Counting with 3-valued truth tables of bracketed formulae connected by implication
aa r X i v : . [ m a t h . G M ] O c t Counting with 3-valued truth tables of bracketed formulaeconnected by implication
Volkan Yildiz [email protected]
Abstract
In this paper we investigate the combinatorical structure of the Kleene type truth tables of allbracketed formulae with n distinct variables connected by the binary connective of implication.Keywords: Propositional logic, implication, Catalan numbers, asymptoticAMS classification: 05A15, 05A16, 03B05. Notations • p , ..., p n , and φ, ψ are all distinct propositional variables. • ‘True’ will be denoted by 1 • ‘False’ will be denoted by 0 • ‘Unknown’ will be denoted by 2 • The set of counting numbers is denoted by N • : such that • ν is the valuation function : ν ( φ ) = 1 if φ is true, ν ( φ ) = 0 if φ is false, and ν ( φ ) = 2 if φ isunknown. • ∧ , ∨ are the conjunction and disjunction operators. • ⇒ the implication operator • ¬ the negation operator • c denotes the case number in t cn • For the coefficient of x n in G ( x ) [ x n ] G ( x ) = [ x n ] (cid:18) P n ≥ g n x n (cid:19) = g n We have written in former papers about counting in truth tables in 2012. This paper was written in2013, but never been published due to time constraints and changes in my living conditions. Nowduring the pandemic period, I have some time in my hands to return to my incomplete work. For2-valued truth tables counting arguments one can refer back to my paper ‘General combinatoricalstructure of truth tables of bracketed formulae connected with implication’, [1].
Recall that the number of bracketings of a product of n terms is the Catalan number: C n = 1 n n − n − ! and its generating function C ( x ) = 1 − √ − x . Here we define the implication operator in the following way, “Kleene’s way” ⇒ Proposition 1.
Let g n be the total number of rows in all Kleene truth tables for bracketed implicationwith n distinct variables p , ..., p n . Then g n = n − X i =1 g i g n − i , g = 3 Proof. g n = 3 n C n = 3 n n − X i =1 C i C n − i = n − X i =1 (3 i C i )(3 n − i C n − i ) = n − i X i =1 g i g n − i ⋆ Thus it has the following generating function. G ( x ) = (1 − √ − x ) / roposition 2. Let f n , t n and u n be the number of rows with the value “false” , ”true”, and”unknow” in the Kleene truth tables of all bracketed formulae with n distinct propositions p , ..., p n connected by the binary connective of implication. Then u n has the following reccurence relation,and generating function U ( x ) . u n = n − X i =1 u i g n − i = n − X i =1 u i n − i C n − i , u = 1 U ( x ) = 1 − √ − x Proof. u n = t i u n − i + u i f n − i + u i u n − i = t i u n − i + u i ( g n − i − t n − i )Summing over n, gives us U ( x ) = x + U ( x ) G ( x )Solving this for U ( x ) gives us the required result. ⋆ Corollary 3.
The number of unknown entries in bracketed Kleene’s truth table connected by theimplication is given by u n = 3 n − n n − n − ! (2) Proposition 4. f n has the following reccurence relation, f n = n − X i =1 f i (cid:18) C n − i n − i − − f n − i (cid:19) , f = 1 and f n has the following generating function F ( x ) = − − √ − x + p x + 4 √ − x Proof.
Consider f n = f i t n − i = f i ( g n − i − f n − i − u n − i )= f i n − i C n − i − f i f n − i − f i n − i − C n − i = f i (2 C n − i n − i − − f n − i )Summing over n, gives us F ( x ) = 2 F ( x ) U ( x ) − F ( x ) + x. Solving it for F ( x ) gives us the required result. ⋆ Corollary 5. t n has the following generating function T ( x ) = 4 − √ − x − p x + 4 √ − x Theorem 6.
Let f n , t n and u n be number of rows with the value false, true and unknown in theKleene truth tables of all the bracketed implications with n variables. Then we have the followingasymptotics f n ∼ (cid:18) − √ (cid:19) n − √ πn t n ∼ (cid:18) √ (cid:19) n − √ πn u n ∼ (cid:18) (cid:19) n − √ πn g n ∼ n − √ πn roof. Recall F ( x ) = − − √ − x + p x + 4 √ − x r = and F ( ) = 0.So let A ( x ) = F ( x ) − F ( ).lim x → A ( x ) f ( x ) = lim x → −√ − x + p x + √ − x − √ √ − x = lim x → − p x + 4 √ − x + 2 √ − x − p x + 4 √ − x = 7 − √ − f n ∼ − √ − n − / n !(cid:18) (cid:19) − n ∼ (cid:18) − √ (cid:19) n − √ πn With similar arguments we have the above asymptotics for t n and u n . ⋆ Corollary 7.
The number of rows with unknown in the Kleene truth tables is the average of thenumber of rows with true and false. f n + t n u n , ∀ n ≥ . Proof.
Since u n = 3 n − C n ∀ n ≥
1, we have f n + t n = 23 g n f n + t n ) = 2 g n f n + t n = 2 g n − f n − t n f n + t n = 2 u n u n = f n + t n . ⋆ Note here 7 − √ ≈ . , √ ≈ . , ≈ . − √
721 + 7 + 2 √
721 = 23The below table shows the sequences which we have discussed so far, up to n = 9. n t n f n u n g n Since g n = t n + f n + u n , ∀ n ≥
1, we have g n = X g i g n − i = X ( t i + f i + u i )( t n − i + f n − i + u n − i )= X t i t n − i | {z } t n + X t i f n − i | {z } f n + X t i u n − i | {z } u n + X f i t n − i | {z } t n + X f i f n − i | {z } t n + X f i u n − i | {z } t n + X u i t n − i | {z } t n + X u i f n − i | {z } u n + X u i u n − i | {z } u n e can generate nine more sequences: t n , t n , t n , t n , t n , f n , u , u n , and u n , exceptfrom f n , all other sequences equals to 0 when n = 1. Each of these sequences, (and their generatingfunctions) counts different rows of the corresponding truth table. E.g. Let φ and ψ be propositionalvariables, then ν ( φ ⇒ ψ ) = 1 : ( ν ( φ ) = 1 = ν ( ψ )) ν ( φ ⇒ ψ ) = 1 : ( ν ( φ ) = 0 ∧ ν ( ψ ) = 1) ν ( φ ⇒ ψ ) = 1 : ( ν ( φ ) = 0 = ν ( ψ )) ν ( φ ⇒ ψ ) = 1 : ( ν ( φ ) = 0 ∧ ν ( ψ ) = 2) ν ( φ ⇒ ψ ) = 1 : ( ν ( φ ) = 2 ∧ ν ( ψ ) = 1) . In each case we are interested in formulae obtained from p ⇒ ... ⇒ p n by inserting brackets suchthat the valuation of the first i bracketing and the rest ( n − i ) bracketing both give 1, ‘true’; suchthat the valuation of the first i bracketing is 0 and the rest ( n − i ) bracketing gives 1; such that thevaluation of the first i bracketing and the rest ( n − i ) bracketing both give 0; such that the valuationof the first i bracketing is 0 and and the rest ( n − i ) bracketing gives 2; such that the valuationof the first i bracketing is 2 and and the rest ( n − i ) bracketing gives 1, respectively. To get thecorresponding generating function for t n we can make the following calculations t n = n − X n =1 t i t n − Summing over n gives us T ( x ) = T ( x ) . Using the same method we can obtain the following gen-erating functions: T ( x ) = F ( x ) T ( x ) , T ( x ) = F ( x ) , T ( x ) = F ( x ) U ( x ), and T ( x ) = U ( x ) T ( x ).A few terms for these fresh sequences:( t n ) n> = 0 , , , , , , , , , , , ... ( t n ) n> = 0 , , , , , , , , , , , ... ( t n ) n> = 0 , , , , , , , , , , , ... ( t n ) n> = 0 , , , , , , , , , , , ... ( t n ) n> = 0 , , , , , , , , , , , ... ( t n ) n> = 1 , , , , , , , , , , , ... Note that ∀ n ≥ t n = P i =1 t in . With similar arguments we can get the rest of the generatingfunctions and their corresponding sequences. ν ( φ ⇒ ψ ) = 0 : ( ν ( φ ) = 1 ∧ ν ( ψ ) = 0) ν ( φ ⇒ ψ ) = 2 : ( ν ( φ ) = 1 ∧ ν ( ψ ) = 2) ν ( φ ⇒ ψ ) = 2 : ( ν ( φ ) = 2 ∧ ν ( ψ ) = 0) ν ( φ ⇒ ψ ) = 2 : ( ν ( φ ) = 2 = ν ( ψ )) F ( x ) and U ( x ) have been studied in former chapters. Here we want to get more sequences fromthe original sequence u n , i.e. we want to break u n into u , u , and u . Moreover the followingcorresponding generating function, and their sequences exist: U ( x ) = T ( x ) U ( x ), U ( x ) = U ( x ) F ( x ),and U ( x ) = U ( x ) .( u n ) n> = 0 , , , , , , , , , , , ... ( u n ) n> = 0 , , , , , , , , , , , ... ( u n ) n> = 0 , , , , , , , , , , , ... ( u n ) n> = 1 , , , , , , , , , , , ... Consequently, the following observation is essential.
Corollary 8. g n = X i =1 t in + X i =1 u in + f n X i =1 t in + 2( t n + t n ) + u n + f n Asymptotics 2
In this part we will be exploring the asymptotics of the sequences which we have seen in the formerchapter.
Lemma 9.
Consider the sequences that we have discussed in the former chapter: t n , t n , t n , t n , t n , and u n , then we have the following asymptotics t n ∼ (cid:18)
14 + √ (cid:19) n − √ πn , f n = t n ∼ (cid:18) − √ (cid:19) n − √ πn ,t n ∼ (cid:18) √ − (cid:19) n − √ πn , u n = t n ∼ (cid:18) √ − (cid:19) n − √ πn ,u n = t n ∼ (cid:18) − √ (cid:19) n − √ πn , u n ∼ (cid:18) (cid:19) n − √ πn . where (cid:18)
14 + √ (cid:19) ≈ . , (cid:18)
14 + √ (cid:19) ≈ . (cid:18) √ − (cid:19) ≈ . , (cid:18) √ − (cid:19) ≈ . (cid:18) − √ (cid:19) ≈ . , (cid:18) (cid:19) ≈ . . Proof.
Proofs are similar to the proof of theorem 6. No need to repeat the same type of calculations. ⋆ t n : 12 → (cid:18)
14 + √ (cid:19) decreased ≈ . t n = f n : 3 − √ → (cid:18) − √ (cid:19) decreased ≈ . t n : 2 √ − → (cid:18) √ − (cid:19) decreased ≈ . References [1] Volkan Yildiz.
General Combinatorical Structure of Truth Tables of Bracketed Formulae Con-nected by Implication . Arxiv: https://arxiv.org/abs/1205.5595Her sabah yeni bir g¨un do˜garkenBir g¨un de eksilir ¨om¨urdenHer ¸safak bir hırsız gibidirElinde bir fenerle gelen.¨O Hayyam.. Arxiv: https://arxiv.org/abs/1205.5595Her sabah yeni bir g¨un do˜garkenBir g¨un de eksilir ¨om¨urdenHer ¸safak bir hırsız gibidirElinde bir fenerle gelen.¨O Hayyam.