An elementary proof that P ≠ NP

Bhupinder Singh Anand1

We show that first-order Peano Arithmetic has no non-standard models, and that

this implies P ≠ NP.

1. Introduction

In a paper presented to ICM 2002, Ran Raz comments [Ra02]:

“A Boolean formula f(x1, ..., xn) is a tautology if f(x1, ..., xn) = 1 for every x1, ..., xn. A

Boolean formula f(x1, ..., xn) is unsatisfiable if f(x1, ..., xn) = 0 for every x1, ..., xn.

Obviously, f is a tautology if and only if ~f is unsatisfiable.

Given a formula f(x1, ..., xn), one can decide whether or not f is a tautology by

checking all the possibilities for assignments to x1, ..., xn. However, the time needed

for this procedure is exponential in the number of variables, and hence may be

exponential in the length of the formula f. …

P ≠ NP is the central open problem in complexity theory and one of the most

important open problems in mathematics today. The problem has thousands of

equivalent formulations. One of these formulations is the following:

1 The author is an independent scholar. E-mail: re@alixcomsi.com; anandb@vsnl.com. Postal address: 32,

Agarwal House, D Road, Churchgate, Mumbai - 400 020, INDIA. Tel: +91 (22) 2281 3353. Fax: +91 (22)

2209 5091.

Created: Saturday, 25th

March 2006, 8:21:29 AM IST. Updated Wednesday, 27th

June 2007, 11:14:41 AM

IST by re@alixcomsi.com.

2

Is there a polynomial time algorithm A that gets as input a Boolean formula f and

outputs 1 if and only if f is a tautology? P ≠ NP states that there is no such

algorithm.”

Now, it follows from Gödel’s reasoning [Go31, Theorem VI, p25, eqn.12] that we can

define a PA-formula, [R(x)]2,3, such that:

(i) [R(x)] is constructible in a standard, first-order, Peano Arithmetic, PA

(Appendix A);

(ii) we can prove, meta-mathematically, that [R(x)] translates as an arithmetical

tautology, R(x), under the standard interpretation (Appendix A), M, of the

Arithmetic;

(iii) [R(x)] is not formally provable4 in the Arithmetic.

Now, it also follows from Gödel’s reasoning [Go31, Theorem VI (1), p25-26], that R(x)

is Turing-decidable as always TRUE in M, where:

Definition 1: A total number-theoretical relation, say R'(x1, x2, ..., xn), when treated

as a Boolean function, is Turing-decidable in M if, and only if, it is instantiationally

equivalent to a number-theoretic relation, S(x1, x2, ..., xn), and there is a Turing-

2 In his reasoning ([Go31], Theorem VI, p25, eqn.12), Gödel only refers to the formula [R(x)] indirectly as

the “CLASS EXPRESSION r”, where r is the Gödel-number of the PA-formula [R(x)].

3 We use square brackets to indicate when the expression within the brackets is intended to be viewed

purely as a syntactical string of mathematical symbols, devoid of any meaning whatsoever. In other words,

when it is not to be seen as representing any concept under an intuitive interpretation of the symbols.

4 A formal proof in a system of, say, Peano Arithmetic, is a sequence of well-formed formulas of PA, each

of which is either an axiom of PA, or which can be shown to be an immediate consequence - by the

mechanical application of the rules of deduction of PA - of any one, or more, of the preceding formulas of

the sequence.

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machine T such that, for any given natural number sequence, (a1, a2, ..., an), T will

compute S(a1, a2, ..., an) as either TRUE, or as FALSE, in M.

Definition 2: A total number-theoretical relation, R'(x1, x2, ..., xn), when treated as a

Boolean function, is Turing-computable in M if, and only if, there is a Turing-

machine T such that, for any given natural number sequence, (a1, a2, ..., an), T will

compute R'(a1, a2, ..., an) as either TRUE, or as FALSE, in M.

The question arises: Is R(x) also Turing-computable as always TRUE in M?5

If we assume, first, that every total arithmetical relation that is Turing-computable as

always TRUE in M is PA-provable, then R(x) is Turing-decidable as always TRUE in M,

but not Turing-computable as always TRUE6 in M, and, so, P ≠ NP7.

5 We note that, although Turing-decidability (which is, essentially, constructive verifiability in the classical

sense) of a number-theoretic relation necessarily implies Turing-computability, the converse need not hold

unless we assume the standard formulation of the Church-Turing Thesis [Ch36], [Tu36].

However, the arguments of this paper suggest that CT may need to be weakened from a strong identity -

between intuitively computable functions (and relations) and recursive (Turing-computable) functions (and

relations) - to an instantiational equivalence only.

6 The reason, in this case, that R(x) may be Turing-decidable as always TRUE in M, but not Turing-

computable as always TRUE in M, lies in the fact that Gödel’s peculiar, but constructive, definition of the

arithmetical relation R(x) - more accurately, R(x, p) - is in terms of a primitive recursive relation Q(x) -

more accurately, Q(x, p) - and involves a circularity.

This follows from Gödel’s definition of Q(x, y), a primitive recursive relation whose domain includes the

Gödel-number of the PA-formula [R(x, y)]. Hence, the definition of Q(x, y) implicitly references the range

of values of R(x, y).

As a consequence, there may be some value of n such that, when computing R(n, p), any Turing-machine

that computes R(x, p) does not halt, but goes into a perpetual (machine-generated, but not program-

generated) loop, on the input n since, as in the case of Turing's Halting function, an instantaneous tape

description repeats itself.

7 This follows only on a strict interpretation of the SAT problem as expressed by Raz’s assertion, since, in

this case, R(x) is instantiationally equivalent to a Turing-computable primitive recursive relation. We

assume that this interpretation follows from [Cook].

4

If we assume, however, that there is a total arithmetical relation that is Turing-

computable as always TRUE in M, but which is not PA-provable, then this implies that

there is a non-standard8 model of PA9.

We conclude that, if PA has no non-standard model, then, under a strict interpretation of

the above expression of the PvNP problem, P ≠ NP.

2. Standard, first-order, PA has no non-standard model

We give an elementary proof that, if PA is a standard, first-order, Peano Arithmetic

(Appendix A) then PA has no non-standard model (i.e., a model whose domain contains

an element that is not a successor of 0).

We denote by G(x) the PA-formula:

[x=0 v ¬(∀y)¬(x=y')].

This translates, under every interpretation10 of PA, as:

8 We define a non-standard model, say M', of first-order Peano Arithmetic as one in which there is some s

in the domain of M' that is not a successor of 0; hence s is not a natural number.

9 This follows since, if [R(x)] is unprovable in PA, but Turing-computable as always TRUE in M, then, by

Gödel’s Completeness Theorem ([Me64], p68), there is some M' such that ~R(s) must hold for some s in

the domain of M' that is not a natural number.

10

The word “interpretation” may be used both in its familiar, linguistic, sense, and in a mathematically

precise sense; the appropriate meaning is usually obvious from the context.

Mathematically, following Tarski ([Me64], §2, p49): “An interpretation consists of a non-empty set D,

called the domain of the interpretation, and an assignment to each predicate letter Ajn of an n-place relation

in D, to each function letter fjn of an n-place operation in D (i.e., a function from D

n into D), and to each

individual constant ai of some fixed element of D. Given such an interpretation, variables are thought of as

ranging over the set D, and ¬, →, and quantifiers are given their usual meaning. (Remember that an n-

place relation in D can be thought of as a subset of Dn, the set of all n-tuples of elements of D.)”

Loosely speaking, the interpreted relation R'(x) is obtained from the formula [R(x)] of a formal system P by

replacing every primitive, undefined, non-logical, symbol of P in the formula [R(x)] by an interpreted

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Either x is 0, or x is a ‘successor’11.

Now, in every interpretation of PA, we have that:

(a) G(0) is true;

(b) If G(x) is true, then G(x') is true.

It follows, from Gödel's completeness theorem12, that:

(c) [G(0)] is provable in PA;

(d) [G(x) => G(x')] is provable in PA.

We also have, by Generalisation (Appendix A), that:

(e) [(∀x)(G(x) => G(x'))] is provable in PA;

From the Induction axiom S9 (Appendix A), we thus have that:

(f) [(∀x)G(x)] is provable in PA.

We conclude that, except 0, every element in the domain of any interpretation of PA is a

‘successor’.

mathematical symbol. So the P-formula [(∀x)R(x)] interprets as the sentence (∀x)R'(x), and the P-formula

[¬(∀x)R(x)] as the sentence ¬(∀x)R'(x).

11

Note that, by virtue of axiom S4, x can only be a ‘successor’ of a unique element in the interpretation. It

follows that an ordinal such as the first infinite ordinal, ω, is not the ‘successor’ of any ordinal in the sense

required by the PA axioms.

12

“In any first-order predicate calculus, the theorems are precisely the logically valid wfs.” ([Me64], p68).

6

Now, it follows, first, that, since there are no infinitely descending sequences of

ordinals13, every set-theoretical interpretation of PA must be restricted to the domain that

consists only of the ordinal 0, and the ordinals that are the ‘successors’ of the ordinal 0.

By definition, this domain corresponds to the natural numbers only.

Second, if we accept that [Se91]:

... ZFC exhausts our intuition except for things like consistency statements, so a

proof means a proof in ZFC ... all of us are actually proving theorems in ZFC.

then we must conclude that all non-standard models of PA are isomorphic, and that there

are no non-standard models of a standard, first-order, Peano Arithmetic.

3. Conclusions

(i) A total arithmetical relation is Turing-computable as always TRUE under the

standard interpretation of a Peano Arithmetic if, and only if, it is PA-provable.

(ii) There are no non-standard models of PA.

(iii) P ≠ NP.

Appendix A

1: Logical Symbols, Axioms, and Rules of Inference of PA

The primitive logical symbols of PA are:

→ & v ~ ∀ x, y, ... a, b, ...

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As would be required if there were a non-standard model of PA in, say, ZF, in which all the elements in

the domain of the interpretation, except the ordinal 0, are ‘successors’, but not necessarily the ‘successors’

of the ordinal 0.

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implies and or not for all variables constant terms

If A, B, C are well-formed formulas of PA, then the logical axioms of PA are:

(1) A → (B → A);

(2) (A → (B → C)) → ((A → B) → (A → C));

(3) (~B → ~A) → ((~B → A) → B);

(4) (∀x)A(x) → A(t), if A(x) is a well-formed formula of PA, and t is a term of PA

free for x in A(x);

(5) (∀x)(A → B) → (A → (∀x)B), if A is a well-formed formula of PA containing

no free occurrences of x.

The rules of inference of PA are:

(i) Modus Ponens: B follows from A and A → B;

(ii) Generalisation: (∀x)A follows from A.

By Tarski's definitions of the satisfiability and truth of the formulas of PA under an

interpretation, when the rules of inference are applied to true well-formed formulas of PA

under a given interpretation, then the results of these applications are also true (i.e., every

theorem of PA is true in any model of PA).

2: Primitive Symbols and Proper Axioms of PA

(a) PA has a single predicate letter, A2

1 (as usual, we write “ t = s ” for A2

1(t, s);

(b) PA has one individual constant a1 (written, as usual, “ 0 “);

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(c) PA has three function letters f1

1, f2

1, f22. We shall write “ t' ” instead of f

11(t); “

t+s ” instead of f2

1(t, s); and “ t*s ” instead of f2

2(t, s).

The proper axioms of PA are:

(S1) (x1 = x2) → ((x1 = x3) → (x2 = x3));

(S2) (x1 = x2) → (x1' = x2');

(S3) 0 ≠ (x1)';

(S4) ((x1)' = (x2)') → (x1 = x2);

(S5) (x1 + 0) = x1;

(S6) (x1 + x2') = (x1 + x2)';

(S7) (x1*0) = 0

(S8) (x1*(x2')) = ((x1* x2) + x1);

(S9) For any well-formed formula F(x) of PA:

F(0) → ((∀x)(F(x) → F(x')) → (∀x)F(x)).

3: The standard interpretation, M, of PA

The standard interpretation, M, of PA, is taken to be the intuitive arithmetic of the natural

numbers, as expressed by Dedekind's semi-axiomatic formulation of the Peano

Postulates, where:

(i) the integer 0 is the interpretation of the PA-symbol “ 0 ”;

(ii) the successor operation (addition of 1) is the interpretation of the

PA-symbol “ ' ”;

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(iii) ordinary addition and multiplication are the interpretations of the

PA-symbols “ + ” and “ * ”;

(iv) the interpretation of the PA-symbol “ = ” is the identity relation.

References

[Ch36] Church, A. 1936. An unsolvable problem of elementary number theory. Am. J.

Math., Vol. 58, pp. 345-363. Also in M. Davis (ed.). 1965. The Undecidable.

Raven Press, New York.

[Cook] Cook, S. The P versus NP Problem. Official description provided for the Clay

Mathematical Institute, Cambridge, Massachusetts.

<http://www.claymath.org/millennium/P_vs_NP/Official_Problem_Description.pdf>

[Go31] Gödel, Kurt. 1931. On formally undecidable propositions of Principia

Mathematica and related systems I. Translated by Elliott Mendelson. In M.

Davis (ed.). 1965. The Undecidable. Raven Press, New York.

[Me64] Mendelson, Elliott. 1964. Introduction to Mathematical Logic. Van Norstrand,

Princeton.

[Ra02] Raz, Ran. 2002. P ≠ NP, Propositional Proof Complexity, and Resolution

Lower Bounds for the Weak Pigeonhole Principle. ICM 2002, Vol. III, 1–3.

[Se91] Shelah, Saharon. 1991. The Future of Set Theory. Proceedings, Israel

Mathematical Conference, 6.

<http://shelah.logic.at/future.html>

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[Tu36] Turing, Alan. 1936. On computable numbers, with an application to the

Entscheidungsproblem. Proceedings of the London Mathematical Society, ser.

2. vol. 42 (1936-7), pp.230-265; corrections, Ibid, vol 43 (1937) pp. 544-546.

<http://www.abelard.org/turpap2/tp2-ie.asp - index>