ON SOME PROPERTIES OF SEVERAL PROOF

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Abstract. In this work we investigate the relations between the proofs complexities of minimal tautologies and of results of substitutions in them in some systemsΒ ...
ON SOME PROPERTIES OF SEVERAL PROOF SYSTEMS FOR 2-VALUED AND 3-VALUED PROPOSITIONAL LOGIC Chubaryan Anahit Doctor of Physical and Mathematical Sciences, Full Professor of Yerevan State University and Russian-Armenian University E-mail: [email protected]

PetrosyanGarik Master Student of Department of Informatics and Applied Mathematics of Yerevan State University and Scientific Assistant in of Russian-Armenian University E-mail: [email protected]

Abstract In this work we investigate the relations between the proofs complexities of minimal tautologies and of results of substitutions in them in some systems of 2-valyed classical logic an 3-valued logic.. We show that the result of substitution can be proved easier, than corresponding minimal tautology, therefore the systems, which are considered in this paper, are no monotonous neither by lines nor by size. Keywords: minimal tautology; Frege systems; substitution Frege systems; Sequent proof systems; proof complexity measures; monotonous system, 3-valued logic.

1. Introduction The minimal tautologies, i.e. tautologies, which are not a substitution of a shorter tautology, play main role in proof complexity area. Really all hard propositional formulaes, proof complexities of which are investigated in many well known papers, are minimal tautologies. There is traditional assumption that minimal tautology must be no harder than any substitution in it. This idea was revised at first by Anikeev in [1]. He had introduced the notion of monotonous proof system and had given two examples of no complete propositional proof systems: monotonous system, in which the proof lines of all minimal tautologies are no more, than the proof lines for results of a substitutions in them, and no monotonous system, the proof lines of substituted 1

formulas in which can be less than the proof lines of corresponding minimal tautologies. We introduce for the propositional proof systems the notions of monotonous by lines and monotonous by sizes of proofs. In [2] it is proved that Frege systems 𝓕 are no monotonous neither by lines nor by size. In this paper we prove that substitution Frege systems S𝓕, the well known propositional sequent systems 𝑷𝑲, π‘·π‘²βˆ’ and corresponding systems with substitution rule 𝐒𝑷𝑲, π’π‘·π‘²βˆ’ are no monotonous neither by lines nor by size. Analogous result we prove for some systems of Łukasiewicz’s 3-valued propositional logic also. This work consists from 4 main sections. After Introduction we give the main notion and notations as well as some auxiliary statements in Preliminaries. The main results are given in the last two sections.

2.Preliminaries We will use the current concepts of a propositional formula, a classical tautology, sequent, Frege proof systems, sequent systems for classical propositional logic and proof complexity [3]. Let us recall some of them.

2.1.The considered systems of 2-valued propositional logic. Follow [3] we give the definition of main systems, which are considered in this point. 2.1.1. A Frege system 𝓕 uses a denumerable set of 2-valued variables, a finite, complete set of propositional connectives; 𝓕 has a finite set of inference rules defined by a figure of the form

A1 A2  Am (the rules of inference with zero hypotheses are the axioms schemes); 𝓕 must be sound B and complete, i.e. for each rule of inference

A1 A2  Am every truth-value assignment, satisfying B

A1 A2  Am , also satisfies B , and 𝓕 must prove every 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

2

the propositional variables p, q and pi ,qi (𝑖 β‰₯ 1), logical connectives Β¬, βŠƒ, ∨, ∧ and parentheses (,). Note that some parentheses can be omitted in generally accepted cases. We assume also that 𝓕 has well known inference rule modus ponens. 2.1.2 A substitution Frege system Sβ„± consists of a Frege system β„± augmented with the substitution rule with inferences of the form

πœ‘π‘–1 for any substitution 𝜎 = ( 𝑝 AΟƒ

πœ‘ 𝑖2 … πœ‘ 𝑖𝑠 𝑝𝑖2 … 𝑝𝑖𝑠 ), s β‰₯ 1,

𝐴

𝑖1

consisting of a mapping from propositional variables to propositional formulas, and 𝐴𝜎 denotes the result of applying the substitution to formula A, which replaces each variable in A with its image under Οƒ. This definition of substitution rule allows to use the simultaneous substitution of multiple formulas for multiple variables of A without any restrictions. 2.1.3. π‘·π‘²βˆ’ system uses the denotation of sequentΞ“ β†’ Ξ” where Ξ“ is antecedent and Ξ” is succedent. The axioms of π‘·π‘²βˆ’ system are 1) p β†’ p, 2) β†’ T, where p is propositional variable and 𝑇denotes Β«truthΒ». For every formulas 𝐴 , 𝐡 and for any sequence of formulas Ξ“, Ξ” the logic ruls are. βŠƒβ†’ βˆ¨β†’

Ξ“β†’Ξ”, A

B, Ξ“β†’Ξ”

β†’βŠƒ

π΄βŠƒπ΅, Ξ“β†’Ξ”

A, Ξ“β†’Ξ”

B, Ξ“β†’Ξ”

Ξ“β†’Ξ”, A

β†’βˆ¨ Ξ“β†’Ξ”,

𝐴∨𝐡, Ξ“β†’Ξ” A, Ξ“β†’Ξ”

βˆ§β†’ 𝐴∧𝐡,

Ξ“β†’Ξ”

B, Ξ“β†’Ξ”

and βˆ§β†’ 𝐴∧𝐡,

β†’βˆ§

Ξ“β†’Ξ”

Ξ“β†’Ξ”, A

Β¬β†’ ¬𝐴,

𝐴∨𝐡

A, Ξ“β†’Ξ”, B Ξ“β†’Ξ”, π΄βŠƒπ΅ Ξ“β†’Ξ”, B

and β†’βˆ¨ Ξ“β†’Ξ”, Ξ“β†’Ξ”, A

Ξ“β†’Ξ”, B

Ξ“β†’Ξ”, 𝐴∧𝐡 𝐴, Ξ“β†’Ξ”

β†’ Β¬ Ξ“β†’Ξ”,

Ξ“β†’Ξ”

The only structured inference rule is

3

𝐴∨𝐡

¬𝐴

Ξ“β†’Ξ” Ξ“β€²β†’Ξ”β€²

Str.r., where Ξ“β€²(Ξ”β€²) contains Ξ“(Ξ”) as a set.

2.1.4. The system 𝑷𝑲 is π‘·π‘²βˆ’ augmented with cut-rule

Ξ“1 β†’Ξ”1 , A

A, Ξ“2 β†’Ξ”2

Ξ“1 ,Ξ“2 β†’Ξ”1 ,Ξ”2

2.1.5. Substitution sequent calculus 𝑺𝑷𝑲(π‘Ίπ‘·π‘²βˆ’ ) defined by adding to 𝑷𝑲 (π‘·π‘²βˆ’ ) the following rule of substitution Ξ“β†’Ξ”, A (p) Ξ“β†’Ξ”, A (B)

,

where simultaneous substitution of the formula B is allowed for the variable p, and where p does not appear in Ξ“, Ξ”.

Note (1) LetΞ“ β†’ Ξ” be some sequent, where Ξ“ is a sequence of formulas 𝐴1 , 𝐴2 , … , 𝐴𝑙 (𝑙 β‰₯ 0) and Ξ”is a sequence of formulas 𝐡1 , 𝐡2 , … , π΅π‘š (π‘š β‰₯ 0). The formula form of sequent Ξ“ β†’ Ξ” is the formula πœ‘Ξ“β†’Ξ” , which is defined usually as follows: 1) 𝐴1 ∧ 𝐴2 ∧ … ∧ 𝐴𝑙 βŠƒ 𝐡1 ∨ 𝐡2 ∨ … ∨ π΅π‘š 𝑙, π‘š β‰₯ 1 2) 𝐴1 ∧ 𝐴2 ∧ … ∧ 𝐴𝑙 βŠƒβŠ₯ for 𝑙 β‰₯ 1 , π‘š = 0 βŠ₯is false 3) 𝐡1 ∨ 𝐡2 ∨ … ∨ π΅π‘š

for 𝑙 = 0 and π‘š β‰₯ 1.

2.2. Proof complexity measures By | πœ‘| we denote the size of a formula πœ‘, defined as the number of all logical signs in it. It is obvious that the full size of a formula, which is understood to be the number of all symbols is bounded by some linear function in |πœ‘ |. In the theory of proof complexity two main characteristics of the proof are: t- complexity (length), defined as the number of proof steps, 𝑙-complexity (size), defined as sum of sizes for all formulas in proof (size) [3]. πœ™

πœ™

Let Ο• be a proof system and Ο† be a tautology. We denote by 𝑑φ (𝑙φ )the minimal possible value of t-complexity (𝑙 -complexity) for all Ο•-proofs of tautology Ο†. 4

Definition 2.2.1. A tautolgy is called minimal if it is not a substitution of a shorter tautology. We denote by S(πœ‘) the set of all formulas, every of which is result of some substitution in a minimal tautology πœ‘. Definition 2.2.2. The proof system Ο• is called t-monotonous (𝒍-monotonous) if for every πœ™

πœ™

πœ™

πœ™

minimal tautology Ο† and for all formulas πœ“βˆˆS(Ο†) 𝑑φ ≀ π‘‘Οˆ (𝑙φ β‰€π‘™Οˆ ). Definition 2.2.3. Sequent Ξ“ β†’ Ξ” is called minimal valid if its formula form πœ‘Ξ“β†’Ξ” is minimal tautology. Definition 2.2.4. Sequent proof system Ξ¦ is called t-monotonous (𝑙 -monotonous) if for every πœ™

πœ™

valid sequent Ξ“ β†’ Ξ” and for every sequent Ξ“1 β†’ Ξ”1 such that πœ‘Ξ“1 β†’Ξ”1 ∈S(πœ‘Ξ“β†’Ξ” ) 𝑑Γ→Δ ≀ 𝑑Γ1 β†’Ξ”1 πœ™

πœ™

(𝑙Γ→Δ ≀𝑙Γ1 β†’Ξ”1 ). 2.3. Essential subformulas of tautologies For proving the main results we use the notion of essential subformulas, introduced in [4]. Let F be some formula and Sf (F ) be the set of all non-elementary subformulas of formula

F. For every formula F , for every οͺ οƒŽ Sf (F ) and for every variable p by Fοͺp is denoted the result of the replacement of the subformulas οͺ everywhere in F by the variable p . If οͺ  Sf (F ) , then Fοͺp is F .ᡞ We denote by Var (F ) the set of all variables in F . Definition 2.3.1. Let p be some variable that p Var (F ) and οͺ οƒŽ Sf (F ) for some tautology

F . We say that οͺ is an essential subformula in F iff Fοͺp is non-tautology. The set of essential subformulas in tautology F we denote by Essf(F), the number of essential subformulas – by Nessf(F) and the sum of sizes of all essential subformulas by Sessf(F).

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If F is minimal tautology, then Essf ( F ) = Sf ( F ) . Definition 2.3.2. The subformula οͺ is essential for valid sequent Ξ“ β†’ Ξ” if it is essential for its formula form. In [4] the following statement is proved. Proposition 1. Let F be a tautology and οͺ οƒŽ Essf (F ) , then a) in every 𝓕-proof of F subformula οͺ must be essential either at least in some axiom, used in proof or in formulae A1βŠƒ (A2βŠƒ (β€¦βŠƒAm)…)βŠƒB for some used in proof inference rule

A1 A2  Am , B

b) in every S𝓕-proof of F , where

𝐴

,

𝐴

𝐴

, … ,AΟƒk (kβ‰₯1) are used substitution rules,

Aσ1 Aσ2

subformula οͺ must be essential either at least in some axiom, used in proof, or in formulae A1βŠƒ (A2βŠƒ (β€¦βŠƒAm)…)βŠƒB for some used in proof inference rule

A1 A2  Am , B

and or must be result of successive substitutions 𝛔i1, 𝛔i2, …, 𝛔ir for 1≀ i1, i2, … , ir≀k in them. Note (2) that for every Frege system the number of essential subformulas both in every axiom and in formulae A1βŠƒ (A2βŠƒ (β€¦βŠƒAm)…)βŠƒB for every inference rule

A1 A2  Am is bounded B

with some constant. Note (3). It is not difficult to prove that all above statements are true for every formula form of valid sequent and axioms and rules of above mentioned sequent systems. 2.4. Main Formulas It is known, that in the alphabet, having 3 letters, for every 𝑛 > 0 a word with size 𝑛 can be constructed such, that neither of its subwords repeats in it twice one after the other [5]. Let 𝛼1 , 𝛼2 , … , 𝛼𝑛 be some of such words in the alphabet {π‘Ž, 𝑏, 𝑐}. In [4] the formulas πœ“π‘› are 6

obtained as follow: For 𝑛 > 0 let πœ“π‘›+1,𝑛 = (𝑝0 βŠƒ ¬¬𝑝0 ). Let the formula πœ“π‘–+1,𝑛 for the subwords 𝛼𝑖+1 , 𝛼𝑖+2 , … , 𝛼𝑛 (1 ≀ 𝑖 ≀ 𝑛) be constructed then: 1) if 𝛼𝑖 = π‘Ž then πœ“π‘–,𝑛 = (𝑝𝑖 βŠƒ 𝑝𝑖 ) ∧ πœ“π‘–+1,𝑛; 2) if 𝛼𝑖 = 𝑏 then πœ“π‘–,𝑛 = (¬𝑝𝑖 ∨ 𝑝𝑖 ) βŠƒ πœ“π‘–+1,𝑛 ; 3) if 𝛼𝑖 = 𝑐 then πœ“π‘–,𝑛 = (¬𝑝𝑖 ∧ 𝑝𝑖 ) ∨ πœ“π‘–+1,𝑛 ; As formula πœ“π‘› (𝑝0 , 𝑝1 , 𝑝2 , … , 𝑝𝑛 )we take the formula πœ“1,𝑛 (𝑝0 , 𝑝1 , 𝑝2 , … , 𝑝𝑛 ), for which we have

|πœ“π‘› |= Θ(𝑛) and all subformulas πœ“π‘–,𝑛 (1 ≀ 𝑖 ≀ 𝑛 + 1)are essential for πœ“π‘› , therefore

2 Nessf(πœ“π‘› ) = Ξ©(𝑛) and Sessf(πœ“π‘› ) = Ξ©(𝑛 ).

Note (4) As neither of these essential subformulas of formulas πœ“π‘› cannot be obtained from other by substitution rule, it is proved in [4] that formulas πœ“π‘› require more or equal than 𝑛 steps and 𝑛2 size both in the Frege systems and substitution Frege systems. It is obvious, that this statement is valid for sequent β†’ πœ“π‘› for the all mentioned sequent systems also. 3. Main results for 2-valued logic. Here we give the main theorem, but at first we must give the following easy proved auxiliary statements. Lemma 1.a) The minimal t-complexity and 𝑙 -complexity of Frege proofs and thus of substitution Frege proofs for formula pβŠƒ p are bounded by constant. b) For each formulae A and B the minimal t-complexity of Frege proofs and thus of substitution Frege proofs for formula 𝐴 βŠƒ (¬𝐴 βŠƒ 𝐡) is bounded by constant, just as its minimal 𝑙 -

complexity is bounded by cΒ·max( | 𝐴|, | 𝐡|) for some constant c. Lemm𝒂 𝟐. a) If 𝐹 is minimal tautology and 𝑝 is some variable that 𝑝 βˆ‰ π‘‰π‘Žπ‘Ÿ(𝐹), then the formula 𝑝 βŠƒ 𝐹 is also minimal tautology, and all essential subfomulas of formula 𝐹 are essential for formula 𝑝 βŠƒ 𝐹. 7

b) If β†’ 𝐹 is minimal valid sequent and 𝑝 is some variable that 𝑝 βˆ‰ π‘‰π‘Žπ‘Ÿ(𝐹) then sequent β†’ 𝑝 βŠƒ 𝐹 is also minimal valid sequent, and all essential subfomulas of sequent β†’ 𝐹 are essential for sequent β†’ 𝑝 βŠƒ 𝐹.

Theorem 1. Every Frege system 𝓕, substitution Frege system Sβ„±, sequent systems𝑷𝑲, π‘·π‘²βˆ’ , 𝐒𝑷𝑲, π’π‘·π‘²βˆ’ are neither t-monotonous nor 𝑙 -monotonous. Proof. Let us consider the tautologies 𝝋𝐧 = pβŠƒπœ“π‘› ,

𝛂n=Β¬ (pβŠƒp)βŠƒπœ“π‘›

𝛃n=(pβŠƒp)βŠƒΞ±π‘› ,

and

where πœ“π‘› are the formulas from previous section and variable p is not belong to Var(πœ“π‘› ). Note that for every n formula 𝛂n (sequent →𝛂n) belongs to S(𝝋𝐧 ) (S(β†’ 𝝋𝐧 )). According to statement of Lemma 2. every formula 𝝋𝐧 is minimal tautology and the sequentβ†’ 𝝋𝐧 is minimal valid sequent. For every n the formula 𝛂n can be deduced in every Frege system 𝓕 as follows: at first we deduce formula pβŠƒp, then 𝛃n=(pβŠƒp)βŠƒΞ±π‘› and lastly 𝛂n by modus ponens. From statement of Lemma 1 we obtain for lengths and sizes of proofs in every Frege system for the set of formulas 𝛂n the bounds O(1) and O(𝑛) accordingly. It is obvious, that for more β€œstronger” substitution systems Frege the bounds are no more. By statements of Lemma 2 and Note (4) we obtain for lengths and sizes of proofs in every Frege system and substitution systems Frege for the set of formulas 𝝋𝐧 the bounds Ξ©(𝑛) and Ξ©(𝑛2 ) accordingly. Now we give deduction of the sequent β†’ Β¬ (pβŠƒp) βŠƒπœ“π‘› in the system π‘·π‘²βˆ’ as follows: 𝑝→𝑝 β†’(π‘βŠƒπ‘) Β¬(π‘βŠƒπ‘)β†’ Β¬(π‘βŠƒπ‘)β†’πœ“π‘›

β†’Β¬ (π‘βŠƒπ‘) βŠƒπœ“π‘›

.

It is obvious, that the bounds for lengths and sizes of proofs in system π‘·π‘²βˆ’ for the set of sequents →𝛂n are O(1) and O(𝑛) accordingly, therefore for more β€œstronger” systems 𝑷𝑲,

8

𝐒𝑷𝑲,π’π‘·π‘²βˆ’ the bounds are no more. By statements of Lemma 2 and Note (4) we obtain for lengths and sizes of proofs in the systems 𝑷𝑲, π‘·π‘²βˆ’ ,𝐒𝑷𝑲,π’π‘·π‘²βˆ’

for the set of sequents β†’ 𝝋𝐧 the bounds Ξ©(𝑛) and Ξ©(𝑛2 )

accordingly. β§  Remark .Above results only for Frege systems are parts of publications [6,7]. 4. Main results for 3-valued logic. Many-valued logic (MVL) as a separate subject was created and developed first by Łukasiewicz [6]. At present many interesting applications of MVL were found in such fields as logic, mathematics, hardware design, artificial intelligence and some other area soft information technologies, therefore the investigations in area of MVL are very actual. 4.1.Main Definitions Let E3 be the set {0,1/2,1}. We use the well-known notions of propositional formula, which defined as usual from 3-valued propositional variables p, q, pi (i β‰₯1,) with values from E3, may be also propositional constants, parentheses (,), and logical connectives & , οƒš ,  ,~ defined as follow: 𝒑 ∨ 𝒒 = π‘šπ‘Žπ‘₯(𝑝, π‘ž), π’‘βŠƒπ’’={

1, 1 βˆ’ 𝑝 + π‘ž,

𝒑&π‘ž = π‘šπ‘–π‘›(𝑝, π‘ž), π‘“π‘œπ‘Ÿ 𝑝 ≀ π‘ž π‘“π‘œπ‘Ÿ 𝑝 > π‘ž

and negation, defined as follow: ~𝒑 = 1 βˆ’ 𝑝

(Łukasiewicz’s negation). 𝑖

For propositional variable p and 𝛅=2(0≀i≀2) we define additionally: p𝛅 as (𝑝 βŠƒ Ξ΄)& (Ξ΄ βŠƒ 𝑝) (exponent function). Note, that exponent is no new logical function and note also that exponent for 𝛅=1 is 𝒑, and for 𝛅=0 is ~𝒑. In considered logics we fix 1 as designated value, so a formula Ο† with variables p1,p2,…pn is called k-tautology if for every 𝛿̃ = (𝛿1 , 𝛿2 , … , 𝛿𝑛 ) ∈ 𝐸3𝑛 assigning 𝛿 j (1≀j≀n) to each pj gives the value 1 of Ο†. 4.2. Łukasiewicz’s proof system L for 3-Valued Logic 9

For every formula A, B, C of 3-valued logic the following formulas are axioms schemes of L [7]: 1. 𝐴 βŠƒ (𝐡 βŠƒ 𝐴)

2. (𝐴 βŠƒ (𝐡 βŠƒ 𝐢)) βŠƒ (𝐡 βŠƒ (𝐴 βŠƒ 𝐢))

3. (𝐴 βŠƒ 𝐡) βŠƒ ((𝐡 βŠƒ 𝐢) βŠƒ (𝐴 βŠƒ 𝐢)) 4. (𝐴 βŠƒ (𝐴 βŠƒ 𝐡)) βŠƒ ((∼ 𝐡 βŠƒ (∼ 𝐡 βŠƒβˆΌ 𝐴)) βŠƒ (𝐴 βŠƒ 𝐡)) 5. (𝐴 βŠƒ 𝐡) βŠƒ (∼ 𝐡 βŠƒβˆΌ 𝐴)

6. 𝐴 βŠƒβˆΌβˆΌ 𝐴

7. ∼∼ 𝐴 βŠƒ 𝐴

8. 𝐴&𝐡 βŠƒ 𝐡

9. 𝐴&𝐡 βŠƒ 𝐴

10. (𝐢 βŠƒ 𝐴) βŠƒ ((𝐢 βŠƒ 𝐡) βŠƒ (𝐢 βŠƒ 𝐴&𝐡))

11. 𝐴 βŠƒ 𝐴 ∨ 𝐡

12. 𝐡 βŠƒ 𝐴 ∨ 𝐡

13. (𝐴 βŠƒ 𝐢) βŠƒ ((𝐡 βŠƒ 𝐢) βŠƒ (𝐴 ∨ 𝐡 βŠƒ 𝐢))

Inference rule is modus ponens /m.p./

𝐴,π΄βŠƒπ΅ 𝐡

.

4.3. A substitution system SL consists of a system L augmented with the substitution rule with inferences of the form

πœ‘ 𝑖1 for any substitution 𝜎 = ( 𝑝𝑖1 AΟƒ 𝐴

πœ‘ 𝑖2 … 𝑝𝑖2 …

πœ‘ 𝑖𝑠 𝑝𝑖𝑠 ), s β‰₯ 1, consisting of a

mapping from 3-valued propositional variables to 3-valued propositional formulas, and 𝐴𝜎 denotes the result of applying the substitution to formula A, which replaces each variable in A with its image under Οƒ. This definition of substitution rule allows to use the simultaneous substitution of multiple formulas for multiple variables of A without any restrictions. 4.4. Every definitions and statements of points 2.2. and 2.3. relate to 2-valued tautologies, to essential subformulas, to Frege systems and to substitution Frege systems are valid for 3-valued tautologies, systems L and SL. The Main Formulas define here as follow: For 𝑛 > 0 let 𝛾𝑛+1,𝑛 = (𝑝0 βŠƒ ~~𝑝0 ). Let the formula 𝛾𝑖+1,𝑛 for the subwords 𝛼𝑖+1 , 𝛼𝑖+2 , … , 𝛼𝑛 (1 ≀ 𝑖 ≀ 𝑛) be constructed then: 1) if 𝛼𝑖 = π‘Ž then 𝛾𝑖,𝑛 = (𝑝𝑖 βŠƒ 𝑝𝑖 ) ∧ 𝛾𝑖+1,𝑛 ; 1/2

∨ 𝑝𝑖1 ) βŠƒ 𝛾𝑖+1,𝑛 ;

1/2

∧ 𝑝𝑖1 ) ∨ 𝛾𝑖+1,𝑛 ;

2) if 𝛼𝑖 = 𝑏 then 𝛾𝑖,𝑛 = (𝑝𝑖0 ∨ 𝑝𝑖 3) if 𝛼𝑖 = 𝑐 then 𝛾𝑖,𝑛 = (𝑝𝑖0 ∧ 𝑝𝑖

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As formula 𝛾𝑛 (𝑝0 , 𝑝1 , 𝑝2 , … , 𝑝𝑛 )we take the formula 𝛾1,𝑛 (𝑝0 , 𝑝1 , 𝑝2 , … , 𝑝𝑛 ), for which we have |𝛾𝑛 |= Θ(𝑛) and all subformulas 𝛾𝑖,𝑛 (1 ≀ 𝑖 ≀ 𝑛 + 1)are essential for 𝛾𝑛 , therefore Nessf(𝛾𝑛 ) = Ξ©(𝑛) 2 and Sessf(𝛾𝑛 ) = Ξ©(𝑛 ).

4.5. Main Lemmas for 3-valued tautologies. It is not difficult to prove the following statements. Lemma 4.5.1.a) The minimal t-complexity and 𝑙 -complexity of L proof and thus of SL proof for formula pβŠƒ p are bounded by constant. b) For each formulae A and B the minimal t-complexity of L proof and thus of SL proof for formula AβŠƒ ( AΒ½ βŠƒ ( A0 βŠƒ 𝐡)) is bounded by constant, just as its minimal 𝑙 -complexity is bounded by cΒ·max( | 𝐴|, | 𝐡|) for some constant c. Lemm𝒂 πŸ’. πŸ“. 𝟐. If 𝐹 is minimal tautology and 𝑝 is some variable that 𝑝 βˆ‰ π‘‰π‘Žπ‘Ÿ(𝐹), then the formula p1/2 βŠƒ(p0 βŠƒF) also is minimal tautology, and all essential subfomulas of formula 𝐹 are essential for formula p1/2 βŠƒ(p0 βŠƒF) . Theorem 2. The systems L and SL are neither t-monotonous nor 𝑙 -monotonous. Proof. Let us consider the tautologies 𝝋𝐧 = p1/2 βŠƒ(p0 βŠƒ 𝛾𝑛 ), 𝛂n= (pβŠƒp)1/2 βŠƒ((pβŠƒp)0 βŠƒ 𝛾𝑛 )) and

𝛃n=(pβŠƒp)βŠƒΞ±π‘› ,

where 𝛾𝑛 are the formulas from previous section and variable p is not belong to Var(𝛾𝑛 ). Note that for every n formula 𝛂n belongs to S(𝝋𝐧 ). According to statement of Lemma 4.5.2. every formula 𝝋𝐧 is minimal tautology and as above we obtain for lengths and sizes of proofs in the systems L and SL for the set of formulas 𝝋𝐧 the bounds Ξ©(𝑛) and Ξ©(𝑛2 ) accordingly. For every n the formula 𝛂n can be deduced in the system L as follows: at first we deduce formula pβŠƒp, then 𝛃n=(pβŠƒp)βŠƒΞ±π‘› and lastly 𝛂n by modus ponens. From statement of Lemma 4.5. 1 we obtain for lengths and sizes of proofs in the system L for the set of formulas 𝛂n the bounds O(1) and O(𝑛) accordingly. It is obvious, that for more β€œstronger” system SL the bounds are no more. β§  Acknowledgments 11

This work arose in the context of propositional proof complexities research supported by the Russian-Armenian University from founds of MES of RF. References 1. S.A.Cook, A.R.Reckhow: The relative efficiency of propositional proof systems, Journal of Symbolic logic, vol. 44, 1979, 36-50. 2. А.Π‘. АникССв, О Π½Π΅ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ классификации Π²Ρ‹Π²ΠΎΠ΄ΠΈΠΌΡ‹Ρ… ΠΏΡ€ΠΎΠΏΠΎΠ·ΠΈΡ†ΠΈΠΎΠ½Π°Π»ΡŒΠ½Ρ‹Ρ… Ρ„ΠΎΡ€ΠΌΡƒΠ», ΠœΠ°Ρ‚Π΅ΠΌΠ°Ρ‚ΠΈΡ‡Π΅ΡΠΊΠΈΠ΅ Π·Π°ΠΌΠ΅Ρ‚ΠΊΠΈ, 11, Π²Ρ‹ΠΏ.2, 1972, 165-174. 3. Ан.Чубарян,О слоТности Π²Ρ‹Π²ΠΎΠ΄ΠΎΠ² Π² Π½Π΅ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… систСмах классичСского исчислСния

высказываний, β€œΠœΠ°Ρ‚Π΅ΠΌΠ°Ρ‚ΠΈΡ‡Π΅ΡΠΊΠΈΠ΅ вопросы кибСрнСтики”, Π²Ρ‹ΠΏ. 14, 2005, М., Π€ΠΈΠ·ΠΌΠ°Ρ‚Π»ΠΈΡ‚, 49-56. 4. Aдян Π‘.И., ΠŸΡ€ΠΎΠ±Π»Π΅ΠΌΡ‹ БСрнсайда ΠΈ тоТдСства Π² Π³Ρ€ΡƒΠΏΠΏΠ°Ρ…. М., Наука, 1975. 5. Anahit Chubaryan, GarikPetrosyan, Frege systems are no monotonous, Evolutio, ЕстСствСнныСнауки, Π’Ρ‹ΠΏ 3, 2016, 12-14. 6. J.Lukasiewicz, OLogiceTrojwartosciowej,Ruchfiloseficzny(Lwow), Vol.5,1920,169-171. 7. I.D. Zaslawskij, Symmetrical Konstructive Logic, Press. ASAR, (in Russian),1978.

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