Boolean Function Analogs of Covering Systems
aa r X i v : . [ m a t h . C O ] J a n Boolean Function Analogs of Covering Systems
Anthony ZALESKI and Doron ZEILBERGER
Abstract : Bob Hough recently disproved a long-standing conjecture of Paul Erd˝os regardingcovering systems. Inspired by his seminal paper, we describe analogs of covering systems to Booleanfunctions, and more generally, the problem of covering discrete hyper-boxes by non-parallel lowerdimensional hyper-subboxes. We point out that very often primes are red herrings. This is definitelythe case for covering system, and who knows, perhaps also for the Riemann Hypothesis.
Prime Numbers are Sometimes Red Herrings
The great French mathematical columnist Jean-Paul Delahaye [D] recently posed the followingbrain-teaser, adapting a beautiful puzzle, of unknown origin, popularized by Peter Winkler in hiswonderful book [W] (pp. 35-43).Here is a free translation from the French.
Enigma: Nine Beetles and prime numbers
One places nine beetles on a circular track, where the nine arc distances, measured in meters,between two consecutive beetles are the first nine prime numbers, 2,3,5,7,11,13,17,19 and 23. Theorder is arbitrary, and each number appears exactly once as a distance.At starting time, each beetle decides randomly whether she would go, traveling at a speed of 1meter per minute, clockwise or counter-clockwise. When two beetles bump into each other, theyimmediately do a “U-turn,” i.e. reverse direction. We assume that the size of the beetles isnegligible. At the end of 50 minutes, after many collisions, one notices the distances between thenew positions of the beetles. The nine distances are exactly as before, the first nine prime numbers!How to explain this miracle?Before going on to the next section, we invite you to solve this lovely puzzle all by yourself.
Solution of the Enigma
Note that the length of the circular track is 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 = 100 meters.Let each beetle carry a flag, and whenever they bump into each other, let them exchange flags.Since the flags always move in the same direction, and also move at a speed of 1 meter per minute,after 50 minutes, each flag is exactly at the “antipode” of its original location; hence, the distancesare the same! Of course, this works if the original distances were any sequence of numbers: Allthat they have to obey is that their sum equals 100, or more generally, that half the sum of thedistances divides the product of the speed (1 meter per minute in this puzzle) and the elapsed time(50 minutes in this puzzle). 1his variation, due to Delahaye, is much harder than the original version posed in [W], where alsothe initial distances were arbitrary. In Delahaye’s rendition, the solver is bluffed into trying to usethe fact that the distances are primes. Something analogous happened to the great Paul Erd˝os,the patron saint of combinatorics and number theory, who introduced covering systems . Covering Systems
In 1950, Paul Erd˝os [E1], introduced the notion of covering systems . A covering system is afinite set of arithmetical progressions { a i ( mod m i ) | ≤ i ≤ N } , whose union is the set of all non-negative integers. For example { mod } , is such a (not very interesting) covering system, while { mod , mod } , and { mod , mod , mod , mod , mod } , are other, almost as boring examples. A slightly more interesting example is { mod , mod , mod } . A covering system is exact if all the congruences are disjoint (like in the above boring examples).It is distinct if all the moduli are different. [From now on, let a ( b ) mean a ( mod b ).]Erd˝os gave the smallest possible example of a distinct covering system: { , , , , } . Of course, the above covering system is not exact since, for example, 0 ( 2 ) and 0 ( 3 ) both containany multiple of 6. A theorem proved by Mirsky and (Donald) Newman, and independently byDavenport and Rado (described in [E2]) implies that a covering system cannot be both exact and distinct. Even a stronger statement holds. Assuming that our system { a i ( m i ) } Ni =1 is written innon-decreasing order of the moduli m ≤ m ≤ . . . ≤ m N , the Mirsky-Newman-Davenport-Radotheorem asserts that m N − = m N , in other words, the two top moduli are equal (and hence anexact covering system can never be distinct). See [Zei1] for an exposition of their snappy proof.While their proof was nice, it was not as nice as the combinatorial-geometrical proof that was foundby Berger, Felzenbaum, Fraenkel ([BFF1][BFF2]), and exposited in [Zei1]. In fact, they proved themore general Znam theorem that asserts that the highest moduli shows up at least p times, where p is the smallest prime dividing lcm ( m , . . . , m N ). Jamie Simpson ([S]) independently found asimilar proof, but unfortunately chose not to express it in the evocative geometrical language.2 he Berger-Felzenbaum revolution: From Number Theory to Discrete Geometry viathe Chinese Remainder Theorem While it is a sad truth that the set of positive integers is an infinite set, a covering system is a finite object. In order to verify that a covering system, { a i ( m i ) } Ni =1 is indeed one, it suffices tocheck that it covers all the integers n between 0 and M −
1, where M = lcm ( m , m , . . . , m N ) . By the fundamental theorem of arithmetic M = p r p r · · · p r k k , where p , . . . , p k are primes and r , . . . , r k are positive integers.For the sake of simplicity, let’s assume that M is square-free, i.e. all the exponents r , . . . , r k equal1. The same reasoning, only slightly more complicated, applies in the general case.Now we have M = p p · · · p k . The ancient, but still useful,
Chinese Remainder Theorem tells you that there is a bijectionbetween the set of integers between 0 and M −
1, let’s call it [0 , M − , p i − i = 1 . . . k . f := [0 , M − → k Y i =1 [0 , p i − , defined by f ( x ) := [ x mod p , x mod p , . . . , x mod p k ] . So each integer in [0 , M −
1] is represented by a point in the p × p × . . . × p k k -dimensionaldiscrete box Q ki =1 [0 , p i − a ( m ) is a member of our covering system, since m is a divisor of M , it can be written as aproduct of some of the primes in { p , . . . , p k } , say m = p i p i · · · p i s . Let m i = a mod p i , m i = a mod p i , . . . , m i s = a mod p i s . It follows that the members of the congruence a ( m ) correspond to the points in the k − s -dimensional subbox { ( x , . . . , x k ) ∈ [0 , p − × . . . × [0 , p k − | x i = m i , . . . , x i s = m i s } . M = 30 = 2 · ·
5, the congruence class 7(10), corresponds to the one-dimensionalsubbox (since 7 mod mod { ( x , x , x ) | x = 1 , ≤ x ≤ , x = 2 } . In other words a covering system (with square-free M ) is nothing but a way of expressing a certain k -dimensional discrete box as a union of sub-boxes. This was the beautiful insight of Marc Berger,Alex Felzenbaum, and Aviezri Fraenkel, nicely exposited in [Zeil1]. Erd˝os’s Famous Problem and Bob Hough’s Refutation
Erd˝os ([E2]) famously asked whether there exists a distinct covering system a i ( mod m i ) , ≤ i ≤ N , m < m < . . . < m N , with the smallest modulo, m , arbitrarily large.As computers got bigger and faster, people (and their computers) came up with examples thatprogressively made m larger and larger, and many humans thoughts that indeed m can be madeas large as one wishes. This was brilliantly refuted by Bob Hough ([H]) who proved that m ≤ .This is definitely not sharp, and the true largest m is probably less than 1000.Let’s now move on from number theory to something apparently very different: logic! Boolean Functions
Let’s recall some basic definitions. A
Boolean function (named after George Boole ([Bo])) of n variables is a function from { F alse, T rue } n to { F alse, T rue } . Altogether there are 2 n Booleanfunctions of n variables. Any Boolean function f ( x , . . . , x n ), is determined by its truth table , orequivalently, by the set f − ( T rue ), one of the 2 n subsets of { F alse, T rue } n .The simplest Boolean functions are the constant
Boolean functions
True (the tautology ) cor-responding to the whole of { F alse, T rue } n , and False (the anti-tautology ) corresponding to the empty set .In addition to the above constant Boolean functions, there are three atomic functions. The simplestis the unary function
NOT , denoted by ¯ x , that is defined by¯ x = (cid:26) F alse , if x = T rue ; T rue , if x = F alse .
The two other fundamental Boolean functions are the (inclusive) OR , denoted by ∨ and AND ,denoted by ∧ . x ∨ y is True unless both x and y are false, and x ∧ y is true only when both x and y are true. 4y iterating these three operations on n variables, one can get many Boolean expressions , and eachBoolean function has many possible expressions.From now on we will denote, as usual, true by 1 and false by 0. Also let x = x and x = ¯ x = 1 − x .One particularly simple type of expression is a conjunction (also called term ). It is anything ofthe form, for some t , called its size , x j i ∧ · · · ∧ x j t i t , where 1 ≤ i < . . . < i t ≤ n and j i ∈ { , } for all 1 ≤ i ≤ t .Of interest to us in this article is the type of expression called the Disjunctive Normal Form (DNF)(featured prominently, along with its dual , Conjunctive Normal Form , (CNF), in Norbert Blum’sbrave attempt ([Bl], see also [Zeil2])). It simply has the form N _ i =1 C i , where each C i are pure conjunctions.Every Boolean expression corresponds to a unique function, but every function can be expressed inmany ways, and even in many ways that are DNF. One way that is the most straightforward wayis the canonical DNF form _ { v ∈ f − (1) } n ^ i =1 x v i i . Note that a pure conjunction of length t x j i ∧ · · · ∧ x j t i t corresponds to a sub-cube of dimension n − t , namely to { ( x , . . . , x n ) | x i = j , . . . , x i t = j t } . Hence, one can view a DNF as a (usually not exact) covering of the set f − (1) of truth-vectorsby sub-cubes. In particular, a DNF tautology is a covering of the whole n -dimensional unit cubeby lower-dimensional sub-cubes. Digression: DNFs and the Million Dollar Problem
The most fundamental problem in theoretical computer science, the question of whether P is notNP (of course it is not, but proving it rigorously is another matter), is equivalent to the questionof whether there exists a polynomial time algorithm that decides if a given Disjunctive NormalForm expression is the tautology (i.e. the constant function 1). Of course, there is an obvious algorithm : For each term, find the truth-vectors covered by it, take the union, and see whetherit contains all the 2 n members of { , } n . But this takes exponential time and exponentialmemory . 5 he Covering System Analog Input a system of congruences a i ( mod m i ) , ≤ i ≤ N , and decide, in polynomial time , whether it is a covering system. Initially it seems that we needto check infinitely many cases, but of course (as already noted above), it suffices to check whetherevery integer between 1 and lcm ( m , . . . , m N ) belongs to at least one of the congruences. Thisseems fast enough! Alas, the size of the input is the sum of the number of digits of the a i ’s and m i ’sand this is less than a constant times the logarithm of lcm ( m , . . . , m N ), so just like for Booleanfunctions, the naive algorithm is exponential time (and space) in the input size .We next consider Boolean function analogs of covering systems. The first one to consider suchanalogs was Melkamu Zeleke ([Zel]). Here we continue his pioneering work. Boolean Function Analogs of Covering Systems
We saw that a DNF tautology is nothing but a covering of the n -dimensional unit cube { , } n bysub-cubes. So it is the analog of a covering system.The analog of exact covering systems is obvious: all the terms should cover disjoint sub-cubes.For example, when n = 2, (from now on xy means x ∧ y ) x x ∨ x ¯ x ∨ ¯ x x ∨ ¯ x ¯ x ,x ∨ ¯ x x ∨ ¯ x ¯ x , are such.In order to define distinct DNF, we define the support of a conjunction as the set of the variablesthat participate. For example , the support of the term ¯ x ¯ x x x is the set { x , x , x , x } . In otherwords, we ignore the negations. For each t -subset of { x , . . . , x n } there are 2 t conjunctions withthat support. Geometrically speaking, two terms with the same support correspond to sub-cubeswhich are “parallel” to each other.Note that the supports correspond to the modulo, m , and the assignments of negations (or nonegation) corresponds to a residue class modulo m .A DNF tautology is distinct if it has distinct supports.An obvious example of a distinct DNF tautology in n variables is n _ i =1 x i ∨ ∧ ni =1 ¯ x i . More generally, for every 1 ≤ t ≤ n , ( t = n/
2) the following is a distinct DNF tautology:6 _ ≤ i
Taking n to be odd, the above DNF tautology with t = ( n − / n − /
2, and that can be made as large as one wishes.
First Challenge
This leads to a more challenging problem: For each specific n , how large can the minimum clausesize, let’s call it k , in a distinct DNF tautology, be?An obvious necessary condition , on density grounds , is that n X i = k (cid:18) ni (cid:19) i ≥ . (Each subset of size i of { , . . . , n } can only show up once and covers 2 n − i vertices of the n -dimensional unit cube. Now use Boole’s inequality that says that the number of elements of aunion of sets is ≤ than the sum of their cardinalities).Let A n be the largest such k . The first 14 values of A n are1 , , , , , , , , , , , , , . We were able to find such optimal distinct DNF tautologies for all n ≤
14 except for n = 10,where the best that we came up with was one that covers 1008 out of the 1024 vertices of the10-dimensional unit cube, leaving 16 points uncovered, and for n = 14, where 276 out of the2 = 16384 points were left uncovered. 7ee the output file http://sites.math.rutgers.edu/~zeilberg/tokhniot/odt2.txt . Second Challenge
Another challenge is to come up with distinct DNF tautologies with all the terms of the same size.By density arguments a necessary condition for the existence of such a distinct DNF tautology (cid:18) nm (cid:19) m ≥ . Let B m be the largest such m . The first 14 values are0 , , , , , , , , , , , , , . Obviously for n = 3, where B = 1, it is not possible, since x ∨ x ∨ x can’t cover everything.We were also unable to find such optimal DNF tautologies for n = 5, where B = 3 and we hadto leave one vertex uncovered, n = 9, (with B = 6), where 13 vertices were left uncovered, and n = 13 (with B = 9) where 2 − n ≤
14, we met the challenge. See the output file http://sites.math.rutgers.edu/~zeilberg/tokhniot/odt1.txt . Supporting Maple Packages and Output
Many more examples can be gotten from the Maple package • ,whose output files are available from the front of this article . The General Problem: Covering a Discrete Box by Non-Parallel Sub-boxes
Let { a i } ∞ i =1 be a weakly increasing sequence of positive integers, with a ≥ m there exists an n such that the box [1 , a ] × . . . [1 , a n ] can be covered by non-parallel sub-boxes, each of dimension ≤ n − m ?.We saw that for the Boolean case, with a i = 2 for each i (and analogously, for each constantsequence), the answer is obviously yes .On the other hand, if ∞ X i =1 a i < ∞ , no , since ∞ Y i =1 (1 + 1 a i ) < ∞ , and by a trivial density argument, all tails of the product will eventually be less than 1, so there isnot enough room.Regarding the original Erd˝os problem, Hough ([H]) proved the answer is no in the case with a i = p i ,the sequence of prime numbers. (In fact, Hough proved the slightly harder result where the moduliare not necessarily square-free.) Here the sum of the reciprocals almost converges. The very naiveBoole’s inequality does not suffice to rule out a positive answer to the Erd˝os problem, but theLov´asz Local Lemma [that is also fairly weak; for example, it barely improves the lower bounds forthe Ramsey numbers] suffices to do the job.So prime numbers were indeed red herrings . All that was needed was their asymptotic behavior. Itwould be interesting to see to what extent Hough’s proof of impossibility extends to other sequences( a i ) for which the answer is neither an obvious Yes, nor an obvious No. We also desperately needmore powerful sieves than Lov´asz, Brun, Selberg, and the other known sieves, which, with lots ofingenuity, got close to the twin-prime conjecture (Yitang Zhang); but even the still open twin-primeconjecture, and the Goldbach conjecture, are much weaker than the true state of affairs.Even though the Bonferroni sieve is fairly weak, it can often decide satisfiability (both positivelyand negatively). See the article [Za] by the first-named author. References [BFF1] Marc A. Berger, Alexander Felzenbaum and Aviezri Fraenkel,
A nonanalytic proof of theNewman-Znam result for disjoint covering systems , Combinatorica (1986), 235-243.[BFF2] Marc A. Berger, Alexander Felzenbaum and Aviezri Fraenkel, New results for coveringsystems of residue sets , Bull. Amer. Math. Soc. (N.S.) (1986), 121–126.[Bl] Norbert Blum, A solution of the P versus NP problem , https://arxiv.org/abs/1708.03486 .[Bo] George Boole, L.L.D., “ Investigations of THE LAWS OF THOUGHT, Of Which Are FoundedThe Mathematical Theories of Logic and Probabilities ”, Macmillan, 1854. Reprinted by Dover,1958.[D] Jean-Paul Delahaye,
Cinq ´enigmes pour la rentr´ee , Logique et Calcul column,
Pour La Science ,No. (Sept. 2017), 80-85.[E1] Paul Erd˝os,
On integers of the form k + p and some related problems , Summa Brasil. Math. (1950), 113-123.[E2] Paul Erd˝os, On a problem concerning covering systems (Hungarian, English summary), Mat.9apok (1952), 122-128.[H] Bob Hough, Solution of the minimum modulus problem for covering systems , Ann. Math. (2015), 361-382. https://arxiv.org/abs/1307.0874 .[S] Jamie Simpson,
Exact covering of the integers by arithmetic progressions , Discrete Math. (1986), 181-190.[SZel] Jamie Simpson and Melkamu Zeleke, On disjoint covering systems with exactly one repeatedmodulus , Adv. Appl. Math. (1999), 322-332.[W] Peter Winkler, “ Mathematical Mind-Benders ,” A.K. Peters/CRC Press, 2007.[Za] Anthony Zaleski,
Solving satisfiability using inclusion-exclusion , http://sites.math.rutgers.edu/~az202/Z/sat/sat.pdf . Maple package: http://sites.math.rutgers.edu/~az202/Z/sat/sat.txt .[Zei1] Doron Zeilberger, How Berger, Felzenbaum and Fraenkel revolutionized Covering Systemsthe same way that George Boole revolutionized Logic , Elect. J. Combinatorics (2001) (specialissue in honor of Aviezri Fraenkel), A1. http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/af.html .[Zei2] Doron Zeilberger,
CNF-DNF and all that (Videotaped Lecture) , available from http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/CNFDNFLecture.html .[Zel] Melkamu Zeleke,
Ph.D. dissertation , Temple University, 1998.Anthony Zaleski, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.Email: az202 at math dot rutgers dot edu .Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.Email: