On some universal proof system for all versions of many-valued - PowerPoint PPT Presentation

ANAHIT CHUBARYAN, ARTUR KHAMISYAN On some universal proof system for all versions of many-valued logics Department of Informatics and Applied Mathematics Yerevan State University 1

Main notions of k-valued logic. 1 k−2 Let E k be the set {0, k−1 , … , k−1 , 1} . Definitions of main logical functions are: 𝒒 ∨ 𝒓 = 𝑛𝑏𝑦(𝑞, 𝑟) (1) disjunction or 𝒒 ∨ 𝒓 = [(𝑙 − 1)(𝑞 + 𝑟)](𝑛𝑝𝑒 𝑙)/(𝑙 − 1) (2) disjunction, 𝒒&𝑟 = 𝑛𝑗𝑜(𝑞, 𝑟) (1) conjunction or 𝒒&𝑟 = max (𝒒 + 𝑟 − 1, 0) (2) conjunction 2

For implication we have two following versions: 𝒒 ⊃ 𝒓 = {1, 𝑔𝑝𝑠 𝑞 ≤ 𝑟 1 − 𝑞 + 𝑟, 𝑔𝑝𝑠 𝑞 > 𝑟 (1) Łukasiewicz’s implication or p ⊃ 𝒓 = {1, 𝑔𝑝𝑠 𝑞 ≤ 𝑟 𝑟, 𝑔𝑝𝑠 𝑞 > 𝑟 (2) Gödel’s implication And for negation two versions also: ¬𝒒 = 1 − 𝑞 (1) Łukasiewicz’s negation or ¬𝒒 = ((𝑙 − 1)𝑞 + 1)(𝑛𝑝𝑒 𝑙)/(𝑙 − 1) (2) cyclically permuting negation. ̅ instead of ¬𝒒. Sometimes we can use the notation 𝒒 3

𝑗 k−1 (0≤ i≤k - 1) we define For propositional variable p and 𝛆 = additionally “exponent” functions: p 𝛆 as ( 𝑞 ⊃ δ)& (δ ⊃ 𝑞) with (1) implication (1) exponent, p 𝛆 as p with ( k- 1 ) – i (2) negations. (2) exponent. 4

We use the well-known notions of propositional formula, which defined as usual from propositional variables with values from E k, (may be also propositional constants), parentheses (,), and logical connectives & ,  ,  , ¬ . Additionaly we use two modes of exponential function p 𝛕 and introduce the additional notion of formula: for every formulas A and B the expression 𝑩 𝑪 (for both modes) is formula also. 5

1 𝒋 2 ≤ 𝐥−𝟐 ≤ 1 In the considered logics either only 1 or every of values can be fixed as designated values . 1 𝑗 If we fix “ 1” (every of values 2 ≤ k−1 ≤ 1 ) as designated value, so a formula φ with variables p 1 ,p 2 ,…p n is called 1 - k -tautolog y ( ≥1/2 -k- ̃ = (𝜀 1 , 𝜀 2 , … , 𝜀 𝑜 ) ∈ 𝐹 𝑙 𝑜 assigning 𝜀 j (1≤ j≤n ) to each tautology ) if for every 𝜀 1 𝑗 k−1 ≤ 1 ) of φ. 2 ≤ p j gives the value 1 (or some value Sometimes we call 1 - k -tautolog y or ≥1/2 -k-tautology simply k - tautolog y. 6

Definitions of universal system for MVL and some properties of them. Sequent type system US for all versions of MVL. Sequent system uses the denotation of sequent Γ ⊢ Δ where Γ (antecedent) and Δ (succedent) are finite (may be empty) sequences (or sets) of propositional formulas. 1 k−1 k−2 k−1 ⁄ ⁄ For every propositional variable 𝑞 in k-valued logic 𝑞 0 , 𝑞 ,…, 𝑞 and 𝑞 1 in sense of both exponent modes are the literals 7

For every literal 𝐷 and for any set of literals Γ the axiom sxeme of propositional system US is Γ , 𝐷 → 𝐷. For every formulas 𝐵 , 𝐶, for any sets of literals Γ , each 𝜏 1 , 𝜏 2 , 𝜏 from the set E k and for ∗∈ {&,∨, ⊃} the logical rules of US are: Γ ⊢𝐵 𝜏1 and Γ ⊢𝐶 𝜏2 Γ ⊢𝐵 𝜏1 and Γ ⊢𝐶 𝜏2 ⊢∗ Γ ⊢(𝐵 ∗ 𝐶) 𝜒∗(𝐵,𝐶,𝜏1,𝜏2) ) , ⊢ exp Γ ⊢(𝐵 𝐶 ) 𝜒exp(𝐵,𝐶,𝜏1,𝜏2) ) Γ ⊢𝐵 𝜏 ⊢ ¬ Γ ⊢ (¬𝐵) 𝜒¬(𝐵,𝜏) literals elimination 1 𝑙−2 Γ ,𝑞 0 ⊢𝐵, Γ ,𝑞 𝑙−1 ⊢𝐵, Γ ,𝑞 1 ⊢𝐵 𝑙−1 ⊢𝐵, … , Γ ,𝑞 ⊢ , Γ ⊢𝐵 8

where many-valued functions 𝜒 ∗ (𝐵, 𝐶, 𝜏 1 , 𝜏 2 ) , 𝜒 exp (𝐵, 𝐶, 𝜏 1 , 𝜏 2 ) , 𝜒 ¬ (𝐵, 𝜏), must be defined individually for each version of MVL such, that 1) formulas 𝐵 𝜏 1 ⊃ (𝐶 𝜏 2 ⊃ (𝐵 ∗ 𝐶) 𝜒 ∗ (𝐵,𝐶,𝜏 1 ,𝜏 2 ) ) , 𝐵 𝜏 1 ⊃ (𝐶 𝜏 2 ⊃ (𝐵 𝐶 ) 𝜒 exp (𝐵,𝐶,𝜏 1 ,𝜏 2 ) ) and 𝐵 𝜏 ⊃ (¬𝐵) 𝜒 ¬ (𝐵,𝜏) must be k -tautolog y in this version, 2) if for some 𝜏 1 , 𝜏 2 , 𝜏 the value of 𝜏 1 ∗ 𝜏 2 ( 𝜏 1𝜏 2 , ¬𝜏 ) is one of designed values in this version of MVL , then (𝜏 1 ∗ 𝜏 2 ) 𝜒 ∗ (𝜏 1 ,𝜏 2 ,𝜏 1 ,𝜏 2 ) = 𝜏 1 ∗ 𝜏 2 𝜏 2 ) 𝜒 exp (𝜏 1 ,𝜏 2 ,𝜏 1 ,𝜏 2 ) = 𝜏 1𝜏 2 , (¬𝜏) 𝜒 ¬ (𝜏,𝜏) = ¬𝜏). ( (𝜏 1 9

We say that formula A is derived in US iff the sequent ⊢ 𝑩 is deduced in US. 10

Completeness of US Here we give at first for the system US some generalization of Kalmar’s proof of deducibility for two-valued tautologies in the classical propositional logic . Lemma . Let 𝑄 = {𝑞 1 , 𝑞 2 , … , 𝑞 𝑜 } be the set of all variables of any formula A, ̃ = (𝜀 1 , 𝜀 2 , … , 𝜀 𝑜 ) ∈ 𝐹 𝑙 𝑜 the following sequent is proved in US . then for every 𝜀 𝜀 𝑜 ⊢ 𝐵 𝐵(𝜀 1 ,𝜀 2 ,…,𝜀 𝑜 ) 𝜀 1 , 𝑞 2 𝜀 2 , … , 𝑞 𝑜 𝑞 1 11

̃ = (𝜀 1 , 𝜀 2 , … , 𝜀 𝑜 ) ∈ 𝐹 𝑙 𝑜 in US Corollary. If A is k -tautology, then for every 𝜀 is proved the sequent 𝜀 𝑜 ⊢ 𝐵. 𝜀 1 , 𝑞 2 𝜀 2 , … , 𝑞 𝑜 𝑞 1 Really we must use the properties 2) of the functions 𝜒 ∗ (𝐵, 𝐶, 𝜏 1 , 𝜏 2 ) , 𝜒 exp (𝐵, 𝐶, 𝜏 1 , 𝜏 2 ) and 𝜒 ¬ (𝐵, 𝜏) . 15

Theorem Any formula is derived in US iff it is k -tautology. 16

Examples of US for some versions of MVL For the first of constructed systems LN k ( Łukasiewicz’s negation ) with fixed a) “1” as designated value, use conjunction, disjunction, (1) implication, (1) 𝒋 𝒍−𝟐 (1≤i≤k -2) for using negation and (1) exponent, and constants 𝛆 = (1)exponent the functions 𝝌 ∗ (𝑩, 𝑪, 𝝉 𝟐 , 𝝉 𝟑 ) , 𝝌 𝒇𝒚𝒒 (𝑩, 𝑪, 𝝉 𝟐 , 𝝉 𝟑 ) , 𝝌 ¬ (𝑩, 𝝉) are defined as follows: 𝝌 ∗ (𝑩, 𝑪, 𝝉 𝟐 , 𝝉 𝟑 ) = 𝝉 𝟐 ∗ 𝝉 𝟑 𝝌 𝒇𝒚𝒒 (𝑩, 𝑪, 𝝉 𝟐 , 𝝉 𝟑 ) = 𝝉 𝟐𝝉 𝟑 𝝌 ¬ (𝑩, 𝝉) = ¬𝝉 . 20

For the second systems CN 3 (cyclically permuting negation) with fixed “1” as b) designated value, use conjunction, disjunction,(2)implication, (2)negation and (2)exponent the functions 𝝌 ∗ (𝑩, 𝑪, 𝝉 𝟐 , 𝝉 𝟑 ) , 𝝌 𝒇𝒚𝒒 (𝑩, 𝑪, 𝝉 𝟐 , 𝝉 𝟑 ) , 𝝌 ¬ (𝑩, 𝝉) are defined as follows: ̿ ⊃ 𝐶))⋁(¬(𝐵⋁𝐵̿)&¬(𝐶⋁𝐶 ̿)) , 𝜒 ⊃ (𝐵, 𝐶, 𝜏 1 , 𝜏 2 ) = (𝜏 1 ⊃ 𝜏 2 )&(¬(𝐵⋁𝐵̅)⋁(𝐶 ̅⋁𝐶 ̿))⋁(¬(𝐵̅⋁𝐵̿)&(𝐶 ⊃ 𝐶 ̅)) , 𝜒 ∨ (𝐵, 𝐶, 𝜏 1 , 𝜏 2 ) = (𝜏 1 ⋁𝜏 2 )⋁((𝐵 ⊃ 𝐵̅)&¬(𝐶 ̅))⋁((𝐵&𝐵̅)⋁(𝐶&𝐶 ̿) 𝜒 & (𝐵, 𝐶, 𝜏 1 , 𝜏 2 ) = (𝜏 1 &𝜏 2 )⋁((𝐵&𝐵̿)⋁(𝐶&𝐶 ̅ 𝜏 2 )⋁¬¬(𝐵 𝜏 1 &𝐶 ̅ 𝜏 2 ))) 𝜒 exp (𝐵, 𝐶, 𝜏 1 , 𝜏 2 ) = 𝜏 1𝜏 2 ⋁(¬(𝜏 1𝜏 2 )&¬(¬(𝐵 𝜏 1 &𝐶 φ ¬ (A, σ) = ¬σ. 21

c) For LN 3,2 - Łukasiewicz’s logic with fixed “1/2” and “1” as designated values, and with (1) conjunction, (1) disjunction, (1) implication, (1) negation and (1) exponent, and constants 0, ½ and 1 for using (1)exponent we have 𝜒 ∗ (𝐵, 𝐶, 𝜏 1 , 𝜏 2 ) = ((𝐵 𝜏 1 & 𝐶 𝜏 2 )&¬(𝐵 ∗ 𝐶)) ⊃ ¬((𝐵 𝜏 1 &𝐶 𝜏 2 )&¬(𝐵 ∗ 𝐶)) 𝜒 𝑓𝑦𝑞 (𝐵, 𝐶, 𝜏 1 , 𝜏 2 ) = ((𝐵 𝜏 1 & 𝐶 𝜏 2 )&¬(𝐵 𝐶 )) ⊃ ¬((𝐵 𝜏 1 & 𝐶 𝜏 2 )&¬(𝐵 𝐶 )) φ ¬ (A, σ) = (𝐵 &𝜏) ⊃ ¬(𝐵& 𝜏) The work with other version of MVL is in progress. 22

Thank you for attention 23