2. OBJETIVOS
4.2 DESCRIPCIÓN GENERAL DE LAS ACTIVIDADES
4.2.4 Pegas obras civiles
Let us consider a functionf :N→Nsuch thatf(n)≥nand functiong:N→N such that g(1) = 1, g(n) = g(n−1) +f(n−1). We define now a geometric environmentE(g), that is defined by the set of vertices{(x, y)⊆N×N|x≥g(y)}. The example of an environment is shown on Figure 2.
Theorem 1. Given a numberp∈N. If for alln > p a functionf satisfies the conditionf(n)≡p (modn)then an environment E(g) is non-efficient.
Proof. The proof is done by reduction of the reachability problem for 2-counter automaton (Minsky machine) to the reachability problem for walking finite state automaton in the above environment. In particular we show how to perform operation of multiplication and division in the environmentE(g) that allow us to perform the proposed reduction. We show the simulation of multiplication by 2 in the environmentE(g) in details. It is easy to apply this method to simulate the multiplication by 3 and division by 2 or 3 by walking automaton inE(g).
Universality of Walking Automata on a Class of Geometric Environments 127 1 2 3 4 5 ... 1 2 3 4 5 ... (g(2),2) (g(3),3) (g(4),4) (g(5),5) (g(1),1)
Fig. 2.The example of an environment represented by functionf(x) =x.
Let us assume that an automaton starts from a point (g(y), y) in the envi- ronmentE(g). The vertex of environmentE(g) in this point has a label
0 0 00 1 1
1 1 1
.
Now we show that a finite state automaton on environment E(g) can reach a vertex (g(2y),2y) from a vertex (g(y), y) and stop on it. Note that all movements (computations) of a finite state automaton that interacts with an environment E(g) is done using only labels of environments that are available to our walking automaton and its finite number of states.
BySn= (1,0)n ∈O∗, n∈N we denote the movement of automaton from a
vertex (x, y) to a vertex (x+n, y), that isnmovements to the right.
Another kind of movements that we are going to use for simulation are move- ments of typeC :C1 orC0. 1 2 3 4 5 ... 1 2 3 4 5 ... (g(1),1) (g(2),2) (g(3),3) (g(4),4) C0
Fig. 3.The example of a movementC0.
Ifg(y)≤x < g(y+ 1)−y byC0= (1,−1)y−1(1,0)(0,1)y−1∈O∗ we denote
128 O. Kurganskyy, I. Potapov 1 2 3 4 5 ... 1 2 3 4 5 ... (g(1),1) (g(2),2) (g(3),3) (g(4),4) C1
Fig. 4.The example of a movementC1.
Ifg(y+ 1)−y≤x < g(y+ 1) byC1= (1,−1)y−1(1,0)(0,1)y∈O∗we denote
the movement of automaton from vertex (x, y) to a vertex (x+y, y+ 1) . The examples of movementsC0andC1 are shown on Figures 3, 4.
There are several useful properties of C-movements. Let us assume that a walking automaton is in the bounded vertex (x, y) of an environmentE(g), where g(y)≤x < g(y+ 1), then it can make only a finite number of consecutive moves of type C0. If f(y) = iy+p (note that p is a constant), then the number of
consecutive moves of typeC0is bounded byi. So finally after a finite number of
C0 movements a walking automaton doesC1 movement. It can also determine
the fact of changing from C0 to C1 by checking evenness/oddness of second
component of the vertex coordinate using a finite number of its internal states. Next type of movements that we use for simulation isT. Let a walking au- tomaton is at the boundary vertex (x, y),g(y)≤x < g(y) +yof an environment E(g), in which the following propertyf(y) =iy+pholds. ByT we denote the sequence of movements from a vertex (x, y) to (x+p, y) bySp and then a finite
number ofC0 movements that end by C1 . Let us prove now thatT will move
an automaton from a vertex (x, y) to a vertex (x0, y+ 1), where
x−g(y) =x0−g(y+ 1) (1) Since we have thatf(y) =iy+p, a walking automaton should make exactly i−1 movements of type C0 to reach vertex (x+ (i−1)y+p, y) from vertex
(x+p, y). Then after movementC1it reaches vertex (x+ (i−1)y+y+p, y+ 1)
that is (x+iy+p, y+1). From it follows thatx0−g(y+1) =x+iy+p−g(y+1) = x+iy+p−g(y)−f(y) =x−g(y). In other words, we proved that after a movement T the distance from a corner (g(y), y) of the environment to a vertex (x, y) is the same as the distance from a vertex (x0, y+ 1) to a corner (g(y+ 1), y+ 1). So we can keep some information of the computations during the walk of automaton as a distance from its location to the nearest left corner of the environment.
Let us show how to move an automaton from a vertex (g(y), y) to a vertex (g(2y),2y) that corresponds to the multiplication of y by 2. The automaton starts in a vertex (g(y), y) by making a movement (T S2)yT until the automaton
Universality of Walking Automata on a Class of Geometric Environments 129 reaches a vertex (g(2y+ 1),2y+ 1), i.e. the first vertex with the label
0 0 00 1 1
1 1 1
and then by making a movement (0,−1)(−1,0)f(2y)to the first vertex with the
label
0 0 00 1 1 1 1 1
.
This is exactly the vertex (g(2y),2y).
Let us track the sequence of above movements. The first movementT S2 will
change the position of automaton from (g(y), y) to (g(y+ 1) + 2, y+ 1). The position of automaton after next movementsT S2 will be (g(y+ 2) + 4, y+ 2).
Thus, the position of automaton will be changed to (g(y +y) + 2y, y+y) = (g(2y) + 2y,2y) exactly aftery of such movements. After the last movement T of the movement (T S2)yT the position of automaton will be changed exactly to
(g(2y+ 1),2y+ 1) because of the equation 1. The example of such movements that correspond to the multiplication by 2 is shown on Figure 5.
(g(7),7)
(g(5),5)
(g(4),4)
(g(3),3)
Fig. 5.The example of a multiplicationy×2, wherey= 3, in the environment repre- sented by functionf(x) =x.
By analogy with a case of multiplication by 2 we can show that a sequence of movements
(T S3)2yT(0,−1)(−1,0)f(3y)
corresponds to multiplication by 3 and changes the location of automaton from (g(y), y) to (g(3y),3y). The sequence of movements that guarantees the division by 2 and 3 can be easy derived from the operation of multiplications. It follows that the walking automaton interacting with environment E(g) can simulate the computation of two counter machine where one of the counters is used as a scratchpad. Thus the reachability problem of a walking automaton in environ- mentE(g) is undecidable and the model is universal.
130 O. Kurganskyy, I. Potapov
Corollary 1. Let g : N → N be an integral polynomial function of degree n, wheren≥2. The environmentE(g) is non-efficient.
Proof. Letg(x) =anxn+. . .+a1x+a0 and f(x)= g(x+1)-g(x). Sincef is also
polynomial, i.e.f(x) =bnxn+. . .+b1x+b0, thenf(x)≡b0 (modx). It follows
form Theorem 1 that the environmentE(g) is non-efficient.
Moreover we can change the main theorem, but using the same ideas of computations with respect to some modulo.
Corollary 2. Given a constant p∈ N. Let function f fulfil condition f(n) ≡ n−p (modn)for alln > p then an environment E(g)is non-efficient.
Corollary 3. Given a constantp∈N. Let functionffulfil conditionf(n)≡ bn 2c
(mod n) for alln∈Nthen an environment E(g)is non-efficient.
Corollary 4. Given a function f : N → N that can be represented as f1(x)·
f2(x), where f1(n) ≡0 (mod n) and f2(n) is any nondecreasing function. An
environmentE(g) is non-efficient.
We can also use Theorem 1 to show that for any environment limited by unbounded nondecreasing function there exist a sub-environment where a finite automaton exhibits universal behaviour:
Corollary 5. Let E(h) = {(x, y) ⊆ N×N|x ≥ h(y)} and h : N → N is an unbounded integral nondecreasing function. Then there is a function g:N→N, such that E(g)⊆E(h)andE(g) is non-efficient.
Proof. Letg:N→Nis a function such thatg(1) = 1,g(x) =g(x−1) +f(x−1) and let f(x) = x·c(x), where c(x) : N → N is large enough to satisfy to the following property:
f(x)> h(x+ 1)−h(x).
From the above construction follows thatE(g)⊆E(h). On the other hand the fact that f(x)≡0 (modx) gives us that the environmentE(g) is non-efficient by Theorem 1.
5
Conclusion
The decidability of reachability problem for automaton interacting with envi- ronments defined by other functions is still an open question. Our conjecture is that the reachability problem forE(g) is undecidable for any everywhere defined function f :N→N, (i.e., not only for those wheref(n)≡constant (modn)).
Though we show in Corollary 5 that bounds on increase rate of the envi- ronment width does not have any influence on universality, it is not clear what would be the cornerstones of undecidability for some specific environment, e.g. when environment is limited by a logarithmic function:
Universality of Walking Automata on a Class of Geometric Environments 131
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