Linear read-once functions constitute a remarkable subclass of several important classes of Boolean functions and appear in the literature frequently under various names. In theoretical computer science, they are known as nested  or linear read-once . The latter term is justified by the fact that a linear read-once function is a read-once function admitting a Boolean formula that can be constructed inductively in a “linear” fashion. In , it was also proved that linear read-once functions are equivalent to -decision lists. Later and independently this class was introduced in  under the name nested canalyzing functions, as a subclass of canalyzing functions, which appear to be important in biological applications . The significance of nested canalyzing functions in the applications motivated their further theoretical study [9, 12]. In particular,  establishes the equivalence between nested canalyzing and unate cascade functions, which have been studied in the design of logic circuits and binary decision diagrams. All mentioned terms refer to the same class of Boolean functions which we call linear read-once.
In , it was shown that the class of linear read-once functions is the intersection of the classes of read-once and threshold functions. Generalizing this result we show that the class of linear read-once functions is the intersection of read-once and Chow functions. We also find the set of minimal read-once functions which are not linear read-once and the set of minimal threshold functions which are not linear read-once. In other words, we characterize the class of linear read-once functions by means of minimal forbidden subfunctions within the universe of read-once and the universe of threshold functions.
Within the universe of threshold functions the importance of linear read-once functions is due to the fact that they attain the minimum value of the specification number, i.e. of the number of Boolean points that uniquely specify a function in this universe. To study the range of values of specification number of threshold functions one can be restricted to positive threshold functions depending on all their variables, in which case the functions can be completely specified by their sets of extremal points, i.e. maximal zeros and minimal ones. In other words, the specification number of a positive threshold function is upper bounded by the number of its extremal points. For a linear read-once function, these numbers coincide and equal . In 1995 Anthony et al.  conjectured that for all other threshold functions the specification number is strictly greater than .
Our result about the minimal threshold non-linear read-once functions seems to support this conjecture, since all these functions have specification number . One more result proved in this paper, which can be viewed as a supporting argument for the conjecture, states that a positive function depending on variables has exactly extremal points if and only if it is linear read-once. Nevertheless, rather surprisingly, we show that the conjecture is not true by exhibiting a positive threshold non-linear read-once function depending on variables whose specification number is .
The organization of the paper is as follows. All preliminary information related to the topic of the paper, including definitions and notation, is presented in Section 2. Section 3 is devoted to the number of extremal points in positive functions. In Section 4 we show that the class of linear read-once functions is the intersection of the classes of read-once and Chow functions, and identify the set of minimal read-once functions which are not linear read-once. In Section 5, we characterize the class of linear read-once functions in terms of minimal threshold functions which are not linear read-once and give a counterexample to the conjecture of Anthony et al.
Let . For a Boolean -dimensional hypercube we define sub-hypercube as the set of all points of for which coordinate is equal to for every . For a point we denote by the point in with if and only if for every .
For a Boolean function on , , and we denote by the Boolean function on defined as follows:
For and we denote by the function . We say that is the restriction of to . We also say that a Boolean function is a restriction (or subfunction) of a Boolean function if there exist and such that .
A variable is called irrelevant for if , i.e., for every . Otherwise, is called relevant for . If is irrelevant for we also say that does not depend on .
2.1 Positive functions and extremal points
By we denote a partial order over the set , induced by inclusion in the power set lattice of the -set. In other words, if implies . In this case we will say that x is below y. When and we will sometimes write .
A Boolean function is called positive (also known as positive monotone or increasing) if and imply .
For a positive Boolean function , the set of its false points forms a down-set and the set of its true points forms an up-set of the partially ordered set . We denote by
the set of maximal false points,
the set of minimal true points.
We will refer to a point in as a maximal zero of and to a point in as a minimal one of . A point will be called an extremal point of if it is either a maximal zero or a minimal one of . We denote by
the number of extremal points of .
2.2 Threshold functions
A Boolean function on is called a threshold function if there exist weights and a threshold such that, for all ,
The inequality is called threshold inequality representing function
. The hyperplaneis called separating hyperplane for the function . It is not hard to see that there are uncountably many different threshold inequalities (and separating hyperplanes) representing a given threshold function, and if there exists an inequality with non-negative weights, then is a positive function.
Let . A Boolean function on is -summable if, for some , there exist (not necessarily distinct) false points and (not necessarily distinct) true points such that (where the summation is over ). A function is asummable if it is not -summable for all .
 A Boolean function is a threshold function if and only if it is asummable.
It is known (see e.g. Theorem 9.3 in ) that the class of threshold functions is closed under taking restrictions, i.e. any restriction of a threshold function is again a threshold function.
2.3 Chow functions
An important class of Boolean functions was introduced in 1961 by Chow  and is known nowadays as Chow functions. This notion can be defined as follows.
The Chow parameters of a Boolean function are the integers , where is the number of true points of and is the number of true points of where is also true. A Boolean function is a Chow function if no other function has the same Chow parameters as .
The importance of the class of Chow functions is due to the fact that it contains all threshold functions, which was also shown by Chow in .
2.4 Read-once, linear read-once and canalyzing functions
A Boolean function is called read-once if it can be represented by a Boolean formula using the operations of conjunction, disjunction, and negation in which every variable appears at most once. We say that such a formula is a read-once formula for .
A read-once function is linear read-once (lro) if it is either a constant function, or it can be represented by a nested formula defined recursively as follows:
both literals and are nested formulas;
, , , are nested formulas, where is a variable and is a nested formula that contains neither , nor .
It is not difficult to see that an lro function is positive if and only if a nested formula representing does not contain negations.
In , it has been shown that the class of lro functions is precisely the intersection of threshold and read-once functions.
A Boolean function is called canalyzing111The notion of canalyzing functions was introduced in  and is widely used in biological applications of Boolean networks. The lro functions form a subclass of canalyzing functions and are known in this context as nested canalyzing. if there exists such that or is a constant function.
It is easy to see that if is a positive canalyzing function then or , for some .
2.5 Specifying sets and specification number
Let be a class of Boolean functions of variables, and let .
A set of points is a specifying set for in if the only function in consistent with on is itself. In this case we also say that specifies in the class . The minimal cardinality of a specifying set for in is called the specification number of (in ) and denoted .
Also, in  it was shown that the lower bound is attained for lro functions.
 For any lro function depending on all its variables, .
2.6 Essential points
In estimating the specification number of a threshold Boolean functionit is often useful to consider essential points of defined as follows.
A point x is essential for (with respect to class ), if there exists a function such that and for every , .
Clearly, any specifying set for must contain all essential points for . It turns out that the essential points alone are sufficient to specify in . Therefore, we have the following well-known result.
 The specification number of a function is equal to the number of essential points of .
The following result is a restriction of Theorem 4 in  (proved for threshold functions of many-valued logic) to the case of Boolean threshold functions.
 A zero of a threshold function is essential if and only if there is separating hyperplane containing it.
Thus, the set of all essential zeros (resp. ones) of is the union of all points in belonging to at least one separating hyperplane for the function (resp. ).
2.7 The number of essential points vs the number of extremal points
It was observed in  that in the study of specification number of threshold functions, one can be restricted to positive functions. To prove Theorem 3, the authors of  first showed that for a positive threshold function depending on all its variables the set of extremal points specifies . Then they proved that for any positive lro function of relevant variables the number of extremal points is .
 If has the specification number , then is linear read-once.
In the next section, we show that this conjecture becomes a true statement if we replace ‘specification number’ by ‘number of extremal points’. Nonetheless, in spite of this result supporting the conjecture, we conclude the paper with a counterexample disproving it.
3 Positive functions and the number of extremal points
The main goal of this section is to prove the following theorem.
Let be a positive function with relevant variables. Then the number of extremal points of is at least . Moreover has exactly extremal points if and only if is lro.
We will prove Theorem 6 by induction on . The statement is easily verifiable for . Let and assume that the theorem is true for functions of at most variables. In the rest of the section we prove the statement for -variable functions. Our strategy consists of three major steps. First, we prove the statement for canalyzing functions in Section 3.2. This case includes lro functions. Then, in Section 3.3, we prove the result for non-canalyzing functions such that for each variable both restrictions and are canalyzing. Finally, in Section 3.4, we consider the case of non-canalyzing functions depending on a variable such that at least one of the restrictions and is non-canalyzing. In Section 3.1, we introduce some terminology and prove a preliminary result.
3.1 A property of extremal points
We say that a maximal zero (resp. minimal one) y of corresponds to a variable if (resp. ). It is not difficult to see that for any relevant variable , there exists at least one minimal one and at least one maximal zero corresponding to . We say that an extremal point of corresponds to a set of variables if it corresponds to at least one variable in .
For every set of relevant variables of a positive function , there exist at least extremal points corresponding to this set.
Let be a minimal counterexample and let be the set of extremal points corresponding to the variables in . Without loss of generality we assume that consists of the first variables of the function, i.e. . Due to the minimality of we may also assume that and for every proper subset of there exist at least extremal points corresponding to . This implies, by Hall’s Theorem of distinct representatives , that there exists a bijection between and mapping variable to a point corresponding to .
Let a be any maximal zero in . We denote by b the point which coincides with a in all coordinates beyond the first , and for each we define the -th coordinate of b to be if is a maximal zero, and to be if is a minimal one.
Assume first that and let c be any maximal zero above b (possibly ). If , then , contradicting that a is a maximal zero. Therefore, for some and hence c is a maximal zero corresponding to . Moreover, c is different from any maximal zero because the -th coordinate of is 0, while the -th coordinate of c is 1.
Suppose now that and let c be any minimal one below b (possibly ). If , then , contradicting the positivity of . Therefore, for some and hence c is a minimal one corresponding to . Moreover, c is different from any minimal one because the -th coordinate of is 1, while the -th coordinate of c is 0.
A contradiction in both cases shows that there is no counterexamples to the statement of the lemma. ∎
3.2 Canalyzing functions
Let be a positive canalyzing function with relevant variables. Then the number of extremal points of is at least . Moreover has exactly extremal points if and only if is lro.
The case is trivial, and therefore we assume that .
Let be a variable of such that (the case is similar). Let and . Clearly, is a relevant variable of , otherwise , that is, . Since every relevant variable of is relevant for at least one of the functions and , we conclude that has relevant variables.
The equivalence implies that for every extremal point of , the corresponding point is extremal for . For the same reason, there is only one extremal point of with the -th coordinate being equal to , namely, the point with all coordinates equal to , except for the -th coordinate. Hence, .
If is lro, then is also lro, since can be expressed as . By the induction hypothesis and therefore .
If is not lro, then is also not lro, which is easy to see. By the induction hypothesis and therefore .
3.3 Non-canalyzing functions with canalyzing restrictions
In this section, we study non-canalyzing positive functions such that for each variable both restrictions and are canalyzing.
First we remark that all variables of those functions are relevant. Indeed, if such a function has an irrelevant variable then the function is canalyzing.
Let be a positive non-canalyzing function such that for each variable both restrictions and are canalyzing. Then all variables of are relevant.
Let be irrelevant, then . But , are canalyzing, hence there exists such that or . In the former case , in the latter case . In any case is canalyzing. Contradiction. ∎
Let be a positive non-canalyzing function such that for each variable both restrictions and are canalyzing. Then for each ,
there exists a maximal zero that contains ’s in exactly two coordinates one of which is ,
there exists a minimal one that contains ’s in exactly two coordinates one of which is .
Fix an and denote , . Since is canalyzing, there exists such that or . We claim that the latter case is impossible. Indeed, the positivity of and imply , and therefore . This contradicts the assumption that is non-canalyzing. Thus, . Now we claim that the Boolean point y with exactly two ’s in coordinates and is a maximal zero. Indeed, if in at least one of three points above y is , then, by positivity of , or , which contradicts the assumption that is non-canalyzing.
Similarly, one can show that for some implying that the Boolean point with exactly two ’s in coordinates and is a minimal one. ∎
Let be a positive non-canalyzing function such that for each variable both restrictions and are canalyzing. Then there is a minimal one y of Hamming weight such that is a maximal zero, unless in which case has extremal points.
Consider a graph (resp. ) with vertex set every edge of which represents a maximal zero (resp. minimal one) that contains ’s (resp. ’s) in exactly two coordinates and . By Claim 2, every vertex in is covered by an edge and every vertex in is covered by an edge. From this it follows in particular that each graph , has at least edges.
In terms of the graphs and , the claim is equivalent to saying that and have a common edge. It is not difficult to see that for the graphs and necessarily have a common edge. Let us show that this is also the case for .
Assume that and have no common edges, i.e. every edge of is a non-edge (a pair of non-adjacent vertices) in . Let us prove that
every edge of forms a vertex cover in , i.e. every edge of shares a vertex with either or (and not with both according to our assumption).
Indeed, let be an edge of and assume that contains an edge such that is different from and is different from . Then the minimal one corresponding to the edge of is below the maximal zero corresponding to the edge of . This contradicts the positivity of and proves (*).
Consider an edge in . Since , then has at least edges, hence from (*) we get that at least one of covers at least two edges of , say covers and . Let be an edge of covering . If , then does not cover the edge of which contradicts to (*). If , let be any vertex different from . The vertex must be covered by some edge in . If is different from then does not cover in . If is different from then does not cover in . In both cases we get a contradiction to (*), hence for the graphs and necessarily have a common edge and hence the result follows in this case.
It remains to analyze the case . Up to renaming variables, the only possibility for and to avoid a common edge is for to have edges and and for to have edges and . In other words, and are maximal zeros and and are minimal ones. By positivity, this completely defines the function , except for two points and . However, regardless of the value of in these points, both of them are extremal and hence has extremal points. ∎
Let be a positive non-canalyzing function such that for each variable both restrictions and are canalyzing. Let y be a minimal one of Hamming weight such that is a maximal zero. Denote the two coordinates of y containing ’s by and , and let and .
Variable is relevant for both functions and .
If a point is an extremal point of , , then is an extremal point of .
First, we note that since y is a minimal one, . Similarly, since is a maximal zero, .
To prove (a), suppose to the contrary that does not depend on . Then , and therefore , which contradicts the assumption that is non-canalyzing. Similarly, one can show that is relevant for .
Now we turn to (b) and prove the statement for . For the arguments are symmetric.
Assume first that . Since y is a minimal one, we have for all with . Due to the extremality of , all its components besides are zeros. It follows that , which is a minimal one by assumption.
It remains to assume that . Let a be a maximal zero for the function . If is not a maximal zero for , then there is with . Since and , the -th component of is . By its removal, we obtain a zero of that is strictly above a in contradiction to the minimality of the latter.
Let a be a minimal one for the function . If is not a minimal one for , then there is with . The -th component of must be , since otherwise by its removal we obtain a one for strictly below a. Also, the -th component of must be , since this component equals in a. But then with and , a contradiction. ∎
Let be a positive non-canalyzing function such that for each variable both restrictions and are canalyzing. Then the number of extremal points of is at least .
By Claim 3 we may assume that there is a minimal one y that contains ’s in exactly two coordinates, say and , such that is a maximal zero. Denote and .
Let , , and be the sets of relevant variables of , and , respectively. By Claim 1, is the set of all variables. Since any relevant variable of is relevant for at least one of the functions , and, by Claim 4 (a), is a relevant variable of both of them, we have
3.4 Non-canalyzing functions containing a non-canalyzing restriction
Due to Lemmas 2 and 3 it remains to show the bound for a positive non-canalyzing function such that for some at least one of and is non-canalyzing. Let be the number of relevant variables of and let us prove that the number of extremal points of is at least .
Consider two possible cases:
is a irrelevant variable of ;
is a relevant variable of .
In case (a) the function is non-canalyzing and has the same number of extremal points and the same number of relevant variables as . By induction, the number of extremal points of is at least .
Now let us consider case (b). Assume without loss of generality that , and let and . We assume that is non-canalyzing and prove that has at least extremal points, where is the number of relevant variables of . The case when is canalyzing, but is non-canalyzing is proved similarly.
Let us denote the number of relevant variables of by . Clearly, . Exactly of relevant variables of are irrelevant for the function . Note that these variables are necessarily relevant for the function . By the induction hypothesis, the number of extremal points of is at least .
We introduce the following notation:
– the set of maximal zeros of corresponding to ;
– the set of all other maximal zeros of , i.e., ;
– the set of minimal ones of corresponding to ;
– the set of all other minimal ones of , i.e., .
For a set we will denote by the restriction of to the first coordinates, i.e., .
By definition, the number of extremal points of is
We want to express in terms of the number of extremal points of and . For this we need several observations. First, we observe that if is an extremal point for , the point is extremal for . Furthermore, we have the following straightforward claim.
is the set of minimal ones of and is the set of maximal zeros of .
In contrast to the minimal ones of , the set of maximal zeros of in addition to the points in may contain extra points, which we denote by . In other words, . Similarly, besides , the set of minimal ones of may contain additional points, which we denote by . That is, .
The set is a subset of the set of maximal zeros of . The set is a subset of the set of minimal ones of .
We will prove the first part of the statement, the second one is proved similarly. Suppose to the contrary that there exists a point , which is a maximal zero for , but is not a maximal zero for . Notice that , as otherwise would be a maximal zero for , which is not the case, since . Since a is not a maximal zero for , there exists a maximal zero for such that . But then we have and , which contradicts the positivity of function . ∎
Using the induction hypothesis we conclude that . To derive the desired bound , in the rest of this section we show that contains at least points.
Let , , be a relevant variable for , which is irrelevant for . Then every maximal zero for corresponding to belongs to and every minimal one for corresponding to belongs to .
Let and assume . Then by changing in x the -th coordinate from to we obtain a point with , since x is a maximal zero for . This contradicts the assumption that is irrelevant for . Therefore, and hence no maximal zero for corresponding to belongs to , i.e. every maximal zero for corresponding to belongs to
Similarly, one can show that no minimal one for corresponding to belongs to , i.e. every minimal one for corresponding to belongs to . ∎
Recall that there are exactly variables that are relevant for and irrelevant for . Lemma 1 implies that there are at least extremal points for corresponding to these variables. By Claim 7, all these points belong to the set . This conclusion establishes the main result of this section.
Let be a positive non-canalyzing function with relevant variables such that for some at least one of the restrictions and is non-canalyzing. Then the number of extremal points of is at least .
4 Chow and read-once functions
In this section, we look at the intersection of the classes of Chow and read-once functions and show that this is precisely the class of lro functions. Thus, our result generalizes a result from  showing that the class of lro functions is the intersection of the classes of read-once and threshold functions.
There are two read-once functions that play a crucial role in our characterization of read-once Chow functions:
Functions and all the functions obtained from them by negation of variables are not Chow.
Function is not Chow, because is different from (e.g. they have different values at the point , , , ), but both functions have the same Chow parameters . In a similar way, one can show that neither nor any function obtained from or by negation of variables is Chow. ∎
The following lemma shows that the class of Chow functions is closed under taking restrictions.
If is a Chow function, then any restriction of is also Chow.
Suppose to the contrary that has a restriction which is not a Chow function, namely,
for some , and is not a Chow function. Then there exists a function with the same Chow parameters as . We define function as follows:
Since , we conclude that . Similarly, for every , the equality implies that . Consequently, and have the same Chow parameters, which contradicts the fact that is Chow. ∎
Any canalyzing read-once function , which is not lro, has a non-constant non-canalyzing read-once function as a restriction.
Let be a minimum counterexample to the claim. Since is canalyzing, there exists such that . We assume that , i.e. , in which case (the other cases are similar).
Clearly, is read-once, since any restriction of a read-once function is read-once. Also, is not lro, since otherwise is lro, and hence is not a constant function. Since is a counterexample, is canalyzing and has no non-constant non-canalyzing read-once restrictions. But then we have a contradiction to the minimality of . ∎
For a read-once function the following statements are equivalent:
is an lro function;
is a Chow function;