Proving P!=NP in first-order PA

05/19/2020
by   Rupert McCallum, et al.
0

We show that it is provable in PA that there is an arithmetically definable sequence {ϕ_n:n ∈ω} of Π^0_2-sentences, such that - PRA+{ϕ_n:n ∈ω} is Π^0_2-sound and Π^0_1-complete - the length of ϕ_n is bounded above by a polynomial function of n with positive leading coefficient - PRA+ϕ_n+1 always proves 1-consistency of PRA+ϕ_n. One has that the growth in logical strength is in some sense "as fast as possible", manifested in the fact that the total general recursive functions whose totality is asserted by the true Π^0_2-sentences in the sequence are cofinal growth-rate-wise in the set of all total general recursive functions. We then develop an argument which makes use of a sequence of sentences constructed by an application of the diagonal lemma, which are generalisations in a broad sense of Hugh Woodin's "Tower of Hanoi" construction as outlined in his essay "Tower of Hanoi" in Chapter 18 of the anthology "Truth in Mathematics". The argument establishes the result that it is provable in PA that P ≠ NP.

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1. The language of arithmetic

Let us now state what our lemma will be in the context of the first-order language of arithmetic.

Lemma 1.1.

Assume PA. On those assumptions, it is provable that there exists an arithmetically definable sequence of -sentences with the property that


(1) There exists a polynomial function with integer coefficients and positive leading coefficient such that the length in symbols of is eventually always bounded above by the value of this polynomial function at ;
(2) PRA+ is -sound and -complete
(3) PRA+ always proves 1-consistency of PRA+, and the length of the shortest proof is polynomially bounded as a function of ;
(4) The provably total general recursive functions of PRA+ have growth rates cofinal in the set of growth rates of all the total general recursive functions.

Proof.

Our background metatheory is PA. Choose a positive integer such that, given a -sentence of length symbols, a -sentence of length at most symbols exists which implies 1-consistency of PRA+. The construction of the desired sequence is as follows.


Define to be a polynomial with constant term 100, degree at least 2, and such that for all . Select the sentence , starting from and proceeding by induction on , assuming at each step that all previous sentences have already been constructed, as follows. If we are in the case , then consider the set of all true -sentences with at most symbols which (together with PRA) imply 1-consistency of PRA+, otherwise just consider the set of all true -sentences with at most 50 symbols; this set is non-empty and finite. Each sentence in is true in the standard model, and so is associated with a particular general recursive function, of which the sentence asserts totality. The total general recursive functions are partially ordered by comparison of growth rates.


Our sentence will be a conjunction of two sentences, the first one a -sentence chosen as follows. We choose our -sentence by selecting a maximal element of this partially ordered set at each stage, also strictly dominating growth-rate-wise, in the case , each general recursive function provably total from PRA+ via a proof of length at most , where again is an appropriately chosen polynomial. With the polynomial fixed, this is indeed possible for an appropriate choice of the polynomial .


Our second conjunct is chosen (where possible) from the set of true -sentences of length bounded so as to make the length of the conjunction at most . For appropriate choices of all the polynomial bounds involved, this gives rise to a sequence with the desired set of properties. The sequence constructed in this way is clearly arithmetically definable. ∎


We now arrive at our main result.

Theorem 1.2.

It is provable in PA that .

Proof.

Work in PA, and denote by the sequence constructed in the previous lemma, and suppose for a contradiction that . Then there must be a -sentence witnessing and provable in PRA+ for sufficiently large , which will identify a specific polynomial-time solver of an -complete problem which according to PRA+ is always guaranteed to work. Denote by the exponent on the polynomial bound for the number of steps the solver takes to halt.


Choose any positive integers , and let . Consider the set of formulas of length at most defining PRA+-provably total general recursive functions. The sequence has been constructed in such a way that every sentence in the sequence is indeed true in the standard model, but we will frame the following discussion in such a way that we can dispense with that assumption if need be.


We wish to prove that, for sufficiently large , there is at least one formula in with the property that, if we let denote the function which the formula defines, then, for each natural number , (1) it is PRA+-provable that there is a proof in PRA+

of the halting of the Turing machine for computing

, and (2) it is PRA+-provable on the assumption that is sufficiently large that if we let be the number of symbols in the shortest proof from PRA+ of halting of the Turing machine for , then the number of steps in which the Turing machine actually halts is between and and (3) it is PRA+-provable that if we let denote the halting time for the Turing machine for computing , then for any Turing machine which has halting time , if we let denote the number of symbols in the shortest proof from PRA+ of halting of , then . (If we were allowing ourselves to make use of the fact that PRA+ is arithmetically sound, then of course the reference to what is provable in PRA+ would be dispensible, however we wish to organise the discussion so that it will be easily generalisable to the cases where it is not known that and are true in the standard model.)


We just now made reference to a particular Turing machine for computing ; our choice of Turing machine will be determined by our choice of defining formula for the function , which again must be of length at most . It is not immediately apparent that a defining formula for with the appropriate properties can always be found for appropriate choices of and . Let us now give consideration to that question.


The possibility of proving the existence of a defining formula for for sufficiently large and for an appropriate choice of and depends on it always being the case when is sufficiently large that a -sentence witnessing is provable from PRA+. On the assumption that does indeed have this property, let us describe a PRA+-provably total function with the required properties, given that the appropriate choices for and have been made.


In preparation for the description of the algorithm for computing , we will need to define an increasing sequence of theories whose union is PRA, with the property that the set of provably total general recursive functions of is equal to the -th set in the Grzegorczyk hierarchy of primitive recursive functions. is defined to be the deductive closure in the predicate calculus for of the set of all consequences of the axioms of Q which can be formulated in together with bounded induction for that language. is defined to be the deductive closure in of Presburger arithmetic. is Bounded Induction Arithmetic. For , extend the language to include a symbol for exponentiation and include the defining recursion equations for exponentation, and then add bounded induction axioms for that language, and take the set of consequences of this axiom set in . Similarly with , except that we include a symbol for super-exponentiation as well as exponentiation. And with we have symbols for exponentiation, super-exponentiation, and super-super-exponentiation. And so on. This completes the specification of the hierarchy of theories to which we will refer in what follows.


Recall that a polynomial-time solver of any -complete problem is available, which is indeed PRA+-provably a correct algorithm, and recall that PRA+ proves the 1-consistency of PRA+.


Our algorithm for computing is as follows. For each natural number which serves as the argument to the function , we search through the space of Turing machines with a description of length bounded above by some polynomial function of , to be chosen later. We then use the polynomial-time algorithm for solving -complete problems to determine whether a proof in PRA++ is 1-consistent” can be found, with a given polynomial bound on its length, of the existence of a proof of the halting of the Turing machine under consideration (with empty input). Note that the proof is polynomially bounded in length but the proof is not. We then use the same algorithm to determine whether a proof of the desired lower and upper bounds given earlier in (2), on the number of steps for Turing machine to halt, relative to the length of , exists in PRA+, with some appropriately chosen polynomial bound on the length of .


There always exists some Turing machine for which the proofs and are available, for which there also exists a PRA+-proof , with some appropriate polynomial bound on the length of the proof, that if we let be any other Turing machine which has halting time less than or equal to that of , and if we let be the shortest length of a proof from PRA+ of halting of , then for all such Turing machines . To see this, let the Turing machine be such that its halting time is optimized over all Turing machines for which and for which the same is also true of any other Turing machine with a description of no greater length than . (The proof of the existence of such a Turing machine can be carried out in PRA+ with polynomially bounded length, but we are not assuming that the search through all Turing machines with description of no greater length is actually part of the running time of .) Furthermore, as tends towards infinity, the value of for such a Turing machine , for which the proofs and are available, may be chosen to be arbitrarily large (since the maximum value of over all Turing machines with a given bound on the length of their description, does go to infinity as that bound goes to infinity).


Then the value of the function for the value of under consideration will be the maximum number of steps taken to halt for any Turing machine in the set considered for which the appropriate proofs and , and can be found. (We spoke of “searching through the space of Turing machines”, but actually the availability of a polynomial-time solver for an -complete problem will make possible a speed-up of the computation required to determine a description of an algorithm which will output the maximum number of steps, so that a complete search will not be necessary. We will assume that some such speed-up is indeed used.)


With the assumption that some such “speed-up” is used, it can then be seen that a -formula for the function may be chosen so that the corresponding Turing machine for is such that the requisite proofs and will be available for that Turing machine, with polynomially bounded length, when is sufficiently large, possibly with a slightly higher polynomial bound needed on the number of steps taken to find them. For checking that the requisite proofs will indeed still be available given that the -formula is appropriately chosen, the fact that PRA+ proves the correctness of a specific polynomial-time solver of any -complete problem with exponent is crucial. This guarantees that, for an appropriate choice of -formula, the requisite proofs and will be available for the Turing machine in the case where is a lot larger than . The possibility of choosing the -formula in such a way that the proof will be available depends on being a lot larger than . This is the point in the argument at which the assumption is used.


We are essentially using the fact that the functions such that are (1) closed under functional composition (2) dominated in growth rate by the exponential function, and (3) are further such that for each the function is dominated in growth rate by some for sufficiently large . If we try to define the algorithm for with reference to some different family of functions which lack properties (1)-(3), then there will no longer be a guarantee that the proofs and are available for the Turing machine . The reason why property (2) is necessary is to ensure that the growth rates of the functions end up being cofinal in the growth rates of all total general recursive functions, as discussed below. If it were not the case that property (2) held, then the observation made previously about the possibility of choosing the value of for the Turing machine to be arbitrarily large as tends towards infinity would no longer hold, and this would end up blocking the conclusion argued for later that the growth rates of are cofinal in the set of growth rates of all total general recursive functions.


This completes the description of the algorithm for computing . Making an appropriate choice for all the polynomial bounds involved, we can now see that has the required properties provided and are chosen appropriately. Note that the defining formula for is a polynomial-time computable function of the sentences and . For which are too small we simply choose with a least defining formula in the lexicographic ordering (and we do assume that and are chosen so that is indeed always non-empty). We have shown that a choice of and is available such that has the required properties when is sufficiently large. We have spoken of actual total general recursive functions here, thereby helping ourselves to the assumption that PRA+ and PRA+ are indeed arithmetically sound; however the discussion can be re-framed in terms of Turing machines and -formulas instead, and we can remain neutral on the question of whether the Turing machines described actually always halt, and we shall later move to a context where we shall need to frame things this way. In that context, there will be a formula in the place of the defining formula for which will be a polynomial-time computable function of the two sentences (not assumed any longer to be arithmetically sound).


We will let be a formula in the first-order language of arithmetic with free variables and , of length at most , defining the function . Again, this is a polynomial-time computable function of and . Note that the totality of the function will be provable in PRA+, but the growth rate of will dominate that of any provably total general recursive function of PRA+ (because of the observation made before about the possibility of choosing for the Turing machine to be arbitrarily large as goes to infinity). Thus the total general recursive functions are cofinal growth-rate-wise in the set of all total general recursive functions.


Note that, if we selected some recursive set such that and repeated the entire foregoing discussion, dealing with bounds on lengths of proofs from the axiom sets mentioned in the previous discussion together with an axiom set consisting of all statements of facts about membership or failure of membership of a given natural number in , rather than bounds of lengths of proofs from the axiom sets mentioned previously just by themselves, then the possibility of proving the existence of an with the required properties for sufficiently large is no longer clear, because the proofs and may not always be available for with polynomially bounded length, and so we do not necessarily end up with a sequence of general recursive functions cofinal growth-rate-wise in the set of all total general recursive functions. This is how we avoid the objection that if successful our proof should also show for all .


Consider the sentence , shortly to be described, which can be constructed by means of the diagonal lemma. This construction is inspired by Hugh Woodin’s “Tower of Hanoi” sentence constructed in [1]. The sentence says “Either the Turing machine for halts and I’m refutable from PRA+ in at most steps, or the Turing machine for does not halt and I’m refutable from PRA+”, where the sentences are given by actually writing them out, and the function is given by means of the formula , and the Turing machine for is again determined by this formula , and is given by a numeral of polynomially bounded length. Note that the growth of the length of as a function of is polyonmially bounded.


Recall that by construction of the sequence , PRA+ is arithmetically sound, and recall that we are assuming, for a contradiction, the existence of a polynomial-time solver for every -complete problem with an exponent of , so there is always a refutation of from in at most steps for some sufficiently large , provided is sufficiently large. It is also apparent that the shortest refutation has more than steps. We are now going to obtain a contradiction by constructing a total general recursive function , whose growth rate dominates every . Since the sequence of functions is cofinal growth-rate-wise in the set of all total general recursive functions, by construction of the , including the previously mentioned fact that dominates any PRA+-provably total general recursive function growth-rate-wise, and the properties of the sequence stated in the previous lemma, this will indeed be a contradiction and so our conclusion will be that is indeed provable in PA, completing the proof.


Let us now describe the construction of . We described how to construct the sentence given the sentences and . Some reflection will reveal that, given any quadruple where and are -sentences and is a proof of 1-consistency of PRA+ from PRA+, is a proof of the -sentence witnessing from PRA+, one may construct a corresponding sentence in a similar way which is a polynomial-time-computable function of the quadruple . (The candidate for the such that will be unique given the way the sequence was constructed.)


So, given an for which we wish to evaluate , one now searches through all such quadruples such that the length of each component of the quadruple is bounded above by an appropriate polynomial function of , and then, for each such quadruple in the resulting finite set, one embarks on a search for either a proof in PRA+ of arithmetical unsoundness – a proof in PRA+ of -unsoundness for some – of PRA++, or a refutation of from PRA+, and the maximum length of the shortest such proof or refutation found (the maximum being taken over all the quadruples whose components have length appropriately polynomially bounded) is the value of . We must prove that the search always terminates.


This search is (provably from -induction arithmetic) always guaranteed to halt, as can be seen when we recall the construction of the sentence . We must argue this point in detail. Suppose that the search never terminates. Then PRA++{“PRA+ is -sound”: is consistent and PRA+ is consistent. Let be a model of . is a model for (schematic) arithmetical soundness of PRA+, and PRA+ predicts (with a proof of standard length) that PRA+ will eventually refute . Thus in the model is a proof in PRA+ of .


We will begin speaking of “standard” and “non-standard” elements of a model shortly, so it may be desirable to clarify what is meant by “standard”. We will eventually want to claim that all of our reasoning can be formalised in which is -conservative over PRA, but currently we are assuming for a contradiction the consistency of and then implies the existence of a model of . There is a cut of such a model (a -definable class relative to ) which is the least cut that models PRA+“PRA is 1-consistent”. Elements of will be said to be standard if they belong to this cut. Thus it will be okay in what follows to assume that the standard fragment of is closed under Ackermann’s function. So, with that understanding of what “standard” means, could be a standard element of the model ? If that were the case, then with the further assumption that Ackermann’s function is total (in the meta-theory) we could conclude that the search of which we spoke in the previous paragraph terminates. Our final conclusion will be that , which is -conservative over PRA, proves either the termination of the search or the termination of the search on the assumption that Ackermann’s function is total. That will be enough for the function to be PA-provably total (in fact provably total from -induction arithmetic). Given this consideration, we will be able to assume without loss of generality in what follows that is a non-standard element of .


Denote by the theory PRA++ is 1-consistent”. The least cut of satisfying satisfies PRA+ since is , and it can be seen – recalling that the functions were constructed so that a proof of halting of could always be found in in a number of steps bounded above by a polynomial function of – that may be chosen so that , and we can assume without loss of generality that is non-standard by the foregoing. Denote by be the (standard) Turing machine which plays in the part that was played by the Turing machine for in .


Since is a cut of a model of , and is a non-standard element of , the halting time of in is non-standard. Then the shortest proof of halting of from PRA+ has non-standard length, and when raised to the power of a sufficiently large standard natural number, it exceeds the halting time for (as we see when we recall the way in which was constructed). One can then also conclude that for each standard natural number , the length of the shortest proof from PRA+ of “PRA+ proves PRA+ proves PRA+ proves… halts”, going levels deep, is also non-standard. Let us discuss how to argue for this point.


We note that the shortest proof in PRA+ of “ halts” is “close to” the actual halting time of in the sense that if is the length of the shortest proof and is the halting time then for a standard . Further for any each Turing machine of halting time is such that the length of the shortest proof in PRA+ of halting of is such that for a standard – as we can see when we recall the construction of the algorithm for earlier, particularly the requirement that the proof must exist, and also recall that we are working in a cut of a model for the theory . It follows that the length of the shortest proof in PRA+ of “PRA+ proves halts” must also be non-standard. We see this as follows. Let be the Turing machine which searches for the shortest proof in PRA+ of halting of . Its halting time is such that where for a standard . So the shortest proof of halting of is such that raising the length of the proof to the power of a standard integer yields a result that dominates the non-standard halting time , so therefore the length of the shortest proof of halting of is itself non-standard. So, the length of the shortest proof in PRA+ of “ halts” is non-standard, and going two levels deep, the length of the shortest proof in PRA+ of “PRA+ proves halts” is non-standard. We can keep on going in this way by induction, and see as claimed before that for any standard the length of the shortest proof from PRA+ of “PRA+ proves PRA+ proves PRA+ proves… halts”, going levels deep, is non-standard. We want to argue that given the way the Turing machine is constructed, this gives rise to a contradiction. The nature of the argument for this point is basically proof-theoretic.


One can introduce function symbols for every primitive recursive function and also for the following total general recursive functions (relative to our model which is a model for with no proper cuts which model ): the function whose totality is asserted by and the each of the functions for whose totality is asserted by the sentence “+ is 1-consistent”. Then one can construct a “quantifier-free” system in a language with these function symbols, with open formulas as axioms, with the first-order theory being -conservative over this quantifier-free system in an appropriate sense (where here we are dealing entirely with proofs of standard length).


We can now consider that there will be a closed term of standard length in this language which denotes the halting time of in every model of . If we recall the construction of , we see that such a term of standard length can be obtained computationally by searching through the space of proofs in of standard length bounded above by a certain fixed standard upper bound. Furthermore, it can be proved in PRA+“PRA is 1-consistent” – which we are allowed to use here given the previous discussion of how we are using “standard” in such a way that it is okay to assume that the standard fragment of any model of or models PRA+“PRA is 1-consistent” – that every such term is bounded above (in every model of ) by another term of standard length, where nested occurrences of in the first term are replaced by a single occurrence of for a larger , such that for every occurrence of with in the term, the argument to is a term with the property that it denotes a standard natural number in every model of . Having noted this, we then see that this term of standard length can in turn be proved in PRA+“PRA is 1-consistent” to be bounded above (in every model of ) by another term of standard length in which only the primitive recursive functions and occur. From the existence of such a term of standard length one can now obtain the result that there exists some standard natural number such that PRA+ proves PRA+ proves… halts, where we only go levels deep. But this contradicts the observation previously made.


We are now at the point where we have derived a contradiction (in +“Ackermann’s function is total”) from supposing that the theory has a model and also that PRA+ is consistent, but this supposition was seen to be a consequence of the assumption that the search never termiantes. Our conclusion is that the search does provably terminate in -induction arithmetic. Hence the function is provably total in -induction arithmetic as claimed.


So, we do indeed have a provably total general recursive function here, and computer code for this function is perfectly feasible to construct. But the assumption that , together with -induction, can now be seen to give rise to a problematic conclusion, namely that this general recursive function grows faster than any . This means that it grows faster than the Busy Beaver function, which is impossible for a general recursive function. In this way, from we can derive a contradiction in PA, and in fact in -induction arithmetic. Since we have now derived a contradiction in PA from the assumption that , this completes the proof in PA that . ∎

2. The language of set theory

The argument in the previous section was inspired by the initial observation that it is provable in ZFC+“=Ultimate-” that there exists a certain set-theoretically definable sequence of -sentences in the first-order language of set theory with a certain set of properties. Here by “set-theoretically definable” we mean, definable in the first-order language of set theory without parameters. Let us discuss this result.


Recall the statement of the axiom “=Ultimate-”, formulated by Hugh Woodin.

Definition 2.1.

The axiom =Ultimate- is defined to be the assertion that


(1) There is a proper class of Woodin cardinals.
(2) Given any -sentence which is true in , there exists a universally Baire set of reals , such that, if is defined to be the least ordinal such that there is no surjection from onto in , then the sentence is true in .


Now we state the main lemma of this section.

Lemma 2.2.

Assume ZFC+“=Ultimate-”. On those assumptions, it is provable that there exists a set-theoretically definable sequence of -sentences with the property that


(1) There exists a polynomial function with integer coefficients and positive leading coefficient such that the length in symbols of is eventually always bounded above by the value of this polynomial function at ;
(2) ZFC+ is -sound and -complete;
(3) ZFC+ always proves -soundness of ZFC+.


Here, by saying that the sequence

is set-theoretically definable, we mean that the set of ordered pairs

such that is the Gödel code of is a set which is definable by a formula in the first-order language of set theory without parameters.

Proof.

Choose a positive integer such that, given a -sentence of length symbols, a -sentence of length at most symbols exists which implies -soundness of ZFC+. The construction of the desired sequence is as follows.


Define . Select the sentence , starting from and proceeding by induction on , assuming at each step that all previous sentences have already been constructed, as follows. If we are in the case , then consider the set of all true -sentences with at most symbols which (together with ZFC) imply -soundness of ZFC+, otherwise just consider the set of all true -sentences with at most 50 symbols; this set is non-empty and finite. Each sentence in is true in , and recall that we are assuming =Ultimate-. Recall also that the Borel degrees of universally Baire sets are well-ordered. For each sentence in , we can select a minimal Borel degree correponding to such that there is a universally Baire set of that Borel degree witnessing condition (2) of Definition 2.1 for the sentence . Now consider the set consisting of all sentences from for which the associated Borel degree is as large as possible. Choose a universally Baire set from the Borel degree corresponding to all sentences , and then for each sentence , select the least possible such that the sentence is true in . In this way an ordinal is associated to each sentence in ; then choose a sentence in with the largest possible associated ordinal. Let be the maximum integer such that the conjunction of a sentence of length and a sentence of length has length at most . Now form the conjunction of the previously mentioned sentence chosen from with the -sentence of length at most least in the lexicographical ordering which has not occurred as a conjunct in any sentence so far, if there is any such sentence (otherwise simply leave the chosen sentence from as is). This completes our account of how we select the sentence . This construction will produce a sequence with the desired list of properties. It is clear from our construction that the sequence is set-theoretically definable. ∎


One could of course note that the same lemma is provable from assumptions much weaker than ZFC+“=Ultimate-” without making any essential use of the machinery about universally Baire sets; the point of interest about this particular construction is that in some sense the -sentences grow in logical strength as fast as possible while still remaining only polynomially bounded in their growth in length. I am not currently aware of any interesting application of this particular construction of a sequence of sentences, but it may be a topic that would repay further thought.

References

  • [1] Hugh Woodin. The Tower of Hanoi, Chapter 18 of [2].
  • [2] Truth in Mathematics, eds. H. G. Dales and G. Oliveri, Oxford Science Publications 1998.