## Is P versus NP formally independent (2003)

Venue: | Bulletin of the European Association for Theoretical Computer Science |

Citations: | 8 - 0 self |

### BibTeX

@ARTICLE{Fortnow03isp,

author = {Lance Fortnow and Scott Aaronson},

title = {Is P versus NP formally independent},

journal = {Bulletin of the European Association for Theoretical Computer Science},

year = {2003},

volume = {81},

pages = {109--136}

}

### OpenURL

### Abstract

I have moved back to the University of Chicago and so has the web page for this column. See above for new URL and contact informaion. This issue Scott Aaronson writes quite an interesting (and opinionated) column on whether the P = NP question is independent of the usual axiom systems. Enjoy!

### Citations

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Citation Context ...s no result, there would then also be no reason to think further about the problem. In the seventies, when serious work on P vs. NP began following the discovery of NP-completeness by Cook [14], Karp =-=[30]-=-, and Levin [36], the problem continued to be seen in logical terms. People tried to separate P and NP using the same ideas that worked in recursion theory; this is the context of the Baker-Gill-Solov... |

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Citation Context ...chine yields no result, there would then also be no reason to think further about the problem. In the seventies, when serious work on P vs. NP began following the discovery of NP-completeness by Cook =-=[14]-=-, Karp [30], and Levin [36], the problem continued to be seen in logical terms. People tried to separate P and NP using the same ideas that worked in recursion theory; this is the context of the Baker... |

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Citation Context ... = nc . Why is this considered implausible? In the case C = P/poly (that is, the lower bound proof works against polynomial-size circuits), the evidence comes cryptographic reductions. H˚astad et al. =-=[27]-=- showed that given any one-way function, we can construct a pseudorandom generator that’s roughly as hard to break. Earlier, Goldreich, Goldwasser, and Micali [21] had shown that given any pseudorando... |

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Citation Context ...tographic reductions. H˚astad et al. [27] showed that given any one-way function, we can construct a pseudorandom generator that’s roughly as hard to break. Earlier, Goldreich, Goldwasser, and Micali =-=[21]-=- had shown that given any pseudorandom generator, we can construct a pseudorandom function that’s roughly as hard to break. Putting these together, we find that our lower bound proof would give us a w... |

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Citation Context ...ulmuley and Sohoni [39, 57] have proposed to tackle P vs. NP using algebraic geometry—an approach that, they think, might escape the jaws of Razborov and Rudich. Using the deep result that MIP = NEXP =-=[3]-=-, Buhrman, Fortnow, and Thierauf [8] showed that MAEXP, the exponential-time analogue of MA, doesn’t have polynomial-size circuits. This result is notable because it avoids both the Baker-Gill-Solovay... |

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Citation Context ...nt upon which axioms we choose. In the eighties, the combinatorial view became dominant, in the wake of the circuit lower bounds of Furst-Saxe-Sipser [18], Ajtai [2], Razborov [50, 52], and Smolensky =-=[61]-=-. On this view, P vs. NP is no more likely to be formally independent than Goldbach’s Conjecture, the Riemann Hypothesis, or any other ‘natural’ mathematical problem. Proving P �= NP (and indeed the s... |

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Citation Context ... [35]. If C is a smaller class, for instance the class TC 0 of constant-depth polynomial-size threshold circuits, then the reductions of [27] and [21] can’t be carried out. However, Naor and Reingold =-=[40]-=- gave a direct construction of a pseudorandom function in TC 0 , which is provably as hard as factoring and discrete logarithm. It follows that any natural proof that a function isn’t in TC 0 would yi... |

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Citation Context ...ing fame, has argued that even mathematical statements are ultimately about physical reality—since mathematics is rooted in computation, and the laws of physics determine what is and isn’t computable =-=[16]-=-. 13 Whether or not you agree with this, it does suggest the following “physical process criterion” for mathematical truth: We should expect a mathematical question to have a definite answer, if and o... |

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Citation Context ...r unknowable, or dependent upon which axioms we choose. In the eighties, the combinatorial view became dominant, in the wake of the circuit lower bounds of Furst-Saxe-Sipser [18], Ajtai [2], Razborov =-=[50, 52]-=-, and Smolensky [61]. On this view, P vs. NP is no more likely to be formally independent than Goldbach’s Conjecture, the Riemann Hypothesis, or any other ‘natural’ mathematical problem. Proving P �= ... |

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Citation Context ...t 2nε-secure pseudorandom generators. From there it’s easy to show that the tautologies {� Circuitn} n≥1 (assuming they are tautologies!) don’t have polynomial-size proofs in S. As an example, Pudlák =-=[45]-=- showed that the Cutting Planes proof system satisfies the EIP, and this immediately implies that Cutting Planes can’t prove NP � P/poly, again under the pseudorandomness assumption. So EIP is a usefu... |

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Citation Context ...ypothesis—its truth forever unknowable, or dependent upon which axioms we choose. In the eighties, the combinatorial view became dominant, in the wake of the circuit lower bounds of Furst-Saxe-Sipser =-=[18]-=-, Ajtai [2], Razborov [50, 52], and Smolensky [61]. On this view, P vs. NP is no more likely to be formally independent than Goldbach’s Conjecture, the Riemann Hypothesis, or any other ‘natural’ mathe... |

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Citation Context ...sistency. As Shelah [59] put it, consistency strength makes the universe of logical theories ‘taller,’ while relative consistency makes it ‘fatter.’ Relative consistency was what Gödel [20] and Cohen =-=[13]-=- used to prove the independence of the Axiom of Choice (AC) and Continuum Hypothesis (CH) from ZF. Recall that AC is the assertion that, given a set x of nonempty, pairwise disjoint sets, there exists... |

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Citation Context ...is relative consistency. As Shelah [59] put it, consistency strength makes the universe of logical theories ‘taller,’ while relative consistency makes it ‘fatter.’ Relative consistency was what Gödel =-=[20]-=- and Cohen [13] used to prove the independence of the Axiom of Choice (AC) and Continuum Hypothesis (CH) from ZF. Recall that AC is the assertion that, given a set x of nonempty, pairwise disjoint set... |

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Citation Context ...formalizable in S1 2!). Whether a theorem can be proven in a weak fragment of arithmetic might not say much about the theorem’s conceptual difficulty. In one of the many ironies of P vs. NP, 12 Pratt =-=[44]-=- showed that Primes ∈ NP ; that is, every prime has a succinct proof of primality. An obvious question is whether the deterministic primality test of Agrawal, Kayal, and Saxena [1] might eliminate the... |

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Citation Context ...ability to prove circuit lower bounds. Thus, many researchers hoped to show the EIP for stronger proof systems, especially the so-called Extended Frege (EF) system. Unfortunately, Krajíček and Pudlák =-=[31]-=- showed that the EIP fails for this system—ironically, under a cryptographic assumption like those used to prove lower bounds on proof size! The idea is simple: they created a disjunction ϕ = A (z) ∧ ... |

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Citation Context ...My favorite reference for the material of Section 2 is the book of Cohen [13]. For a definition of Cutting Planes and other proof systems mentioned in Section 4.1, see the survey of Beame and Pitassi =-=[5]-=-. Finally, there are interesting discussions about the logical status of P vs. NP on the Foundations of Mathematics (FOM) mailing list; see 8 Acknowledgments http://www.cs.nyu.edu/pipermail/fom/2001-A... |

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Citation Context ...s that S1 2 can’t even formalize the Shannon counting argument, which shows that almost all Boolean functions require exponential-size circuits. As a consequence, S1 2 can’t formalize Kannan’s result =-=[29]-=- that NP EXP � P/poly, which uses the Shannon counting argument; or the Buhrman-Fortnow-Thierauf result [8] that MAEXP � P/poly, which in turn uses Kannan’s result. It’s clear, then, that there exist ... |

56 | Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic
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Citation Context ... object is one that doesn’t exist in the recursion-theory world.sefficient ways to solve the very problems they were supposed to prove intractable! Subsequently, as we’ll see in Section 4.1, Razborov =-=[55]-=- used this idea to prove, under cryptographic assumptions, that P vs. NP is independent of certain theories of ‘bounded arithmetic.’ I don’t know if it’s possible any longer to discern a unifying tren... |

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Citation Context ..., and moved that one there. . . darn, still doesn’t work!” Resolution isn’t powerful enough to count the pigeons and the holes, and see that the former are more numerous than the latter. Recently Raz =-=[47]-=- improved Haken’s result to show that even formulas based on the so-called ‘Weak Onto Pigeonhole Principle’—that m ≫ n pigeons can’t be assigned to n holes, with exactly one pigeon in every hole—requi... |

51 | The history and status of the P versus NP question
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Citation Context ... mathematical question to have a definite answer. 1 Introduction The P vs. NP problem has been called “one of the most important problems in contemporary mathematics and theoretical computer science” =-=[60]-=-. That is an understatement. Not only is P vs. NP the defining question of our field; it’s one of the deepest questions ever asked for which we’d know how to recognize an answer. 2 (In other words, on... |

49 | Bounded arithmetic and lower bounds in Boolean complexity
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Citation Context ...or k = ε log n. Taking a step back, do these results tell us anything about P vs. NP—for example, whether the problem is likely to be independent of ZF or some other strong theory? In the appendix of =-=[48]-=-, Razborov argues that all known “explicit” techniques for circuit lower bounds can be formalized in S1 2. Therefore, the inability even of S2 2 to prove NP � P/poly under a pseudorandomness assumptio... |

47 | Positive vacuum energy and the N-bound
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Citation Context ...really gotten into the spirit of it and asked: what if the universe is finite? Recent evidence for a positive cosmological constant (e.g. [43]), together with arguments from black hole thermodynamics =-=[7]-=-, imply an upper bound of about 10 122 on the maximum number of bits accessible by any one observer. Intuitively, even if the universe is spatially infinite, most of it recedes too quickly from any on... |

44 | Nonrelativizing separations
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Citation Context ...posed to tackle P vs. NP using algebraic geometry—an approach that, they think, might escape the jaws of Razborov and Rudich. Using the deep result that MIP = NEXP [3], Buhrman, Fortnow, and Thierauf =-=[8]-=- showed that MAEXP, the exponential-time analogue of MA, doesn’t have polynomial-size circuits. This result is notable because it avoids both the Baker-Gill-Solovay and the Razborov-Rudich limitations... |

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Citation Context ... a new clause. For example, given the clauses (a∨�b) and (�a ∨ c ∨ d) we can derive (�b ∨ c ∨ d). The length of a resolution proof is the number of clauses it derives prior to the empty clause. Haken =-=[23]-=- showed the first superpolynomial lower bound on the lengths of resolution refutations. The unsatisfiable formulas he used were based on the Pigeonhole Principle; they encoded that n + 1 pigeons are e... |

43 | Sohoni: Geometric complexity theory: An approach to the P vs. NP and related problems
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Citation Context ...e, under cryptographic assumptions, that P vs. NP is independent of certain theories of ‘bounded arithmetic.’ I don’t know if it’s possible any longer to discern a unifying trend. Mulmuley and Sohoni =-=[39, 57]-=- have proposed to tackle P vs. NP using algebraic geometry—an approach that, they think, might escape the jaws of Razborov and Rudich. Using the deep result that MIP = NEXP [3], Buhrman, Fortnow, and ... |

42 | Resolution lower bounds for perfect matching principles
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Citation Context ... one does need to show that there are more ‘yes’ instances of SAT than there are clauses in the DNF—and that requires the Weak Onto Pigeonhole Principle! Improving on Raz’s result, Razborov has shown =-=[54]-=- that � Circuitn has no succinct resolution proofs even when the circuits in the encoding have bounded fan-in. Using a different reduction— based on the Nisan-Wigderson pseudorandom generator [41], ra... |

41 | Pseudorandom generators hard for k-dnf resolution and polynomial calculus resolution. Manuscript availbale at author’s webpage
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Citation Context ... the circuits in the encoding have bounded fan-in. Using a different reduction— based on the Nisan-Wigderson pseudorandom generator [41], rather than the pigeonhole principle— Razborov has also shown =-=[53]-=- that � Circuitn has no succinct proofs in several extensions of resolution, including PCR (Polynomial Calculus and Resolution) and k-DNF resolution, or resolution with k-DNF’s instead of clauses, for... |

36 | Reingold.Number-theoretic construc-tions of efficient pseudo-random functions - Naor, O - 2004 |

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Σ1 1-formulae on finite structures. Annals of Pure and Applied Logic, 24:1–48
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Citation Context ...s truth forever unknowable, or dependent upon which axioms we choose. In the eighties, the combinatorial view became dominant, in the wake of the circuit lower bounds of Furst-Saxe-Sipser [18], Ajtai =-=[2]-=-, Razborov [50, 52], and Smolensky [61]. On this view, P vs. NP is no more likely to be formally independent than Goldbach’s Conjecture, the Riemann Hypothesis, or any other ‘natural’ mathematical pro... |

30 |
Universal search problems, Problemi Peredachi Informatsii 9
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Citation Context ...re would then also be no reason to think further about the problem. In the seventies, when serious work on P vs. NP began following the discovery of NP-completeness by Cook [14], Karp [30], and Levin =-=[36]-=-, the problem continued to be seen in logical terms. People tried to separate P and NP using the same ideas that worked in recursion theory; this is the context of the Baker-Gill-Solovay [4] result th... |

29 |
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Citation Context ...r unknowable, or dependent upon which axioms we choose. In the eighties, the combinatorial view became dominant, in the wake of the circuit lower bounds of Furst-Saxe-Sipser [18], Ajtai [2], Razborov =-=[50, 52]-=-, and Smolensky [61]. On this view, P vs. NP is no more likely to be formally independent than Goldbach’s Conjecture, the Riemann Hypothesis, or any other ‘natural’ mathematical problem. Proving P �= ... |

27 | bounds for propositional proofs and independence results in bounded arithmetic, Automata, languages, and programming: 23rd international colloquium, icalp ’96
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Citation Context ...l n, even if we could prove � Circuit50, � Circuit100, and so on. 11 To do justice to the second-generation independence results would require another survey altogether, and indeed such surveys exist =-=[10, 46, 49]-=-. Here I’ll just sketch the main ideas. Many of the results involve a set of theories called bounded arithmetic [9, 10, 11]. In these theories, the objects are natural numbers, and we have available t... |

21 |
On the restricted ordinal theorem
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Citation Context ...rated, obviously PA +Π1 is an unrealistically powerful theory. Despite its power, though, it turns out to be unable to prove even some simple Π2-sentences.sAs an example, consider Goodstein’s Theorem =-=[22]-=-. We write a positive integer n (say 40) as a sum of powers of 2, then write the exponents as sums of powers of 2, and so on: 40 = 2 5 + 2 3 = 2 22 +2 0 + 2 2+20 . (This is called the base-2 hereditar... |

21 | An elementary problem equivalent to the Riemann hypothesis
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Citation Context ...ecture by including an explicit lower bound—say, that there are at least log log log n twin primes less than n. The complexity of the Riemann Hypothesis (RH) is less obvious. But a result of Lagarias =-=[34]-=- shows that RH is equivalent to the assertion that for all positive integers n, the sum of the divisors of n is at most α (n) = Hn + e Hn ln Hn, where Hn = 1+ 1 2 1 +· · ·+ n is the nth harmonic numbe... |

16 | Super-Polynomial versus half-exponential circuit size in the exponential hierarchy
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Citation Context ...n a finite statement, such as 10 For t (n) to be half-exponential means that t (t (n)) grows like 2 n . Such functions exist, but are difficult to describe (see Miltersen, Vinodchandran, and Watanabe =-=[37]-=-).s� Circuitn := “ SATn requires circuits of size n log n ” where SATn is the set of SAT instances of size n. The goal is to show (perhaps under a complexity assumption) that in some proof system, � C... |

14 |
A mathematical incompleteness
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Citation Context ...1 each time. Surprisingly, by an argument involving countable ordinals, Goodstein showed that no matter what n we start with, this process will always converge to 0. Subsequently Paris and Harrington =-=[42]-=- showed that Goodstein’s Theorem is independent of PA—meaning that the use of countable ordinals was necessary. Clearly Goodstein’s Theorem is Π2: for all n, there exists a t such that the process con... |

11 | Bounded Arithmetic and Propositional Proof Complexity
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(Show Context)
Citation Context ...l n, even if we could prove � Circuit50, � Circuit100, and so on. 11 To do justice to the second-generation independence results would require another survey altogether, and indeed such surveys exist =-=[10, 46, 49]-=-. Here I’ll just sketch the main ideas. Many of the results involve a set of theories called bounded arithmetic [9, 10, 11]. In these theories, the objects are natural numbers, and we have available t... |

11 |
The Continuum Hypothesis, Part I
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Citation Context ...ce and for all whether intermediate cardinalities exist. Cohen predicted that CH might come to be seen as obviously false—a prediction that’s partly come to pass (see, for example, a survey by Woodin =-=[62]-=-). But even if we reject CH, there remains the question: how many intermediate cardinalities are there? It turns out thatsany number would be consistent with ZF. Cohen thought perhaps there are uncoun... |

6 | On the Independence of P versus NP
- Ben-David, Halevi
- 1992
(Show Context)
Citation Context ...east a large chunk of it). But there’s a reason that’s probably impossible with current techniques, which was pointed out by Kurtz, O’Donnell, and Royer [33], and (independently) Ben-David and Halevi =-=[6]-=-. Let’s define a Π1-sentence to be any sentence of the form, “For all x, P (x),” where P is function that can be proven to be recursive in Peano Arithmetic, PA. (That is, there’s a Turing machine M th... |

6 | How to prove representation-independent independence results
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(Show Context)
Citation Context ...le O, such that P O vs. NP O is independent of ZF, no matter which Turing machine is used to specify O? The answer turns out be yes, as shown by Hartmanis [25] and also by Kurtz, O’Donnell, and Royer =-=[33]-=-. Skipping technicalities, the intuition is as follows. We construct O so that for almost all input lengths, O collapses P and NP. But for a few, widely-separated input lengths, call them f (1) , f (2... |

6 |
Lenstra Jr. The Development of the Number Field Sieve
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- 1993
(Show Context)
Citation Context ...reak any one-way function (including factoring, discrete logarithm, etc.), in time 2O(nε ) for any ε > 0. By comparison, the best known factoring algorithm is conjectured to run in time roughly 2n1/3 =-=[35]-=-. If C is a smaller class, for instance the class TC 0 of constant-depth polynomial-size threshold circuits, then the reductions of [27] and [21] can’t be carried out. However, Naor and Reingold [40] ... |

6 |
AND 32 OTHERS (SUPERNOVA COSMOLOGY PROJECT): Measurements of Ω and Λ from 42 high-redshift supernovae
- PERLMUTTER
- 1999
(Show Context)
Citation Context ...al be different.sIf you’re still reading this, perhaps you’ve really gotten into the spirit of it and asked: what if the universe is finite? Recent evidence for a positive cosmological constant (e.g. =-=[43]-=-), together with arguments from black hole thermodynamics [7], imply an upper bound of about 10 122 on the maximum number of bits accessible by any one observer. Intuitively, even if the universe is s... |

5 |
The P=?NP poll
- Gasarch
- 2002
(Show Context)
Citation Context ... But we may not be able to prove which way it goes, and we may not be able to prove that we can’t prove it. 7 Further Reading For the current consensus on P vs. NP, see William Gasarch’s “P=?NP poll” =-=[19]-=-—or go to http://www.ideosphere.com/fx-bin/Claim?claim=P!NP to bet on whether P �= NP will be proved by 2010. See my Complexity Zoo at http://www.cs.berkeley.edu/˜aaronson/zoo.html for more about the ... |

5 |
Results about Context-Free Languages and Lower Bounds
- Hartmanis
- 1984
(Show Context)
Citation Context ...ally time to say something about P vs. NP. We’ll start with some results from the seventies and eighties: in Section 3.1, oracle independence theorems due to Hartmanis and Hopcroft [26] and Hartmanis =-=[25]-=-; and in Section 3.2, non-oracle independence theorems (concerning weak logical theories) due to DeMillo and Lipton [15] and Sazanov [58]. 3.1 Oracles Given a Turing machine M, let L (M) be the langua... |

5 | Understanding the Mulmuley-Sohoni Approach to P vs
- Regan
(Show Context)
Citation Context ...e, under cryptographic assumptions, that P vs. NP is independent of certain theories of ‘bounded arithmetic.’ I don’t know if it’s possible any longer to discern a unifying trend. Mulmuley and Sohoni =-=[39, 57]-=- have proposed to tackle P vs. NP using algebraic geometry—an approach that, they think, might escape the jaws of Razborov and Rudich. Using the deep result that MIP = NEXP [3], Buhrman, Fortnow, and ... |

5 | A logical approach to the problem "P=NP - Sazanov - 1980 |

3 | Bounded arithmetic, cryptography, and complexity, Theoria 63:147–167
- Buss
- 1997
(Show Context)
Citation Context ...ults would require another survey altogether, and indeed such surveys exist [10, 46, 49]. Here I’ll just sketch the main ideas. Many of the results involve a set of theories called bounded arithmetic =-=[9, 10, 11]-=-. In these theories, the objects are natural numbers, and we have available the constants 0 and 1 and the functions +, ×, ≤, ⌊x/2⌋, |x|, and #, where |x| = ⌈log 2 (x + 1)⌉ and x#y = 2 |x|·|y| . We can... |

3 |
The consistency of P=NP and related problems within fragments of number theory
- DeMillo, Lipton
- 1979
(Show Context)
Citation Context ..., oracle independence theorems due to Hartmanis and Hopcroft [26] and Hartmanis [25]; and in Section 3.2, non-oracle independence theorems (concerning weak logical theories) due to DeMillo and Lipton =-=[15]-=- and Sazanov [58]. 3.1 Oracles Given a Turing machine M, let L (M) be the language accepted by M. The following was shown in [24, 26]. Theorem 2 (Hartmanis-Hopcroft) There exists a Turing machine M th... |

3 |
Independence results in computer science
- Joseph, Young
- 1981
(Show Context)
Citation Context ... else, that would at least rule out any approach to P vs. NP that can be formalized within those theories. Already in the seventies and early eighties, there was a good deal of work in this direction =-=[15, 28, 58]-=-; we’ll look at two examples. DeMillo and Lipton [15] show P �= NP unprovable in a fragment of number theory they call ET. The objects of ET are integers, while the language consists of the functions ... |

3 |
On the concepts of completeness and interpretation of formal systems, Fundamenta Mathematicae 39:103–127
- Kreisel
- 1952
(Show Context)
Citation Context ... actually showed was that Goodstein’s Theorem is independent of PA +Π1—in other words, even if we threw in all true Π1-sentences for free, the theorem would still be unprovable in PA. Indeed, Kreisel =-=[32]-=- showed the following (see also [6]). Lemma 3 (Kreisel) A function is provably recursive in PA +Π1 if and only if it’s provably recursive in PA. Now suppose P vs. NP were independent of PA +Π1. Then N... |