Automata and Arithmetics in Canonical Number Systems
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsPlease see the attached
Comments for author File: 
Comments.pdf
Author Response
Response:
Thank you very much for taking the time to review our manuscript and for your positive evaluation. We truly appreciate your encouraging feedback and are delighted that you found our work valuable.
Your support is highly motivating, and we are confident that your endorsement will contribute to the successful dissemination of our findings.
Reviewer 2 Report
Comments and Suggestions for Authors- Some of the technical results and some mathematical proofs have to be revised in detail.
-There is a doubt about Theorem 2. If all the roots of P(x) fulfill modulus(x)>1 then all the real roots are less than -1. So, the second part (real roots less than -1) of the statement is already included in the first one and it is not needed to specify it.
-Definition 2 can be illustrated with some simple example by applying the given formula in the following cases for A distinct of P: a) degree of A< deg P, deg A=deg P, deg A< deg P.
-Definition 2: The meaning of the notation of the special case PbarA has to be specified. Also, is the zero value of I(A) assigned to this case also obtained from the above generic formula in the definition?.
-Proof of Lemma 1, Part (i): Why is there at least a root alpha of the irreducible divisor P´ of the CNS polynomial P ( then , It seems that modulus (alpha)>1) such that modulus ( A(alpha))>=1?- This implies that the evaluation of the formula given below is non-negative as said. But, again, why for “alpha” it follows that modulus (A (alpha))>=1.?.
-Proof of Lemma 1, Part (iii): if alpha is a root of P then modulus ( A(alpha)>= modulus ( B(Alpha) is correct but, in fact , the inequality becomes an equality since A(alpha)=B(alpha) as said in the Part (ii) of the proof.
-The sentence before Lemma 2 has an error syntax at the end.
-Lemma 2, Lemma 3: Why is it necessary the use of both sets of positive integers b(.) and a(.)?.
-Proof of Lemma 3: Check in detail the correctness of the last formula.
-Proof of Lemma 5: Revise it. aj=1 iff "formula >0" is the same as both implications below hold together:
a) “aj=1” implies “formula >0” ( this is proved with the first part of the given proof leading to formula>=1 so formula >0)
b) “formula>0” implies “aj=1”---- This is equivalent to its contrapositive logic proposition-- “aj distinct from 1” implies “formula <=0” . This has only been proved for just one value “aj distinct from 1”, which is aj=0, not still proved for any “aj distinct from 1”.
- Theorem 3 and Algorithm 1: It seems that "c " in the Theorem and "c" in the algorithm is the minimum real constant which sartisfies the upper-bound given in the statement of the theorem. This should be reflected in the theorem statement and in that of the algorithm. In the way that the theorem is stated , there are infinitely many real constannts "c" which satisfy the given upper-bound and the algorithm finds the minimum one among such constants being comparible with the inequality in the thoerem statement.
- In order to faciliate the reading: a) the introduction should reflect more concrete details on the rest of the paper organization ; and b) a section of conclusions with the main developed ideas would be welcome.
Author Response
Response:
Thank you very much for your detailed feedback on our manuscript. We have carefully addressed the points you raised and made the necessary corrections to improve the quality of the paper. You can find our responses to your comments below. Your insights were incredibly helpful, and we truly appreciate your time and effort.
Comments 1: Some of the technical results and some mathematical proofs have to be revised in detail.
Response 1: Thank you for your comment. In response to the subsequent remarks, we will address each point in detail, providing clarifications and revisions.
Comments 2: There is a doubt about Theorem 2. If all the roots of P(x) fulfill modulus(x)>1 then all the real roots are less than -1. So, the second part (real roots less than -1) of the statement is already included in the first one and it is not needed to specify it.
Response 2: Thank you for your observation. Theorem 2 is cited in a slightly modified form from the original paper. The independence of the two statements can be seen if we accept that the CNS polynomial is not necessarily irreducible. In this way, we may consider the polynomial x2-4. It has two roots: 2, -2. The first property is true, but the second is false for the polynomial. However, the theorem states that x2-4 is not a CNS polynomial. We also added another reference where the displayed result is more similar to the one we stated.
Comments 3: Definition 2 can be illustrated with some simple example by applying the given formula in the following cases for A distinct of P: a) degree of A< deg P, deg A=deg P, deg A< deg P.
Response 3: Thank you very much for your advice. On page 4 (in the revised manuscript), below Definition 2, Example 1 is added with three cases.
Comments 4: Definition 2: The meaning of the notation of the special case PbarA has to be specified. Also, is the zero value of I(A) assigned to this case also obtained from the above generic formula in the definition?.
Response 4: Thank you for your comment. The notation PbarA is replaced by the explanation of the notation. For the second part of the comment we may observe that if A is divisible by P than any root x of P is a root of A, whence A(x)=0 and log|A(x)| is not defined.
Comments 5: Proof of Lemma 1, Part (i): Why is there at least a root alpha of the irreducible divisor P´ of the CNS polynomial P ( then , It seems that modulus (alpha)>1) such that modulus ( A(alpha))>=1?- This implies that the evaluation of the formula given below is non-negative as said. But, again, why for “alpha” it follows that modulus (A (alpha))>=1.?.
Response 5: Thank you for calling our attention to the gaps in the proof. We elaborated the proof in more detail.
Comments 6: Proof of Lemma 1, Part (iii): if alpha is a root of P then modulus ( A(alpha)>= modulus ( B(Alpha) is correct but, in fact , the inequality becomes an equality since A(alpha)=B(alpha) as said in the Part (ii) of the proof.
Response 6: Thank you for your insightful comment. We have extended the proof of Lemma 1, Part (iii) to include an explanation of the properties of A(alpha) and B(alpha). This modification clarifies that the inequality indeed becomes an equality, as stated in Part (ii) of the proof.
Comments 7: The sentence before Lemma 2 has an error syntax at the end.
Response 7: Thank you very much for pointing this out. We corrected the error syntax.
Comments 8: Lemma 2, Lemma 3: Why is it necessary the use of both sets of positive integers b(.) and a(.)?.
Response 8: Thank you for raising this point. We rewrote and clarified the assumptions of Lemma 2 and Lemma 3 (and for similar reasons, in Lemma 4 and Lemma 5, as well). The reason we use two different sets of coefficients is that in Lemma 2, the set a0,..., a(n-1) is fixed, while the set b0,..., b(n-1) is a parameter set, running through all possible values, on which we take the maximum of the sums in the formula. In the proof of Lemma 3, the set a0,..., a(n-1) is chosen to fulfil the first equality in line 126 (line 91 in the original manuscript), but during the proof, it is assumed to be fixed, and again, the set b0,..., b(n-1) is a parameter set.
Comments 9: Proof of Lemma 3: Check in detail the correctness of the last formula.
Response 9: Thank you for bringing our attention to the issue of the proof lacking sufficient detail. We have extended the last inequality in the proof of Lemma 3 to provide a clearer and more thorough explanation.
Comments 10: Proof of Lemma 5: Revise it. aj=1 iff "formula >0" is the same as both implications below hold together:
a) “aj=1” implies “formula >0” ( this is proved with the first part of the given proof leading to formula>=1 so formula >0)
b) “formula>0” implies “aj=1”---- This is equivalent to its contrapositive logic proposition-- “aj distinct from 1” implies “formula <=0” . This has only been proved for just one value “aj distinct from 1”, which is aj=0, not still proved for any “aj distinct from 1”.
Response 10: We appreciate your insights. We acknowledge that the proof is ambiguous. We have revised the second sentence in the proof of Lemma 5 to emphasize that the value of aj can only be 0 or 1.
Comments 11: Theorem 3 and Algorithm 1: It seems that "c " in the Theorem and "c" in the algorithm is the minimum real constant which sartisfies the upper-bound given in the statement of the theorem. This should be reflected in the theorem statement and in that of the algorithm. In the way that the theorem is stated , there are infinitely many real constannts "c" which satisfy the given upper-bound and the algorithm finds the minimum one among such constants being comparible with the inequality in the thoerem statement.
Response 11: Thank you very much for your valuable comment, which we strongly agree with. We have added Remark 2 below Theorem 3 in line 128 in the revised version of the manuscript. We have revised the statements of Theorem 3 and Algorithm 1 accordingly to clarify that "c" is indeed the minimum real constant that satisfies the upper bound given in the theorem. The modification explicitly reflects this, ensuring that the relationship between the theorem and the algorithm.
Comments 12: In order to faciliate the reading: a) the introduction should reflect more concrete details on the rest of the paper organization ; and b) a section of conclusions with the main developed ideas would be welcome.
Response 12: Thank you for pointing these out.
a) We extended the introduction according to your comment. The paragraph on page 2 from line 38 to 45 is replaced by the paragraphs from line 38 to 62.
b) Section 5 (Conclusion) is appended to the manuscript.
Reviewer 3 Report
Comments and Suggestions for AuthorsThe paper is an interesting addition to the topic of generalized positional number systems/arithmetic.
My main concern is the presentation of the argument, especially its accessibility.
For instance, an example right after Definition 1 could increase the number of possible readers by an order of magnitude.
The lemmas needed for Theorem 3 completely lack any narrative.
Algorithm 1 should be more self-contained (e.g., describe what the input/result are).
Other comments:
Line 79 seems to be redundant.
Line 84 - it is suggested that the information quantity is an optional construction. However, it is used later in Theorem 5.
Section 4 mentions computational experiments, but does not give any details. It vaguely mentions the space complexity of the algorithm and the data representation. What are these?
Regarding Theorem 6. Does it really follow that automaton is minimal from the fact that in the given construction none of the states can be eliminated? There might be totally different constructions.
Also, an interesting question, what do the authors think about the fundamental limitation they exposed?
Author Response
Response:
Thank you very much for your detailed feedback on our manuscript. We have carefully addressed the points you raised and made the necessary corrections to improve the quality of the paper. You can find our responses to your comments below. Your insights were incredibly helpful, and we truly appreciate your time and effort.
Comments 1: An example right after Definition 1 could increase the number of possible readers by an order of magnitude.
Response 1: We appreciate the suggestion and have included three examples from Line 72 to Line 123 and from Line 131 to Line 159 to help clarify the concept of canonical number systems. These examples aim to illustrate the practical applications of the theory, making it more accessible and engaging for a wider audience. Thank you for your valuable feedback, which has greatly contributed to improving the clarity and reach of this section.
Comments 2: The lemmas needed for Theorem 3 completely lack any narrative.
Response 2: Thank you very much for pointing this out. We added for the better understanding some remarks and explanations for instance Remark 5. from Line 218 to Line 256. We also added some explanations before the lemmas leading up Theorem 3. For the more detailed explanations we extended the proof of Lemma 4 at Line 3 of page 11. We added geometric and algebraic description of the result before Lemma 5 from Line 278 to Line 284. For the better understanding, we added some comments before Corollary 3. from Line 290 to Line 295. We also extend with some clarification the first sentence of the proof of Corollary 3. We gave a detailed observation with a new Definition 3. and Remark 7. and 8. before Algorithm 1. Here we described the operations of the algorithm with some sentences. We added some comments before Theorem 4.
Comments 3: Algorithm 1 should be more self-contained (e.g., describe what the input/result are).
Response 3: Thank you very much for your advice. We added the explanatory comments to the algorithm.
Comments 4: Line 79 seems to be redundant.
Response 4: Thank you very much for your comment. We deleted Line 79.
Comments 5. Line 84 - it is suggested that the information quantity is an optional construction. However, it is used later in Theorem 5.
Response 5: Thank you for raising this point. We called the attention before Definition 2. for the importance of information quantity and referred to the later usage.
Comments 6: Section 4 mentions computational experiments, but does not give any details. It vaguely mentions the space complexity of the algorithm and the data representation. What are these?
Response 6: Thank you for your valuable comment. We extended Section 4. by an example CNS. We described how to execute addition and how to determine the addition automata based on the example. We also detailed the circumstances of the experiment. We transformed the motion of space and time complexity to memory usage and running time.
Comments 7: Regarding Theorem 6. Does it really follow that automaton is minimal from the fact that in the given construction none of the states can be eliminated? There might be totally different constructions.
Response 7: We appreciate your insights. We added Remark 12. before Theorem 6. to explain the relation between the state and the output of an automaton. We also detailed more carefully the proof of Theorem 6.
Comments 8: Also, an interesting question, what do the authors think about the fundamental limitation they exposed?
Response 8: Thank you for your question. We explained the effect of the limitation we exposed at the end of Section 5.
Round 2
Reviewer 2 Report
Comments and Suggestions for AuthorsThe paper has been improved. There are original results on the subject. The material adjusts well to the scope of this journal. We have no specific technical comments on this revised version.
Author Response
Response:
Thank you very much for taking the time to review our revised manuscript and for your positive evaluation. We truly appreciate your encouraging feedback and are delighted that you found our work valuable.
Your support is highly motivating, and we are confident that your endorsement will contribute to the successful dissemination of our findings.
