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

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ANAHIT CHUBARYAN, ARTUR KHAMISYAN

On some universal proof system for all versions

• f many-valued logics

Department of Informatics and Applied Mathematics Yerevan State University

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Main notions of k-valued logic. Let Ek be the set {0,

1 k−1 , … , k−2 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

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For implication we have two following versions: 𝒒 ⊃ 𝒓 = {1, 𝑔𝑝𝑠 𝑞 ≤ 𝑟 1 − 𝑞 + 𝑟, 𝑔𝑝𝑠 𝑞 > 𝑟 (1) Łukasiewicz’s implication

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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

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¬𝒒 = ((𝑙 − 1)𝑞 + 1)(𝑛𝑝𝑒 𝑙)/(𝑙 − 1) (2) cyclically permuting negation. Sometimes we can use the notation 𝒒 ̅ instead of ¬𝒒.

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For propositional variable p and 𝛆=

𝑗 k−1(0≤i≤k-1) we define

additionally “exponent” functions: p𝛆 as (𝑞 ⊃ δ)& (δ ⊃ 𝑞) with (1) implication (1) exponent, p𝛆 as p with (k-1)–i (2) negations. (2) exponent.

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

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In the considered logics either only 1 or every of values

1 2 ≤ 𝒋 𝐥−𝟐 ≤ 1

can be fixed as designated values. If we fix “1” (every of values

1 2 ≤ 𝑗 k−1 ≤ 1) as designated value,

so a formula φ with variables p1,p2,…pn is called 1-k-tautology (≥1/2-k- tautology) if for every 𝜀 ̃ = (𝜀1, 𝜀2, … , 𝜀𝑜) ∈ 𝐹𝑙

𝑜 assigning 𝜀j (1≤j≤n) to each

pj gives the value 1 (or some value

1 2 ≤ 𝑗 k−1 ≤ 1) of φ.

Sometimes we call 1-k-tautology or ≥1/2-k-tautology simply k- tautology.

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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. For every propositional variable 𝑞 in k-valued logic 𝑞0, 𝑞

1 k−1 ⁄

,…, 𝑞

k−2 k−1 ⁄

and 𝑞1in sense of both exponent modes are the literals

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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 𝜏1, 𝜏2, 𝜏 from the set Ek and for ∗∈ {&,∨, ⊃} the logical rules of US are: ⊢∗

Γ⊢𝐵𝜏1 and Γ⊢𝐶𝜏2 Γ⊢(𝐵 ∗ 𝐶)𝜒∗(𝐵,𝐶,𝜏1,𝜏2)) , ⊢ exp Γ⊢𝐵𝜏1 and Γ⊢𝐶𝜏2 Γ⊢(𝐵𝐶)𝜒exp(𝐵,𝐶,𝜏1,𝜏2))

⊢ ¬

Γ⊢𝐵𝜏 Γ⊢ (¬𝐵)𝜒¬(𝐵,𝜏)

literals elimination ⊢

Γ,𝑞0⊢𝐵, Γ,𝑞

1 𝑙−1⊢𝐵, … , Γ,𝑞 𝑙−2 𝑙−1⊢𝐵, Γ,𝑞1⊢𝐵

Γ⊢𝐵

,

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where many-valued functions 𝜒∗(𝐵, 𝐶, 𝜏1, 𝜏2), 𝜒exp(𝐵, 𝐶, 𝜏1, 𝜏2), 𝜒¬(𝐵, 𝜏), must be defined individually for each version of MVL such, that 1) formulas 𝐵𝜏1 ⊃ (𝐶𝜏2 ⊃ (𝐵 ∗ 𝐶)𝜒∗(𝐵,𝐶,𝜏1,𝜏2)), 𝐵𝜏1 ⊃ (𝐶𝜏2 ⊃ (𝐵𝐶)𝜒exp(𝐵,𝐶,𝜏1,𝜏2)) and 𝐵𝜏 ⊃ (¬𝐵)𝜒¬(𝐵,𝜏) must be k-tautology in this version, 2) if for some 𝜏1, 𝜏2, 𝜏 the value of 𝜏1 ∗ 𝜏2 (𝜏1𝜏2, ¬𝜏) is one of designed values in this version of MVL , then (𝜏1 ∗ 𝜏2)𝜒∗(𝜏1,𝜏2,𝜏1,𝜏2) = 𝜏1 ∗ 𝜏2 ((𝜏1

𝜏2)𝜒exp(𝜏1,𝜏2,𝜏1,𝜏2) = 𝜏1𝜏2, (¬𝜏)𝜒¬(𝜏,𝜏) = ¬𝜏).

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

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Completeness of US

Here we give at first for the system US some generalization of Kalmar’s proof

• f deducibility for two-valued tautologies in the classical propositional logic .

Lemma . Let 𝑄 = {𝑞1, 𝑞2, … , 𝑞𝑜} be the set of all variables of any formula A, then for every 𝜀 ̃ = (𝜀1, 𝜀2, … , 𝜀𝑜) ∈ 𝐹𝑙

𝑜 the following sequent is proved in US.

𝑞1

𝜀1, 𝑞2 𝜀2, … , 𝑞𝑜 𝜀𝑜 ⊢ 𝐵𝐵(𝜀1,𝜀2,…,𝜀𝑜)

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• Corollary. If A is k-tautology, then for every 𝜀

̃ = (𝜀1, 𝜀2, … , 𝜀𝑜) ∈ 𝐹𝑙

𝑜 in US

is proved the sequent 𝑞1

𝜀1, 𝑞2 𝜀2, … , 𝑞𝑜 𝜀𝑜 ⊢ 𝐵.

Really we must use the properties 2) of the functions 𝜒∗(𝐵, 𝐶, 𝜏1, 𝜏2), 𝜒exp(𝐵, 𝐶, 𝜏1, 𝜏2) and 𝜒¬(𝐵, 𝜏).

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

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Examples of US for some versions of MVL

a) For the first of constructed systems LNk (Łukasiewicz’s negation ) with fixed “1” as designated value, use conjunction, disjunction, (1) implication, (1) negation and (1) exponent, and constants 𝛆=

𝒋 𝒍−𝟐 (1≤i≤k-2) for using

(1)exponent the functions 𝝌∗(𝑩, 𝑪, 𝝉𝟐, 𝝉𝟑), 𝝌𝒇𝒚𝒒(𝑩, 𝑪, 𝝉𝟐, 𝝉𝟑), 𝝌¬(𝑩, 𝝉) are defined as follows: 𝝌∗(𝑩, 𝑪, 𝝉𝟐, 𝝉𝟑) = 𝝉𝟐 ∗ 𝝉𝟑 𝝌𝒇𝒚𝒒(𝑩, 𝑪, 𝝉𝟐, 𝝉𝟑) = 𝝉𝟐𝝉𝟑 𝝌¬(𝑩, 𝝉) = ¬𝝉.

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b) For the second systems CN3 (cyclically permuting negation) with fixed “1” as designated value, use conjunction, disjunction,(2)implication, (2)negation and (2)exponent the functions 𝝌∗(𝑩, 𝑪, 𝝉𝟐, 𝝉𝟑), 𝝌𝒇𝒚𝒒(𝑩, 𝑪, 𝝉𝟐, 𝝉𝟑), 𝝌¬(𝑩, 𝝉) are defined as follows: 𝜒⊃(𝐵, 𝐶, 𝜏1, 𝜏2) = (𝜏1 ⊃ 𝜏2)&(¬(𝐵⋁𝐵̅)⋁(𝐶 ̿ ⊃ 𝐶))⋁(¬(𝐵⋁𝐵̿)&¬(𝐶⋁𝐶 ̿)), 𝜒∨(𝐵, 𝐶, 𝜏1, 𝜏2) = (𝜏1⋁𝜏2)⋁((𝐵 ⊃ 𝐵̅)&¬(𝐶 ̅⋁𝐶 ̿))⋁(¬(𝐵̅⋁𝐵̿)&(𝐶 ⊃ 𝐶 ̅)), 𝜒&(𝐵, 𝐶, 𝜏1, 𝜏2) = (𝜏1&𝜏2)⋁((𝐵&𝐵̿)⋁(𝐶&𝐶 ̅))⋁((𝐵&𝐵̅)⋁(𝐶&𝐶 ̿) 𝜒exp(𝐵, 𝐶, 𝜏1, 𝜏2) = 𝜏1𝜏2⋁(¬(𝜏1𝜏2)&¬(¬(𝐵𝜏1&𝐶 ̅𝜏2)⋁¬¬(𝐵𝜏1&𝐶 ̅𝜏2))) φ¬(A, σ) = ¬σ.

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c) For LN3,2 - 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, 𝜏1, 𝜏2) = ((𝐵𝜏1& 𝐶𝜏2)&¬(𝐵 ∗ 𝐶)) ⊃ ¬((𝐵𝜏1&𝐶𝜏2)&¬(𝐵 ∗ 𝐶)) 𝜒𝑓𝑦𝑞(𝐵, 𝐶, 𝜏1, 𝜏2) = ((𝐵𝜏1& 𝐶𝜏2)&¬(𝐵𝐶)) ⊃ ¬((𝐵𝜏1& 𝐶𝜏2)&¬(𝐵𝐶)) φ¬(A, σ) = (𝐵 &𝜏) ⊃ ¬(𝐵& 𝜏) The work with other version of MVL is in progress.

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Thank you for attention