Computing the density of tautologies in propositional logic by solving system of quadratic equations of generating functions
aa r X i v : . [ m a t h . L O ] M a r COMPUTING THE DENSITY OF TAUTOLOGIES IN PROPOSITIONAL LOGIC BYSOLVING SYSTEM OF QUADRATIC EQUATIONS OF GENERATING FUNCTIONS
TAEHYUN EOMA
BSTRACT . In this paper, we will provide a method to compute the density of tautologies among the setof well-formed formulae consisting of m variables, a negation symbol and an implication symbol, whichhas a possibility to be applied for other logical systems. This paper contains computational numericalvalues of the density of tautologies for two, three, and four variable cases. Also, for certain quadraticsystems, we will introduce the s -cut concept to make a better approximation when we compute the ratioby brute-force counting, and discover a fundamental relation between generating functions’ values onthe singularity point and ratios of coefficients, which can be understood as another intepretation of theSzeg˝o lemma for such quadratic systems. With this relation, we will provide an asymptotic lower bound m − − (7 / m − / + O ( m − ) of the density of tautologies as m goes to the infinity. C ONTENTS
1. Introduction 12. Basic Definitions 23. The case with more than one variables 44. Numerical Theory 95. Estimated results for the multivariable cases 19Acknowledgement 34References 341. I
NTRODUCTION
In propositional logic systems, theorems, which are determined by axioms and inference rules, andtautologies, which are determined by valuation, are important objects. The positional logic systemon which modern mathematics usually relies is well known to be sound and complete, which meanstautologies and theorems are equivalent. Even though tautologies and theorems are equivalent, theyare quite different in structure. Tautologies can be checked by the usual buttom-up way, using truthtables recursively, whereas theorems cannot be determined by its subformulae. Hence, we are goingto count tautologies.In fact, just counting tautologies is not interesting, since they are countably infinite. What maybe interesting is the probability that a given well-formed formula is a tautology, but this probabilityshould be specified further since the number of total well-formed formulae is infinite. Thus, wewill consider the density, which means we will consider the limit of the portion of tautologies inthe set of all well-formed formulae with a fixed length. Also, to make this density nonzero, we willconsider the case that the number of variables is finite. There are some preceding studies such as [1],which computes the density of tautologies in a logic system with implication and negation on onevariable, [2], which computes the asymptotic density, as the number of variables goes to infinity,
Mathematics Subject Classification.
Key words and phrases. tautologies, density, propositional logic, generating functions, analyticity. of tautologies in a logic system with implication and negative literals, and [3], which computes thedensities in several logic systems based on one variable.In this paper, we will consider the logic system based on a finite number of variables with implica-tion and negation mainly, but methods in this paper may be used for other logic systems such as ‘and’with negation, or ‘or’ with negation, etc. Basic definitions are in
Section 2 . This section includes acorrespondence from a well-formed formula to a subset of the power set of variables which makeslogical symbols as a set-theoretical operator.
Section 3 will provide a method to solve the equationsand compute the limit ratios completely, by introducing well-organized partitions, which is coset-likefor set operators, to reduce the number of equations by merging, which is quotient-like among powersets. We will solve the system of generating functions of well-formed formulae by hierarchical clus-tering, and apply Szeg˝o lemma to compute the ratio of coefficients. In
Section 4 , we will define the‘ s -cut’, which can be used to estimate the limit ratios for coefficients of generating functions to a fixedgenerating function by computing first s values when fixed generating function has at-most-quadraticstructure and other generating functions consist a quadratic system with a common factor. This gen-eralization can be applied to any finite partitioning of well-formed formulae where logical symbolsbetween partitioned sets are well-defined. With this concept we will prove a relation between ratiosof coefficients and values of generating functions on the first singularity, which also can be deducedby Szeg˝o lemma. Moreover, when generating functions are partitioning fixed generating functionwith specific conditions, we will prove that there is a possibility that s -cut concept will give a validestimation to the real values. Lastly, Section 5 will give a computational result, using the result ofSection 2, of the density of tautologies for the cases that the number of variables are two, three, orfour. Also, it will give a computational evidence that s -cut can be more efficient to estimate for thecases with 1, 2, or 3 variables compare to counting and direct dividing. Lastly, this section will usethe relation between ratios and values of generating functions, to give an asymptotic lower bound m − − (7 / m − / + (5 / m − + O ( m − / ) for limit ratios as the number of variables, whichis denoted as m , goes to the infinity. This result has some reasonable evidences to be conjecturedcarefully as the largest order term is correct asymptotically.2. B ASIC D EFINITIONS
Consider the logic system with a ( m -element) set of variables X , ¬ (negation) and → (implica-tion). The well-formed formulae of the logic system are defined recursively as follows: • Every variable is a well-formed formula. • If φ is a well-formed formula, then ¬ φ is a well-formed formula. • If φ and ψ are well-formed formulae, then [ φ → ψ ] is a well-formed formula.In any well-formed formula, if the left and right ends are parentheses, they can be omitted simultane-ously. The length ℓ of a well-formed formula is defined recursively as ℓ ( x ) = 1 when x is a variable ,ℓ ( ¬ φ ) = ℓ ( φ ) + 1 ,ℓ ([ φ → ψ ]) = ℓ ( φ → ψ ) = ℓ ( φ ) + ℓ ( ψ ) + 1 . Also, if we have trueness or falseness for each variable, we may extend trueness or falseness to well-formed formulae naturally and uniquely. Hence, we have a trivial bijection between a set of variables T ⊆ X which are assigned to be true, called an (truth) assignment , and a valuation v T which isa function mapping each well-formed formula to true or false. If we use 0 for false and 1 for true,then we have v T ( ¬ φ ) = 1 − v T ( φ ) and v T ( φ → ψ ) = 1 − v T ( φ )(1 − v T ( ψ )) . Practically, ifwe consider m variables x , x , · · · , x m − , then we have a natural bijection between P ( X ) and { n ∈ Z | ≤ n < m } by the binary representation. OMPUTING THE DENSITY OF TAUTOLOGIES 3
Now, for any well-formed formula φ , we may define the falsity set F φ of φ , which is the set ofassignments on variables that make φ false. In other words, F φ = { T ⊆ X | v T ( φ ) = false } . Again, for any possible falsity sets, we may consider its integer representation in { n ∈ Z | ≤ n < m } based on the binary representation, when the logic system has m variables.A well-formed formula φ is a tautology if and only if F φ = ∅ ; it is an antilogy if and only if F φ = P ( X ) . Let W be the set of well-formed formulae and from now on, we will fix the number ofvariables to be m , so | X | = m .Let W ( z ) be the generating function of all well-formed formulae, i.e., W ( z ) = X φ ∈W z ℓ ( φ ) , and I A ( z ) be the generating function of all well-formed formulae φ such that F φ = A , i.e., I A ( z ) = X φ ∈W ,F φ = A z ℓ ( φ ) . Note that I ∅ ( z ) is the generating function of the set of tautologies.From the generating rules for well-formed formulae, we get W ( z ) = mz + zW ( z ) + zW ( z ) . To find a similar formula for I A = I A ( z ) , it is enough to find the condition that F ¬ φ = A and F φ → ψ = A . From F ¬ φ = F cφ and F φ → ψ = F ψ \ F φ , we obtain the following. Proposition 2.1.
For any A ⊆ P ( X ) , if A = F x = { T ⊆ X | x T } for some x ∈ X , then wehave I A = z + zI A c + X C \ B = A zI B I C , and if there is no such x , then we have I A = zI A c + X C \ B = A zI B I C . In practice, if we consder the binary integer intepretation, then F x i corresponds to the number m − i +1 = (2 i − Q m − j = i +1 (2 j + 1) = Q m − j =0 ,j = i (2 j + 1) and F A c corresponds to the number m − − F A , for ≤ i < m .Note that this construction of a system of equations also can be applied for logic systems withdifferent logical symbols. Also, we may add false variable ⊥ , which is nothing but a variable with F ⊥ = P ( X ) . Of course, these variations of the system will give different densities of tautologies.Now, we may regard logical symbols as set operators defined on P ( P ( X )) , ¬ F φ := F ¬ φ = F cφ and F φ → F ψ := F φ → ψ = F ψ \ F φ . Since F φ is defined by falsity, if we extend ∨ and ∧ to P ( P ( X )) , we have A ∨ B = A ∩ B and A ∧ B = A ∪ B . Note that this does not match with the usual convention where A ∨ B correspondsto A ∪ B and A ∧ B to A ∩ B . We want to find the limit of the portion of tautologies in the set of allwell-formed formulae with a fixed length n , i.e. lim n →∞ [ z n ] I ∅ ( z )[ z n ] W ( z ) . T. EOM
This limit is not easy to compute since the recursion for I ∅ ( z ) is difficult to solve. To illustrate howcomplicated it can be, we first consider the case when m = 1 , i.e. just one variable. For the onevariable case, algebraic analysis on generating functions and limits of ratio between their coefficientsare already done in [1] with ¬ and → , and in [3] with ¬ and ∨ , etc. But since the equation for I ∅ isnot written anywhere, we present it without a proof. Proposition 2.2.
The generating function, I ∅ = I ∅ ( z ) , for tautologies in the logic system with ¬ , → and one variable satisfies the following equation. z I ∅ − z I ∅ + z (27 + 2 z − z + z (1 − z ) W ) I ∅ − z (50 + 12 z − z + 6 z (1 − z ) W ) I ∅ + z (55 + 28 z − z − z + z + z (2 z + 15)(1 − z ) W ) I ∅ − z (36 + 32 z − z − z + 4 z + z (8 z + 20)(1 − z ) W ) I ∅ + z (13 + 18 z − z − z + 4 z + z ( z + 12 z + 15)(1 − z ) W ) I ∅ − (2 + 4 z − z − z + z (2 z + 8 z + 6)(1 − z ) W ) I ∅ +( − z − z ) + ( z + 2 z + 1)(1 − z ) W = 0
3. T
HE CASE WITH MORE THAN ONE VARIABLES
Given a graded structure U = ` ∞ n =0 U n , the disjoint union of U n ’s, with | U n | < ∞ for all n , wemay consider two concepts of density of a subset A of U . The first one is µ ( A ) = lim n →∞ | A ∩ U n || U n | and the second one is µ ( A ) = lim n →∞ | A ∩ S nk =0 U k || S nk =0 U k | . Although they are different, if {| U n |} n ≥ is nondecreasing then µ is stronger than µ , by the fol-lowing proposition. Proposition 3.1. If a n , b n are nonnegative for all n and satisfy P ∞ n =0 b n = ∞ and lim n →∞ a n b n = r ,then lim n →∞ P nk =0 a k P nk =0 b k = r. This proves that | U | = ∞ and µ ( A ) = r imply µ ( A ) = r , so it is enough to consider only µ .Hence, we will try to compute lim n →∞ [ z n ] I A ( z )[ z n ] W ( z ) , as done in [3] and [1]. Definition 3.2.
A partition { P , · · · , P n } of P ( P ( X )) is said to be well-organized for a logicalsymbol if the logical symbol is well-defined on the partition in the following sense. For instance, apartition is well-organized for → , if for any i and j , there exists a unique k depending only on i and j such that F φ ∈ P i and F ψ ∈ P j imply F φ → ψ = F φ → F ψ ∈ P k , so P i → P j is well-defined as P k .This is reminiscent of a quotient group in group theory. Hence, we will give a reminiscent conceptfor the coset as the following. Definition 3.3.
For any disjoint subsets
A, B ⊆ P ( X ) , the subset P A ; B = { A ∪ Y | Y ⊆ B } iscalled the standard subclass of P ( P ( X )) associated to ( A, B ) , with | P A ; B | = 2 | B | . OMPUTING THE DENSITY OF TAUTOLOGIES 5
Proposition 3.4.
Given B ⊆ P ( X ) , the family of subsets of P ( X ) P ; B = { P A ; B | A ∩ B = ∅} is a partition of P ( P ( X )) such that | P ; B | = 2 m −| B | . We call this the standard subclass partitionassociated to B .Proof. Since Y ∈ P A ; B if and only if A = Y \ B , P ; B is a partition of P ( P ( X )) . Clearly, we have | P ; B | = |{ C ⊆ P ( X ) | C ∩ B = ∅}| = 2 |P ( X ) |−| B | = 2 m −| B | . (cid:3) It is easy to check that this partition is based on the equivalence relation A ∼ B C ⇔ A \ B = C \ B ⇔ A ∪ B = C ∪ B ⇔ ( A \ C ) ∪ ( C \ A ) ⊆ B . From this, we may extend the definition of thestandard subclass to non-disjoint A, B as P A ; B := P A \ B ; B = { C ⊆ P ( X ) | A \ B ⊆ C ⊆ A ∪ B } . Proposition 3.5.
Standard subclasses satisfy the following equalities, whenever A ∩ B = C ∩ B = ∅ .(a) { Y c | Y ∈ P A ; B } = P ( A ∪ B ) c ; B = P A c \ B ; B ,(b) { Y ∩ Z | Y ∈ P A ; B , Z ∈ P C ; B } = P A ∩ C ; B ,(c) { Y ∪ Z | Y ∈ P A ; B , Z ∈ P C ; B } = P A ∪ C ; B ,(d) { Y \ Z | Y ∈ P A ; B , Z ∈ P C ; B } = P A \ C ; B ,(e) P A ; B ∪ P A ∪{ y } ; B = P A ; B ∪{ y } if y A ∪ B .Proof. (a) For every Y ∈ P A ; B , we have Y c \ B = Y c ∩ B c = ( Y ∪ B ) c = ( A ∪ B ) c , and so Y c ∈ P ( A ∪ B ) c ; B . The first equality is obtained by | P A ; B | = | P ( A ∪ B ) c ; B | , or by a direct argument thatfor any Y with Y ⊆ B ⊆ A c , we have ( A ∪ B ) c ∪ Y = ( A c ∩ B c ) ∪ Y = A c ∩ ( B c ∪ Y ) = ( A ∪ ( B \ Y )) c and A ∪ ( B \ Y ) ∈ P A ; B .(b), (c), (d) can be done similarly.(e) P A ; B ∪{ y } = { A ∪ Y, A ∪ Y ∪ { y } | Y ⊆ B } = P A ; B ∪ P A ∪{ y } ; B . (cid:3) From (a) to (d) are saying that P ; B and P ( P ( X ) \ B ) are isomorphic under set operations that tak-ing the complement( c ), intersection( ∩ ), union( ∪ ), and set difference( \ ), by the natural isomorphism P ( P ( X ) \ B ) ∋ A P A ; B . Proposition 3.6.
For any B ⊆ P ( X ) , P ; B is a well-organized partition for ¬ and → .Proof. Since F ¬ φ = F cφ and F φ → ψ = F ψ \ F φ , it directly follows from the above proposition. (cid:3) By the same reason, standard subclass partitions are well-organized for ∨ and ∧ as well. Hence,we may apply a similar method to the case where ∨ or ∧ is a basic logical symbol instead of → .Also, the converse of this proposition is true. For any given finite set Y , if a partition P of P ( Y ) is awell-organized partition for \ alone, c with ∪ , or c with ∩ , then P is a well-organized partition for allof those four operators \ , c , ∩ , ∪ , and P = P ; B for some B ⊆ Y .Let I A ; B be the generating function of P A ; B , i.e., I A ; B ( z ) := X Y ∈ P A ; B I Y ( z ) . Since P ∅ ; P ( X ) = P ( P ( X )) , I ∅ ; P ( X ) = W . Proposition 3.7.
For any disjoint
A, B ⊆ P ( X ) , I A ; B satisfies the following recursion. I A ; B ( z ) = { x ∈ X | F x ∈ P A ; B } z + zI ( A ∪ B ) c ; B ( z ) + X C \ D = A C ∩ B = D ∩ B = ∅ zI C ; B ( z ) I D ; B ( z ) . This system of equations satisfies the conditions of Drmota-Lalley-Woods Theorem ( [4], p.489),so we have proper analytic functions h A ; B such that I A ; B ( z ) = h A ; B ( p − z/r ) for some r > .This shows that our generating functions are ready to apply Szeg˝o lemma as [3] and [1], whichstates that for two generating functions P ∞ n =0 σ n z n = P ∞ n =0 ˆ σ n (1 − z/z ) n/ and P ∞ n =0 τ n z n = T. EOM P ∞ n =0 ˆ τ n (1 − z/z ) n/ with unique singularity at z in the disk | z | ≤ | z | , we have lim n →∞ σ n /τ n =ˆ σ / ˆ τ if ˆ τ = 0 .Now, to prove and solve the equation, we begin with the following. Proposition 3.8.
For any disjoint
A, B ⊆ P ( X ) , we have the following:(a) I A ; B ( z ) is a linear combination of elements of the set { I ∅ ; B ( z ) } ∪ { I C ; B ′ ( z ) | C ( A, C ∩ B ′ = ∅ , | B ′ | = | B | + 1 , B ⊆ B ′ } , where the coefficient of I ∅ ; B is ± .(b) I A ; B ( z ) is a linear combination of elements of the set { I B c ; B ( z ) } ∪ { I C ; B ′ ( z ) | A ⊆ C, C ∩ B ′ = ∅ , | B ′ | = | B | + 1 , B ⊆ B ′ } , where the coefficient of I B c ; B is ± .Proof. (a) We will induct on | A | . It is trivial for A = ∅ . Suppose it is true for every A such that | A | = n and A ∩ B = ∅ . Then, if | A | = n + 1 and A ∩ B = ∅ , choose y ∈ A . Now, from Proposition 3.5 (e), I A ; B = I A \{ y } ; B ∪{ y } − I A \{ y } ; B , which proves the proposition.(b) It follows from I A ; B = I A ; B ∪{ y } − I A ∪{ y } ; B for y A ∪ B . (cid:3) Corollary 3.9.
For any fixed A and A ′ such that A ∩ B = A ′ ∩ B = ∅ , I A ′ ; B is a linear combinationof elements of the set { I A ; B } ∪ { I C ; B ′ | C ∩ B ′ = ∅ , | B ′ | = | B | + 1 , B ⊆ B ′ } , where the coefficient of I A ; B is ± .Proof. It directly follows from the above proposition, since the coefficient of I ∅ ; B is ± . (cid:3) Theorem 3.10.
For any disjoint
A, B ⊆ P ( X ) , I A ; B ( z ) is obtained by arithmetic operations andtaking quadratic roots. In particular, so is I A ( z ) .Proof. Note that I A ; ∅ = I A for every A ⊆ P ( X ) . We will induct on | B | in reverse direction, fromthe largest to the smallest. If | B | = |P ( X ) | , then I ∅ ; P ( X ) = W , so W ( z ) = − z − √ (1 − z ) − mz z ,which is a composition of arithmetic operations and taking quadratic roots.Now, assume that it holds for every B with | B | = n + 1 . Then, for the case | B | = n , by Corollary 3.9 , for any A ∩ B = ∅ , the equation for I A ; B in Proposition 3.7 can be written as an atmost quadratic equation, where coefficients consist of z , integers and I A ′ ; B ′ ’s for every A ′ , B ′ suchthat A ′ ∩ B ′ = ∅ and | B ′ | = n + 1 . Moreover, after we simplify the equation for I A ; B , we find thatthe coefficient of I A ; B is 1 modulo z . In particular, nonzero. Hence, the equation is not trivial, andso, I A ; B is a composition of arithmetic operations and taking quadratic roots. Thus, by mathematicalinduction, it is true for every A, B such that A ∩ B = ∅ , so is I A . (cid:3) For fixed disjoint
A, B ⊆ P ( X ) , we will count the number of pairs ( C, D ) such that ( C ∪ D ) ∩ B = ∅ and C \ D = A . This is equivalent to counting pairs ( C ′ , D ′ ) such that D ′ ∩ ( A ∪ B ) = ∅ and C ′ ⊆ D ′ since ( C, D ) ( C ′ , D ′ ) = ( C \ A, D ) and ( C ′ , D ′ ) ( C, D ) = ( A ∪ C ′ , D ′ ) OMPUTING THE DENSITY OF TAUTOLOGIES 7 are bijections between such pairs ( C, D ) ’s and ( C ′ , D ′ ) ’s. We may easily count the number of ( C ′ , D ′ ) pairs by choosing D ′ as a subset of P ( X ) \ ( A ∪ B ) and C ′ as a subset of D ′ , as follows. m −| B |−| A | X k =0 k X l =0 (cid:18) m − | B | − | A | k (cid:19)(cid:18) kl (cid:19) = 3 m −| B |−| A | . This is nothing but partitioning elements of P ( X ) \ ( A ∪ B ) into C ∩ D, D \ C and ( C ∪ D ) c .Hence, it is easy to write the equation for I A ; B when | A | + | B | is big. Thus, we may consider I B c ; B as special, and introduce a simpler notation I − ; B := I B c ; B . Proposition 3.11.
For any A ∩ B = ∅ , I A,B ( z ) is a linear combination of { I − ; B ′ ( z ) | A ∩ B ′ = ∅ , B ⊆ B ′ } . Proof.
This follows from
Proposition 3.8 (b). (cid:3)
The following proposition gives a way to compute exact coefficients when we write I A ; B ( z ) as alinear combination of I − ; B ′ ( z ) ’s. Proposition 3.12.
For any disjoint
A, B ⊆ P ( X ) , I A ; B ( z ) = ( − | A | X B ⊆ B ′ ⊆ A c ( − | B ′ | I − ; B ′ ( z ) Proof.
First, C ∈ P − ; B ′ if and only if B ′ c ⊆ C which is equivalent to C c ⊆ B ′ . Hence, P − ; B ′ ∩ P − ; B ′′ = P − ; B ′ ∩ B ′′ is satisfied for any B ′ and B ′′ . Now, we have P A ; B = P − ; A c \ [ y ∈ ( A ∪ B ) c P − ;( A ∪{ y } ) c . Thus, by the inclusion-exclusion principle, we get I A ; B ( z ) = m −| A |−| B | X i =0 X Y ⊆ ( A ∪ B ) c , | Y | = i ( − i I − ;( A ∪ Y ) c ( z )= X Y ⊆ ( A ∪ B ) c ( − | Y | I − ;( A ∪ Y ) c ( z )= X B ⊆ B ′ ⊆ A c ( − | B ′ c \ A | I − ; B ′ ( z )= ( − | A | X B ⊆ B ′ ⊆ A c ( − | B ′ | I − ; B ′ ( z ) . (cid:3) We consider an equation for I − ; B , obtained from Proposition 3.7 , I − ; B = { x | F x ∈ P − ; B } z + zI ∅ ; B + zI ∅ ; B I − ; B . Since we have I ∅ ; B = X B ⊆ B ′ ( − | B ′ | I − ; B ′ = ( − | B | I − ; B + X B ( B ′ ( − | B ′ | I − ; B ′ , T. EOM adopting short notations m B = { x | F x ∈ P − ; B } , σ B = ( − | B | and I ↑ ; B = P B ( B ′ ( − | B ′ | I − ; B ′ ,we can rewrite the equation as I − ; B = z ( m B + I ↑ ; B ) + z ( σ B + I ↑ ; B ) I − ; B + zσ B I − ; B . This gives I − ; B = 1 − ( σ B + I ↑ ; B ) z − p (1 − ( σ B + I ↑ ; B ) z ) − σ B ( m B + I ↑ ; B ) z (2 σ B ) z , since I − ; B (0) = 0 .Naturally, the coefficients of I − ; B as a formal power series are always nonnegative and smallerthan the corresponding coefficients of W . So the coefficients of I − ; B cannot have a larger growthrate than those of W . Since √ m +1 is the closest singularity of W by Theorem IV.7 in [4], I − ; B cannot have a singularity closer to than √ m +1 . If I − ; B has singularity at s = √ m +1 , then wemay apply Szeg˝o lemma as [3] and [1], by computing lim z → s − I − ; B ( z ) − I − ; B ( s ) p − z/s or, equivalently, lim z → s − − s I ′− ; B ( z ) p − z/s to get the ratio lim n →∞ [ z n ] I − ; B ( z )[ z n ] W ( z ) . If I − ; B has no singularity at s , as long as singularities of I − ; B are not closer to 0 than s , the limitof I ′− ; B ( z ) p − z/s exists and it will give the value 0, where for such a case the growth rate ofcoefficients of I − ; B has strictly slower than that of W , so the ratio will be 0. Hence, this computationalso matches for such cases. Now, let α B = I − ; B ( s ) ,α ↑ B = X B ( B ′ ( − | B ′ | α B ′ and β B = lim z → s − s I ′− ; B ( z ) p − z/s ,β ↑ B = X B ( B ′ ( − | B ′ | β B ′ . Moreover, to simplify, denote f B = (1 − ( σ B + I ↑ ; B ) z ) − σ B z ( m B + I ↑ ; B ) . Then, we have I ′− ; B ( z ) = − ( σ B + I ↑ ; B ( z )) − zI ′↑ ; B ( z ) − f ′ B ( z )2 √ f B ( z ) (2 σ B ) z − I − ; B ( z ) z . Note that the second part becomes when we consider the limit after multiplying p − z/s . Bysimple computation, f ′ B ( z ) = − σ B + I ↑ ; B + I ′↑ ; B z )(1 − ( σ B + I ↑ ; B ) z ) − σ B z ( m B + I ′↑ ; B ) − σ B z ( m B + I ↑ ; B ) . From above, if we define d B = f B ( s ) /s = (2 √ m + 1) f B ( s ) = (2 √ m + 1 − σ B − α ↑ B ) − σ B ( m B + α ↑ B ) , OMPUTING THE DENSITY OF TAUTOLOGIES 9 we obtain α B = 2 √ m + 1 − σ B − α ↑ B − √ d B σ B and if d B = 0 , then β B = β ↑ B × − √ m +1+ σ B − α ↑ B √ d B σ B . Moreover, since β B cannot go to the infinite, so d B = 0 only if β ↑ B = 0 . In particular, when B = P ( X ) , we obtain by simple computation α P ( X ) = √ m,β P ( X ) = q m + √ m. After computing every values of β B , we can compute the density of well-formed formulae φ suchthat F φ = A . Corollary 3.13.
For any A ⊆ P ( X ) , we have lim n →∞ [ z n ] I A ( z )[ z n ] W ( z ) = ( − | A | P B ⊆ A c ( − | B | β B p m + √ m . In particular, lim n →∞ [ z n ] I ∅ ( z )[ z n ] W ( z ) = P B ⊆P ( X ) ( − | B | β B p m + √ m . Proof.
It directly follows from
Proposition 3.12 and Szeg˝o’s lemma. (cid:3)
Lastly, for different propositional logic systems, there may be some cases in which it is better toconsider I ∅ ; B rather than I − ; B . In such cases, we have a variant of Proposition 3.12 , which is I A ; B ( z ) = ( − | A | + | B | X B ⊆ B ′ ⊆ A ∪ B ( − | B ′ | I ∅ ; B ′ ( z ) . This is from P ∅ ; B ′ ∩ P ∅ ; B ′′ = P ∅ ; B ′ ∩ B ′′ and P A ; B = P ∅ ; A ∪ B \ [ y ∈ A P ∅ ;( A \{ y } ) ∪ B where A, B are disjoint. 4. N
UMERICAL T HEORY
In this section, we will focus on systems of at-most-quadratic equations of generating functionswhich is more general. Consider a power series Z ( x ) = P ∞ n =0 Z n x n with limit ratio lim n →∞ Z n +1 Z n = 1 r > , satisfying at-most-quadratic relation of the form Z ( x ) = f ( x ) + g ( x ) Z ( x ) + h ( x )( Z ( x )) , where g, h are polynomials, and f is a power series with a ratio condition lim n →∞ [ x n ] f ( x )[ x n ] Z ( x ) = lim n →∞ f n Z n = γ. With this generating function Z as a base, we will consider a system of quadratic equations of gener-ating functions related to Z . We will consider power series A i ( x ) = P ∞ n =0 A i n x n for i = 1 , · · · , N .First, these generating functions are comparable with Z ( x ) by lim n →∞ [ x n ] A i ( x )[ x n ] Z ( x ) = lim n →∞ A i n Z n = β i , the limit ratios of coefficients. Second, these generating functions satisfy the following system ofat-most-quadratic relations A i ( x ) = f i ( x ) + N X j =1 g ij ( x ) A j ( x ) + N X j,k =1 h ( x ) h ijk ( x ) A j ( x ) A k ( x ) where g ij , h ijk ’s are some polynomials, and f i ( x ) = P ∞ n =0 f i n x n are power series. Here, h is thepolynomial in the equation of Z ( x ) . Lastly, f i ’s have ratio conditions lim n →∞ [ x n ] f i ( x )[ x n ] Z ( x ) = lim n →∞ f i n Z n = γ i , for i = 1 , · · · , N . Also, for any formal power series F ( x ) = P ∞ n =0 F n x n , define F ≤ s ( x ) = s X n =0 F n x n . Definition 4.1.
With the above conditions on r, γ, γ i , g, h, g ij , and h ijk , the s -cut solution ( β ( s )1 , · · · , β ( s ) N ) for the system ( Z, A , · · · , A N ) is defined as an N -tuple of numbers satisfying β ( s ) i = γ i + N X j =1 g ij ( r ) β ( s ) j + N X j,k =1 h ( r ) h ijk ( r ) (cid:16) A ≤ sj ( r ) β ( s ) k + A ≤ sk ( r ) β ( s ) j (cid:17) + N X j,k =1 ζ s h ijk ( r ) β ( s ) j β ( s ) k , where ζ s = 1 − γ − g ( r ) − h ( r ) Z ≤ s ( r ) , and the s -cut operator is the map C s : ( x , · · · , x N ) ( c , · · · , c N ) where for i = 1 , , . . . , N , c i = γ i + N X j =1 g ij ( r ) x j + N X j,k =1 h ( r ) h ijk ( r ) (cid:16) A ≤ sj ( r ) x k + A ≤ sk ( r ) x j (cid:17) + N X j,k =1 ζ s h ijk ( r ) x j x k . In other words, an s -cut solution is a fixed point of the s -cut operator. Also, note that we may add Z as A to the system of equations for A , · · · , A N freely. So, we may consider Z as one of A i ’s ifwe need. Moreover, we may deal with cubic terms of form h ( x ) A i ( x ) A j ( x ) A k ( x ) by introducing B ij ( x ) = h ( x ) A j ( x ) A j ( x ) , and it can be generalized to any degree.The following is a motivation to consider the s -cut concept. Theorem 4.2.
Suppose formal power series
Z, A , · · · , A N satisfy Z ( x ) = f ( x ) + g ( x ) Z ( x ) + h ( x ) Z ( x ) ,A i ( x ) = f i ( x ) + N X j =1 g ij ( x ) A j ( x ) + N X j,k =1 h ( x ) h ijk ( x ) A j ( x ) A k ( x ) where g, h, g ij , h ijk are polynomials for i, j, k = 1 , · · · , N . Also, assume that there exist r , β , . . . , β N , γ , γ , · · · , γ N such that lim n →∞ Z n +1 Z n = 1 r > , lim n →∞ A in Z n = β i , lim n →∞ f n Z n = γ, lim n →∞ f in Z n = γ i , and let β be N -tuple ( β , · · · , β N ) . Lastly, assume that Z, h have only non-negative coefficients and Z ( r ) is bounded. Then β = lim s →∞ C s ( β ) . OMPUTING THE DENSITY OF TAUTOLOGIES 11
Proof.
Since f n Z n + deg g X u =0 g u Z n − u Z n + deg h X u =0 h u n − u X v =0 Z v Z n − u − v Z n , lim n →∞ Z n − m Z n = r m , and lim n →∞ f n Z n = γ , we have lim n →∞ deg h X u =0 h u n − u − s − X v = s +1 Z v Z n − u − v Z n = 1 − γ − g ( r ) − h ( r ) Z ≤ s ( r ) = ζ s . Then, using the notation | g | ( x ) to denote P n | g n | x n , we have | ζ s | ≤ | γ | + | g | ( r ) + 2 h ( r ) Z ( r ) , which implies that ζ s is bounded. If we let C s ( β ) = ( c ( s )1 , . . . , c ( s ) N ) , then by a similar argument, ifwe apply lim n →∞ to the formula of A in after dividing by Z n , we get β i − c ( s ) i = lim n →∞ N X j,k =1 deg h ijk X t =0 ( h ijk ) t deg h X u =0 h u n − t − u − s − X v = s +1 ∆ njktuv where ∆ njktuv = A jv A k,n − t − u − v Z n − β j β k Z v Z n − t − u − v Z n . Then, for any ǫ > , choose s so that s ≤ min { u, v } implies | β j β k − A jv A ku Z u Z v | < ǫ for any j, k ; andchoose n so that n > s + deg h ijk + deg h for every i, j, k . Then, for any v such that s < v Z, f, g, h, f i satisfy the following three conditions: (1) Z, f, f i are formal power series, (2) g, h are polynomials and (3) r, γ, γ i are well-defined. Proposition 4.3. Suppose that all the coefficients of Z, g, h are nonnegative, h is nonzero or γ isnonnegative, and f ( r ) , Z ( r ) converge. Then we have ≤ p (1 − g ( r )) − f ( r ) h ( r ) − γ ≤ ζ s ≤ − γ − g ( r ) and lim s →∞ ζ s = p (1 − g ( r )) − f ( r ) h ( r ) − γ = 1 − γ − g ( r ) − h ( r ) Z ( r ) . In this case, we set ζ ∞ = lim s →∞ ζ s and call it the impurity of the equation Z = f + gZ + hZ .Moreover, if f has no singularity in { z ∈ C | | z | < r + ǫ } for some ǫ > , then both γ and theimpurity, ζ ∞ , are zero. Proof. First, we have ζ s = 1 − γ − g ( r ) − h ( r ) Z ≤ s ( r ) ≤ − γ − g ( r ) . Moreover, ζ s = lim n →∞ deg h X u =0 h u n − u − s − X v = s +1 Z v Z n − u − v Z n gives ζ s ≥ always. If h = 0 , then we have ζ s = 1 − γ − g ( r ) = p (1 − g ( r )) − f ( r ) h ( r ) − γ = 0 where − g ( r ) ≥ is from γ ≥ . Now, if h is nonzero, then we have Z ( x ) = 1 − g ( x ) − p (1 − g ( x )) − f ( x ) h ( x )2 h ( x ) . Since f ( r ) , Z ( r ) converge, it gives Z ( r ) = 1 − g ( r ) − p (1 − g ( r )) − f ( r ) h ( r )2 h ( r ) . Hence, ζ s = 1 − g ( r ) − h ( r ) Z ≤ s ( r ) − γ ≥ − g ( r ) − h ( r ) Z ( r ) − γ and ≤ lim s →∞ ζ s = 1 − g ( r ) − h ( r ) Z ( r ) − γ = p (1 − g ( r )) − f ( r ) h ( r ) − γ. Now, consider the case that f has no singularity in { z ∈ C | | z | < r + ǫ } . By Theorem IV.7 in [4], r is the closest singularity to zero of Z . If h is zero, then Z ( x ) = f ( x )1 − g ( x ) . Since f has no singularity in { z ∈ C | | z | < r + ǫ } , it means g ( r ) = 1 . Hence, ζ s = 1 − γ − g ( r ) = − γ ≤ , so the impurity and γ are zero. For the case that h is nonzero, f has no singularity in { z ∈ C | | z | < r + ǫ } and g, h are polynomials, so (1 − g ( r )) − f ( r ) h ( r ) = 0 . Then − g ( r ) − h ( r ) Z ( r ) = 0 and lim s →∞ ζ s = − γ . Hence, it is enough to prove that γ = 0 .This can be induced from again Theorem IV.7 in [4], which gives lim sup( f n ) /n ≤ r + ǫ . (cid:3) Combining Theorem 4.2 and Proposition 4.3 , we directly obtain the following. Theorem 4.4. Suppose that our system of equations on formal power series Z ( x ) , A ( x ) , · · · , A N ( x ) satisfies that the coefficients of Z, g, h are nonnegative, h is nonzero, f ( r ) , Z ( r ) , A ( r ) , · · · , A N ( r ) converge and the impurity is zero. Then β i = γ i + N X j =1 g ij ( r ) β j + N X j,k =1 h ( r ) h ijk ( r )( A j ( r ) β k + A k ( r ) β j ) . whenever β i = lim n →∞ [ x n ] A i ( x )[ x n ] Z ( x ) exists for every i . Note that this result can be understood as an application of Szeg˝o’s lemma, just differentiate andmultiply p − x/r and take limit. Moreover, this is linear on β j ’s when A j ( r ) ’s are given, andlinear on A j ( r ) ’s when β j ’s are given. Note that if A j ( r ) are given and γ i ’s are zero, then it is ahomogeneous linear system on β j ’s, in which case we need more conditions to solve completely.This theorem gives an alternative practical method to compute lim n →∞ [ z n ] I A ( z )[ z n ] W ( z ) OMPUTING THE DENSITY OF TAUTOLOGIES 13 which uses α B = I \ ; B ( s ) , since we have natural additional condition X A ⊆P ( X ) lim n →∞ [ z n ] I A ( z )[ z n ] W ( z ) = 1 . From the equation Z = f + gZ + hZ , g and h show recursive structures of the object countedby Z , and f counts basic elements. Hence it is natural that basic elements do not form so large aportion among objects to focus on its recursive structures, which means γ , the limit portion of thebasic elements among the whole objects, is natural to be 0.Since for any polynomial δ , δZ is also a formal power series which satisfies the ratio condition,we may define new f as f + δZ to modify the value of γ . Also, from equation Z = f + gZ + hZ ,we can make a different equation by multiplying the constant c and rewrite as Z = ( cf + (1 − c ) Z ) + cgZ + chZ . We will give a name to these conversions, and show that even when we convert γ bythese conversions, the impurity is a kind of an invariant, so the zeroness of the impurity is preserved,and hence we can change γ safely. Definition 4.5. (a) If γ = 1 , γ − b γ conversion of equation Z = f + gZ + hZ is defined as Z = b f + b gZ + b hZ where b f = 1 − b γ − γ f + b γ − γ − γ Z, b g = 1 − b γ − γ g, b h = 1 − b γ − γ h. (b) If δ ( x ) is a polynomial, δ conversion is defined as Z = e f + e gZ + e hZ where e f = f + δZ, e g = g − δ, e h = h. Proposition 4.6. (a) For γ − b γ conversion, we have lim n →∞ b f n Z n = b γ. (b) For γ − b γ conversion, ζ s − γ is invariant and, moreover, if Z ( r ) converges, then ζ ∞ − γ is invari-ant.(c) For δ conversion, we have e γ = γ + δ ( r ) . (d) For δ conversion, ζ s is invariant and if Z ( r ) converges, then ζ ∞ is invariant.Proof. (a),(c) are simple computation.(b) We have b ζ s = 1 − b γ − b g ( r ) − b h ( r ) Z ≤ s ( r )= 1 − b γ − γ (cid:0) − γ − g ( r ) − h ( r ) Z ≤ s ( r ) (cid:1) = 1 − b γ − γ ζ s . (d) We have e ζ s = 1 − e γ − e g ( r ) − e h ( r ) Z ≤ s ( r )= 1 − γ − δ ( r ) − g ( r ) + δ ( r ) − h ( r ) Z ≤ s ( r )= ζ s . (cid:3) Now, we are going to compute the numerical estimations of the ratio β i ’s by computing s -cutsolutions, which means we expect that lim s →∞ β ( s ) i = lim n →∞ A in Z n = β i is satisfied. Since the equation for s -cut solution is quadratic, existence and uniqueness are not guar-anteed. Hence, we will provide some condition for existence, uniqueness and above convergence of s -cut solution. Definition 4.7. Suppose that a formal power series Z satisfies Z ( x ) = f ( x ) + g ( x ) Z ( x ) + h ( x ) Z ( x ) with a formal power series f and polynomials g, h . Then, A , · · · , A N satisfying A i ( x ) = f i ( x ) + N X j =1 g ij ( x ) A j ( x ) + N X j,k =1 h ( x ) h ijk ( x ) A j ( x ) A k ( x ) are a natural partition of Z if Z ( x ) = N X i =1 A i ( x ) ,f ( x ) = N X i =1 f i ( x ) ,g ( x ) = N X i =1 g ij ( x ) , N X i =1 ( h ijk ( x ) + h ikj ( x )) . Also, a natural partition system ( Z, A , · · · , A N ) is nonnegative if all the coefficients of Z , A i , g , h , g ij , h ijk , and γ, γ i are nonnegative. From nonnegativity, we have [ x n ] Z ( x ) ≥ [ x n ] f ( x ) and [ x n ] A i ( x ) ≥ [ x n ] f i ( x ) which imply γ, γ i ≤ , and since sum of h ijk ( x ) ’s is a constant polynomial,every h ijk ( x ) is also a constant polynomial. Proposition 4.8. Let ( c , · · · , c N ) be a fixed point of the s -cut operator C s for a natural partitionsystem ( Z, A , · · · , A n ) with nonzero ζ s . Then ( c , · · · , c N ) is on the hyperplane x + · · · + x N = γζ s or x + · · · + x N = 1 in R N . OMPUTING THE DENSITY OF TAUTOLOGIES 15 Proof. N X i =1 c i = N X i =1 γ i + N X i =1 N X j =1 g ij ( r ) c j + N X i =1 N X j,k =1 h ( r ) h ijk ( r )( A ≤ sj ( r ) c k + A ≤ sk ( r ) c j )+ N X i =1 N X j,k =1 ζ s h ijk ( r ) c j c k = γ + N X j =1 g ( r ) c j + 12 N X j,k =1 N X i =1 h ( r )( h ijk ( r ) + h ikj ( r ))( A ≤ sj ( r ) c k + A ≤ sk ( r ) c j )+ 12 N X j,k =1 N X i =1 ζ s ( h ijk ( r ) + h ikj ( r )) c j c k = γ + g ( r ) N X j =1 c j + N X j,k =1 h ( r )( A ≤ sj ( r ) c k + A ≤ sk ( r ) c j ) + N X j,k =1 ζ s c j c k = γ + g ( r ) N X j =1 c j + 2 h ( r ) Z ≤ s ( r ) N X j =1 c j + ζ s N X j =1 c j . Hence, ( ζ s + γ ) N X j =1 c j = γ + ζ s N X j =1 c j , which proves the proposition. (cid:3) Proposition 4.9. The s -cut operator C s of a nonnegative natural partition system ( Z, A , · · · , A N ) has a fixed point in H := { ( x , · · · , x N ) ∈ R N : 0 ≤ x i ≤ , N X i =1 x i = 1 } . Proof. Let C s ( x , · · · , x N ) = ( c , · · · , c N ) . If ( x , · · · , x N ) ∈ H , then as in the proof of Proposi-tion 4.8 , we have N X i =1 c i = γ + g ( r ) · h ( r ) Z ≤ s ( r ) · ζ s · = 1 . In the proof of Proposition 4.3 we obtain ζ s ≥ from the fact that coefficients of Z and h are nonneg-ative. Hence, we have c i ≥ for every i . Then, P Ni =1 c i = 1 implies c i ≤ , so ( c , · · · , c N ) ∈ H .Now, H is a convex compact set in R N , so by Brouwer fixed point theorem, C s has a fixed pointin H . (cid:3) By simple computation, we have ∂c i ∂x j = g ij ( r ) + N X k =1 h ( r )( h ijk ( r ) + h ikj ( r )) A ≤ sk ( r ) + ζ s N X k =1 ( h ijk ( r ) + h ikj ( r )) x k . From this, we have the following result. Proposition 4.10. For the Jacobian J of the s -cut operator C s of a nonnegative natural partitionsystem ( Z, A , · · · , A n ) , k J ( x , · · · , x N ) k = 1 − ζ s − γ + 2 ζ s N X i =1 x i ! on [0 , ∞ ) N , where k · k denotes the -norm of a matrix. In particular, k J k = 1 − γ + ζ s on H .Proof. Note that k B k = max ≤ j ≤ n P mi =1 | b ij | for any m × n matrix B . Since the system isnonnegative, ∂c i ∂x j ≥ on [0 , ∞ ) N . Then, N X i =1 ∂c i ∂x j = N X i =1 g ij ( r ) + N X k =1 h ( r ) N X i =1 ( h ijk ( r ) + h ikj ( r )) A ≤ sk ( r )+ ζ s N X k =1 N X i =1 ( h ijk ( r ) + h ikj ( r )) x k = g ( r ) + N X k =1 h ( r )2 A ≤ sk ( r ) + ζ s N X k =1 x k = g ( r ) + 2 h ( r ) Z ≤ s ( r ) + 2 ζ s N X k =1 x k =1 − ζ s − γ + 2 ζ s N X k =1 x k , which proves the result. (cid:3) This result is also true when nonnegative condition is weakened: For instance γ and γ i may not benonnegative. Moreover, we have the following result for general p -norms. Proposition 4.11. For the Jacobian J of the s -cut operator C s of a natural partition system, k J ( x , · · · , x N ) k p ≥ | − γ + ζ s | on H . Note that | − γ + ζ s | = 1 − γ + ζ s when the given system is nonnegative, since we have γ ≤ and ζ s ≥ .Proof. Let J T denote the transpose of the Jacobian. We have J T ... = (1 − γ + ζ s ) ... on H , from P Ni =1 ∂c i ∂x j = 1 − ζ s − γ + 2 ζ s P Nk =1 x k = 1 − γ + ζ s . Hence, − γ + ζ s is an eigenvalueof J T , so is an eigenvalue of J . Thus, we get k J k p ≥ | − γ + ζ s | . (cid:3) Since the norm of the Jacobian of the s -cut operator C s can be larger than 1, especially when γ = 0 , this fact may induce some convergence problem when we try to find an s -cut solution byapplying fixed point iteration method on C s . Hence, we may consider modification. Definition 4.12. The σ -shifted s -cut operator f C σs is defined as f C σs ( x ) = C s ( x ) − σ N X i =1 x i − ! · (1 , , · · · , . OMPUTING THE DENSITY OF TAUTOLOGIES 17 Since f C σs ( x ) = C s ( x ) for all x ∈ H , fixed points of C s on H are fixed points of f C σs . Moreover, e J = J − σ σ · · · σσ . . . · · · ...... ... . . . ... σ · · · · · · σ = J − σ , where e J is the Jacobian for f C σs .From the Banach contraction principle, we deduce the following. Proposition 4.13. If the Jacobian e J of f C σs satisfies k e J k < for a matrix norm k · k on H , then f C σs is a contraction on H , and C s has the unique fixed point on H . Note that since H is compact, k e J k < is enough to apply the Banach contraction principle ratherthan the condition that there exists K < such that k e J k ≤ K . From k A − B k ≥ |k A k − k B k| , itwould be best to choose σ satisfying k J k = k σ k , and one of such choice is σ = − γ + ζ s N , which isfrom the 1-norm. Hence, we will call the s -cut operator shifted by this value as the standard shifted s -cut operator . Corollary 4.14. The Jacobian ˜ J of the standard shifted s -cut opertor f C s of a nonnegative naturalpartition system satisfies k ˜ J k < on H if − γ + 2 ζ s > and ∂c i ∂x j < − γ + ζ s N + max (cid:26) − γ + ζ s N (1 − γ + 2 ζ s ) , N − (cid:27) . Note that − γ + 2 ζ s ≤ implies − γ + ζ s ≤ , which means k J k < is already satisfiedwithout shifting.Proof. Since k e J k = max nP Ni =1 (cid:12)(cid:12)(cid:12) ∂c i ∂x j − − γ + ζ s N (cid:12)(cid:12)(cid:12) | j = 1 , · · · , N o and we have P Ni =1 ∂c i ∂x j = 1 − γ + ζ s already, it is enought to prove that a i s arranged as max { − γ + ζ s N (1 − γ +2 ζ s ) , N − } + − γ + ζ s N >a ≥ a ≥ · · · ≥ a m ≥ − γ + ζ s N ≥ a m +1 ≥ · · · ≥ a N ≥ satisfying P Ni =1 a i = 1 − γ + ζ s satisfies P Ni =1 | a i − − γ + ζ s N | < . Since − γ + ζ s N is the mean of a i s, we may assume m < N . Easily, N X i =1 (cid:12)(cid:12)(cid:12)(cid:12) a i − − γ + ζ s N (cid:12)(cid:12)(cid:12)(cid:12) = m X i =1 ( a i − − γ + ζ s N ) + N X i = m +1 ( 1 − γ + ζ s N − a i )= 1 − γ + ζ s N ( N − m ) + m X i =1 a i − N X i = m +1 a i = 1 − γ + ζ s N ( N − m ) + 2 m X i =1 a i − (1 − γ + ζ s )= 2 m X i =1 a i − mN (1 − γ + ζ s ) . If N − ≥ − γ + ζ s N (1 − γ +2 ζ s ) , we have a i < N − + − γ + ζ s N , so m X i =1 a i − mN (1 − γ + ζ s ) < m ( 12( N − 1) + 1 − γ + ζ s N ) − mN (1 − γ + ζ s ) = mN − ≤ . For the other case, if m > N (cid:16) − − γ + ζ s ) (cid:17) , we have m X i =1 a i − mN (1 − γ + ζ s ) ≤ N X i =1 a i − mN (1 − γ + ζ s ) ≤ − γ + ζ s ) − mN (1 − γ + ζ s )= 2(1 − γ + ζ s )(1 − mN ) < , and if m ≤ N (1 − − γ + ζ s ) ) , we have a i < − γ + ζ s N (1 − γ +2 ζ s ) + − γ + ζ s N , so m X i =1 a i − mN (1 − γ + ζ s ) < m (cid:18) − γ + ζ s N + 1 − γ + ζ s N (1 − γ + 2 ζ s ) (cid:19) − mN (1 − γ + ζ s )= mN − γ + ζ s )1 − γ + 2 ζ s ≤ (cid:18) − − γ + ζ s ) (cid:19) − γ + ζ s )1 − γ + 2 ζ s = 1 . (cid:3) This corollary gives a condition to have the unique s -cut solution by computing the 1-norm of theshifted s -cut operator. Finding the best choice to shift based on the matrix 1-norm of the Jacobianis equivalent to find σ from given nonnegative sequences a (1) , · · · , a ( N ) satisfying P i a (1) i = · · · = P i a ( N ) i such that minimizes the max nP i (cid:12)(cid:12)(cid:12) a ( j ) i − σ (cid:12)(cid:12)(cid:12) | j = 1 , · · · , N o . For each j , it is well-knownthat the median minimizes P i (cid:12)(cid:12)(cid:12) a ( j ) i − σ (cid:12)(cid:12)(cid:12) , compare with that mean minimizes P i ( a ( j ) i − σ ) , wherethe standard shift operator is defined as to choose σ as the mean, which is easier to compute than themedian. Hence, it may possible to refine the condition to have unique s -cut solution by consideringthe median rather than the mean. In such case, we may have to use some variant of the iterationmethod, which uses different iteration function for each iteration.Lastly, we will prove the following. Theorem 4.15. Suppose that a nonnegative natural partition system ( Z, { A i } ) satisfies the following: • lim n →∞ Z n +1 Z n = r > , • lim n →∞ A in Z n = β i , • there exist a common contraction factor K < and a sequence of proper shifting factor { σ s } of the s -cut operator C s satisfying (cid:12)(cid:12)(cid:12) g C σ s s ( x ) − g C σ s s ( y ) (cid:12)(cid:12)(cid:12) ≤ K | x − y | for any x, y ∈ H except for finitely many s .Then, there exists a sequence of s -cut solutions on H , β ( s ) = ( β ( s )1 , · · · , β ( s ) N ) , converging to β =( β , · · · , β N ) as s → ∞ .Proof. From Theorem 4.2 , lim s →∞ C s ( β ) = β is satisfied. We may assume s is large enough tohave a common contraction constant K . Then, we have | β − β ( s ) | ≤ | β − C s ( β ) | + | C s ( β ) − β ( s ) | = | β − C s ( β ) | + | C s ( β ) − C s ( β ( s ) ) | . Since C s = f C σs on H , C s is also a contraction on H with same contraction constant. Hence, | β − β ( s ) | ≤ | β − C s ( β ) | + | C s ( β ) − C s ( β ( s ) ) | ≤ | β − C s ( β ) | + K | β − β ( s ) | . OMPUTING THE DENSITY OF TAUTOLOGIES 19 Thus, | β − β ( s ) | ≤ − K | β − C s ( β ) | → as s → ∞ . (cid:3) Note that except for finitely many s ’s, each β ( s ) is uniquely determined.5. E STIMATED RESULTS FOR THE MULTIVARIABLE CASES We now go back to the original problem, computing the density of tautologies. Even if our logicsystem has more than variable, we have a method to get an exact formula of the density of tautolo-gies and of antilogies. But the formulae will include nearly m nested quadratic roots, which makesvisualization difficult. Hence, in the following table we only provide numerical results for densitiesof tautologies and antilogies, when the number of variables is two, three and four computed by Sage.density tautologies antilogies m = 2 m = 3 m = 4 F φ , the largest class is the class of tautologies and the second largest is the class of antilogies whenthe number of variables is two, three or four, which is false when there is only one variable. But it iseasy to prove that the density of tautologies is Ω( m ) and the density of antilogies is Ω( m √ m ) , wherethe number of classes is m . The next paragraph proves it.Note that if ψ is a tautology, then ¬¬ ψ is a tautology and that for any well-formed formula φ , p → [ φ → p ] is a tautology for any variable p ∈ X m . Since these two types have no commonelements, we have [ z n ] I ∅ ( z ) ≥ [ z n − ] I ∅ ( z ) + m · ([ z n − ] W ( z )) . Now we deduce [ z n ] W ( z ) ≃ r m + √ m πn (2 √ m + 1) n from the expression W ( z ) = 1 − z − p (1 − (2 √ m + 1) z )(1 + (2 √ m − z )2 z = √ m − q m + √ m q − (2 √ m + 1) z + O (1 − (2 √ m + 1) z ) . Hence, lim n →∞ [ z n − ] W ( z )[ z n ] W ( z ) = √ m +1) and lim n →∞ [ z n − ] W ( z )[ z n ] W ( z ) = √ m +1) . Thus, lim n →∞ [ z n ] I ∅ ( z )[ z n ] W ( z ) ≥ (cid:18) √ m + 1) lim n →∞ [ z n − ] I ∅ ( z )[ z n − ] W ( z ) (cid:19) + m (2 √ m + 1) and so, lim n →∞ [ z n ] I ∅ ( z )[ z n ] W ( z ) ≥ √ m √ m + 1)(2 √ m + 1) , where we have √ m √ m +1)(2 √ m +1) = Θ( m ) . Thus, the density of tautologies is Ω( m ) . For antilogies,we can get Ω( m √ m ) from [ z n ] I P ( X ) ( z ) ≥ [ z n − ] I ∅ ( z ) , since every ¬ φ is an antilogy for any tautology φ .Now, we will show that using s -cut solution is efficient to compute the approximation of the limitvalue. The following table compares exact values, the ratios at s , which are [ z s ] I A [ z s ] W , and s -cut solutionsfor the density of tautologies and of antilogies when the number of variables and s change. s = 10 s = 50 s = 200 value ratio cut-sol ratio cut-sol ratio cut-sol m = 1 taut 0.4232 0.3102 0.4243 0.4142 0.4233 0.4210 0.4233anti 0.1632 0.1868 0.1642 0.1612 0.1634 0.1628 0.1633 m = 2 taut 0.3321 0.2374 0.3345 0.3206 0.3323 0.3293 0.3322anti 0.0971 0.0996 0.0982 0.0947 0.0972 0.0965 0.0971 m = 3 taut 0.2700 0.1913 0.2732 0.2581 0.2703 0.2670 0.2701anti 0.0663 0.0637 0.0673 0.0641 0.0663 0.0657 0.0663Values for m = 1 is from Theorem 22 of [1]. The s -cut solution is computed by Sage using fixedpoint iteration, starting with (1 , , · · · , . This shows that s -cut solution converges faster than justcomputing ratio, so we can compute more accurate values with less exact numbers of well-formed for-mulae in each class. Since for quadratic generating functional equations, we need every a , · · · , a n − values to compute a n and computing a n itself is also time-consuming, so even though computing s -cut solution takes more time than just dividing, computing s -cut solution gives advantages in memory,also possible in time, for fixed accuracy.The above deduction of the density of tautologies, Ω( m ) , is similar to results in [2] and [5], whichgive that in the logic system with → and negative literals the density of tautologies is asymptoticallysame as the density of simple tautologies, i.e., m + O ( m ) . A simple tautology, which is defiendin [2], is a tautology of the form φ → [ φ → [ · · · → [ φ n → p ] · · · ]] , which can be simplified with the canonical form of an expression, defined in [2], as φ , · · · , φ n p where each φ i is a well-formed formula and p is a variable, with condition φ i = p for some i , or forsome distinct pair i and j , φ i is a variable and φ j = ¯ φ i . Here, ¯ x means negative literal of x . Theformer is called a simple tautology of the first kind, and the latter is called a simple tautology of thesecond kind. But there are some differences between our case and the given cases. Firstly, for thecase of implication with negative literals, there are no antilogies. Secondly, we have to negate, ratherthan using negative literals, which increases the length of the formula. It introduces the factor √ m inasymptotic ratio, which changes the order.With these facts in mind, we will try to compute the asymptotic density of tautologies as thenumber of variables goes to the infinity. In the following, our m -element variable set X is consideredas the set of variables { x , x , · · · , x m − } , so it will generate a chain sturcture as the number ofvariable changes. Definition 5.1. Let X = { x , x , · · · } be a countably infinite set of variables, and W be the setof well-formed formulae of X . For any σ ∈ S { , , ,... } =: S ∞ , the set of all permutations of { , , . . . } with a finite support, we have a natural action on W defined as σx i = x σ ( i ) ,σ ¬ φ = ¬ σφ,σ [ φ → ψ ] = σφ → σψ. A formula φ ∈ W is a type formula if for every occrurence of x i , there must exist occurrences of x , · · · , x i − before it. The type of a well-formed formula ψ is the type formula φ such that there OMPUTING THE DENSITY OF TAUTOLOGIES 21 exists σ ∈ S ∞ satisfying ψ = σφ . It is easy to prove that the type of a well-formed formula existsuniquely. For any well-formed formula ψ , [ ψ ] is the set of well-formed formulae with the same typeas ψ , and [ ψ ] m be the elements in [ ψ ] consisting of x , · · · , x m − . Note that [ ψ ] is just the S ∞ -orbitin W , and for any ψ consisting of x , · · · , x m − , the set [ ψ ] m is nothing but the S m -orbit.For any formula φ ∈ W , k φ k is the number of distinct variables in φ . In other words, this is theminimum m such that the type of φ consists of x , · · · , x m − . Lastly, | φ | is defined as | φ | = k φ k − ℓ ( φ ) . From the definition of the action, we obtain the following. Proposition 5.2. For any σ ∈ S ∞ , we have F σφ = { σT | T ∈ F φ } where σT = { x σi | x i ∈ T } .In particular, φ is a tautology or an antilogy if and only if its type is a tautology or an antilogy,respectively. The following is a motivation for | · | . Proposition 5.3. For any type formula φ and m ≥ k φ k , we have X ψ ∈ [ φ ] m z ℓ ( ψ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = √ m +1 = m k φ k (2 √ m + 1) ℓ ( φ ) = Θ( m | φ | ) . where m k is the falling factorial m ( m − · · · ( m − k + 1) . From Theorem 4.4 , we have relation between the generating function value at the singularity pointand the limit ratio of coefficients, so it can be expected that tautologies with large | · | values dominatethe density of tautologies.Now, we will prove basic properties of | · | . We begin with the following lemma. Lemma 5.4. (a) If φ has no ¬ ’s, then φ is true when the rightmost variable is true.(b) If φ has no repeated variables, then there is an assignment that makes φ true, and there is anassignment that makes φ false.(c) Suppose that φ has no repeated variables, no ¬ ’s and that p is a variable in φ with φ = p .Then, there are an assignment that p is true and an assignment that p is false, which make φ true.(d) Suppose that φ has no repeated variables, no ¬ ’s and that p is not the rightmost variable of φ . Then, there is an assignment that makes φ false and p is true.Proof. For any assignment set T ⊆ X and a well-formed formula φ , let T φ be the set of variables in T which appear in φ . By definition of the valuation, v T ( φ ) = v T φ ( φ ) is natural.(a) We will induct on the length of φ . If φ is a variable, then done. Otherwise, since φ = ψ → η and η is true by induction hypothesis, so is φ .(b) By induction, if φ is a variable, then done. Otherwise, it is trivial when φ = ¬ ψ for some ψ ,since ¬ reverses trueness and falseness. Now, if φ = ψ → η , then there are an assignment T on ψ that makes ψ true and an assignment S on η that makes η false, by induction hypothesis. Since φ hasno repeated variables, we have ( T ψ ∪ S η ) ψ = T ψ and ( T ψ ∪ S η ) η = S η It gives v T ψ ∪ S η ( φ ) = 1 − v T ψ ( ψ )(1 − v S η ( η )) = 0 . Hence, T ψ ∪ S η is an assignment that makes φ false. Similarly, there is an assignment that makes φ true.(c) Since φ has no ¬ ’s and φ is not p , φ = ψ → η for some ψ, η . If ψ has p , then there existsan assignment T that makes η true by (b). Then, T η is an assignment that makes φ true and does not contain p , so p is false, and T η ∪ { p } is an assignment that makes φ true and contains p , so p is true.Similarly, when η has p , it is done by using an assignment that makes ψ false.(d) Since φ has no ¬ ’s and p is not the rightmost variable, clearly we have φ = p . If p is not avariable of φ , then the case (b) is applicable. So we consider the case that φ = ψ → η for some ψ and η . First, suppose ψ has p . Then, by (b), there is an assignment T that makes η false. Now, if ψ = p ,then by (c), there is an assignment S that makes ψ true and contains p . Then, T η ∪ S ψ contains p andmakes φ false, since φ has no repeated variables. If ψ = p , then for any assignment T that makes η false, T ∪ { p } is an assignment that makes φ false.Now, if η has p , then by the induction hypothesis, there is an assignment T that contains p andmakes η false. By (b), there is an assignment S that makes ψ true, so there is an assignment T η ∪ S ψ that makes φ false. (cid:3) Then, we have the following. Proposition 5.5. (a) For any well-formed formula φ , | φ | ≤ .(b) For any tautology φ , | φ | ≤ − . Moreover, | φ | = − if and only if φ does not contains ¬ symbol, φ has unique variable appears twice, and every other variable in φ appears onlyonce.(c) For any antilogy φ , | φ | ≤ − .Proof. (a) Induction on the length. At first, | x i | = 1 − = , and |¬ φ | = | φ | − ≤ . Finally, wehave | φ → ψ | ≤ k φ k + k ψ k − 12 ( ℓ ( φ ) + ℓ ( ψ ) + 1) = | φ | + | ψ | − ≤ . (b) For a well-formed formula φ , the number of occurrences of variables is exactly the number ofoccurrences of → ’s plus 1. Let R be the number of variables in φ that do not appear first time in φ , Y be the number of occurrences of → ’s, and N be the number of occurrences of ¬ ’s. Then, we have | φ | = k φ k − ℓ ( φ )= Y + 1 − R − 12 ( Y + ( Y + 1) + N )= 12 − R − N. Hence, | φ | ≥ implies R = 0 , so φ has no repeated variables. Then, by Lemma 5.4 (b), φ is not atautology. The remaining part follows from the fact that R ≥ and | φ | = − imply N = 0 .(c) By Lemma 5.4 (a) and (b), any antilogy φ needs at least one ¬ and repeated variables. Hence, R ≥ and N ≥ , so | φ | ≤ − . (cid:3) It is remarkable that every simple tautology φ of the first kind with exactly one repetition andwithout ¬ ’s has | φ | = − , and every simple tautology φ of the second kind with exactly one repetitionand one ¬ has | φ | = − . Actually, the converse holds, for the first kind. Proposition 5.6. Suppose φ is a tautology and | φ | = − . Then, there are well-formed formu-lae ψ , · · · , ψ k , η without ¬ ’s, pairwise common variables, and repeated variables such that φ is ψ , · · · , ψ k , p η where p is the rightmost variable of η . Here, k = 0 is possible.Proof. First, φ has no ¬ ’s and has the unique repeated variable p which appears twice, by aboveproposition. Hence, φ = ψ → η for some ψ, η .Suppose ψ and η have no common variables. Then, by Lemma 5.4 (a), there is an assignment T that makes ψ true. Hence, if there is an assignment S that makes η false, T ψ ∪ S η makes φ = ψ → η false. Thus, there is no assignment that makes η false, so η is again a tautology. This implies that p must in η , since every tautology has at least one repeated variable. Hence, η is again a tautology with OMPUTING THE DENSITY OF TAUTOLOGIES 23 | η | = − . Then, by induction on length, η is ψ , · · · , ψ k , p η ′ and so, φ is ψ, ψ , · · · , ψ k , p η ′ .Thus, done.Now, assume that ψ and η have a common variable. Then, from the uniqueness of the repeatedvariable of φ , it must be p . If ψ = p , then by Lemma 5.4 (b), there is an assignment T on η thatmakes η false. If p ∈ T , then by Lemma 5.4 (c), there is an assignment S on ψ that makes ψ trueand p ∈ S . Also, if p T , then we have an assignment S on ψ that makes ψ true and p S . Then, T η ∪ S ψ makes φ = ψ → η false, which is a contradiction. So ψ = p .Then, we have φ = p → η . If p is not the rightmost variable of η , then by Lemma 5.4 (d), there isan assignment T on η that makes η false and p ∈ T . Hence, T makes φ false, which is a contradiction.Thus, p is the rightmost variable of η . (cid:3) From these propositions and Proposition 5.3 we can guess that the density of tautologies is of m order: since the maximum | · | of well-formed formulae is and the maximum | · | of tautologies is − , we may expect m − m = m order. Similarily, for antilogies, we may expect m √ m order. Definition 5.7. In the following, k ≥ .(a) A well-formed formula φ is a simple tautology of the first kind , if there exist well-formedformulae ψ , · · · , ψ k and a variable p such that φ is ψ , · · · , ψ k p with ψ i = p for some i . Let S be the set of simple tautologies of the first kind.(b) A well-formed formula ψ , · · · , ψ k p is a strict simple tautology of the first kind , if ψ = p and ψ , · · · , ψ k = p . Let S c be the set of strict simple tautologies of the first kind.(c) A well-formed formula φ is a simple tautology of the second kind , if there exist well-formedformulae ψ , · · · , ψ k +2 and a variable p so φ is ψ , · · · , ψ k +1 ψ k +2 where ψ k +2 is not η → η form well-formed formula, and there exists distinct i, j ≤ k + 1 such that ψ i = p and ψ j = ¬ p . Here, if ψ k +2 is η → η , then ψ , · · · , ψ k +1 ψ k +2 issame as ψ , · · · , ψ k +1 , η η . So actually, the restriction for ψ k +2 is only for determining ψ , · · · , ψ k +2 uniquely. Let S be the set of simple tautologies of the second kind. Proposition 5.8. (a) The generating function S of S is mz (1 + z − zW ( z ))(1 − zW ( z )) and lim n →∞ [ z n ] S ( z )[ z n ] W ( z ) = m (4 m + 6 √ m + 3)( √ m + 1) (2 m + 3 √ m + 2) = 1 m − m √ m + 7 m + O ( 1 m √ m ) . (b) The generating function S c of S c is mz z − zW ( z ) and lim n →∞ [ z n ] S c ( z )[ z n ] W ( z ) = m (2 m + 3 √ m + 2) = 14 m − m √ m + 1916 m + O ( 1 m √ m ) . Proof. (a) The generating function of well-formed formulae of the form ψ , · · · , ψ k p is mz ( zW ( z )) + mz ( zW ( z )) + mz ( zW ( z )) + · · · = mz W ( z )1 − zW ( z ) . Here, mz term is for the variable p , and zW ( z ) term is for the ψ i with → symbol. Now we selectthose of the first kind by using the fact that a given well-formed formula is not a simple tautologyof the first kind if and only if every ψ i is not p . We induce that the generating function of suchwell-formed formulae of the first kind is mz ( z ( W ( z ) − z )) + mz ( z ( W ( z ) − z )) + mz ( z ( W ( z ) − z )) + · · · = mz ( W ( z ) − z )1 + z − zW ( z ) . Hence, we have S ( z ) = mz W ( z )1 − zW ( z ) − mz ( W ( z ) − z )1 + z − zW ( z ) = mz (1 + z − zW ( z ))(1 − zW ( z )) . Then, by Szeg˝o’s lemma, when we take s = √ m +1 , we have lim n →∞ [ z n ] S ( z )[ z n ] W ( z ) = lim z → s − S ′ ( z ) p − z/s lim z → s − W ′ ( z ) p − z/s . Now, we have S ′ ( z ) = − mz ((1 + z − zW ( z ))( − zW ′ ( z )) + (1 − zW ( z ))( − zW ′ ( z ))(1 + z − zW ( z )) (1 − zW ( z )) + R ( z )= mz (2 + z − zW ( z )) W ′ ( z )(1 + z − zW ( z )) (1 − zW ( z )) + R ( z ) where lim z → s − R ( z ) p − z/s = 0 . Thus, we have lim n →∞ [ z n ] S ( z )[ z n ] W ( z ) = ms (2 + s − s W ( s ))(1 + s − s W ( s )) (1 − s W ( s )) = m (4 m + 6 √ m + 3)( √ m + 1) (2 m + 3 √ m + 2) . Also, from S ( z )(1 + z − zW ( z ))(1 − zW ( z )) = mz and z ( W ( z )) = W ( z ) − mz − zW ( z ) ,we have an equation S ( z ) = mz + ( m − z S ( z ) + z (1 + z + z ) W ( z ) S ( z ) , and if we use Theorem 4.4 , we get lim n →∞ [ z n ] S ( z )[ z n ] W ( z ) = s (1 + s + s ) S ( s )1 − ( m − s − s (1 + s + s ) W ( s ) = m (4 m + 6 √ m + 3)( √ m + 1) (2 m + 3 √ m + 2) . which corresponds to the result from Szeg˝o’s lemma.(b) This can be done similarly as (a). (cid:3) This lower bound of the density of tautologies from (a) of the above proposition is quite improvedfrom the first result √ m √ m + 1)(2 √ m + 1) . To improve more, we will consider the following. Definition 5.9. Let B be a set of tautologies.(a) The strong B -category is a partition of well-formed formulae consisting of strong B -tautologies( T ∗ ), B -unknowns ( U ∗ ), and B -antilogies ( A ∗ ) determined by B such that • φ ∈ T ∗ if and only if φ ∈ B ; φ is ¬ ψ form where ψ ∈ A ∗ ; or φ is ψ → η form where η ∈ T ∗ . • φ ∈ A ∗ if and only if φ is ¬ ψ form where ψ ∈ T ∗ ; or φ is ψ → η form where ψ ∈ T ∗ and η ∈ A ∗ . • φ ∈ U ∗ if and only if φ 6∈ T ∗ ∪ A ∗ .The following table shows this recursive classification. OMPUTING THE DENSITY OF TAUTOLOGIES 25 T ∗ U ∗ A ∗ ¬ A ∗ U ∗ T ∗ T ∗ → T ∗ U ∗ A ∗ U ∗ → T ∗ U ∗ U ∗ A ∗ → T ∗ U ∗ U ∗ (b) The weak B -category is a partition of well-formed formulae consisting of strong B -tautologies( T ∗ ), B -unknowns ( U ∗ ), and B -antilogies ( A ∗ ) determined by B such that • φ ∈ T ∗ if and only if φ ∈ B ; φ is ¬ ψ form where ψ ∈ A ∗ ; or φ is ψ → η form where η ∈ T ∗ or ψ ∈ A ∗ . • φ ∈ A ∗ if and only if φ is ¬ ψ form where ψ ∈ T ∗ ; or φ is ψ → η form where ψ ∈ T ∗ and η ∈ A ∗ . • φ ∈ U ∗ if and only if φ 6∈ T ∗ ∪ A ∗ .The following table shows this recursive classification. T ∗ U ∗ A ∗ ¬ A ∗ U ∗ T ∗ T ∗ → T ∗ U ∗ A ∗ U ∗ → T ∗ U ∗ U ∗ A ∗ → T ∗ T ∗ T ∗ (c) A well-formed formula φ is weak (resp. strong) B -basic if φ is a weak (resp. strong) B -tautology and φ is not a weak (resp. strong) ( B \ { φ } ) -tautology.(d) The set B is weak (resp. strong) basic if every φ ∈ B is weak (resp. strong) B -basic.(e) A set of B -tautologies B ′ is a weak (resp. strong) basis of B if B ′ is weak (resp. strong) basicand every φ ∈ B is a weak (resp. strong) B ′ -tautology.This proposition is true for both weak and strong category. Proposition 5.10. Suppose B is a set of tautologies.(a) For a basis e B of B , e B -category is same as B -category.(b) Every well-formed formula φ of a basis e B of B is B -basic.(c) Every B has a basis and a well-formed formula φ is in a basis e B if and only if φ is B -basic.In particular, there is a unique basis e B of B , which is a subset of B .Proof. (a) With induction on the length of well-formed formulae, it comes from the recursive struc-ture of categories.(b) If φ is not B -basic, then φ is ( B \ { φ } ) -tautology. Now, for every well-formed formula ψ with ℓ ( ψ ) < ℓ ( φ ) , B -category, ( B \ { φ } ) -category, e B -cateogory and ( e B \ { φ } ) -category are all same.Hence, φ is a ( e B \ { φ } ) -tautology, contradicting that e B is basic.(c) It is enough to show that every B -basic φ is in e B and the set of B -basic well-formed formulaeis a basis. If φ is B -basic, then φ is not a ( B \ { φ } ) -tautology and so not a ( e B \ { φ } ) -tautology. Since e B is a basis, φ is a e B -tautology, and so φ ∈ e B .Let b B be the set of B -basic well-formed formulae. Then, by the definition of basic well-formedformula, b B ⊆ B . Since φ ∈ b B is not a ( B \ { φ } ) -tautology, it is not a ( b B \ { φ } ) -tautology, and so b B is basic. Let ψ be a shortest B -tautology that is not a b B -tautology. Then, for every shorter well-formed formula ψ ′ than ψ , ( B \ { ψ } ) -category is same as b B -category. Now, ψ is not B -basic, so is a ( B \ { ψ } ) -tautology, and hence ψ is a b B -tautology, which is a contradition. So every B -tautology isa b B -tautology, and so, b B is a basis of B . (cid:3) Proposition 5.11. For a set B of tautologies, every strong B -tautology is a weak B -tautology, andevery weak B -tautology is a tautology. Hence, every weak B -basic well-formed formula is strong B -basic. The following system of equations naturally follows from the structure of B -categories. Proposition 5.12. Let B be a set of tautologies.(a) Let B ∗ , T ∗ , U ∗ , A ∗ be the generating functions of the strong basis of B , strong B -tautologies,strong B -unknowns, and strong B -antilogies, respectively. Then the following system of equa-tions is satisfied. T ∗ ( z ) = B ∗ ( z ) + zA ∗ ( z ) + zT ∗ ( z ) W ( z ) ,U ∗ ( z ) = mz − B ∗ ( z ) + zU ∗ ( z ) + z [ U ∗ ( z ) W ( z ) + A ∗ ( z ) W ( z ) − A ∗ ( z ) T ∗ ( z )] ,A ∗ ( z ) = zT ∗ ( z ) + zA ∗ ( z ) T ∗ ( z ) . (b) Let B ∗ , T ∗ , U ∗ , A ∗ be the generating functions of the weak basis of B , weak B -tautologies,weak B -unknowns, and weak B -antilogies, respectively. Then the following system of equa-tions is satisfied. T ∗ ( z ) = B ∗ ( z ) + zA ∗ ( z ) + z [ T ∗ ( z ) W ( z ) + A ∗ ( z ) W ( z ) − A ∗ ( z ) T ∗ ( z )] ,U ∗ ( z ) = mz − B ∗ ( z ) + zU ∗ ( z ) + zU ∗ ( z ) W ( z ) ,A ∗ ( z ) = zT ∗ ( z ) + zA ∗ ( z ) T ∗ ( z ) . Note that these systems of equations have fixed number of equations whenever m , the numberof variables of the propositional logic system, changes, so it makes easy to analyze an asymptoticbehavior as m → ∞ . Also, for fixed B , we have lim n →∞ [ z n ] I ∅ ( z )[ z n ] W ( z ) ≥ lim n →∞ [ z n ] T ∗ ( z )[ z n ] W ( z ) ≥ lim n →∞ [ z n ] T ∗ ( z )[ z n ] W ( z ) so computing lim n →∞ [ z n ] T ∗ ( z )[ z n ] W ( z ) or lim n →∞ [ z n ] T ∗ ( z )[ z n ] W ( z ) will give a lower bound for the asymptoticdensity of tautologies. Proposition 5.13. (a) S c is the strong basis of S .(b) The weak basis of S is the set of well-formed formulae of the form ψ → [ · · · → [ ψ k → p ] · · · ] where ψ = p , and ψ , · · · , ψ k are not p nor S -antilogy. Its generating functionsatisfies B ∗ ( z ) = mz z − zW ( z ) + zA ∗ ( z ) . which naturally satisfies B ∗ ( z ) = mz − z B ∗ ( z ) + z [ B ∗ ( z ) W ( z ) − B ∗ ( z ) A ∗ ( z )] . Now, we may solve the equation for S -strong case algebraically, by using the identity A ∗ ( z ) = zT ∗ ( z )1 − zT ∗ ( z ) , to obtain T ∗ ( z ) = 1 − z + zS c ( z ) − zW ( z ) − p (1 − z + zS c ( z ) − zW ( z )) − zS c ( z )(1 − zW ( z ))2 z (1 − zW ( z )) A ∗ ( z ) = zT ∗ ( z )1 − zT ∗ ( z ) ,U ∗ ( z ) = mz − S c ( z ) + zA ∗ ( z ) − z − z ( W ( z ) + A ∗ ( z )) = mz − S c ( z ) + zA ∗ ( z )( W ( z ) − T ∗ ( z ))1 − z − zW ( z ) . OMPUTING THE DENSITY OF TAUTOLOGIES 27 For s = √ m +1 , we have T ∗ ( s ) = √ m (2 m + 4 √ m + 3)2 m + 3 √ m + 2 − (2 √ m + 1) √ m + 1 s m (4 m + 24 m √ m + 60 m + 84 m √ m + 70 m + 33 √ m + 7)(2 √ m + 1) (2 m + 3 √ m + 2) . and it is also possible to compute A ∗ ( s ) and U ∗ ( s ) . Note that if we substitute /y for √ m , then yT ∗ ( s ) , yU ∗ ( s ) and yA ∗ ( s ) are analytic about y near 0. So we have series expansions T ∗ ( s ) = 12 √ m − m + 178 m √ m + O ( 1 m ) ,A ∗ ( s ) = 14 m − m √ m + O ( 1 m ) ,U ∗ ( s ) = √ m − √ m + 1 m − m √ m + O ( 1 m ) . Then, by Theorem 4.4 , if we let γ = lim n →∞ [ z n ] S c ( z )[ z n ] W ( z ) , we have lim n →∞ [ z n ] T ∗ ( z )[ z n ] W ( z ) = ( T ∗ ( s ) − /s )( T ∗ ( s ) + γ/s ) T ∗ ( s )(1 /s − √ m ) + A ∗ ( s ) + √ m/s − /s + 1= ( T ∗ ( s ) − /s )( T ∗ ( s ) + γ/s ) T ∗ ( s )( √ m + 1) + A ∗ ( s ) − √ m (2 √ m + 3)= 1 m − m √ m + 314 m + O ( 1 m √ m ) , which gives a slight improvement from lim n →∞ [ z n ] S ( z )[ z n ] W ( z ) .To use this method of undetermined coefficients of power series for weak class case, we need toprove that yT ∗ ( s ) , yU ∗ ( s ) , yA ∗ ( s ) and yB ∗ ( s ) are also analytic about y = √ m near 0. Wewill prove that our equations have analytic solutions near y = 0 , and there are unique solutions for B, T, U in a bounded region for fixed small y , so our analytic solutions match with real solutions thatwe want.We will consider the general case, i.e., the case with arbitrary B ∗ ( z ) . First, the equation U ∗ ( z ) = mz − B ∗ ( z ) + zU ∗ ( z ) + zU ∗ ( z ) W ( z ) is actually equivalent to mz − B ∗ ( z ) = U ∗ ( z )(1 − z − zW ( z )) = mzU ∗ ( z ) W ( z ) = mz (cid:18) − T ∗ ( z ) + A ∗ ( z ) W ( z ) (cid:19) . Moreover, it is easy to check that a system of equations T ∗ ( z ) = B ∗ ( z ) + zA ∗ ( z ) + z [ T ∗ ( z ) W ( z ) + A ∗ ( z ) W ( z ) − A ∗ ( z ) T ∗ ( z )] ,A ∗ ( z ) = zT ∗ ( z ) + zA ∗ ( z ) T ∗ ( z ) , is actually equivalent to W ( z ) B ∗ ( z ) = mz ( T ∗ ( z ) + A ∗ ( z )) ,A ∗ ( z ) = zT ∗ ( z ) + zA ∗ ( z ) T ∗ ( z ) . Then, with s = 12 √ m + 1 = y y = y − y y − y 16 + · · · ,m = 1 y ,W ( s ) = √ m = 1 y , we have the system of equations T ∗ ( s ) = B ∗ ( s ) + yy + 2 A ∗ ( s ) + yy + 2 (cid:20) T ∗ ( s ) y + A ∗ ( s ) y − A ∗ ( s ) T ∗ ( s ) (cid:21) ,A ∗ ( s ) = yy + 2 T ∗ ( s ) + yy + 2 A ∗ ( s ) T ∗ ( s ) , which is equivalent to ( y + 2) B ∗ ( s ) = T ∗ ( s ) + A ∗ ( s ) ,A ∗ ( s ) = yy + 2 T ∗ ( s ) + yy + 2 A ∗ ( s ) T ∗ ( s ) . Note that since B ∗ , T ∗ , A ∗ are generating functions, which are bounded by W , the values of T ∗ ( s ) , B ∗ ( s ) , A ∗ ( s ) satisfy yB ∗ ( s ) , yT ∗ ( s ) , yA ∗ ( s ) ≤ for each y = m − / where m is a positiveinteger. Then, we need to solve(1) ( y + 2)[ yB ∗ ( s )] = [ yT ∗ ( s )] + [ yA ∗ ( s )] , [ yA ∗ ( s )] = y + [ yA ∗ ( s )] y + 2 [ yT ∗ ( s )] . in [0 , . Now, assume that we have an equation B ∗ ( z ) = Θ( B ∗ ( z ) , T ∗ ( z ) , A ∗ ( z ); m, z, W ( z )) ,and define θ ( b, t, a ; w ) = w Θ( b/w, t/w, a/w ; w , ww +2 , w ) . We define λ ( b, t, a ; w ) = (cid:18) θ ( b, t, a ; w ) , ( w + 2) b − a, w + aw + 2 t (cid:19) , ˜ λ ( b, t, a ; w ) = (cid:18) θ ( b, t, a ; w ) , b w + 1) a + ( w + 3) t − at w + 2) , w + aw + 2 t (cid:19) . As we said, the set of fixed points of λ and ˜ λ are same. Now, solving our original system of equations(1) for yB ∗ ( s ) , yT ∗ ( s ) , yA ∗ ( s ) is equivalent to finding a fixed point of λ when w is fixed as y . Assume that we have a unique solution b , t , a in { ( b, t, a ) ∈ C | | b | , | t | , | a | ≤ } satisfying ( b , t , a ) = λ ( b , t , a ; 0) , in other words, a fixed point at w = 0 . Since we have a = a t and t = 2 b − a , this gives t = 2 b and a = 0 . Then, for ǫ > , we say D ⊆ C is a proper ǫ -region if it satisfies following: • D is closed and bounded, i.e. compact. • D contains an open neighborhood of ( b , t , a ) , • θ is analytic about w, b, t, a when | w | < ǫ and ( b, t, a ) ∈ D , • ˜ λ ( D ; w ) ⊆ D when | w | < ǫ , • if y < ǫ , then every solution ( yB ∗ ( s ) , yT ∗ ( s ) , yA ∗ ( s )) in [0 , of (1) is in D .For the last condition, it is sufficient to show that if ( b, t, a ) = λ ( b, t, a ; y ) and | b | , | t | , | a | ≤ , then ( b, t, a ) ∈ D . Hence, by the analytic implicit function theorem, we will get the existence of analyticsolution when the determinant of the Jacobian det J = det ∂ ( id − λ ) ∂ ( b, t, a ) OMPUTING THE DENSITY OF TAUTOLOGIES 29 is nonzero at ( b , t , a ) where w = 0 , and by the Banach contraction principle, we will get theuniqueness of solution for fixed w = y = m − / when the Jacobian J = ∂ ˜ λ∂ ( b, t, a ) has norm value less than 1 whenever | w | < ǫ and ( b, t, a ) ∈ D for some fixed norm. Here, we areusing ˜ λ since the Jacobian of λ contains w + 2 entry, which makes hard to get small norm. By simplecomputation, we have det J ( b, t, a ; w ) = 2 + 2 w + a − t w − w + a − t w ∂θ∂b − (2 + w − t ) ∂θ∂t − ( a + w ) ∂θ∂a . and J ( b, t, a ; w ) = ∂θ∂b ∂θ∂t ∂θ∂a w +3 − a w +2) w +1 − t w +2) w + aw +2 tw +2 . Moreover, if Θ is a function of A ∗ only, then we may reduce the number of variables by considering b λ ( a ; w ) = w + aw + 2 (( w + 2) θ ( a ; w ) − a ) = ( w + a ) θ ( a ; w ) − a ( w + a ) w + 2 , which gives b J ( a ; w ) = 2 w + 2 a + 2 w + 2 − aθ ′ ( a ; y ) − θ ( a ; y ) = 1 − b J ( a ; w ) , b J ( a ; w ) = aθ ′ ( a ; w ) + θ ( a ; w ) − w + 2 aw + 2 . We have free to choose J or b J to check the existence of analytic solution, and J or b J to checkthe uniqueness of solution. Of course, we need to variate the definition of proper region and chooseproperly to use b J . Lastly, for the proper ǫ -region with ǫ < , suppose ( b, t, a ) is a solution of ( b, t, a ) = λ ( b, t, a ; w ) satisfying | b | , | t | , | a | ≤ where | w | < ǫ . A proper ǫ -region must containevery such ( b, t, a ) , and we want to find ǫ -region as narrow as possible to get uniqueness easily. Notethat we have a = w + aw +2 t , which gives | a | = (cid:12)(cid:12)(cid:12)(cid:12) wt w − t (cid:12)(cid:12)(cid:12)(cid:12) ≤ | w || t || w | − | t | ≤ ǫ | t | − ǫ − | t | ≤ ǫ | t | − ǫ , so it is reasonable to try to take proper ǫ -region as a subset of { ( b, t, a ) | | b | , | t | , | a | ≤ , | a | ≤ ǫ − ǫ | t |} .Now, consider S -weak case. We have two choices of Θ( B ∗ ( z ) , T ∗ ( z ) , A ∗ ( z ); z, m, W ( z )) . Oneis Θ( B ∗ ( z ) , T ∗ ( z ) , A ∗ ( z ); z, m, W ( z )) = mz − z B ∗ ( z ) + z [ B ∗ ( z ) W ( z ) − B ∗ ( z ) A ∗ ( z )] , and the other is Θ( B ∗ ( z ) , T ∗ ( z ) , A ∗ ( z ); z, m, W ( z )) = mz z − zW ( z ) + zA ∗ ( z ) . Note that the latter is a function of A ∗ only. If we take the latter as our Θ , then we have θ ( b, t, a ; w ) = w w + 2 · w + 3 w + 2 + ( w + 2) a . Since θ ( b, t, 0; 0) = 0 always, so b = t = a = 0 is a unique solution. Now, if ǫ ≤ , | w | < ǫ and | a | ≤ , then we have | θ ( b, t, a ; w ) | ≤ ǫ − | w | · − | w | − | w | − | w || a | ≤ − · − − − (2 + ) = 284335 . Hence, if we define D = { ( b, t, a ) | | b | ≤ , | t | ≤ , | a | ≤ } , then D is closed, bounded region containing an open neighborhood of ( b , t , a ) = (0 , , . More-over, if ( b, t, a ) ∈ D , then | θ ( b, t, a ; w ) | ≤ , (cid:12)(cid:12)(cid:12)(cid:12) b w + 1) a + ( w + 3) t − at w + 2) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ) + (3 + ) + − ) = 144335 + 3235 ≤ < (cid:12)(cid:12)(cid:12)(cid:12) w + aw + 2 t (cid:12)(cid:12)(cid:12)(cid:12) ≤ + − = 17 , and so ˜ λ ( D ; w ) ⊆ D . Now, if ( b, t, a ) = λ ( b, t, a ; w ) and | b | , | t | , | a | ≤ , then we have | a | ≤ ǫ − ǫ | t | ≤ , | b | = | θ ( b, t, a ; w ) | ≤ , | t | = (cid:12)(cid:12)(cid:12)(cid:12) b w + 1) a + ( w + 3) t − at w + 2) (cid:12)(cid:12)(cid:12)(cid:12) ≤ , and so ( b, t, a ) ∈ D . Thus, D is a proper ǫ -region. Note that we may choose smaller D . For example,from | t | ≤ , we may get | a | ≤ · and from this bound of a , we can get smaller bounds for t and θ ( b, t, a ; w ) . Hence we may repeat this bootstrap process to make D smaller and smaller. Lastly,we will consider Jacobians. We will choose J , J rather than b J and b J . By direct computation, wehave det J ( b, t, a ; w ) = 2 + 2 w + a − t w + ( w + a ) w w + 2 · w + 2(2 w + 3 w + 2 + ( w + 2) a ) ,J ( b, t, a ; w ) = − w (2 w +3 w +2+( w +2) a ) w +3 − a w +2) w +1 − t w +2) w + aw +2 tw +2 . First, det J (0 , , 0; 0) = 1 = 0 , so we have local analytic solution about w from the analytic implicitfunction theorem. Then, for the (1,3)-entry of J , we have | ( J ) | ≤ ǫ (2 − | w | − | w | − | ǫ || a | ) ≤ (cid:0) − − − (cid:0) (cid:1) (cid:1) = (cid:18) (cid:19) , when | w | < ǫ ≤ and ( b, t, a ) ∈ D . Now, sum of the absolute values of the second column isbounded by ǫ + 3 + | a | − ǫ ) + ǫ + | a | − ǫ = 3( ǫ + | a | + 1)2(2 − ǫ ) and of the third column is bounded by (cid:18) (cid:19) + ǫ + 1 + | t | − ǫ ) + | t | − ǫ = (cid:18) (cid:19) + ǫ + 3 | t | + 12(2 − ǫ ) . Here, both of them become less than 1 as ǫ → , so there is ǫ ≤ such that w < ǫ implies k J k ∞ < . Hence, by the Banach contraction principle, we have uniqueness of solutions for eachsuch w , so the values of local analytic solution must match to true values of yB ∗ ( s ) , yT ∗ ( s ) and OMPUTING THE DENSITY OF TAUTOLOGIES 31 yA ∗ ( s ) . Then, yW ( s ) = 1 and yU ∗ ( s ) = yW ( s ) − yT ∗ ( s ) − yA ∗ ( s ) , so it is also true for yU ∗ ( s ) .From this result, we may assume B ∗ ( s ) = b − y + b + b y + b y + · · · ,T ∗ ( s ) = t − y + t + t y + t y + · · · ,U ∗ ( s ) = u − y + u + u y + u y + · · · ,A ∗ ( s ) = a − y + a + a y + a y + · · · , where b − , t − , u − , a − ≥ , since we are considering generating functions. Then, we have a seriesof quadratic equations B ∗ ( z ) = mz − z B ∗ ( z ) + z [ B ∗ ( z ) W ( z ) − B ∗ ( z ) A ∗ ( z )] ,T ∗ ( z ) = B ∗ ( z ) + zA ∗ ( z ) + z [ T ∗ ( z ) W ( z ) + A ∗ ( z ) W ( z ) − A ∗ ( z ) T ∗ ( z )] ,U ∗ ( z ) = mz − B ∗ ( z ) + zU ∗ ( z ) + zU ∗ ( z ) W ( z ) ,A ∗ ( z ) = zT ∗ ( z ) + zA ∗ ( z ) T ∗ ( z ) , and if we write this equation in terms of y , we will get B ∗ ( s ) = y (2 + y ) − y (2 + y ) B ∗ ( s ) + 12 + y B ∗ ( s ) − y y B ∗ ( s ) A ∗ ( s ) ,T ∗ ( s ) = B ∗ ( s ) + y y A ∗ ( x ) + 12 + y T ∗ ( s ) + 12 + y A ∗ ( s ) − y y A ∗ ( s ) T ∗ ( s ) ,U ∗ ( s ) = 1 y (2 + y ) − B ∗ ( s ) + y y U ∗ ( s ) + 12 + y U ∗ ( s ) ,A ∗ ( s ) = y y T ∗ ( s ) + y y A ∗ ( s ) T ∗ ( s ) . Then, the method of undetermined coefficients gives B ∗ ( s ) = 14 √ m − m + 916 m √ m + O ( 1 m ) ,T ∗ ( s ) = 12 √ m − m + 54 m √ m + O ( 1 m ) ,U ∗ ( s ) = √ m − √ m + 34 m − m √ m + O ( 1 m ) ,A ∗ ( s ) = 14 m − m √ m + O ( 1 m ) . Now, by Theorem 4.4 again, we have lim n →∞ [ z n ] B ∗ ( z )[ z n ] W ( z ) = 14 m − m √ m + 98 m + O ( 1 m √ m ) , lim n →∞ [ z n ] T ∗ ( z )[ z n ] W ( z ) = 1 m − m √ m + 298 m + O ( 1 m √ m ) . This is a lower bound of the asymptotic density of weak tautologies from simple tautologies of thefirst kind, so is of tautologies. Finally, we are going to consider both the first and second kind ofsimple tautologies. From Proposition 5.3 and Proposition 5.5 , since all simple tautologies of thesecond kind have ¬ symbol in it, we expect that this does not change the √ m order term of T ∗ ( s ) , but it will give an improvement on m order term. Hence, it will not change the m order term of ratio,but it will give an improvement on m √ m order term of it.We have to start from finding the basis of S ∪ S . Let us consider weak sense partition, and usesimple notations B, T, U, A for generating functions of basis, tautologies, unknowns, and antilogies,respectively. A well-formed formula ψ , · · · , ψ k − ψ k such that ψ k is a variable or ¬ η for awell-formed formula η is ( S ∪ S ) -basic if and only if one of the following is true. • First, k ≥ , and there is a variable p such that ψ , ψ k are p , ψ , · · · , ψ k − are not p , and ψ , · · · , ψ k − are not ( S ∪ S ) -antilogies. • There is a variable p and i < k such that ψ is p , ψ i is ¬ p , ψ k is not p , ψ k is not ¬ η for an ( S ∪ S ) -antilogy η , and for any < j < k , ψ j is not an ( S ∪ S ) -antilogy nor p . • There is a variable p and i < k such that ψ is ¬ p , ψ i is p , ψ k is not p , ψ k is not ¬ η for an ( S ∪ S ) -antilogy η , and for any < j < k , ψ j is not an ( S ∪ S ) -antilogy nor ¬ p .Also, these three conditions are pairwise disjoint. The generating function for the first case is mz − z [ W ( z ) − z − A ( z )] , for the second case is mz (cid:18) ( m − z + z [ W ( z ) − A ( z )]1 − z [ W ( z ) − z − A ( z )] − ( m − z + z [ W ( z ) − A ( z )]1 − z [ W ( z ) − z − z − A ( z )] (cid:19) , and for the third case is mz (cid:18) ( m − z + z [ W ( z ) − A ( z )]1 − z [ W ( z ) − z − A ( z )] − ( m − z + z [ W ( z ) − A ( z )]1 − z [ W ( z ) − z − z − A ( z )] (cid:19) . Deducing these formulae is similar to the proof of Proposition 5.8 .(a). To apply the method tocomputing the density of weak tautologies from S case, we have to consider the existence of properregion D . If Θ is a function of only A ∗ ( z ) and θ ( b, t, 0; 0) = 0 , then to prove the existence of properregion D , it is enough to choose ǫ > such that there exists δ > satisfies • if | w | < ǫ and | a | ≤ ǫ − ǫ , then | θ ( b, t, a ; w ) | ≤ δ , and • δ + − ǫ )(1 − ǫ ) ≤ .If these conditions are satisfied, then D = { ( b, t, a ) | | b | ≤ δ, | t | ≤ , | a | ≤ ǫ − ǫ } will be a proper ǫ -region. Then, we may compute Jacobians and check det J (0 , , 0; 0) is nonzero and a norm of J is less than 1, where we may reduce D by bootstrap argument and ǫ freely, if is needed. By directcomputation, we can show θ ( b, t, 0; 0) = 0 is really true for this case either, and hence, other processto prove analyticity is almost automatic. OMPUTING THE DENSITY OF TAUTOLOGIES 33 After we get the analyticity, we have to consider a system of quadratic equations including thegenerating function of the basis. We have following system of equation. B ( z ) = mz − z B ( z ) + z ( W ( z ) B ( z ) − A ( z ) B ( z )) ,B ( z ) = m ( m − z + mz [ W ( z ) − A ( z )] − z B ( z )+ z [ W ( z ) B ( z ) − A ( z ) B ( z )] ,B ( z ) = m ( m − z + mz [ W ( z ) − A ( z )] − ( z + z ) B ( z )+ z [ W ( z ) B ( z ) − A ( z ) B ( z )] ,B ( z ) = m ( m − z + mz [ W ( z ) − A ( z )] − z B ( z )+ z [ W ( z ) B ( z ) − A ( z ) B ( z )] ,B ( z ) = m ( m − z + mz [ W ( z ) − A ( z )] − ( z + z ) B ( z )+ z [ W ( z ) B ( z ) − A ( z ) B ( z )] ,B ( z ) = B ( z ) + B ( z ) − B ( z ) + B ( z ) − B ( z ) ,T ( z ) = B ( z ) + zA ( z ) + z ( T ( z ) W ( z ) + A ( z ) W ( z ) − A ( z ) T ( z )) ,U ( z ) = mz − B ( z ) + zU ( z ) + zU ( z ) W ( z ) ,A ( z ) = zT ( z ) + zA ( z ) T ( z ) . From this system of equations, we have a series solution B ( s ) = 14 √ m − m + 916 m √ m + O ( 1 m ) ,B ( s ) = √ m − − √ m + 58 m − m √ m + O ( 1 m ) ,B ( s ) = √ m − − √ m + 916 m − m √ m + O ( 1 m ) ,B ( s ) = 18 − √ m + 116 m + 332 m √ m + O ( 1 m ) ,B ( s ) = 18 − √ m + 932 m √ m + O ( 1 m ) ,B ( s ) = 14 √ m − m + 316 m √ m + O ( 1 m ) ,T ( s ) = 12 √ m − m + 12 m √ m + O ( 1 m ) ,U ( s ) = √ m − √ m + 12 m + O ( 1 m ) ,A ( s ) = 14 m − m √ m + O ( 1 m ) , and by Theorem 4.4 , we get lim n →∞ [ z n ] B ( z )[ z n ] W ( z ) = 14 m − m √ m + 516 m + O ( 1 m √ m ) , lim n →∞ [ z n ] T ( z )[ z n ] W ( z ) = 1 m − m √ m + 54 m + O ( 1 m √ m ) , lim n →∞ [ z n ] U ( z )[ z n ] W ( z ) = 1 − m + 54 m √ m − m + O ( 1 m √ m ) , lim n →∞ [ z n ] A ( z )[ z n ] W ( z ) = 12 m √ m − m + O ( 1 m √ m ) , and this result shows only improvement in m √ m order term, as we expected.For the upper bound of the density, we have an upper bound − lim n →∞ [ z n ] A ( z )[ z n ] W ( z ) so we finally conclude m − m √ m + 54 m + O ( 1 m √ m ) ≤ lim n →∞ [ z n ] I ∅ ( z )[ z n ] W ( z ) ≤ − m √ m + 98 m + O ( 1 m √ m ) . We may improve the upper bound slightly by dividing the class unknowns into unknowns and not tau-tologies nor antilogies. In such partitioning, B -tautologies and B -antilogies are not changed, and bysame argument, we may compute, with proper analyticity assumption, the density of not tautologiesnor antilogies has lower bound m − m √ m + 532 m + O ( 1 m √ m ) , and this gives an upper bound − m − m √ m + 3132 m + O ( 1 m √ m ) . But this upper bound is still too far from the lower bound. Moreover, we have reasonable conjecturewith Proposition 5.3 that we cannot improve the first term /m for the limit density and indeed, thisresult is asymptotically correct. In other words, we may expect that m times the density of tautologieswill converge to 1. A CKNOWLEDGEMENT This work was partially supported by the National Research Foundation of Korea (NRF) grantfunded by the Korea government (MSIT) (No. 2019R1F1A1062462).R EFERENCES[1] M. Zaionc, On the asymptotic density of tautologies in logic of implication and negation, Rep. Math. Log. , 67–87(2005).[2] H. Fournier, D. Gardy, A. Genitrini, and M. Zaionc, Tautologies over implication with negative literals, Math. Log.Quart. (4), 388–396 (2010).[3] L. Aszal´os and T. Herendi, Density of tautologies in logics with one variable, Acta Cybern. , 385–398 (2012).[4] P. Flajolet and R. Sedgewick, Analytic Combinatorics (Cambridge University Press, London, 2009).[5] M. Zaionc, Probability distribution for simple tautologies, Theor. Comput. Sci. , 243–260 (2006). E-mail address , T. Eom: [email protected] (T. Eom) D EPARTMENT OF M ATHEMATICAL S CIENCES , KAIST, D AEJEON , 34141, R