[LOGIC] Dificulty Very Satisfatiability

8/12/2019 [LOGIC] Dificulty Very Satisfatiability

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N.C.A. da CostaF.A. Doria

ON THE EXISTENCE OF VERY DIFFICULTSATISFIABILITY PROBLEMS1

Our arguments take place within ordinary informal mathematics. isthe set of natural numbers. We recall that analgorithm is a computational

procedure that can be formalized through Turing machines, partial recur-sive functions and the like; a computable function f : is a functionthat can be computed with the help of an algorithm. Chief results will

be presented in the language of onetape Turing machines over a binaryalphabet A2={0, 1}; the set of all finite binary words is noted A

2. Turingmachine results needed in this paper can be found in either [6] or [9]. Ourbinary Turing machines are defined by tables [6] [9] and a Turing machine

cycleis the set of operations described by a line in the machines table. Foran arbitrary real number x, ]x[ is the integral part ofx. |x| will always de-note thelengthofx, that is, the number of letters from A2 inx, even in thecase when xcan be taken to be a number. To do so we fix an enumeration

for the binary words given by a map : A2 defined as:

{, 0, 1, 00, 01, 10, 11, 000, . . .} {0, 1, 2, 3, 4, 5, . . .},

such that ifx A2(x) , then

|x|=]log((x) + 1)[,

wherelog denotes a base 2 logarithm. Therefore our algorithms can repre-

sent maps from to , form to A2, and so on.

Definition 1.1. Theidentity machine is the (trivial) Turing machinewith zero states and an empty set of instructions; it (trivially) holds onevery input. We agree that, ifx is its input, then x= x.

1Partially supported by grants from FAPESP and from CNPq (Brazil), PhilosophySection.

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Definition 1.2.

1. The function s(M, x), given by:

s(M, x) = #( range in tape squares used when M computes over x),

ifMstops over x, and s(M, x) =, ifMdoesnt stop over x, is thespace budget of M over x.

2. The function t(M, x), defined by:

t(M, x) =|x| + # (cycles performed by M over x),

ifMstops over x, and t(M, x) =, ifM doesnt stop over x, is the

operation time of M over x.

3. IfM stops over x, the function t(M, x), defined by:

t(M, x) = # (cycles performed by M over x),

is the strict time measure of M over x.

For the definition of acceptable universal programming systems P see[6], p.95. Two programming systems P,P arepolynomially related if thereis a 11 total recursive map that sends one P over P and ifs, s and t, t

are the space budget and operation time functions in the programmingsystemsP, P, thens =p(s, t), t =q(s, t), wherep, qare polynomials overthe integers. In this note we restrict our attention to polynomially relatedprogramming systems.

Let [x] be a measure of the size ofx as an input for the Turing machine

M. Letxn, n be a sequence of inputs for M such that [xn]< [xn+1]. (Toclarify matters, we notice that one usually takes [x] =|x|, or if meaningful,[x] = v(x) = number of variables in x. In the last case, it may be thenumber of cities in an instance of the traveling salesman problem, or thenumber of propositional variables in the satisfiability problem.) Finally, let

F : be a function:

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Definition 1.3. (Complexity of computational resources)

1. Let SM([x]) s(M, x). Then SM([]) is a space bound for themachineM.

2. Let TM([x]) t(M, x). Then TM([]) is a time bound for themachineM.

3. GM, J([x]) = min all space (or time) bounds. J = S or T, is thespace (or time) complexity of computational resources whenMacts upon a set of inputs of values [x].

KolmogorovChaitinManin complexity

Main references here are [1] [5] [7] [8] [10].

Remark 1.4. We are going to require a pairing function :2 whichis a total recursive embedding with a recursive inverse that satisfies thefollowing condition:

(k, j) k(j),

for all k, j , a prescribed function. For examples of functions thatsatisfy that condition, see [7], p.226 and [10]. nples are coded by functionsdefined recursively as follows:

2(k, j) =(k, j);n(k1, . . . , kn) =(n1(k1, . . . , kn1), kn).

Therefore, ifMfis a Turing machine, the corresponding partial recur-sive 1function is given by f(x) =Mf(x). For an mfunction.

f(x1, . . . , xn) =Mf(n(x1, . . . , xn)).

Definition 1.5.

(KolmogorovChaitin) The complexity ofywith respectto an algorithm f is given by Kf(y) = miny=f(x)|x|, if there is one such x,orotherwise.

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Corollary 1.6. Ify = f(x), then Kf(y) |x|.

The main result on Kf is:

Proposition 1.7. There is an optimal algorithmusuch that for any otheralgorithmv, Ku(f) Kv(f) +c(u, v).

Remark 1.8. We now fix for once and all an optimal algorithm u whichwill be our reference in all further calculations and proofs dealing with the

concept of complexity.

Definition 1.9.

1. Kh(y) Kf(y) if and only ifKh(y) Kf(y) +c(f, h).

2. Kh Kk if and only ifKh Kk and Kk Kh.

Ifu is our fixed optimal family, then:

Definition 1.10. K(y) =Ku(y).

KCMincompressibility

Definition 1.11. A binary sequence x over the alphabet A2 is KCMincompressibleif:

K(x) K(x)

where is given in Definition 1.1.

Remark 1.12. The idea is that we cannot recursively compress x.

From Definitions 1.9. and 1.11. we get:

Corollary 1.13. For KCMincompressible binary sequencesx,

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|x| +c(g, ) K(x) |x| c(, g)

or

K(x) +c(, g) |x| K(x) c(g, ).

Corollary 1.14. For large|x|, the ratio between all KCMincompressiblesequences of length |x| and the remaining sequences of the same lengthexceeds2c(,g) [10].

Corollary 1.15. There are KCMincompressible sequences.

Posts theorem on truthtablesLet L be a classical propositional calculus. Let F(p1, . . . , pn) be a propo-sitional function (=any expression in L) on the propositional variables

p1, . . . , pn. Let be the Turing machine program over an adequate bi-nary coding that out of any such Fcomputes the truth table for F, whichis here seen as a binary sequence of length 2n. We restrict our attention

to propositional functions in either conjunctive normal form (cnf) or indisjunctive normal form (dnf).

Proposition 1.16. (Post) Let n be any binary sequence of length 2n.Then we can algorithmically obtain a propositional functionFn= F p1, . . . , pnin cnf (or in dnf) withinL such that

(F p1, . . . , pn) =n.

Corollary 1.17. Ifn is KCMincompressible, then

K(F p1, . . . , pn) K(n).

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Main results on the difficulty of computations

We suppose that we have translated propositional expressions in cnf or dnfinto binary sequences (which represent natural numbers) that satisfy the

requirement in Remark 1.4. We will be interested in sequences of propo-sitional functions Fn = F(p1, . . . , pn), n in cnf (or in dnf), where ndenotes both the index for the formula in the succession and the num-ber of different propositional variables in each expression Fn. As already

emphasized, we suppose that the corresponding truth tables n are KCMincompressible. Let S be any Turing machine that takes one such proposi-tional function Fn =F(p1, . . . , pn) as its input and decides whether or notFn is satisfiable [6]. Then:

Proposition 1.18. (Exponential dependence on n) If n is KCMincompressible, and ifFn=n, then:

1. t(S, F p1, . . . , pn) 2v(Fn) c, c a constant which doesnt depend on

Fn.

2. For largen= v(Fn), there is a Turing machineSthat checks the sat-isfiability ofFn such that t(S, Fn) p(|Fn|), p a polynomial functionof the length |Fn|.

In order to prove the first statement we proceed as follows: if is the

truthtable generating Turing machine, then we have thatF(p1, . . . , pn) =n. Now, from Corollary 1.5:

K(n) =minn=F|F(p1, . . . , pn)|. (1)

Therefore we can write:

K(n) |Fn|, (2)

whereFn=F(p1, . . . , pn).

From the fundamental theorem on the logarithmic complexity (Propo-sition 1.7.), ifg is our fixed optimal algorithm,

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K(n) =Kg(n) K(n) +c(g, ). (3)

From Corollary 1.13. we have that:

K(n) |n| c(, g) = 2n c(, g). (4)

If we combine equation (4) with the previous expression, that results fromthe fundamental theorem (3) plus equation (2), we have:

2n (c(g, ) +c(, g)) K(n) |Fn|. (5)

Therefore we get an estimate for |Fn|:

|F(p1, . . . , pn)| 2v(Fn) c, (6)

where the parameter c= c(g, ) + c(, g) is independent of both Fn and n.

Now:t(S, Fn) =|Fn| + # (cycles performed by S over Fn). (7)

But from equation (6),

|Fn| 2n c

given that estimate, from (7)

t(S, Fn) 2v(Fn) c+ # (cycles performed by S over Fn),

then:

t(S, Fn) 2n c. (8)

For the second statement, if n = v(Fn) is large, since n is KCMincompressible, we can use Corollary 1.17. Thus, an optimal binary coding

for Fn would give, at last,

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|Fn|= 2n c. (9)

Suppose the usual, direct construction of the truthtable. For eachline in the truthtable we have that the time t spent in its verification is

approximately

t(S, Fn) (2n c). (10)

(Here is the average time it takes S to check each item in Fn whenproducing a truthvalue in the truthtable line.) As there are 2n lines inthat table, from (7),

t(S, Fn) (2n c) + 2n(2n c), (11)

whereas one gets

t(S, Fn) |Fn|((|Fn| +c) + 1), (12)

which, for any nonoptimal binary coding Fnfor Fn such that

|Fn| |Fn|,

gives the desired polynomial relation.

Corollary 1.19.

1 . For an arbitrary familyFn, the strict time measure

t(S, Fn)

is at least linear on the length |Fn|.

2. For an arbitrary familyFnwhose truthtables are KCMincompressible,the strict time measure t(S, Fn) is at least exponential on n.

Proof. For the first assertion: in order to compute the number of variables

inFn (a necessary step in the construction of a truthtable) we will have to

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scan at least the whole length of Fn since that information is not writtenout on the machine tape. The second assertion follows from the first one.

Density, undecidability and incompleteness

Definition 1.20. Given a function s : R (R are the reals), we saythats(n) is asubexponentialif there exists an n0such that for all n > n0s(n)< e(n), for all e(n) of the form e(n) =abn a >1 and b >0.

Definition 1.21. A set Fn of expressions in cnf (or in dnf) is tractableprovided that there is an n0 so that for every n > n0.

|Fn|

< s(n),

wheres(n) is a subexponential.

Remark 1.22. Notice what goes on here: for each new propositionalvariable added to the expressions in the set Fn, the corresponding lengthsdo not blow up in an exponential way. The polynomial case is strictly

contained in our definition of tractability.

Proposition 1.23.

limn # of tractable Fn# of KCMincompressible Fn

= 0

We can give a more elegant interpretation of the previous result: let

B Ir [0, 1] be the set of binary irrationals in the unit interval. Codethe sequences n by clopen basis sets in B Ir defined by the n as initial

segments. Proposition 1.16. allows us to uniquely code expressions in cnf in

Lby the sequences n, so that we map all such expressions in L over clopenbasis sets inB Ir. The density result follows in a more elegant formulation.Granted that coding plus the preceding definitions and results:

Proposition 1.24. The set of tractable satisfiability problems for cnf for-

mulae inL has zero measure.

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its early versions; both authors wish to thank Prof. P. Suppes who helpedand guided them through the conceptual selva selvaggiaof Kolmogorovscomplexity with many references on algorithmic randomness theory; and

Prof. David Miller, for his kindness in having a preliminary sketch of this

work aptly reviewed.

References

[1] G.J. Chaitin, Information, Randomness and Incompleteness,

World Scientific, 1987.[2] N.C.A. da Costa and F.A. Doria,Int. J. Theor. Phys30, 1041

(1991).[3] N.C.A. da Costa and F.A. Doria, Suppes predicates and the con-

struction of unsolvable problems in the axiomatized sciences, to appear in

Patrick Suppes, Mathematician, Philosopher, P. Humphreys, ed.,Kluwer, 1993.

[4] N.C.A. da Costa and F.A. Doria,Very hard satisfiability problems,preprint CETMC/ECO, 1992.

[5] A.N. Kolmogorov,Problemy Peredachi Informatsii1, 3 (1965).[6] M. Machtey, P. Young, An Introduction to the General The-

ory of Algorithms, NorthHolland, 1979.

[7] Yu.I. Manin,A Course in Mathematical Logic, Springer, 1977.[8] P. MartinLof, Inf. and Control9, 602 (1966).

[9] H. Rogers Jr., Theory of Recursive Functions and EffectiveComputability, McGrawHill, 1967.

[10] A.K. Zvonkin and L.A. Levin, Uspekhi Matem. Nauk 25, 1(1970).

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[LOGIC] Dificulty Very Satisfatiability

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