On some closed classes in partial two-valued logic

The precomplete classes TO, Ti, M, 5, L of two-valued logic PI are considered in the algebra Pj of partial two-valued functions. For each of these classes, the lattice of closed classes containing this class is studied. It is shown that this lattice for the first four classes has 9 dosed classes and L is a set of the power of the continuum. 1. DEFINITIONS AND NOTATIONS We use the following notations: E^ = {0, 1}, El = {(<n, 02, . . . , ο«) | o,· 6 E2 for all i}, ft.» = {/!/: E? -> E2], P2 = U ft.». The functions from P^ are said to be two-valued (or Boolean) functions. The functions from P2 are said to be partial two-valued (or Boolean) functions and * is considered as an undefined value. For / 6 P2*n we define £>*(/) = {a Ε Εξ \ /(a) = *} and £)(/) = £2 \ £>*(/). One can consider D(/) and £*(/) as the domains where / is defined and undefined. Let /(χι,...,χη) Ε P2. A variable x,· is said to be an unessential variable of a function /, if the equality /(αϊ, . . . , a t_b 0, at>1, . . . , an) = /(ai, . . . , at-_i, 1, ot-+1, . . . , an) holds for any αϊ, . . . ,ο,·_ι,ο,·+ι,... ,αη from E2. Otherwise, xtis said to be an essential variable. Let / Ε P2% and {0i> ·· · ,0*} Q A· T^ the formula /(0i(*i> · · · . «n), · · · , 9k(xi, · · · , a?«)) (1) defines a function A(XI,.. . ,xn) which can be calculated as usual with the additional rule: if there exists .; such that <?j(a) = *, then h(a) = *. For 91 C P2* let W be the set of all functions that can be obtained from the functions of 9i by adding and rejecting the unessential variables. A formula is said to be a superposition over 91 if it can be obtained from the functions of W by several operations of renaming variables and operations of type (1), where / E W and # is a formula already constructed or identical to one of the variables. Let 9Ϊ C P2* . The set of all functions from P2* representable by superposition over 91 is said to be the closure of 91 and is denoted by [91]. A set 9Ϊ C P2* is id to be a closed class, if [91] = 91. K the function /(χ) = χ belongs to 91, then to prove that 91 is *UDC 519.716. Originally published in Diskretnaya Matematika (1994) 6, No. 4, 60-80 (in Russian). Translated by the authors. 402 V. B. Alekseev and A. A. Voronenko closed, it is enough to show that after adding and rejecting the unessential variables in functions from 91 one again obtains functions from 91 and that the function given by (1) belongs to 91 if / G 91, #1 G 91,... ,gk G 91. It is known that closed classes play the main role in studying representability of functions by superposition of another functions. All closed classes in P2 are described (see, for example [2]). In P2 aM 8 precomplete classes (i.e., those closed classes that are not contained in other closed classes except P2) are described [1]. One of these classes is P2 U {*}, where {*} is the set of all functions depending on any number of variables and taking only the value * (the set of everywhere undefined functions). It is known [1] that the closed class P2 is contained in P2 U {*} and P2, and is not contained in other closed classes in P2*. It is interesting, what situation will be if we consider precomplete classes in P2 instead of P2 itself? In this paper we study the following problem for each of the classes T0, 7\, Μ, 5, L which are precomplete in P2: how many closed classes in P2 contain this given class and what is the structure of the family of these classes with respect to inclusion. For the first 4 classes we show that this structure is finite and contains 9 closed classes. For the class L we show that it contains a continuum of closed classes. Obviously, all these 5 classes are included in P2, PI U {*}, P2. The following assertion is also obvious. Lemma 1. Let Abe a closed class in P2 and {*} be the set of all everywhere undefined functions. Then A U {*} is a closed class. 2. DESCRIPTION OF CLOSED CLASSES BY RELATIONS Let R be some m-ary relation on E2. It can be considered as a subset of ETM. For any natural number n, we associate with R the following m-ary relation on E2. Let o = (a},..., a*),.. . , a = (af,..., aTM) be strings from Εξ. We say that (a,..., a) G R if (α} ,α£ . . . , α?)€Α&Γ8ΐυ = 1,...,η. Let two m-ary relations R on E2 and R\ on E2o{*} be given. We denote by U(R, #1) the set of all functions /(zi,...,zn) € P2 such that the following implication holds for any strings a,..., a from E2: (a,..., a-) G R => (/(a),..., /(a")) G a. (2) The following assertion can be easily proved. Lemma 2. Lef A and R\ be any m-ary relations on E2 and E2 U {*}, respectively. Let a function f(x\,..., xn, xn-n) d#fer ./rom a function g(x\,..., xn) cwfy fry unessential variable xn+i. TTzerc / G £/(#, AI) i/ am/ onfy if 0 G £/(#, RI). It is known that all classes of type U(R, R) in P2 are closed classes. Moreover, all closed classes in P2 but 8 are classes of type U(R, R). We generalize this construction for P2 in the following way. Let Λ C {!,... ,m}. By WTM we denote the m-ary relation on E2 U {*} containing exactly those strings (ai,...,am) G (E2 U {*}) which satisfy the condition: at = * for all i G Λ (all other a, can be chosen arbitrarily). Let 21 be a family of subsets of the set {1,..., m}. Then we define On some closed classes 403 Lemma 3. Let m be any natural number and 21, 2li, . . . , 2lp be the families of subsets of the set {1, . . . , m} , and let W$ C WJJ for all i = 1, . . . , k. Let R be any relation on E2. Then (1) the class U(R, R U WJ) is cfcwed; (2) the class U = U (R, R U W£) U l/( , W£ ) U . . . U i/( , WTM ) is closed. Proof. (1) We have χ G i/( , U WJ). Hence, by Lemma 2, it is enough to prove that the function h(xi, . . . , xn) given by (1) belongs to i/( , U WJ) if {/, #1, . . . , gk] C U(R, R U W5). Let the last inclusion hold and let the strings a, . . . , a from ££ satisfy the condition (a, . . . , a) G R. Then (^(a), . . . , ̂ (a)) G U WJ for all j = 1, . . . , fc. K there exists j such that (gj(a),...,9j(a)) i R, then (gj(a),...,9j(a)) G Wy for some A C {!,... ,m}. This means that ^-(a) = * for all i G Λ. Then /ι(α') = * for all ι G Λ and (h(a),...,h(a)) G WJ C UWJ. If fe (a ),...,£, (aTM)) G for all j, then (^(a), . . . , fc(a)) G U WJ, since / G U (R, R U W£). Thus, if (o, . . . , o) G #, then (^(a), . . . , (o)) G R U WJ in any case. This means that he(R,R(J W£). (2) By Lemma 2, all classes of the union are closed with respect to adding and rejecting the unessential variables. Also χ G U(R,R\JW%). Hence it is enough to show that superposition (1) cannot lead out of U. Let {/,<7ι,. . . ,<7*} ζ U. We prove that the function h(x\, .. . ,xn) given by (1) also belongs to U. Let there exist j, 1 < j < k, and 5, 1 < 5 < p, such that φ(ζι,. . . ,ζ η) € U(R,W£). Then, for any α , . . . ,**TM from E$ satisfying (a,... ,a) G /Γ, there exists Λ G 21, such that 0>(a') = * for all i G Λ. In this case /i(a') = * for all i G A. Hence /i G U(R, W£) in this case and therefore h G V. Let now ̂ (χα, . . . , xn) G (H, RU W£) for all j = l, . . . , fc. If / e (/(R, RU Wf), then, as in the proof of the first assertion, we have h G U(R, R U W%) and h G i/. Let / G (/( ,W£) and let the strings a,...,a from ^J be such that (aV..,a) G R. TTien (^(a),!..,^^)) € U W? for all ;. If (9j(a^... ,^-(a)) G ̂ for some j, then (Κα), . . . , h(a)) G WJ. But we are given that WJ C WJ, so (^(a), . . . , /i(a)) G W£. If feCa), . . · ,#(a)) G for all j = 1, . . . , t, then (/^a),'. . . , fc(a-)) G WTM, since / G U (R, W£). In any case, if (ο, ... , a) G /Γ, then (^(a), . . . , h(a)) G Wj;. Hence h G t/( , W^) C U. Thus, /ι G U in any case. The lemma is proved. 3. CLOSED CLASSES CONTAINING T0 It is known that the class TO = {/ G ΡΪ \ /(O,..., 0) = 0} is precomplete in P2. Consider the classes of functions in P2* shown in Fig. 1, where {*} is the set of all functions taking the only value *, T0,0 = {/ G P2* | /(Ο,.,.,Ο) = 0}, T0,, = {/ G P2* | /(Ο,.,.,Ο) = *}, TS = {/eP2*|/(0,. . . ,0)C{0,*}}. Lemma 4. The classes in Fig. 1 are closed, different and include each other as it is shown in Fig. 1. Proof. Let R be the unary relation with the only (0) G R. Then we can write 7o,o = ί/(Α,Λ), T0* = U(R,R\JW\\ where A = {!}. Consider the binary relations Κ = {(Ο,Ο),(Ο,Ι)} and A = {!}. Then T0 U TO,. = U(R,R)\JU(R',W\). By Lemma 3, all these classes are closed. The closeness of all other classes follows from Lemma 1. The remaining assertions of Lemma 4 are easily checked. 404 K B. Alekseev and A. A. Voronenko