.11). Let

W 1,-1 (Ω e ) = u ∈ W 1,-1 (Ω e ); |u| = O( 1 |x| d-1 ) as |x| → +∞ . (3.15)
We now state the main result of this subsection.

Theorem 3.2. The forward problem (2.1) has a unique solution in W 1,-1 (Ω e ).

Proof. Multiplying the equation (3.14) by v in W 1,-1 (Ω e ) and integrating by parts we obtain the following variational formulation

∀v ∈ W 1,-1 (Ω e ), ˆΩe ∇u∇vdx + 1 α ˆ∂Ω (u - ∂Ω u)(v - ∂Ω v)dσ = 1 α ˆ∂Ω u i vdσ,
where

ffl ∂Ω udσ = 1 |∂Ω| ´∂Ω udσ.
We claim that the left-hand bilinear form is coercive. In fact, when d = 3 we deduce from [START_REF] Nédélec | Acoustic and electromagnetic equations: integral representations for harmonic problems[END_REF], that the term ´Ωe ∇u∇vdx is coercive in W 1,-1 (Ω e ). When d = 2 it is proved in [START_REF] Bonnetier | On the spectrum of the poincaré variational problem for two close-to-touching inclusions in 2d[END_REF] that the term ´Ωe ∇u∇vdx is also coercive in W 1,-1 (Ω e ).

Then, by Lax-Milgram Theorem we obtain the desired result.

We introduce the fundamental solution Γ to the Laplace operator in R d . Definition 3.2. Denoting by ω d the area of the unit sphere in R d , the fundamental solution to the Laplace operator is given by

(3.16) Γ(x, y) = 1 2π log(|x -y|) d = 2, -1 ω d |x -y| 2-d d ≥ 3.
A direct consequence of theorem 3.1 is the following corollary.

Corollary 3.1. Let y ∈ Ω e be fixed. Then, there exists a unique solution to the system

(3.17)    G(x, y) = δ y in Ω e , G -ξ ∂G ∂ν = Γ -ξ ∂Γ ∂ν on ∂Ω, G -Γ ∈ W 1,-1 (Ω e ).
The singular function G(x, y) is the Green function of the Laplace operator in Ω e with the Robin boundary condition.

The Poincaré Variational Problem

In this section, we introduce the Poincaré variational problem by following the approach in [START_REF] Ammari | Identification of an inclusion in multifrequency electric impedance tomography[END_REF], [START_REF] Bonnetier | On the spectrum of the poincaré variational problem for two close-to-touching inclusions in 2d[END_REF]. We denote by W 1,-1 (Ω e ), the subspace of functions

v ∈ W 1,-1 (Ω e ) satisfying |v(x)| = O( 1 |x| d-1
) as |x| → ∞. It follows from [START_REF] Nédélec | Acoustic and electromagnetic equations: integral representations for harmonic problems[END_REF], [START_REF] Bonnetier | On the spectrum of the poincaré variational problem for two close-to-touching inclusions in 2d[END_REF] that the space W 1,-1 (Ω e ) endowed with the scalar product u, v W 1,-1 (Ωe) := ˆΩe ∇u∇vdx is a Hilbert space. Thus, the following bilinear form defines also an equivalent scalar product on W 1,-1 (Ω e ),

a(u, v) := ˆΩe ∇u∇vdx + 1 ξ ˆ∂Ω ūvdσ,
where ū := u -1 |∂Ω| ´∂Ω udσ.

For u ∈ W 1,-1 (Ω e ), we infer from the Riesz Theorem that there exist a unique T u ∈ W 1,-1 (Ω e ) such that for all v ∈ W 1,-1 (Ω e ), (4.1)

ˆΩe ∇T u∇vdx + 1 ξ ˆ∂Ω T uvdσ = ˆD ∇u∇vdx.
It is easy to check that the operator T : W 1,-1 (Ω e ) → W 1,-1 (Ω e ) is self-adjoint and bounded. The spectral problem for T reads as: Find (w, λ)

∈ W 1,-1 (Ω e ) × R, w = 0, such that ∀v ∈ W 1,-1 (Ω e ), (4.2)
λ ˆΩe ∇w∇vdx + λ ξ ˆ∂Ω wvdσ = ˆD ∇w∇vdx.
Integrating by parts, we obtain that an eigenfunction w is harmonic in D and in D := Ω e \ D. We have the transmission and boundary conditions on ∂D (4.3)

w| + ∂D = w| - ∂D , ∂ ν w| + ∂D = (1 - 1 λ )∂ ν w| - ∂D , w| ∂Ω -ξ∂ ν w| ∂Ω = 1 |∂Ω| ˆ∂Ω wdσ,
where w| ± ∂D (x) = lim t→0 w(x + t ± ν(x)) for x ∈ ∂D. In other words, w is a solution to (2.1) for k = 1 -1 λ < 0, and J s = 0. We define the space H as the space of harmonic functions in

K * D [φ](x) = ˆ∂D ∂ ∂ν x G(x, y)φ(y)dσ(y). Since D is smooth K * D is compact on H -1/2 (∂D) (see lemma 2. (4.6)
13 in [START_REF] Ammari | Polarization and moment tensors: with applications to inverse problems and effective medium theory[END_REF]). Since the function G and the Newtonian potential Γ have equivalent weak singularities as x → y (see Lemma 2.14 in [START_REF] Ammari | Reconstruction of small inhomogeneities from boundary measurements[END_REF] for a sketch of the proof and section 2.5.5 in [START_REF] Nédélec | Acoustic and electromagnetic equations: integral representations for harmonic problems[END_REF] for the regularity), the single layer potential S D satisfies the Plemjel jump relations across the boundary ∂D, that is,

(4.7) ∂ ν S D [φ] ± (x) = (± 1 2 I + K * D )[φ](x)
. We next characterize the spectrum of T .

Theorem 4.1. The Poincaré variational operator T has the following decomposition

(4.8) T = 1 2 I + K,
where K is a compact, self-adjoint operator. Let w ± n , n ≥ 1 be the eigenfunctions of T associated to the eigenvalues (λ ± n ) n≥1 , we have the following min-max principle

λ - 1 = min 0 =w∈H ´D |∇w(x)| 2 dx ´Ωe |∇w(x)| 2 dx + 1 ξ ´∂Ω | w| 2 dx λ - n = min 0 =w∈H ,w⊥w - 1 ,••• ,w - n-1 ´D |∇w(x)| 2 dx ´Ωe |∇w(x)| 2 dx + 1 ξ ´∂Ω | w| 2 dx = min Fn⊂H ,dim(Fn)=n max w∈Fn ´D |∇w(x)| 2 dx ´Ωe |∇w(x)| 2 dx + 1 ξ ´∂Ω | w| 2 dx , (4.9)
and similarly

λ + 1 = max 0 =w∈H ´D |∇w(x)| 2 dx ´Ωe |∇w(x)| 2 dx + 1 ξ ´∂Ω | w| 2 dσ λ + n = max 0 =w∈H ,w⊥w + 1 ,••• ,w + n-1 ´D |∇w(x)| 2 dx ´Ωe |∇w(x)| 2 dx + 1 ξ ´∂Ω | w| 2 dx = max Fn⊂H ,dim(Fn)=n min w∈Fn ´D |∇w(x)| 2 dx ´Ωe |∇w(x)| 2 dx + 1 ξ ´∂Ω | w| 2 dx . (4.10)
Proof. We define the operator R : H → H , for all v ∈ H , (4.11)

2( ˆΩe ∇Ru∇vdx + 1 ξ ˆ∂Ω Ruvdσ) = ˆD ∇u∇vdx - ˆD ∇u∇vdx - 1 ξ ˆ∂Ω ūvdσ.
We observe that K is bounded and self-adjoint. A direct calculation shows that (4.12)

T = 1 2 I + R.
It is shown in [START_REF] Ammari | Polarization and moment tensors: with applications to inverse problems and effective medium theory[END_REF] that the single layer potential S D : H -1/2 (∂D) → H 1/2 (∂D) is invertible in dimension three, and we can modify slightly S D to show the invertibility in dimension two.

Integrating by parts the right-hand side of (4.11), and using the jump relation (4.7), we find

ˆΩe ∇Ru∇vdx + 1 ξ ˆ∂Ω Ruvdσ = ˆ∂D K * D [S -1 D [u| ∂D ]]vdσ. Since K * D is compact the operator R is compact.
From Fredholm's alternative, T is a Fredholm operator of index 0, and the spectral decomposition (4.9), (4.10) follows the min-max principle.

4.13) u(x) = ∞ n=1 u ± n w ± n (x),
where

(4.14) u ± n = ˆΩe ∇u∇w ± n dx + 1 ξ ˆ∂Ω ū w± n dx.

Spectral decomposition of the solution u(x, ω)

Theorem 5.1. Let u(x, ω) be the unique solution to the system (2.1). Then, the following decomposition holds:

u(x, ω) = u 0 (x) + u f (x, k(ω)) = u 0 (x) + 1 ξ ∞ n=1 ´∂Ω u i w ± n dσ 1 + λ ± n (k(ω) -1) w ± n (x), x ∈ Ω e , (5.1)
where u 0 ∈ W 1,-1 (Ω e ) is the unique solution to the system:

(5.2)        v = 0 in Ω e \ D, ∇v = 0 in D, v - ffl ∂Ω vdσ = ξ∂ ν v + u i on ∂Ω, |v| = O( 1 |x| d-1 ) as |x| → ∞,
and u i ∈ L 2 (Ω) is the unique solution to the equation:

(5.3)    v = J s in Ω, ∂ ν v = 0 on ∂Ω, ´∂Ω vdσ = 0.
Proof. Let u be the unique solution to (5.4)

               v = J s in Ω, ∂ ν v| -= 0 on ∂Ω, ´∂Ω v| -dσ = 0, v = 0 in Ω e , v| + - ffl ∂Ω v| + dσ -v| -= ξ∂ ν v| + on ∂Ω, |v| = O( 1 |x| d-1 )
as |x| → ∞.

Denote u := u -u , u is therefore harmonic in D and in Ω e \ D. Moreover, it satisfies

u| + - ∂Ω u| + dσ = ξ∂ ν u| + on ∂Ω.
Then u ∈ H , and admits the following spectral decomposition:

(5.5)

u(x) = ∞ n=1 u ± n w ± n (x),
where

(5.6) u ± n = ˆΩe ∇u∇w ± n dx + 1 ξ ˆ∂Ω u w± n dσ.
By definition, u is the unique solution to

(5.7)

   -div(γ(x, ω)∇u) = div(γ(x, ω)∇u ) in Ω e , u -ξ∂ ν u = 1 |∂Ω| ´∂Ω udσ on ∂Ω, |u| = O( 1 |x| d-1 ) as |x| → ∞.
Multiplying the first equation in (5.7) by w ± n , and integrating by parts over Ω e , we have

(5.8) u ± n = ´Ωe div(γ(x, ω)∇u )w ± n dx 1 + λ ± n (k(ω) -1)
.

The term ´Ωe div(γ(x, ω)∇u )w ± n dx can be understood as a dual product between div(γ(x, ω)∇u ) ∈ (W 1,-1 (Ω e )) * and w ± n ∈ W 1,-1 (Ω e ). It can be simplified by means of integration by parts:

ˆΩe div(γ(x, ω)∇u )w ± n dx = - ˆΩe γ(x, ω)∇u ∇w ± n dx - ˆ∂Ω ∂ ν u w ± n dσ = - ˆΩe ∇u ∇w ± n dx -(k(ω) -1) ˆD ∇u ∇w ± n dx - 1 ξ ˆ∂Ω (u | + -u | -)w ± n dσ = - ˆΩe ∇u ∇w ± n dx -(k(ω) -1) ˆD ∇u ∇w ± n dx - 1 ξ ˆ∂Ω (u | + -u | -) w± n dσ = -[ 1 λ ± n + (k(ω) -1)] ˆD ∇u ∇w ± n dx + 1 ξ ˆ∂Ω u | -w ± n dσ, (5.9)
where u | -is exactly the unique solution to (7.5), it means, u | -= u i . Thus, it follows that (5.10)

u ± n = - ´D ∇u ∇w ± n dx λ ± n + ´∂Ω u i w ± n dσ ξ[1 + λ ± n (k(ω) -1)]
.

Let ũ0 ∈ H be the unique solution to the system (5.11)

       ũ0 = 0 in Ω e \ D, ∇ ũ0 = ∇u in D, ũ0 -ξ ∂ ũ0 ∂ν = 1 |∂Ω| ´∂Ω ũ0 dσ on ∂Ω, | ũ0 | = O( 1 |x| d-1 ) as |x| → ∞.
Since w ± n is an eigenfunction of T and ũ0 ∈ H , we have (5.12)

ˆD ∇u ∇w ± n dx = λ ± n [ ˆΩe ∇ ũ0 ∇w ± n dx + 1 ξ ˆ∂Ω ũ0 w± n dσ],
which gives (5.13)

u ± n = -[ ˆΩe ∇ ũ0 ∇w ± n dx + 1 ξ ˆ∂Ω ũ0 w± n dσ] + ´∂Ω u i w ± n dσ ξ[1 + λ ± n (k(ω) -1)]
.

On the other hand a simple calculations yields

(5.14) u -ũ0 = u 0 .
Combining (5.13) amd (5.14), the decomposition (5.1) follows.

Corollary 5.1. Let x ∈ ∂Ω, the function k → u f (x, k) is meromorphic in C. The poles of u f (x, k) are the real solutions to the equations

(5.15) 1 + λ ± n (k -1) = 0, n ≥ 1
where λ ± n are the eigenvalues of the Poincaré variational operator T . The poles of u f (x, k) in the previous corollary are given by k

± n = (1-1 λ ± n ) ∈ R -, and
they can be ordered as follows:

(5.16)

k - 1 ≤ k - 2 ≤ • • • < -1 < • • • ≤ k + 2 ≤ k + 1 < 0
We remark that -1 is the only accumulation point of the sequence of poles, it means k ± n tends to -1 as n → ∞.

Uniqueness and stability estimates

By modifying slightly the proofs in sections 3 and 4 of [START_REF] Ammari | Identification of an inclusion in multifrequency electric impedance tomography[END_REF], we establish the uniqueness of the inverse weakly electric fish, and the following stability estimate |u -ũ|.

Then, there exist constants C and τ ∈ (0, 1), such that the following estimate holds:

(6.1) |D D| ≤ C 1 ln(ε -1 ) τ .
Here, denotes the symmetric difference and the constants C and τ depend only on J s , Ω, D and Σ = {k(ω); ω ∈ (ω, ω)}.

Moreover, if the boundaries are analytic, we have a Hölder-type stability estimate. |u -ũ|.

Then, there exist constants C and τ ∈ (0, 1), such that the following estimate holds:

(6.2) |D D| ≤ Cε τ .
Here, denotes the symmetric difference and the constants C and τ depend only on J s , Ω, D and Σ.

These results show that the reconstruction of D is improving according to the regularity of its boundary. Precisely, the stability estimates vary from logarithmic to Hölder. They can also be extended to a larger class of as non-star shaped and non-connected domains. In this paper for the sake of simplicity we do not handle such general cases.

The proofs of Theorems 6.1 and 6.2 follow exactly the arguments presented in sections 3-4 of [START_REF] Ammari | Identification of an inclusion in multifrequency electric impedance tomography[END_REF], and we do not reproduce them here. They fundamentally rely on the spectral decompositions of u and ũ. Thus, concerning stability estimates, one of the main contributions of our paper is to show that such spectral decompositions also hold for the solutions to exterior problems, and therefore yield similar estimates. In the proof, the electric potential on the electric fish skin is split into two parts u(x, ω) = u 0 (x) + u f (x, k(ω)), and separates between k-dependent and k-independent parts. In fact u 0 corresponds to the response of the same inclusion filled with a perfect conductor, that is u 0 is the limit of u when k tends to infinity. Since k → u is meromorphic and its poles are simple, u f (x, k)| ∂Ω can be recovered using unique continuation for meromorphic functions. (u 0 , ∂ ν u 0 )| ∂Ω can in turn be retrieved from u(., ω)| ∂Ω , ω ∈ (ω, ω). The problem of determining D from the Cauchy data of u 0 has been studied by several authors (see for instance [START_REF] Al Bukhgeim | Conditional stability in an inverse problem of determining a non-smooth boundary[END_REF][START_REF] Alessandrini | Optimal stability for inverse elliptic boundary value problems with unknown boundaries[END_REF] and references therein).

Numerical identification of the target fish

In this section, we develop a numerical scheme to determine the position and shape of the target fish. Regarding the decay of the solution of (2.1) as |x| → +∞, we first reduce the computational domain by taking a Dirichlet boundary condition on a large disk B containing the fish Ω and the inclusion D. In fact we will show that a spectral decomposition similar to (5.1) holds in the truncated problem. Using this spectral decomposition, our numerical algorithm splits into two main steps. In the first step we retrieve the frequency depend part u f , and in the second step we reconstruct the inclusion from the Cauchy data of u 0 on ∂Ω obtained in the first step. Precisely, we recover the fish target by using an optimization scheme that minimizes a given energy functional on the boundary ∂Ω with respect to the shape of the inclusion.

7.1. The mathematical model in a truncated domain. In order to implement the numerical identification method, we need to reduce the system (2.1) to a bounded domain. Let B be a centered disk large enough such that it contains the electric fish Ω and the target fish D. We substitute (2.1) by the system

               ũ = J s in Ω, div[γ(x, ω)∇ũ] = 0 in B \ Ω, ∂ ν ũ| -= 0 on ∂Ω, ũ| + -ũ| -= ξ∂ ν ũ| + on ∂Ω, ũ = 0 on ∂B, ´∂B ∂ ν ũdσ = 0. (7.1)
The standard theory of elliptic partial differential equations shows that (7.1) admits a unique solution in

W 0 := u ∈ H 1 (B \ Ω) ∪ H 1 (Ω), u = 0 on ∂B, ˆ∂B ∂ ν ũdσ = 0 .
We introduce the equivalent scalar product on W 0 and the associated Poincaré variational operator T .

ã(u, v) := ˆB\Ω ∇u∇vdx + 1 ξ ˆ∂Ω ūvdσ.
For u ∈ H, by the Riez theorem, there exists aunique T u ∈ W 0 such that for all v ∈ W 0 , (

ˆB\Ω ∇ T u∇vdx + 1 ξ ˆ∂Ω T uvdσ = ˆD ∇u∇vdx. 7.2)
We introduce also the space H ⊂ W 0 of the functions which are harmonic in D and in B \ (Ω ∪ D), and which satisfy the zero Robin boundary condition u -ξ∂ ν u = 0 on ∂Ω.

Then, T has similar spectral elements denoted (w ± n , λ± n ) as T . We follow the analysis in the unbounded case to derive a similar spectral decomposition for ũ. Theorem 7.1. Let ũ(x, ω) be the unique solution to the system (7.1). Then the following decomposition holds, for x ∈ B \ Ω, ũ(x, ω) = ũ0 (x) + u f (x, k(ω))

= ũ0 (x) + 1 ξ ∞ n=1 ´∂Ω ũi w± n dσ 1 + λ± n (k(ω) -1)
w± n (x), (7.3) where u 0 ∈ W 0 is the unique solution to the system:

(7.4)            v = 0 in B \ (Ω ∪ D), ∇v = 0 in D, v - ffl ∂Ω vdσ = ξ∂ ν v + u i on ∂Ω, v = 0 on ∂B, ´∂B ∂ ν vdσ = 0,
and u i ∈ L 2 (Ω) is the unique solution to the equation:

(7.5)    v = J s in Ω, ∂ ν v = 0
on ∂Ω, ´∂Ω vdσ = 0. 7.2. Retrieval of the frequency independent part. We consider M frequencies of measurements ω 1 , ω 2 , • • • , ω M . Since 1/2 is the unique accumulation point of the eigenvalues ( λ± n ) n∈N , we only consider the N f first eigenvalues as unknowns, and we approximate the others eigenvalues by 1/2. In fact it has been shown in [START_REF] Miyanishi | Eigenvalues and eigenfunctions of double layer potentials[END_REF] that if D is C β with β ≥ 2 then for any α > -2β + 3, we have

| λ± n -1/2| = o(n α ), n → +∞.
If the boundary of D is C ∞ smooth, then the eigenvalues will decay faster than any power. Recently H. Kang and his collaborators have shown the exponential convergence of the eigenvalues in the case of analytic inclusions [START_REF] Ando | Exponential decay estimates of the eigenvalues for the neumann-poincar\'e operator on analytic boundaries in two dimensions[END_REF]. Therefore, we make the following approximation, for x ∈ Ω, 1 ≤ p ≤ M ,

(7.6) ũ(x, ω p ) ≈ ũ0 (x) + N f n=1 v ± n (x) 1 + λ± n (k(ω p ) -1) + 2 k(ω p ) + 1 v N f +1 (x),
where

v ± n (x) = 1 ξ ˆ∂Ω ũi w± n dσ w± n (x),
and

v N f +1 (x) = 1 ξ n>N f ˆ∂Ω ũi w± n dσ w± n (x).
By a simple integration by parts, we have, for all n ∈ N,

(7.7) 1 ξ ˆ∂Ω ũi w± n dσ = ˆB\Ω ∇ũ ∇ w± n dx + 1 ξ ˆ∂Ω ũ w± n dσ,
where ũ is the unique solution in W 0 to (7.8)

       ũ = 0 in B \ Ω, ũ - ffl ũ dσ -ξ∂ ν ũ = ũi on ∂Ω, ũ = 0 on ∂B ´∂B ∂ ν ũ dσ = 0.
In other words, the function ∞ n=1 v ± n is the orthogonal projection of the function ũ on the space H .

On the other hand ũ0 satisfies, for all n ∈ N, n ≥ 1

ˆB\Ω ∇ũ 0 ∇ w± n dx + 1 ξ ˆ∂Ω ũ0 w± n dσ = ˆB\(Ω∪D) ∇ũ 0 ∇ w± n dx + 1 ξ ˆ∂Ω ũ0 w± n dσ = - ˆ∂Ω ũ0 ∂ ν w± n dσ + 1 ξ ˆ∂Ω û0 w± n dσ - ˆ∂D ũ0 ∂ ν w± n dσ = -ũ 0 ˆ∂D ∂ ν w± n dσ = 0. (7.9)
As ũ -ũ0 ∈ H , the orthogonal projection of ũ on the space H is ũ -ũ0 .

Then, the formula (7.6) becomes

ũ(x, ω p ) ≈ k(ω p ) -1 k(ω p ) + 1 ũ0 (x) + 2 k(ω p ) + 1 ũ (x) + N f n=1 ( 1 1 + λ± n (k(ω p ) -1) - 2 k(ω p ) + 1 )v ± n (x). (7.10)
The previous equation can be formulated with the following matrix form.

For x ∈ ∂Ω, we define the vectors Ũ (x, ω 1 , . . . , ω M ) = (ũ(x, ω j )) ,

V (x) = ũ0 (x), v + 1 (x), v - 1 (x), . . . , v + N f (x), v - N f (x) , and
L( λ ± 1 , . . . , λ ± N f , ω 1 , . . . ω M ) = (L i ) 1≤i≤M , where L i (x) = q 0 (ω i ), q( λ + 1 , ω i ), q( λ - 1 , ω i ), • • • , q( λ + N f , ω i ), q( λ - N f , ω i ) .
Here

ũ(x, ω) = ũ(x, ω) - 2 k(ω) + 1 ũ (x), q 0 (ω) = k(ω) -1 (k(ω) + 1)
, and q( λ, ω) = 1

1 + λ(k(ω) -1) - 2 k(ω) + 1 .
The matrix formulation of (7.10) becomes then

(7.11) Ũ T (x, ω 1 , ω 2 , • • • , ω M ) ≈ L( λ ± 1 , . . . , λ ± N f , ω 1 , . . . ω M )V T (x)
. So, the vector V can be obtained by the formula, (7.12)

V T (x) ≈ (L T L) † L T Ũ T (x, ω 1 , • • • ω M ).
where (L T L) † is the pseudo-inverse of the matrix L T L. The conditioning of the matrix (L T L) † depends on the distance between the frequency sampling values ω 0 . The approximate ũ0 (x) is then given by the first coefficient of the vector V (x).

7.3. Identification of the target fish. In this section, we consider a numerical scheme to identify the inclusion D ∈ D from the knowledge of ũ0 | ∂Ω recovered in the previous section. The scheme is based on minimizing the functional

J(D) = 1 2 ˆ∂Ω P i=1 |ũ 0 -ũ(i) meas | 2 dσ + ε T ˆ∂D dσ,
where ũ is the solution to (7.4). P designs the total number of measurements, we take here P = 10. For 1 ≤ i ≤ 10, we use the fact that the electric fish can swim around the target, ũ(i) meas is the measured Dirichlet data corresponding to the case when the electric fish is located at the i-th position. These quantities are obtained in the previous step by retrieving the frequency dependent part from the multifrequency measurements. The term ε T ´∂D dσ represents the Tikhonov regularization.

7.3.1. Shape derivative. Let D ε be the perturbed domain, given by (7.13)

∂D ε = {x : x = x + εh(x)ν(x), x ∈ ∂D},
where h ∈ C 1 (∂D) and ν denote the unit outward normal vector.

Theorem 7.2. We denote by ũ0 and by ũ0,ε respectively the solutions to the equation (7.4) with the inclusion D (resp. D ε ). Then, the following relation holds, for x ∈ ∂Ω,

(7.14) ũ0,ε (x) = ũ0 (x) + εũ h (x) + o(ε),
where ũh is the solution to the following equation

(7.15)                v = 0 in B \ (Ω ∪ D), ∇v = 0 in D, v = -h∂ ν ũ0 | + on ∂D, v - ffl vdσ -ξ∂ ν v = 0 on ∂Ω, v = 0 on ∂B, ´∂B ∂ ν vdσ = 0,
Proof. The result can be proved using layer potential techniques by following the proof of Theorem 3.1 in [START_REF] Habib Ammari | Numerical determination of anomalies in multifrequency electrical impedance tomography[END_REF]. 7.3.2. Gradient descent algorithms. We assume that our domain D is star shaped, centered at the origin and that its boundary ∂D can be described by the Fourier series:

(7.16) ∂D = {r(θ) cos θ sin θ |θ ∈ [0; 2π)}, r = N n=-N c n f n , where C =      c -N c -N +1 . . . c N      ∈ R 2N +1 , f n (θ) = cos(nθ) for 0 ≤ n ≤ N and f n (θ) = sin(nθ) for -N ≤ n < 0.
Using (7.15) and integration by parts, we deduce the expressions of the shape derivatives corresponding to each Fourier coefficient, for -N ≤ n ≤ N , (7.17)

∂J ∂c n = ˆB\Ω∪D ∇w∇ũ h dx + ε T ˆ∂D κhdσ,
where h(θ) = f n (θ) cos θ sin θ , ν , κ represents the curvature of ∂D and w is the solution of the following equation

(7.18)        w = 0 in B \ Ω ∪ D, ∂ ν w = 0 on ∂D, ∂ ν w = ũ -ũmeas on ∂Ω, w = 0 on ∂B.
Now, we are ready to introduce our iterative algorithm:

(1) Calculate the interior solution ũi .

(2) Using (7.12), we retrieve the frequency independent part ũ0 from the multifrequency measurements. We get the Dirichlet data (û

4 . 1 .

 41 D and in D which are continuous across ∂D, satisfy a Robin boundary condition ū = ξ∂ ν u on ∂Ω and the asymptotic behavior |u| = O( 1 |x| d-1 ) as |x| → ∞, and with a finite energy semiWe remark that T H ⊂ H , thus T defines a bounded operator from H into H . We define now, the single layer potential S D : H -1/2 (∂D) → H and the Neumann-Poincaré operator K * D : H -1/2 (∂D) → H -1/2 (∂D) by Definition Let φ ∈ H -1/2 (∂D), we define, for x ∈ Ω e (4.5) S D [φ](x) = ˆ∂D G(x, y)φ(y)dσ(y), and, for x ∈ ∂D,

Corollary 4 . 1 .

 41 Let u ∈ H , then u has the spectral decomposition.

(

Theorem 6 . 1 .

 61 Let D and D be two inclusions in D. Denote by u and ũ respectively the solution of (2.1) with the inclusion D (resp. D). Let ε = sup x∈∂Ω,ω∈[ω,ω]

Theorem 6 . 2 .

 62 Assume that d = 2, and let D and D be two analytic inclusions in D. Denote by u and ũ respectively the solution of (2.1) with the inclusion D (resp. D). Let ε = sup x∈∂Ω,ω∈[ω,ω]

  meas ) 1≤i≤P .(3) Chose an initial domain D 0 . (4) For each iteration, j > 0:(a) Using (7.4) associated to the domain D j for which the boundary ∂D j is obtained from(7.16). (b) Calculate the shape derivatives ∂J ∂cn for all -N ≤ n ≤ N by (7.17). (c) Update the parameters of the domain C j+1 = C j -α∇ C J(C j ) with α > 0. (d) If the updated domain boundary touches ∂Ω or if J(C j+1 ) > J(C j ), reduce the size of α. (5) When |∇J(C j )| becomes smaller than a fixed threshold, we stop.

Figure 1 .

 1 Figure 1. Interior solution to the equation (7.1)

  (a) without inclusion (b) inclusion presents

Figure 2 .Figure 3 .

 23 Figure 2. Solutions to (7.1) in the presence/absence of inclusion

  Evolution of relative symmetric differences

Figure 4 .

 4 Figure 4. Reconstruction of an ellipse

Figure 5 .

 5 Figure 5. Reconstruction of a triangle

Figure 6 .

 6 Figure 6. Reconstruction of a star-shape domain

Figure 7 .

 7 Figure 7. Reconstruction of a displaced disk

  1,-1 (Ω e ). Next, we show that the decay O( 1 |x| d-1 ) of the exterior solution as |x| → +∞ will uniquely determine the constant c u . Lemma 3.3. Assume that c u is given. Then, the decay O( 1 Proof. Recall the variational formulation for (3.8) with f = 0 and g

			|x| d-1 ) at infinity
	implies (3.13)	1 |∂Ω| ˆ∂Ω	udσ = c u .

Table 1 .

 1 Errors between ũ0reconstruct and ũ0 • ellipse triangle star displaced disk |D i D target |/|D target | 0.07128 0,1988 0.4232 0.16805

	•	ellipse triangle	star	displaced disk
	measure 1 0.10132 0.06892 0.01411	0.05330
	measure 2 0.07466 0.07384 0.03364	0.05010
	measure 3 0.01637 0.05302 0.07036	0.04712
	measure 4 0.00910 0.04580 0.05242	0.04388
	measure 5 0.03460 0.06548 0.02822	0.03792
	measure 6 0.06959 0.08078 0.03558	0.03408
	measure 7 0.05795 0.06175 0.02743	0.03585
	measure 8 0.02060 0.03240 0.03819	0.03891
	measure 9 0.00675 0.02760 0.07660	0.04054
	measure 10 0.03074 0.03623 0.05292	0.04663

Table 2 .

 2 Relative symmetric difference

Numerical experiments. The setting of all numerical tests is as follows:

• We use FreeFem++ for our numerical experiments [START_REF] Hecht | New development in freefem++[END_REF].

• B is a centered ball with the radius R B = 30.

• Ω is an ellipse defined by the equation:

• We assume that the source function is given by a dipole type source, i. e., in the formula (2.2), M = 2, α 1 = 100, α 2 = -100, x

= (-3, 1), and x (2) s = (-3, -1). The interior solution is illustrated in Figure 1.

• We take advantage of the fact that the electric fish can swim around the target. We chose P = 10 different locations to measure the multifrequency electric potentials, those 10 positions are equi-distributed on the circle with a radius 15. Figure 3 shows the 4 locations of them. • The multifrequency conductivity follows the model k(ω) = k r + inω 0 with k r = 5, ω 0 = 0.5 and n are integers from 0 to 7 [START_REF] Habib Ammari | Modeling active electrolocation in weakly electric fish[END_REF]. • Only the first eigenvalues are taken into consideration, and their apriori estimations are settled as

respectively in all cases. • The initial estimation of domain D: a centered disk with a radius 1 2 . • We consider the first 9 Fourier coefficients: N = 9.

• We set the Tikhonov regularization coefficient ε T = 0.01.

• We use P1 finite elements for the numerical resolution of the PDEs.

• At each iteration, we remesh the domain to adapt to the new predicted shape of the domain. • The algorithm stops if |∇J| < 10 -6 or if the number of iterations exceeds 100. We present here several numerical simulations of the algorithm. We first present in Table [START_REF] Robert | Compact sobolev imbeddings for unbounded domains with discrete boundaries[END_REF], errors in the reconstruction method of ũ0 . Here, errors are the L 2 -norm of the difference ũ0reconstruct -ũ0 :

We show in the following figures the targets and the reconstruction result. We calculate also the relative symmetric difference |D i D target |/|D target | during the iterations, and we plot the curves of the symmetric difference in terms of iteration number. We finally give the relative symmetric difference of each case in Table 2.

Conclusion

Using the mathematical model of the weakly electric fish introduced by Ammari and al. in [START_REF] Habib Ammari | Modeling active electrolocation in weakly electric fish[END_REF], we addressed the question of existence and uniqueness of the solution to the direct problem. We have established the uniqueness and stability estimates of the fish target recovery problem using a single measurement, under the assumption that its conductivity profile is known. We provided a gradient descent algorithm to determine the shape and location of the target.

Acknowledgments

The authors were partially supported by the AGIR-HOMONIM grant from Université Grenoble-Alpes, and by the Labex PERSYVAL-Lab (ANR-11-LABX-0025-01).


J Gerard Wolff

Dr Gerry Wolff

Strengths and potential of the SP theory of intelligence in general, human-like artificial intelligence

Keywords: artificial intelligence, information compression, SP theory of intelligence;

This paper first defines general, human-like artificial intelligence (GHLAI) in terms of five principles. In the light of the definition, the paper summarises the strengths and potential of the SP theory of intelligence and its realisation in the SP computer model, outlined in an appendix, in three main areas: the versatility of the SP system in aspects of intelligence; its versatility in the representation of diverse kinds of knowledge; and its potential for the seamless integration of diverse aspects of intelligence and diverse kinds of knowledge, in any combination. There are reasons to believe that a mature version of the SP system may attain full GHLAI in diverse aspects of intelligence and in the representation of diverse kinds of knowledge.

Introduction

The SP theory of intelligence and its realisation in the SP computer model, outlined in Appendix A, is the product of a programme of research aiming to simplify and integrate observations and concepts across artificial intelligence, mainstream computing, mathematics, and human learning, perception, and cognition.

1 Amongst other things, the SP system exhibits features of general, humanlike artificial intelligence (GHLAI). The purpose of this paper is to describe what GHLAI means, and to summarise the strengths and potential of the SP system in the quest for GHLAI.

The next section describes the concept of GHLAI and how it relates to other concepts in AI and mainstream computing. Then the five sections after that describe the strengths and potential of the SP system with respect to five features of GHLAI.

GHLAI and related concepts

The idea that something may have GHLAI or be a GHLAI derives from the concept of a universal framework for the representation and processing of diverse kinds of knowledge (UFK) [24, Section III] but gives weight to the concept of general, human-like intelligence.

The GHLAI concept is similar to the concepts of 'universal artificial intelligence' (UAI) (see, for example, [START_REF] Hutter | One decade of universal artificial intelligence[END_REF][START_REF] Hutter | Universal Artificial Intelligence: Sequential Decisions based on Algorithmic Probability[END_REF]) and 'artificial general intelligence' (AGI) (see, for example, [START_REF]Artificial General Intelligence[END_REF]), but with differences in emphasis, relating to issues that appear to be important.

The concept of GHLAI may at first sight seem to be redundant since it has been recognised for some time that all kinds of computing may be understood in terms of the workings of a UTM or ideas which are recognised as equivalent such as Post's 'canonical system' [START_REF] Post | Formal reductions of the general combinatorial decision problem[END_REF], or Church's 'lambda calculus' [START_REF] Church | The Calculi of Lamda-Conversion[END_REF], or indeed the many conventional computers that are in use today. For the sake of brevity these will be referred to collectively as CCs, short for "conventional computers".

The suggestion here is that, by definition:

1. A GHLAI should demonstrate human-like intelligence without the need for AI-related programming (by contrast with a CC).

2. It should be able to represent any kind of knowledge without the need for additional programming (by contrast with a CC).

3. It should facilitate the seamless integration of diverse aspects intelligence and diverse kinds of knowledge, in any combination.

4. It should, like a CC, provide for any kind of processing within the limits set by computational complexity.

5. It should do these things efficiently.

No attempt has been made here to define 'human [-like] intelligence'. This is because: 1) it would be difficult or impossible to reach agreement about any such attempt at a definition; 2) by contrast, there is likely to be a large measure of agreement about specific features of human intelligence; so that 3) any GHLAI may be assessed in terms of the specific features that it models, concentrating mainly on those that appear to be most challenging.

Versatility in aspects of intelligence via the powerful concept of SP-multiple-alignment

As noted in Appendix A.3, the concept of SP-multiple-alignment has the potential to be the 'double helix' of intelligence, the key to the versatility of the SP system in aspects of intelligence, as summarised here:

• Unsupervised learning via the processing of SP-multiple-alignments. Unsupervised learning appears to be the most fundamental form of learning, with potential as a foundation for other forms of learning such as reinforcement learning, supervised learning, learning by imitation, and learning by being told.

• How other aspects of intelligence flow from the building of SP-multiplealignments. By contrast with the way in which the SP system models unsupervised learning via the processing of already-constructed 'good' SP-multiple-alignments, other aspects of intelligence derive from the building of SP-multiple-alignments (Appendix A.3). These other aspects of intelligence include: the analysis and production of natural language; pattern recognition that is robust in the face of errors in data; pattern recognition at multiple levels of abstraction; computer vision [START_REF] Wolff | Application of the SP theory of intelligence to the understanding of natural vision and the development of computer vision[END_REF]; best-match and semantic kinds of information retrieval; several kinds of reasoning (more under the next bullet point); planning; and problem solving ([19, Chapters 5 to 8], [START_REF] Wolff | The SP theory of intelligence: an overview[END_REF]). • Versatility in the representation of knowledge. The SP system also has versatility in the representation of diverse kinds of knowledge, as summarised in Section 4, below.

Generality in artificial intelligence

Conceivably, the features of human-like intelligence mentioned above are the full extent of the SP system's potential to imitate what people can do. But for reasons given here, it seems likely that the SP system is not constrained in that way: that it provides a relatively firm foundation for the development of GHLAI. The close connection that is known to exist between information compression and concepts of prediction and probability [START_REF] Solomonoff | A formal theory of inductive inference[END_REF][START_REF] Solomonoff | The discovery of algorithmic probability[END_REF][START_REF] Li | An Introduction to Kolmogorov Complexity and Its Applications[END_REF], the central role of information compression in the SP-multiple-alignment framework (Appendix A.3), and the versatility of the SP-multiple-alignment framework in the representation of knowledge (Section 4) and aspects of intelligence (Section 3), suggest that the SP system as it is now will provide a relatively firm foundation for the development of GHLAI.

As noted in Section ??, the SP system differs in several ways from other AI-related systems and, in its strengths and potential, it has advantages compared with alternatives.

How the SP system differs from a CC in aspects of intelligence

With regard to the modelling of human-like intelligence, the main attraction of the SP system compared with CCs, is its versatility in diverse aspects of intelligence (Section 3) and its potential for the seamless integration of diverse aspects of intelligence and diverse kinds of knowledge, in any combination (Section 5, below), both these things without the need for ai-related programming.

Unless a CC has been specifically programmed with SP capabilities-in which case it would be an SP system, not a CC-it would be lacking in the above-mentioned capabilities, and, arguably, for that reason, is likely to fall short of general human-like artificial intelligence.

Versatility in the representation of knowledge via the powerful concept of SP-multiplealignment

Although SP-patterns are not very expressive in themselves, they come to life in the SP-multiple-alignment framework. Within that framework, they may serve in the representation of several different kinds of knowledge, including: the syntax of natural languages; class-inclusion hierarchies (with or without cross classification); part-whole hierarchies; discrimination networks and trees; if-then rules; entity-relationship structures [20, Sections 3 and 4]; relational tuples (ibid., Section 3), and concepts in mathematics, logic, and computing, such as 'function', 'variable', 'value', 'set', and 'type definition' ([19, Chapter 10], [25, Section 6.6.1], [START_REF] Wolff | Software engineering and the SP theory of intelligence[END_REF]Section 2]). With the addition of two-dimensional SP patterns to the SP computer model, there is potential for the SP system to represent such things as: photographs; diagrams; structures in three dimensions [22, Section 6.1 and 6,2]; and procedures that work in parallel [23, Sections V-G, V-H, and V-I, and Appendix C].

Generality in the representation of knowledge

As with the SP system's generality in aspects of intelligence (Section 3.1), it seems likely that the SP system is not constrained to represent only the forms of knowledge mentioned above: that it provides a relatively firm foundation for the development of human-like versatility in the representation of diverse forms of knowledge.

The generality of information compression as a means of representing knowledge in a succinct manner, the central role of information compression in the SP-multiple-alignment framework, and the versatility of that framework in the representation of knowledge, suggest that the SP system may prove to be a means of representing all the kinds of knowledge that may be processed by a GHLAI.

A qualification here is that, in the same way that people can, if necessary, learn and use some over-complex and obscure forms of knowledge, it seems likely that, because information compression lies at the heart of the SP system, a mature version of the system will, like people, have a preference for forms of representation that are relatively simple and direct, in accordance with the principle of efficiency, outlined in Section 7.

How the SP system differs from a CC in the representation of knowledge

With regard to the representation of knowledge, attractions of the SP system compared with CCs are:

• The SP system provides for the succinct representation of knowledge via the powerful concept of SP-multiple-alignment. By contrast, SPmultiple-alignments are not part of CCs and are barely recognised as guides or principles for the representation of knowledge in CCs.1

• The versatility of the SP system in the representation of knowledge is combined with some constraint-knowledge must be represented with SP patterns and processed via the building and manipulation of SP-multiple-alignments (Section 4)-and that constraint seems to be largely responsible for how the system facilitates the seamless integration of different kinds of knowledge (Section 5).

By contrast, the representation of knowledge in a CC is a free-for-all: any kind of structure that may be represented with arrays 0s and 1s is accepted. This relative lack of discipline seems to be largely responsible for the excessive number of formats and formalisms in computing today and the many incompatibilities that exist amongst computer applications today.

The need for some discipline in how computing is done is not a new idea. In the early days of computing by machine, there was much 'spaghetti programming' with the infamous "goto" statement, leading to the creation of programs that were difficult to understand and to maintain. This problem was largely solved by the introduction of 'structured programming' (see, for example, [START_REF] Jackson | Principles of Program Design[END_REF]). Later, it became apparent that there could be more gains in the comprehensibility and maintainability of software via the introduction of 'object-oriented' programming and design, modelling software on real-world objects and classes of object.

Seamless integration

In connection with the potential of the SP system as a GHLAI, an important third feature of the system, alongside its versatility in aspects of intelligence and its versatility in the representation of knowledge, is that there is clear potential for the SP system to provide seamless integration of diverse kinds of knowledge and diverse aspects of intelligence, in any combination. This is because diverse aspects of intelligence and diverse kinds of knowledge all flow from a single coherent and relatively simple source: SP patterns within the SP-multiple-alignment framework.

In this respect, there is a sharp contrast between the SP system and the majority of other AI systems, which are either narrowly specialised for one or two functions or, if they aspire to be more general, are collections or kluges of different functions, with little or no integration. 2 This point is important because it appears that seamless integration of diverse aspects of intelligence and diverse kinds of knowledge, in any combination, are essential pre-requisites for human levels of fluidity, versatility and adaptability in intelligence.

Generality in computing

In addition to the potential generality of the SP system in AI, its potential generality appears to extend to computations of all kinds, not merely those with an AI flavour:

2 Although Allen Newell called for the development of Unified Theories of Cognition [START_REF] Newell | Précis of Unified Theories of Cognition[END_REF][START_REF]Unified Theories of Cognition[END_REF], and researchers in 'Artificial General Intelligence' are aiming for a similar kind of integration in AI, it appears that none of the resulting systems are fully integrated: "We have not discovered any one algorithm or approach capable of yielding the emergence of [general intelligence]." [3, p. 1].

• The SP system may model a NAND gate, as described below.

• Within the SP system, the output of any NAND gate may become the input for any other, as described for 'composite functions' in [START_REF] Wolff | Unifying Computing and Cognition: the SP Theory and Its Applications[END_REF]Section 10.4.3].

• Hence, in principle, NAND gates within the SP system may be connected in any desired combination.

• Since it is widely accepted that, in principle, the computational heart of any general-purpose computer may be constructed entirely from NAND gates [START_REF] Nisan | The Elements of Computing Systems: Building a Modern Computer from First Principles[END_REF], it appears that the SP system has the generality of a CC.

With regard to the first point above, Figure 1 (a) shows four SP-patterns which, together, define the input-output relations in a NAND gate. The nonnumeric symbols are not essential in this context but they have mnemonic value as explained in the caption to the figure, and those kinds of markers are likely to be needed in other context. When the SP computer model is run with those four SP-patterns in the repository of Old SP-patterns, and with the New SP-pattern 'ng i1 1 #i1 i2 0 #i2 #ng', the best SP-multiple-alignment found by the model is the one shown in Figure 1 (b), with the New SP-pattern in row 0 and one of the Old SP-patterns in row 1.

ng i1 0 #i1 i2 0 #i2 op 1 #op #ng ng i1 0 #i1 i2 1 #i2 op 1 #op #ng ng i1 1 #i1 i2 0 #i2 op 1 #op #ng ng i1 1 #i1 i2 1 #i2 op 0 #op #ng (a) 0 ng i1 1 #i1 i2 0 #i2 #ng 0 | | | | | | | | 1 ng i1 1 #i1 i2 0 #i2 op 1 #op #ng 1 (b)
Here, the New pattern may be regarded as input to the NAND gate, with the values '1 0'. The corresponding output may be read off as '1', between the symbols 'op' and '#op' in the Old pattern in row 1. Thus the input '1 0' has produced the output '1', in accordance with any correctly-working NAND gate.

6.1 What about things that the SP system can't do, except with some kind of 'programming' or 'training' ?

In considering the possibility that the SP system might be developed into a GHLAI is that, while the mechanisms for the building and processing of SPmultiple-alignments, yield several different AI-related capabilities, described above, there are lots of things that a newly-created system, without any 'experience', would not be able to do. It would not, for example, have any knowledge of how to hold a pencil, how to climb a ladder, how to negotiate an international treaty, and so on. Is it reasonable to suggest that such a system might be a GHLAI when there are so many shortcomings in what it can do? The answer, of course, is "Yes, such a system can be 'universal' in exactly the same way that a universal Turing machine, or a newborn baby, is universal". This is because, in all three cases, there is the potential to do a wide variety of different things, provided that it has appropriate knowledge, acquired via learning (babies and AI systems) or via programming (non-AI computers).

Since procedures or processes are forms of knowledge, and since we have reason to believe that the SP system has potential to accommodate any kind of knowledge (Section 4), it seems reasonable to believe that the SP system has potential, in principle and with the right knowledge, do any kind of computation that is not ruled out by over-large computational complexity.

Efficiency

The fifth suggested feature of a GHLAI is that it should in some sense be relatively 'efficient' in its ability to support diverse aspects of intelligence, to represent diverse kinds of knowledge, and to provide for seamless integration of diverse kinds of knowledge and diverse aspects of intelligence, in any combination. This section expands on that idea.

It is anticipated that, when the SP system is more fully developed, it is likely to be more 'efficient' than a CC, largely because it contains welldeveloped mechanisms for compression of information via the matching and unification of patterns (ICMUP), expressed via the powerful concept of SPmultiple-alignment. This provides the key to the SP system's versatility in its versatility in aspects of intelligence (Section 3), and in the representation of diverse kinds of knowledge (Section 4), and its potential for the seamless integration of diverse aspects of intelligence and diverse kinds of knowledge in any combination (Section 5).

Although the computational 'core' of a CC is likely to be smaller and simpler than in the SP machine, the SP system has potential for relative advantages like these:

• More intelligence. A CC is likely to fall short of the SP system in modelling the fluidity, versatility, and adaptability of human intelligenceunless the CC has been programmed with all the features of the SP system, in which case it would be an SP system and not a CC.

• Economies in software. Because of the pervasive influence of information compression in the SP system, its 'software' is likely to be relatively compact. By contrast, the absence of well-developed mechanisms for ICMUP in the core of the CC is likely to mean the need for such mechanisms to be repeatedly recreated in different guises and in different applications. This can mean software with a bloating that more than offsets the small size of the central processor.

• Economies in data. Unlike a CC, the SP system is designed to compress its data via unsupervised learning. This would normally mean that data for the SP machine would, after compression, be substantially smaller than data for a CC.

• Dramatic reductions in the variety of formats and formalisms. An enormous variety of formats and formalisms is associated with conventional systems. The SP machine has potential for dramatic simplifications in this area.

• Efficiency in processing. Although CCs, compared with human brains, are extraordinarily effective in such arithmetic tasks as adding up numbers or finding square roots, the advent of big data is creating demands that exceed the capabilities of the most powerful supercomputers [7, p. 9]. But by exploiting statistical information that the SP system gathers as a by-product of how it works, there is potential in the system for substantial gains in the energy efficiency of its computations [START_REF] Wolff | Big data and the SP theory of intelligence[END_REF]Sections VIII and IX].

With regard to the second and third bullet points, all knowledge in the SP system reflects the world outside the system. This may include knowledge of entities and their interrelations-the kind of knowledge that would conventionally be called 'data'-and knowledge of real-world processes or procedures-the kind of knowledge that might conventionally be called 'software'.

All such knowledge is stored as SP patterns without any formal distinctions amongst them. But in a CC, stored knowledge may be seen to comprise two components:

• Knowledge of the system's environment, as in the SP machine. This knowledge may be contained in external 'databases' and also in 'software'.

• Knowledge of processes or procedures, contained largely in 'software', needed to overcome the deficiencies of the core model. This kind of knowledge, such as knowledge of how to search for matching patterns, may be recreated many times in many different guises and in many different applications.

Conclusion

This paper defines general, human-like artificial intelligence (GHLAI) as follows: a GHLAI should demonstrate human-like intelligence without the need for AI-related programming (by contrast with a CC); it should be able to represent any kind of knowledge without the need for additional programming (by contrast with a CC); it should facilitate the seamless integration of diverse aspects intelligence and diverse kinds of knowledge, in any combination; it should, like a CC, provide for any kind of processing within the limits set by computational complexity; and it should do these things efficiently. With regard to GHLAI, the strengths and potential of the SP theory of intelligence (and its realisation in the SP computer model), as it is now, are summarised in three main areas: the versatility of the SP system in aspects of intelligence; its versatility in the representation of diverse kinds of knowledge; and its potential for the seamless integration of diverse aspects of intelligence and diverse kinds of knowledge, in any combination.

There are reasons to believe that, with regard to the versatility of the SP system in aspects of intelligence and in the representation of diverse kinds of knowledge, the potential of the system goes beyond its current capabilities, probably extending to the range of AI-related capabilities, and the representation of kinds of knowledge, that are needed for a fully-developed GHLAI.

A.2 Multiple alignments in bioinformatics

At the heart of the SP system is a process of finding patterns that match each other and merging or 'unifying' multiple instances to make one, a process which is referred to here and elsewhere as 'Information Compression via the Matching and Unification of Patterns' (ICMUP). More specifically, a central part of the SP system is a concept of multiple alignment, borrowed and adapted from bioinformatics. 4The original concept of multiple alignment is an arrangement of two or more DNA sequences or sequences of amino acid residues, in rows or columns, with judicious 'stretching' of selected sequences in a computer to bring matching symbols, as many as possible, into line. An example of such a multiple alignment of five DNA sequences is shown in Figure 3.

A.3 SP-multiple-alignments in the SP system

In the SP system, multiple alignments are sufficiently different from those in bioinformatics for them to be given a different name: SP-multiplealignments. 5 The distinctive features of an SP-multiple-alignment are:

• Normally, one New pattern is shown in row 0. Sometimes there is more than one New pattern in row 0 or column 0.

In this SP-multiple-alignment, a sentence is shown as a New pattern in row 0. The remaining rows show Old patterns, one per row, representing grammatical structures including words. The overall effect is to analyse (parse) the sentence into its parts and subparts. The pattern in row 8 shows the association between the plural subject of the sentence, marked with the symbol 'Np', and the plural main verb, marked with the symbol 'Vp'.

Because, with most ordinary multiple alignments or with SP-multiplealignments, there is an astronomically large number of ways in which patterns may be aligned, discovering good multiple alignments means the use of heuristic methods: building each multiple alignment in stages and discarding all but the best few multiple alignment at the end of each stage. With this kind of technique it is normally possible to find multiple alignments that are reasonably good but it is not normally possible to guarantee that the best possible multiple alignment has been found.

The concept of SP-multiple-alignment has proved to be extraordinarily powerful: in the representation of knowledge (Section 4), in aspects of intelligence (Section 3), and in the seamless integration of diverse kinds of knowledge and diverse aspects of intelligence in any combination (Section 5). It could prove to be as significant for an understanding of intelligence as is DNA for biological sciences: it could be the 'double helix' of intelligence.

There is more about the concept of SP-multiple-alignment in Appendix ??.

A.4 Unsupervised learning

Unsupervised learning in the SP system is described quite fully in [19, Sections 3.9 and 9.2]. The aim with unsupervised learning in the SP system is, for a given set of New patterns, to create one or two grammars-meaning collections of Old SP patterns-that are effective at encoding the given set of New patterns in an economical manner.

The building of SP-multiple-alignments is an integral part of unsupervised learning in the SP system. It provides a means of creating Old patterns via the direct assimilation of New patterns into the set of Old patterns, and via the splitting of New patterns and pre-existing Old patterns to create additional Old patterns. And it provides a means of evaluating candidate grammars in terms of their effectiveness at encoding the given set of New patterns in an economical manner.

As with the building of SP-multiple-alignments, the creation of good grammars requires heuristic search through the space of alternative grammars: creating grammars in stages and discarding low-scoring grammars at the end of each stage.

The SP computer model can discover plausible grammars from samples of English-like artificial languages. This includes the discovery of segmental structures, classes of structure, and abstract patterns.

At present, the program has two main weaknesses outlined in [21, Section 3.3]: it does not learn intermediate levels of abstraction or discontinuous dependencies in data. However, it appears that these problems are soluble, and it seems likely that their solution would greatly enhance the performance of the system in the learning of diverse kinds of knowledge.

To ensure that unsupervised learning in the SP system is robust and useful across a wide range of different kinds of data, it will be necessary for the system, including its procedures for unsupervised learning, to have been generalised for two-dimensional patterns as well as one-dimensional patterns (Appendix A.1).

A.5 The SP machine

As mentioned earlier, it is envisaged that an industrial-strength SP machine will be developed from the SP theory and the SP computer model [START_REF] Palade | Development of a new machine for artificial intelligence[END_REF]. Initially, this will be created as a high-parallel software virtual machine, hosted on an existing high-performance computer. An interesting possibility is to develop the SP machine as a software virtual machine that is driven by the high-parallel search processes in any of the leading internet search engines.

Later, there may be a case for developing new hardware for the SP machine, to take advantage of optimisations that may be achieved by tailoring the hardware to the characteristics of the SP system. In particular, there is potential for substantial gains in efficiency and savings in energy compared with conventional computers by taking advantage of statistical information that is gathered by the SP system as a by-product of how it works ([24, Section IX], [23, Section III], [14, Section 14]).

A.6 Distinctive features and advantages of the SP system

Distinctive features of the SP system and its main advantages compared with AI-related alternatives are described in [START_REF] Wolff | The SP theory of intelligence: its distinctive features and advantages[END_REF]. In particular, Section V of that paper describes thirteen problems with deep learning in artificial neural networks and how, with the SP system, those problems may be overcome. The SP system also provides a comprehensive solution to a fourteenth problem with deep learning-"catastrophic forgetting"-meaning the way in which new learning in a deep learning system wipes out old memories. 6The main strengths of the SP system are in its versatility in the representation of several kinds of knowledge (Section 4), its versatility in several aspects of intelligence (Section 3), and because these things all flow from one relatively simple framework-the SP-multiple-alignment concept-they may work together seamlessly in any combination (Section 5). That kind of seamless integration appears to be essential in any system that aspires to general human-level artificial intelligence.

A.7 Potential benefits and applications of the SP system

Potential benefits and applications of the SP system are described in several peer-reviewed papers, copies of which may be obtained via links from www.cognitionresearch.org/sp.htm: the SP system may help to solve nine problems with big data [START_REF] Wolff | Big data and the SP theory of intelligence[END_REF]; it may help in the development of human-like intelligence in autonomous robots [START_REF] Wolff | Autonomous robots and the SP theory of intelligence[END_REF]; the SP system may help in the understanding of human vision and in the development of computer vision [START_REF] Wolff | Application of the SP theory of intelligence to the understanding of natural vision and the development of computer vision[END_REF]; it may function as a database system with intelligence [START_REF] Wolff | Towards an intelligent database system founded on the SP theory of computing and cognition[END_REF]; it may assist medical practitioners in medical diagnosis [START_REF] Wolff | Medical diagnosis as pattern recognition in a framework of information compression by multiple alignment, unification and search[END_REF]; it provides insights into commonsense reasoning [START_REF] Wolff | Commonsense reasoning, commonsense knowledge, and the SP theory of intelligence[END_REF]; and it has several other potential benefits and applications described in [START_REF] Wolff | The SP theory of intelligence: benefits and applications[END_REF]. And, of course, this paper describes how the SP system may be applied in software engineering.

Figure 1 :

 1 Figure 1: Structures to illustrate how the SP system may simulate a NAND gate, as described in the text. (a) Four SP-patterns which, together, define the NAND function; Key: 'ng' is mnemonic for 'NAND gate'; 'i' stands for 'input'; 'op' stands for 'output'. (b) The best SP-multiple-alignment found by the SP computer model with the New pattern 'ng i1 1 #i1 i2 0 #i2 #ng' and Old patterns as shown in (a).

Figure 2 :

 2 Figure 2: Schematic representation of the SP system from an 'input' perspective. Adapted with permission from Figure 1 in [21].

•

  How several kinds of reasoning flow from the building of SP-multiplealignments. In scientific research and in the applications of science, what is potentially one of the most useful attributes of the SP sys-

	tem is its versatility in reasoning, described in [19, Chapter 7] and [21,
	Section 10]. Strengths of the SP system in reasoning, derived from
	the building of SP-multiple-alignments, include: one-step 'deductive'
	reasoning; chains of reasoning; abductive reasoning; reasoning with
	probabilistic networks and trees; reasoning with 'rules'; nonmonotonic
	reasoning and reasoning with default values; Bayesian reasoning with
	'explaining away'; causal reasoning; reasoning that is not supported by
	evidence; the already-mentioned inheritance of attributes in class hier-
	archies; and inheritance of contexts in part-whole hierarchies. Where
	it is appropriate, probabilities for inferences may be calculated in a
	straightforward manner ([19, Section 3.7], [21, Section 4.4]).
	There is also potential for spatial reasoning [23, Section IV-F.1], and
	for what-if reasoning [23, Section IV-F.2].

For example, none of these ideas is mentioned in "Knowledge representation and reasoning", Wikipedia, bit.ly/2fmKVtP, retrieved

2017-08-07.

This ambitious objective is in keeping with Occam's Razor. And as a means of solving the exceptionally difficult problem of developing general, human-level artificial intelligence, it is in keeping with "If a problem cannot be solved, enlarge it", attributed to President Eisenhower; it chimes with Allen Newell's exhortation that psychologists should work to understand "a genuine slab of human behaviour"[10, p. 303] and his work on Unified Theories of Cognition[START_REF]Unified Theories of Cognition[END_REF]; and it is in keeping with the quest for "Artificial General Intelligence" (Wikipedia, bit.ly/1ZxCQPo, retrieved 2017-08-15).

Six variants of ICMUP and how they may be realised via SP-multiple-alignment are described in Appendix ??.

This name has been introduced fairly recently to make clear that there are important differences between the two kinds of multiple alignment.

A solution has been proposed in[START_REF] Kirkpatrick | Overcoming catastrophic forgetting in neural networks[END_REF] but it appears to be partial, and it is unlikely to be satisfactory in the long run.

A Outline of the SP system

To help ensure that this paper is free standing, the SP system is described here in outline with enough detail to make the rest of the paper intelligible.

The SP theory of intelligence and its realisation in the SP computer model is the product of a unique extended programme of research aiming to simplify and integrate observations and concepts across artificial intelligence, mainstream computing, mathematics, and human learning, perception, and cognition, with information compression as a unifying theme. 3 The latest version of the SP computer model is SP71. Details of where the source code and associated files may be obtained are here: www.cognitionresearch.org/sp.htm#ARCHIVING.

It is envisaged that the SP computer model will provide the basis for the development of an industrial-strength SP machine, described briefly in Appendix A.5, below.

The SP system is described most fully in [START_REF] Wolff | Unifying Computing and Cognition: the SP Theory and Its Applications[END_REF] and quite fully but more briefly in [START_REF] Wolff | The SP theory of intelligence: an overview[END_REF]. Other publications from this programme of research are detailed, many with download links, on www.cognitionresearch.org/sp.htm.

A.1 Overview

The SP theory is conceived as a brain-like system which receives New information via its senses and stores some or all of it in compressed form as Old information, as shown schematically in Figure 2.

Both New and Old information are expressed as arrays of atomic symbols in one or two dimensions called patterns. To date, the SP computer model works only with one-dimensional patterns but it is envisaged that it will be generalised to work with two-dimensional patterns.

In this context, a 'symbol' is simply a mark that can be matched with any other symbol to determine whether they are the same or different. No other result is permitted. Apart from some distinctions needed for the internal workings of the SP system, SP symbols do not have meanings such as 'plus' ('+'), 'multiply' (' * '), and so on. Any meaning associated with an SP symbol derives entirely from other symbols with which it is associated. • The Old patterns are shown in the remaining rows (or columns), one pattern per row (or column).

• As with the original concept of multiple alignment, the aim in building multiple alignments is to bring matching symbols into alignment. More specifically in SP-multiple-alignments, the aim is to create or discover one or more 'good' SP-multiple-alignments that allow the New pattern to be encoded economically in terms of the Old patterns. How this encoding is done is described in [START_REF] Wolff | Unifying Computing and Cognition: the SP Theory and Its Applications[END_REF]Section 2.5] and in [START_REF] Wolff | The SP theory of intelligence: an overview[END_REF]Section 4.1].

An example of an SP-multiple-alignment is shown in Figure 4.

Num PL ; Np Vp 8