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BOUNDS OF PROOF COMPLEXITY MEASURES FOR SOME SEQUENCE OF MANY-VALUED TAUTOLOGIES Tshitoyan A.S. Email: [email protected]
Tshitoyan Arman Samvelovich - PhD Student, DEPARTMENT OF INFORMATICS AND APPLIED MATHEMATICS, YEREVAN STATE UNIVERSITY, YEREVAN, REPUBLIC OF ARMENIA
Abstract: in this paper the main proof complexity characteristics in some proof systems for two versions of many valued propositional logic with more than one designated values are investigated for some class of k-valued(k > 3) tautologies. For many valued logic with Lukasiewicz's negation and negation, defined by permuting the truth values cyclically, we consider the "elimination" system, which is based on the determinative disjunctive normal forms. We suggest some generalization for a family of 2-valued tautologies, which are known as "hard" for some proof systems of classical propositional logic tautologies. For introduced sequence of many-valued tautologies we obtain simultaneously optimal bounds for different proof complexity measures (asymptotically the same upper and lower bounds for each measures). Keywords: Many-valued logic, elimination systems, determinative disjunctive normal form, proof complexity.
ОЦЕНКИ ВЕЛИЧИН СЛОЖНОСТЕЙ ВЫВОДОВ НЕКОТОРОЙ ПОСЛЕДОВАТЕЛЬНОСТИ МНОГОЗНАЧНЫХ ТАВТОЛОГИЙ
Читоян А.С.
Читоян Арман Самвелович - аспирант, фикультет информатики и прикладной математики, Ереванский государственный университет, г. Ереван, Республика Армения
Аннотация: в данной статье исследованы основные сложностные характеристики выводов для определенного класса k-значных (k>3) тавтологий в некоторых системах двух версий многозначных логик с более чем одним выделенным значением. Для многозначных логик, основанных на отрицании Лукасевича и циклическом отрицании, мы рассматриваем «элиминационные» системы, основанные на определяющих дизъюнктивных нормальных формах. Мы предлагаем обобщение некоторого семейства 2-значных тавтологий, которые известны как «трудновыводимые» в ряде пропозициональных систем классической логики. Для введенной последовательности
многозначных тавтологий получены одновременно оптимальные оценки для различных сложностей выводов (ассимптотически одинаковые верхние и нижние оценки для каждой сложности).
Ключевые слова: многозначная логика, системы с правилом удаления, определяющая дизъюнктивная нормальная форма, сложность вывода.
УДК 510.6
1. Introduction.
Many-valued logic (MVL) was created and developed first by Lukasiewicz [1]. Later on many others continued investigation in this area. In the earlier years of development, this caused some doubts about the usefulness of MVL. In the meantime, however, many interesting applications were found in such fields as logic, mathematics, hardware design, artificial intelligence and some other area of soft information technologies, therefore the investigations in area of proof complexity for different systems of MVL are very important. In this paper the main proof complexity characteristics in proof systems for two versions of propositional MVL are investigated for some classes of k-valued (k>3) tautologies. We consider the systems, based on determinative disjunctive normal forms for many-valued logic with two versions of negation (Lukasiewicz's negation and negation, defined by permuting the truth values cyclically) and with more than one designated values. For considered class of tautologies we obtain simultaneously optimal bounds for different proof complexity measures (asymptotically the same upper and lower bounds for each measure).
2. Preliminaries.
2.1. Main Definitions of k-valued Logics
Here we give some well-known notions and notations in area of MVL.
(1 k—2
use the well-known notions of propositional formula, which defined as usual from k-valued propositional variables with
values from ^may be also propositional constants, parentheses (, ), and logical connectives defined as follows:
pV q = max(p , q), p Л q = min (p , q) ,
~ p = 1 — p (Lukasiewicz's negation).
For implication we have two following versions:
{1, for p < q
„' „ ^ (1) Lukasiewicz's implication
1 — p + q, for p > q r
or
{1, for p < q
q for p > q (2) Model's implication.
For propositional variable p and 5 = ( 0 < i < k — 1) we'll define additionally:
ps = (pD 5) & (5 з p ) ((1) exponent).
Note, that (1) exponent is not a new logical function and note also that (1) exponent for is , and for is .
Further we use also negation , defined by permuting the truth values cyclically (cyclical negation) and, using this negation, we
define for propositional variable p and 5 = (0 < i < k — 1 ) p s as p with (k — 1 ) — i negations -i ((2) exponent).
We'll demonstrate the main results only for the 3-valued logics.
In the all considered logics we fix and as designated values, so a formula with variables p x,p 2,. . . ,p„ is called 3-tautology if for every a = ( ai,<72,. . -,ст„) e £„ assigning to each gives the value or of .
2.2. Determinative Disjunctive Normal Form
Here we recall the notions of determinative conjunct and determinative disjunctive normal form, introduced by A.Chubaryan for 2-valued Boolean functions in [2] and generalized in [3] for 3-valued logic.
We will use the current concepts of the unit cube (£") for £3 = { 0, ^.l }, a propositional formula and a classical tautology. The particular choice of a language for presented propositional formulas is immaterial in this consideration. However, because of some technical reasons we assume that the language contains the propositional 3-valued variables p; ( i>1 ) and(or) p; (i>1 ;y>1 ) , logical connectives A, V, ~ ( o r -1 ) and
parentheses (,). For every propositional variable p in 3-valued logic p 0,p and p 1 in sense of (1) exponent ((2) exponent) are the literals.
The conjunct K (term) can be represented simply as a set of literals (no conjunct contains a variable with different exponents simultaneously), and disjunctive normal form (DNF) can be represented as a set of conjuncts.
We call a replacement-rule each of the following trivial identities for a propositional formula //:
0 A 4> = 0, 4> A 0 = 0, 1 A 4> = 4>, 4> A i = 4>,
0Vx/j = ip, ip v 0 = xp, lvi/j = l, I/JV 1 = 1,
~ 0 = 1 , ~ ( V2 ) = y2 , ~ 1 = 0,
= y2, -,(i/2) = 1- = 0,
—Ip = Ip, (-1-1-1'/' = '/')■
Application of a replacement-rule to some word consists in replacing of its subwords, having the form of the left-hand side of one of the above identities, by the corresponding right-hand side.
We introduce the following auxiliary relations for replacement as well: V2aip = ip A1/2 < y2, i/2vip = ipvy2 > y2.
Note, that the replacement-rules and auxiliary relations for replacement can be defined for as well, but they are unnecessary for our consideration.
Let be a propositional formula of 3-valued logic, P = t p 1, p2 ,. . ., p„} be the set of all variables of 9 and P ' = {p;i, p;2,. . ., pjm} (1 < m < n) be some subset of P.
Definition 1: Given ö:=((T1,(T2 ,. . ,,crm) £ ü1™, the conjunct
is called -determinative ( -determinative,
-determinative), if assigning to each and successively using
replacement-rules and, if it is necessary, the auxiliary relations for replacement also, we obtain the value of independently of the values of the remaining variables.
-determinative and -determinative conjuncts are called also -determinative or determinative for .
Definition 2: A DNF is called determinative DNF (dDNF) for if
= D and every conjunct 1 < i < 7) is ^-determinative or 1-determinative for
Remark 1: It is easily proved that
1) if for some tautology <p, the minimal number of literals, containing in -determinative conjunct, is m, then -determinative DNF has at least 3 mconjuncts;
2) if for some tautology there is such that every conjunct with literals is -determinative, then there is -determinative DNF with no more than conjuncts.
By analogy we can define the determinative conjuncts and dDNF for k-valued logic with the mentioned properties. For k-valued logic we must introduce the corresponding replacement-rules and auxiliary relations for replacement.
2.3. Definitions of main systems
Here we recall the main proof systems following [3,4].
The axioms of Elimination systems f L Nfc (f CWfc) aren't fixed, but for every formula < each conjunct from some dDNF of can be considered as an axiom.
For 3-valued logic the elimination rule ( e-rule) infers conjunct KuK 'uK " ' from conjuncts , and , where , and are conjuncts, is a
variable and (1) exponent for fLNfc ((2) exponent for f CNfc) . It is obvious, that this rule can be eazily generalized for k-valued logic.
The proof in is a finite sequence of conjuncts such that every conjunct in
the sequence is one of the axioms of or is inferred from earlier conjuncts in
the sequence by e-rule.
A DNF D = {K,K2,. . .,Kj} is tautological if by using e-rule can be proven the empty conjunct from the axioms .
The completeness of these systems is obvious.
2.4. Proof complexity measures
Four main characteristics of the proof are considered in the theory of proof complexity. Following [5] we give the formal definitions of all proof complexity measures.
If a proof in the system is a sequence of lines, where each line is an axiom, or is derived from previous lines by one of a finite set of allowed inference rules, then a -configuration is a set of such lines. A sequence of «-configurations {D0, D1(.. ., Dr} is said to be -derivation if is empty set and for all the set is obtained from
by one of the following derivation steps:
Axiom Download: , where is an axiom of .
Inference: , for some inferred by one of the inference rules for
from a set of assumptions, belonging to Dt _ 1.
Erasure: .
A «-proof of a tautology is a «-derivation {D0, D^ .. ., Dr} such that 6 Dr, where is empty conjunct in E CVVfc and is in CVVfc-cut-free.
By we denote the size of a formula , defined as the number of all logical signs entries. It is obvious that the full size of a formula, which is understood to be the number of all symbolsis bounded by some linear function in .
The of a -derivation is a sum of the sizes of all lines in a derivation, where
lines that are derived multiple times are counted with repetitions. The of a -
derivation is the number of axioms downloads and inference steps in it. The sp a c e (s ) of a «-derivation is the maximal space of a configuration in a derivation, where the space of a configuration is the total number of logical signs in a configuration, counted with repetitions. The wi dth (w) of a «-derivation is the size of the widest line in a derivation.
Let « be a proof system and be a tautology. We denote by t* ( I *, s*, w*) the minimal possible value of for
all proofs of tautology in «.
3. Main results
Before we'll prove the main theorem, we must give some auxiliary results.
In some papers in area of propositional proof complexity for 2-valued classical logic the following tautologies (Topsy-Turvy Matrix) play key role
m n
TTMn,m = \J f\\J Pij°J (n > 1,1 < m < 2" - 1).
(<T1,<T2,...,<TJt)eS't 7 = 1 i = 1
For all fixed and in above indicated intervals every formula of this kind
expresses the following true statement: given a 0,1-matrix of order n x m we can "topsy-
turvy" some strings (writing 0 instead of 1 and 1 instead of 0) so that each column will contain at least one 1.
The results of proof complexities measures investigations for generalizations of these tautologies in k-valued logics with only 1 as designated value are given in [6] and [7].
In this paper we'll generalize this family of 2-tautologies for 3-valued logic with !-/2 and 1 as designated values.
Definition 3: Given a = (ai, cr2, . . ., <rm) £ E^ and 5 = (0 < i < k — 1 ) we call 5-
(l)-topsy-turvy-result (5-(2)-topsy-turvy-result) the cortege <t5, which contains every a, ( 1 < j < m) with (1) exponent 5 for EL VVfc (with (2) exponent 5 for ECVVfc).
Both 5-(1)-topsy-turvy-result and 5-(2)-topsy-turvy-result we call 5-topsy-turvy-result further.
Given a = (ai, a2 ,. . ., am) £ E™ and 5 = (0 < i < k — 1 ) we denote by | S (5) | the number of 5 occurrence in 8.
Lemma 3.1: In given 3-valued 0, !-/2, 1 -matrix of order n x m we can do 0-(1)-"topsy-turvy" some strings such, that each column will contain at least one ^ or one 1, iff m < 2 n — 1 .
Proof is given by induction on number n of matrix strings. For n = 1 , m = 1 proof is obvious. Suppose that statement is valid for n strings. If the number of strings is n + 1 and the number of columns is no more than , we consider the last string. If
| ^ | + | 1 | , then after 0-(1)-topsy-turvy we obtain in the last string at least 2 n numbers ^ or 1, therefore we'll have at least columns, which contain at least one or 1. If I 0 I < | V2 | + I 1 I we can do nothing (1-(1)-topsy-turvy). In this case also we'll have at least
columns, which contain at least one or 1. For submatrix from the other columns and for the first n strings the statement is valid by induction supposition.
Corollary 1: For every n > 1 and m < 2 n — 1 the following formulas are 3-tautologies:
L 7TMn,m = V (CTl,CT2, . . .,trn)A™L 1 Vn= (where first disjunctions are for all
).
Lemma 3.2: In given 3-valued 0, ^, 1 -matrix of order n x m we can do (2)-"topsy-turvy" some strings such, that each column will contain at least one or one 1, iff m < 3 n — 1 .
Proof is given by induction on number of matrix strings. For , proof is
obvious. Suppose that statement is valid for n strings. If the number of strings is n + 1 and the number of columns is no more, than , we consider the last string. If ,
then we do nothing (1-(2)-topsy-turvy), if then after 0-(2)-topsy-turvy we
obtain in the last string < m/3 , if | 1 | < m/3 , then after V2 -(2)-topsy-turvy we obtain in the last string | 0 | < "y^. After these topsy-turvies for submatrix from first n strings and columns, which contains 0 in the ( )-th string the statement is valid by induction supposition.
Corollary 2: For every n > 1 and m < 3 n — 1 the following formulas are 3-tautologies:
C 77Mnm = V ( . . CTn)Ayi 1Vn= 1P i°> (where first disjunctions are for all ).
Lemma 3.3: The bounds of minimal possible value of s — comp Zexi ty for all proofs of 3-tautology with n variables in EL VV3 and E CVV3 are: s,p = O (n 2 ) and s,p = H (n) .
Proof is given by analogy with the proof of Lemma 3 from [6].
Main Theorem
1) There exists a sequence of 3-tautologies <pn, for the proof complexity measures of which in the systems Я! V3 are valid the following equations:
1 0 a 2 (ф„) = в (п) ;
го02го0з (t(фп)) = в(и); /оа2*оаз(* (Фп)) = б (п) ; гоа2 (s (фи)) = в (П; k>a2 (w (фп)) = в (п .
2) There exists a sequence of 3-tautologies <pn, for the proof complexity measures of which in the systems are valid the following equations:
1 0 a 3 (ф„) = в (п) ;
¿оаз'оаз (t(фп)) = в (п;> ; к^з^О(фп)) = в (п) ; (ф„)) = в (п ; гоаз (w (фп) ) = в (п;> .
Proof:
1) As <pn we take the formulas L TTMnm for every п > 1 and m = 2n — 1. For upper bounds we use the perfect DNF of <pn, and for lower bounds, the properties of <pn-determinative conjuncts.
It is not difficult to see, that number of variables of <pn is п 2 n (2 n — 1 ) , the minimal number of variables in every <pn-determinative conjuncts is 2 n — 1 , therefore by Remark 1 in the end of point 2.2 the minimal number of <pn-determinative conjuncts is 3 2 " 1, hence the number of axioms, using in the system Я! iV3, must be at least 3 2 " 1. So, using these statements and Lemma 3.3, we can obtain all upper and lower bounds.
2) As we take the formulas for every and . For upper bounds we use the perfect DNF of <pn, and for lower bounds, the properties of <pn-determinative conjuncts.
It is not difficult to see, that number of variables of is , the minimal
number of variables in every <pn-determinative conjuncts is 3 n — 1 , therefore by Remark 1 in the end of point 2.2 the minimal number of <pn-determinative conjuncts is 3 3 " 1, hence the number of axioms, using in the system , must be at least . So, using these statements and Lemma 3.3, we can obtain all upper and lower bounds.
Corollary 3: If we take the analogous formulas for k-valued logics for every п > 1 and corresponding , then the analogous results can be obtained for more valued logics. Acknowledgments
I am grateful to Professor of YSU and RAU Anahit Chubaryan, supervisor of my PhD's thesis for her encouragement and productive discussions, for answering to many of questions about many-valued logics and for helpful suggestions, which allowed me to improve and extend the results.
References / Список литературы
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