The definition of the problem P=NP is I believe below:

“If a solution to a problem can be verified as the right solution in polynomial time, the problem can also be solved in polynomial time”

Before going to P & NP, I want to understand the basis for these levels/classes of complexities well. I convey my understanding below and that I do because from that I want to question the question itself.

Assume that there is this hypothetical problem solver program which will try to solve a given problem and find an answer. This program typically takes greater number of machine head moves and tape space in a turing machine to solve a problem if it is given a tougher problem. The length of the time (number of head moves) to solve a problem partly depends on the length of the variables in the problem – number of digits in a number in case of factorization problem or the number of cities in case of travelling salesman problem. For each problem if we come up with an optimal method (program) to solve the problem, still the number of machine head moves and tape space consumed in reaching the solution is a function of the length of the variables – f(lv). The rate at which the value of this function f(lv) increases with the increase in the length of the variables depends on the inherent hardness of the problem. As the length of the variables tend towards infinity almost all the f(lv)s will tend towards infinity. But some tend to infinity faster than the rest or one could reword it as some tend to a bigger infinity than the rest. If we do not accept the different cardinalities in infinities, does this differentiation between the hardness of two problems still hold good?

In other words, consider Limit of (x power n)/ (e power x) as x tends to infinity and n is finite – is it really zero or undefined?

In a stronger version of the above statement consider Limit of (x power n)/ (x power n+1) as x tends to infinity and n is finite – is it really zero or undefined?

In other words, consider two cars, one travelling in uniform velocity and the other with uniform acceleration – would both given infinite time to travel, travel different infinite distances (assuming there is no speed limit like c)?

If say there are three cars covering distances as a function of t (the time taken) – t, t-squared and (e power t). Given infinite time, would they cover markedly different infinite distances?

There is something troubling about these limits when x tends to infinity or zero, cardinalities of infinities, infinity, infinitesimal, the assumption of continuum of the number line itself. Dont you think?

I am not able to convince myself that everything is clear and lucid and “comfortably acceptable” to one’s mind when we deal with these mathematical things. It seems like there is a certain sense of mental delusion and some hacking happening to justify say the continuum of the number line.

Why dont we just be content with those algebraic numbers which involve non infinite process, lets not even include 10/3 in the domain of numbers, definitely not root of 2 and transcendentals like pi and e. For pi and e, its definition depends on I believe on geometry and calculus, both apriori assume continuum and infinite processes. I feel our minds deludes itself that it can hold infinity and infinitesimal in it but it cannot – it truly cannot conceive of an infinite process of continuously reducing epsilon for to do that you will take a lifetime but after a point we just mentally say to ourselves “…” and resort to a delusional induction that enables us to come to terms with infinitesimal and the continuum.

Isnt there something deeply troubling in doing that?

Can we not create an alternate version of maths which doesnt need the continuum. You dont need continuum for calculus or geometry. For all you know space and time might not be continuous. We can create virtual reality mimicking real world geometry and motion (speed & acceleration) using discrete computers. For all you know the physical laws of our world arent based on the continuous. May be a computer can create a reality using quantum mathematics where when you combine laws of gravitation and quantum theory it doesnt result in singularities and infinities as quantum mathematics doesnt include infinity as a valid concept to start with. Somehow I feel the problem of unification of laws of physics doesnt lie in physics but in mathematics, in its assumption of a continuum. Energy is not continuous but comes in quantum packets not because of the laws of nature but the laws of underlying mathematics of nature. The only thing which blocks our understanding of that mathematics is the ability of our mind to conjure up an infinite process but in actuality it doesnt – it thinks it does through induction but it doesnt. I somehow think if we dont get mathematics in order, and cure ourselves of that delusion of infinite induction, we will never be able to unify. I am strongly doubt space-time which surround us is also quantum and discrete and we cannot model it with our mathematics of continuum.

]]>Sorry about that. I think I was fooled by being able to get access to it from a university computer. I’ll try to find a preprint and change the link to that.

]]>fyi decided to ask a similar question on crypto.se, citing this pg: NP complete problems related to permutations of binary vectors or block ciphers

]]>yeah re boaz’ 1st comment above, wonder if there are some proven NP complete problems in crypto wrt block ciphers & permutations/”scramblers” but havent noticed them myself yet over the yrs…. as for P=?NP, the question relating to circuits is stronger and is the P/poly=?NP question. the so called “uniform vs nonuniform” distinction.

here is a tcs.se question that has a formal mechanism for measuring “errors” which might be a way of measuring a “closeness” of functions re your idea of looking at “truly random” vs “pseudo random” cnf/dnf conversion to minimize errors

]]>It’s true that a slight variant of the problem is NP-complete. The basic reason for that is that if you allow some extra inputs and set them all to zero, then you can use 3-bit scramblers to simulate an arbitrary circuit.

]]>another interesting pov that shows how much a TM computation is like bit scrambling & shows ideas for a completeness construction. imagine the TM head moving over a binary tape and making changes as it passes left to right, back/forth, and the series of IDs, instantaneous descriptions aka computational tableau it generates. its similar to a “scrambling composition”, it is computing something very similar to your bit scrambler with somewhat more bits involved (roughly, enough to represent 3 tape squares/symbols)…. now, how to convert all this into a few problems, theorems and lemmas, esp linking up existing theory? there seem to be some key unknown or maybe now-obscure bridge thms waiting to be constructed or highlighted….

]]>hi wtg think that some near-variant of your bit scrambler problem is indeed NP complete. havent found nice formulations that show how distinguishing randomness or PRNGs from “truly random” is NP complete or related to complexity class separations, but have concluded after long study its basically the same phenomenon & the Natural Proofs paper certainly points in that direction. think part of the difficulty here is how you formulated this, what does it technically/formally mean to “distinguish a function from truly random”. how can that be defined mathematically? Razborov/Rudich paper has some ideas along these lines…. unfortunately some of the theory of PRNGs is more in the cryptography literature than the complexity theory literature, and this is a unnatural separation so to speak (in the sense of future hindsight)…. needs to be unified…. there is also an interesting sense in which monotone circuit (slice) functions compute a kind of permutation mapping…. have to write that up sometime…. deep connections….

]]>I can get access to the paper by searching for the title (An almost m-wise independent random permutation of the cube), but, not too surprisingly, I can’t post a working link to it. At some point I’ll try to dig out the preprint from which that paper is derived, but it may take some searching to find it.

I don’t think the problem of distinguishing a random composition of say 3-bit scramblers from a truly random even permutation of the vertices of the cube is an NP-complete problem. Actually, the question there is whether it is a pseudorandom generator, but I don’t believe that the related problem of telling in polynomial time (in ) whether a given even permutation is a composition of at most 3-bit scramblers is NP-complete. However, it is certainly in NP, and I think it is at the hard end of NP. (The problem if you try to show that it is NP-complete is that because you are dealing with permutations, you somehow don’t have enough room to create gadgets. Or at least, that seems to be a problem.)

]]>think there is some possibility that Razborov has already identified a mathematical “property” that may be sufficient to separate P from NP without realizing it…. further details on this idea to anyone who thinks its not outlandish enough to reply =)

Of course, it’s a long way from a proof that a random composition of 3-bit scramblers cannot be efficiently distinguished from a random permutation, but that’s not something we’re going to be able to prove any time soon, since it would imply that P\ne NP.

thats quite a leap there. havent read your 3bit scrambler paper [could you put it on arxiv or online, or does a semievil scientific publishing corporation who shall remain nameless restrict that?], but what you have here is [something close to] a statement/assertion that you have found a different NP complete problem, albeit not really phrased exactly in the typical format. it needs to be quantified a bit better. in particular it appears one needs to bound the # of compositions of the scramblers.

]]>I like Chow’s paper quite a bit, but I don’t think it changes much the message of the natural proof paper. Rather it shows that quantitatively the Razborov-Rudich bound on the “largeness” condition is tight.

I don’t this makes such a difference, because a natural “simplicity” property ought to capture a vanishing fraction of all functions. However, RR actually proved a stronger result, ruling out even simplicity properties that capture a fraction of the inputs. (The way it’s usually described considers the complement of the “simplicity property”, which is called a “useful property”.) Chow’s paper shows that RR’s quantitative bound on the function is essentially tight.

Salil’s proof basically uses the same trivial property I mentioned above – you consider the set H of all functions on bits that compute CLIQUE (or your other favorite NP-hard problem such as SAT) on the first bits of their input for some . The measure of H is but if, as we believe, there is no circuit computing CLIQUE on -bit inputs with size smaller than, say, , then if we take then no polynomial-time computable function is in H and so H is a useful property which is also constructive.

]]>Understanding Chow’s result properly has been on my to-do list for some time. I’ll have a look at it and look out for that paragraph.

]]>The property is apparently obscured by an ambiguity that is implicit in the standard definitions of the Satisfiability problem (SAT), and of the classes P and NP.

These fail to either recognise or explicitly highlight that the assignment of satisfaction and truth values to number-theoretic formulas under an interpretation can be constructively defined in two distinctly different ways (with significant consequences for the foundations of mathematics, logic and computability):

(a) In terms of algorithmically verifiability:

A number-theoretical formula F is algorithmically verifiable under an interpretation (and is therefore in NP) if, and only if, we can define a polynomial-time checking relation R(x, y)—where x codes a propositional formula F and y codes a truth assignment to the variables of F—such that, for any given natural number values (m, n), there is an algorithm which will finitarily decide whether or not R(m, n) holds over the domain of the natural numbers.

(b) In terms of algorithmically computability:

A number-theoretical formula F is algorithmically computable under an interpretation (and is therefore in P) if, and only if, we can define a polynomial-time checking relation R(x, y)—where x codes a propositional formula F and y codes a truth assignment to the variables of F—such that there is an algorithm which, for any given natural number values (m, n), will finitarily decide whether or not R(m, n) holds over the domain of the natural numbers.

Of course we first need to argue that the two concepts are well-defined.

It is fairly straightforward to then argue that there are classically defined arithmetic formulas which are algorithmically verifiable but not algorithmically computable.

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