A General Verification for Functional Completeness by Abstract Operators
aa r X i v : . [ m a t h . L O ] M a y A General Verification for Functional Completeness byAbstract Operators
Yang Tian ∗ Tsinghua University, Beijing, China
Abstract
An operator set is functionally incomplete if it can not represent the full set {¬ , ∨ , ∧ , → , ↔} . The verification for the functional incompleteness highly relies on constructiveproofs. The judgement with a large untested operator set can be inefficient. Givenwith a mass of potential operators proposed in various logic systems, a general ver-ification method for their functional completeness is demanded. This paper offersan universal verification for the functional completeness. Firstly, we propose twoabstract operators b R and ˘ R , both of which have no fixed form and are only definedby several weak constraints. Specially, b R ≥ and ˘ R ≥ are the abstract operators de-fined with the total order relation ≥ . Then, we prove that any operator set R isfunctionally complete if and only if it can represent the composite operator b R ≥ ◦ ˘ R ≥ or ˘ R ≥ ◦ b R ≥ . Otherwise R is determined to be functionally incomplete. This theorycan be generally applied to any untested operator set to determine whether it isfunctionally complete. Keywords:
Functional Completeness, Truth Function Operators, AbstractOperators
1. Introduction
A set of truth function operators (or propositional connectives) is functionallycomplete if and only if all formulas constructed by {¬ , ∨ , ∧ , → , ↔} can also be definedonly based on that operator set. Otherwise the operator set is functionally incomplete[1]. ∗ Corresponding author
Email address: [email protected] (Yang Tian)
Preprint submitted to arXiv (Vision 1) May 12, 2020 etermining whether a given operator set is functionally complete has both the-oretical significance and practical value. For logic theories, verifying the functionalcompleteness helps to analyse the mathematical properties of a given logic systemdefined with non-classical operators [2, 3, 4]. For logic applications, the function-ally complete set acts as the key unit in logic circuits and is the designed to realizedemanded logic operation efficiently [5, 6, 7, 8, 9, 10].However, the definition of functional completeness can not provide a direct andefficient verification method for itself. For an untested operator set, the verificationfor its functional completeness usually relies on constructive proofs. There havebeen abundant researches that demonstrate object-specific verification proofs for thefunctional completeness of various truth function operator families. For a given truthfunction operator family, they develop self-contained and focused method to confirmthe function completeness [11, 12, 13, 14]. As for the much more general cases, asimple characterization method of functionally complete operator sets is proposedby researchers from [15], but the operator set is limited to be one-element.To offer a general verification for the functional incompleteness of any givenoperator set, our work includes following parts: • At first, we define two abstract truth function operators ˘ R and b R in Subsec-tion 3.1 . They are abstract since they have no fixed forms and are definedonly by several weak and elementary constraints. Thus, they can representany operator satisfies those conditions. Specially, if the abstract operators aredefined with the total relation ≥ , we mark them as ˘ R ≥ and b R ≥ . • At second, we analyse the properties of those two abstract operators ˘ R and b R in Subsection 3.2 . Then, in
Subsection 3.3 , we show the connection betweenabstract operators and the known operators, where the relation between b R ≥ and ¬ is confirmed. • At third, we propose a new conception called semi-expressiveness in
Subsec-tion 3.4 . For any given operator set R , it’s semi-expressive for { ˘ R, b R } if andonly if it can represent b R ◦ ˘ R or ˘ R ◦ b R . Moreover, we prove if R is semi-expressivefor { ˘ R, b R } , then it can represent ˘ R and b R . • Finally, we prove that any operator set R is functionally complete if and only ifit is semi-expressive for { ˘ R ≥ , b R ≥ } in Subsection 3.5 , which can offer a generalverification for the functional completeness of any operator set. And in
Sub-section 3.6 , this theory is demonstrated to verify the functional completenessof several operator sets. 2ur theory is demonstrated in a general but elementary form. All definitions andtheorems are proposed to be self-contained.
2. Review of elementary definitions
To provide a clear vision, we propose a brief review for several elementary concep-tions directly relevant with functional completeness. Later in the further introductionof our theory, all notations will follow the sign convention reviewed in this section.
Here we propose the sign convention for truth value set, truth function operatorsand the valuation mapping respectively.The truth value set can be defined in various forms according to the demands ofdifferent logic systems. In our research, we only focus on the basic properties of it,which is given as following:
Definition 2.1.
Let V ⊂ [0 ,
1] be the truth value set, which satisfies • { , } ⊂ V ; • ∀ v ∈ V , there is 1 − v ∈ V .Since we don’t carry out object-specific analysis for any logic system, the truthfunction operator concerned in our research is proposed in a general form. Definition 2.2.
Let R = ∪ n R n be the set of all truth function operators neededto be analysed, where R n = { R | R : V n → V } denotes the set of all n -adic truthfunction operators in R .Given with a set of operators, we can represent formulas (or proportions) basedon them. Thus, the synatx of a logic language can be defined. Definition 2.3.
Let LP ( R ) be the set of all formulas constructed based on thetruth function operators of R .The valuation mapping is used to assign truth value for a given formula anddefine the semantics for the logic language. From the perspective of algebra, it actsas the homomorphic mapping between the formula set to the truth value set. Definition 2.4.
Let v : LP ( R ) → V be the valuation function that determines thetruth value of each formula in LP ( R ). Note that v is the homomorphic mappingbetween LP ( R ) and V , such that for any R of R n , there is ∀ φ , . . . , φ n ∈ LP ( R ) , v ( R ( φ , . . . , φ n )) = R ( v ( φ ) , . . . , v ( φ n )) . (1)3 .2. Functionally complete set After the synatx and semantics of the logic language is reviewed in general forms,we turn to defining the functional completeness.
Definition 2.5.
Any operator set R is functionally complete if and only if for everyformula φ constructed using the full set of operators {¬ , ∨ , ∧ , → , ↔} , there is at leastone formula ψ of LP ( R ) satisfies | = ( φ ↔ ψ ).For convenience, we refer the property proposed in Definition 2.5 as that R canrepresent {¬ , ∨ , ∧ , → , ↔} . The functional completeness of a truth function operatorset is of great significance, since it determines how expressive a logic language definedbased on that operator set is.Moreover, in the studies relevant with functional completeness, the minimumfunctionally complete set often plays an important role, which is given by Definition 2.6.
For a given functionally complete set R , if any proper subset R ′ ofit is functionally incomplete, then it is a minimum functionally complete set.In summary, we have reviewed the elementary conceptions related to the truthfunction operator set and the functional completeness. They will be frequently usedin further analyses.
3. Our work: a general verification for functional completeness
We then turn to introducing our work. In our theory, we start with defining twonew abstract ˘ R and b R , then we propose the sufficient and necessary condition for aoperator set R to be functionally complete step by step. ˘ R and b R A general verification method should be able to fit any truth function operator set.Although we can not enumerate all possible truth function operators, we can still tryto summarize their key properties closely related to the functional completeness. Inour research, we use two abstract operators to contain those properties. An abstractoperator is the truth function operator that has no fixed form and defined only byseveral elementary constraints, which can offer a general and abstract representationfor a family of operators.To make our work easy to follow, we start with giving brief definitions of those twoabstract operators in this subsection. In later analyses, we will show the idea behindthem and the relation between them and functional completeness systematically.Before our analyses, for convenience, we define a sign mark related to the totalorder relation to simplify our introduction process.4 ark 3.1.
For any element x and a non-empty set Y , if there exists a total orderrelation such that x ⊲ y for each y in Y , then we mark it as x ⊲ Y . Dually, we candefine Y ⊲ x .Then, there are two abstract truth function operators defined in our research,they are the choice operator ˘ R and the modification operator b R . Definition 3.2.
We define the choice operator as ˘ R : V n → V with n ≥
2. It hasno fixed form and is given by v (cid:16) ˘ R ( φ , . . . , φ n ) (cid:17) = v ( φ i ) , i ∈ Z ∩ [1 , n ] , (2)where there exists a total order relation ⊲ such that v ( φ i ) ⊲ { v ( φ j ) | j = i } .Based on Definition 3.2 , it can be seen that the choice operator ˘ R is proposedin a very general form. We can simply treat { v ( φ i ) } i ∈ I as a set defined with a totalorder relation, then ˘ R select the greatest element in it. Definition 3.3.
We define the modification operator b R : V → V . It has no fixedform neither, and it satisfies if v (cid:16) ˘ R ( φ , . . . , φ n ) (cid:17) = v ( φ i ) and v ( φ i ) ⊲ { v ( φ j ) | j = i } ,then • n v (cid:16) b R ( φ j ) (cid:17) | j = i o ⊲ v (cid:16) b R ( φ i ) (cid:17) ; • v (cid:16)(cid:16) b R ◦ b R (cid:17) ( φ i ) (cid:17) = v ( φ i );Note that ◦ denotes the composite of functions. For example, (cid:16) b R ◦ b R (cid:17) ( φ i ) = b R (cid:16) b R ( φ i ) (cid:17) . In Definition 3.3 , the modification operator b R is defined dependently.A modification operator needs to correspond to at least one choice operator. Giventhat a choice operator can select the greatest element under a total order relation ⊲ ,the modification operator can propose the inverse relation of ⊲ .Apart of that, for convenience, we propose a sign convention for the modificationoperator b R , which is given as Mark 3.4.
When the modification operator b R is applied to a set of formulas { φ , . . . , φ n } respectively, we mark that b R ( φ , . . . , φ n ) = { b R ( φ ) , . . . , b R ( φ n ) } .5 .2. The properties of abstract truth function operators ˘ R and b R After defining abstract truth function operators ˘ R and b R , here we analyse severalbasic characters of them.Firstly, since the modification operator b R defined in Definition 3.3 can proposethe inverse relation for a given total order relation ⊲ , it’s easy to know the followingtheorem. Theorem 3.5.
If there exists a total order relation ⊲ such that v ( φ ) ⊲ v ( ψ ) , then v (cid:16) b R ( ψ ) (cid:17) ⊲ v (cid:16) b R ( φ ) (cid:17) .Proof. For the choice operator ˘ R : V n → V , assume that n = 2. Based on Defini-tion 3.2 , if v ( φ ) ⊲ v ( ψ ), then v (cid:16) ˘ R ( φ, ψ ) (cid:17) = v ( φ ). Based on Definition 3.3 , thereis v (cid:16) b R ( ψ ) (cid:17) ⊲ v (cid:16) b R ( φ ) (cid:17) .Secondly, being able to define the inverse relation for a given total order relation,the modification operator b R itself can be proved as a bijection. Theorem 3.6.
The modification operator b R : V → V is a bijection from V to V .Proof. Our proof is demonstrated with two steps. • At first, we prove that b R is injective by reduction to absurdity. Assume b R isnot injective, then ∃ v i = v j ∈ V, b R ( v i ) = b R ( v j ) = v k , (3)based on Definition 3.3 , there is v i = (cid:16) b R ◦ b R (cid:17) ( v i ) = b R ( v k ) = (cid:16) b R ◦ b R (cid:17) ( v j ) = v j , (4)which is contradicts with (3) . Thus, b R must be injective. • At second, we prove that b R is a surjection. For convenience, we mark thedomain of definition as dom b R and mark the value range as ran b R . Assume that b R is not a surjection, then there exist v j ∈ ran b R such that for all v i ∈ dom b R , b R ( v i ) = v j . Then, since dom b R = ran b R , there must be ∃ v k ∈ ran b R, |{ v i | b R ( v i ) = v k }| > , (5)based on Definition 3.3 , (5) can implies the same contradiction in (4) . Thus, b R must be a surjection. 6ombine those two conclusions, we can finish our proof. Theorem 3.5 shows an important property of the modification operator b R .Given that it’s a bijection for V to V , there are a lot of theorems can be applied toit to simplify our proofs (such as the proof of Theorem 3.9 ).Thirdly, we also pay special attention to the properties of the composite operators˘ R ◦ b R and b R ◦ ˘ R , in the following theorem, we will analyse what those compositeoperators can imply on the truth value of given formulas. Theorem 3.7. If v (cid:16) ˘ R ( φ , . . . , φ n ) (cid:17) = v ( φ i ) and v ( φ i ) ⊲ { v ( φ j ) | j = i } , then weknow • assume v (cid:16)(cid:16) ˘ R ◦ b R (cid:17) ( φ , . . . , φ n ) (cid:17) = v (cid:16) b R ( φ k ) (cid:17) , then v ( φ i ) ⊲ { v ( φ j ) | j = i and j = k } ⊲ v ( φ k ) ; • there is v (cid:16)(cid:16) b R ◦ ˘ R (cid:17) ( φ , . . . , φ n ) (cid:17) = v (cid:16) b R ( φ i ) (cid:17) .Proof. We respectively prove those two properties. • For the first property, it’s clear that v (cid:16)(cid:16) ˘ R ◦ b R (cid:17) ( φ , . . . , φ n ) (cid:17) = v (cid:16) ˘ R (cid:16) b R ( φ ) , . . . , b R ( φ n ) (cid:17)(cid:17) , (6)= v (cid:16) b R ( φ k ) (cid:17) , (7)based on Definition 3.2 , it’s clear that v (cid:16) b R ( φ k ) (cid:17) ⊲ n v (cid:16) b R ( φ j ) (cid:17) | j = k o .Given Theorem 3.7 , we can know ∀ j = k, v (cid:16)(cid:16) b R ◦ b R (cid:17) ( φ j ) (cid:17) ⊲ v (cid:16)(cid:16) b R ◦ b R (cid:17) ( φ k ) (cid:17) , (8)which implies that n v (cid:16)(cid:16) b R ◦ b R (cid:17) ( φ j ) (cid:17) | j = k o ⊲ v (cid:16)(cid:16) b R ◦ b R (cid:17) ( φ k ) (cid:17) , (9) { v ( φ j ) | j = k } ⊲ v ( φ k ) . (10)Since v ( φ i ) ⊲ { v ( φ j ) | j = i } , there is v ( φ i ) ⊲ { v ( φ j ) | j = i and j = k } ⊲ v ( φ k ) , (11)which finishes the proof. • The proof of the second property can be directly obtained based on that v (cid:16) ˘ R ( φ , . . . , φ n ) (cid:17) = v ( φ i ). 7 .3. The connection between abstract operators and known operators In Subsection 3.1 , we have suggested that ˘ R and b R can be treated as the generaland abstract representation for a family of operators with fixed forms. Up to now,we have analysed several elementary properties of those two abstract operators. Wethen turn to indicating the underlying connection between abstract operators andother known operators.Those two abstract operators are defined based on the total order relation. Sincewe need to talk about different known operators defined in fixed forms, we define asign convention to distinguish between different total order relations. Mark 3.8.
To distinguish between different total order relations, we define if thereexists a total order relation ⊲ such that v (cid:16) ˘ R ( φ , . . . , φ n ) (cid:17) = v ( φ i ) , i ∈ Z ∩ [1 , n ]and v ( φ i ) ⊲ { v ( φ j ) | j = i } , then the choice operator is marked as ˘ R ⊲ . And thecorresponding modification operator is marked as b R ⊲ .We are specially interested in the situation where those two abstract operatorsare defined with ≥ (greater than or equal to) and ≤ (less than or equal to), both ofwhich are the most basic total order relations in the set of real numbers. Theorem 3.9.
If the choice operator is ˘ R ≥ , then the corresponding modify operator b R ≥ must satisfy b R ≥ ( v ) = 1 − v. (12) And the same conclusion can be obtained when the choice operator is ˘ R ≤ .Proof. To prove
Theorem 3.9 , we can prove that (12) is the sufficient and necessarycondition for that b R ≥ satisfies the Theorem 3.5 . • At first, the sufficiency is trivial to prove based on ∀ v i ≥ v j ∈ V, (1 − v j ) ≥ (1 − v i ) . (13) • At second, we prove the necessity by reduction to absurdity. We assume a casewhere ∃ v i ∈ V, k = 0 , b R ≥ ( v i ) = 1 − v i + k ∈ V, (14)which implies that b R ≥ ( v i ) doesn’t satisfy (12) . And based on Definition 2.1 ,it can be known that v i − k ∈ V . Then, we further assume that ∀ v j = v i = v i − k ∈ V, b R ≥ ( v j ) = 1 − v j . (15)8ased on (14) and (15) , we can define a situation where b R ≥ can satisfy (12) when acts on all elements of V , except for v i and v i − k . Since we have provedthat b R ≥ is a bijection from V to V in Theorem 3.6 , given with (14) and (15) , there must be b R ≥ ( v i − k ) = 1 − v i . (16)And it’s easy to know ( b R ≥ ( v i ) ≥ b R ≥ ( v i − k ) , v i ≥ v i − k b R ≥ ( v i − k ) ≥ b R ≥ ( v i ) , v i − k ≥ v i , (17)which implies the contradiction with Theorem 3.5 . Thus, the case assumedin (14) can not happen.To sum up, we know that b R ≥ ( v ) = 1 − v is the sufficient and necessary condition forthat b R ≥ satisfies Theorem 3.5 , which finishes the proof. Following those proposedsteps, we can also prove the conclusion for b R ≤ . Theorem 3.9 shows a simple and clear result: when abstract operators aredefined with ≥ and ≤ , the modification operator has same function as the classicaloperator ¬ (we know ¬ ( v ) = 1 − v ). This inspires us to think about whether thosetwo abstract operators can be used to {¬ , ∨ , ∧ , → , ↔} when they are defined withspecific total order relations. Before analysing the relation between the abstract operators and the functionalcompleteness, here we propose a new conception that is called semi-expressiveness.
Definition 3.10.
A operator set R is semi-expressive for { b R, ˘ R } if there exist atleast one set of operators R , . . . , R k in it satisfy v (cid:16) ( R ◦ . . . ◦ R k ) ( φ , . . . , φ n ) (cid:17) = v (cid:16)(cid:16) b R ◦ ˘ R (cid:17) ( φ , . . . , φ n ) (cid:17) , (18)or v (cid:16) ( R ◦ . . . ◦ R k ) ( φ , . . . , φ n ) (cid:17) = v (cid:16)(cid:16) ˘ R ◦ b R (cid:17) ( φ , . . . , φ n ) (cid:17) . (19)In brief, a operator set R is semi-expressive for { b R, ˘ R } means that merely basedon the operators R contains, the composite operators b R ◦ ˘ R and ˘ R ◦ b R can berepresented.The choice operator ˘ R can select the greatest element in a given subset of V witha given total order relation ⊲ , and the modification operator can propose the inverserelation ⊳ of ⊲ . So we can see that 9 if (18) is satisfied, then greatest element of a given subset of V is selectedunder the total order relation ⊲ and modified to be the least element; • if (19) is satisfied, then least element of a given subset of V is selected underthe total order relation ⊲ (the greatest element under ⊳ is the least elementunder ⊲ ) and modified to be the greatest element.In a word, a operator set R that is semi-expressive for { b R, ˘ R } can realize the trans-formation between the least and greatest element in a given subset of V with a totalorder relation.Just as what have been mentioned for { b R, ˘ R } , the semi-expressiveness is theability to represent the composite operators b R ◦ ˘ R and ˘ R ◦ b R . It’s natural to wanderif it also means the ability to represent b R and ˘ R respectively. Theorem 3.11. If R is semi-expressive for { b R, ˘ R } , then R can represent b R and ˘ R .Proof. The following is our proof. • For ˘ R , – if (18) is satisfied, we assume that ˘ R ( φ , . . . , φ n ) = φ i , then (cid:16) b R ◦ ˘ R (cid:17) (cid:16)(cid:16) b R ◦ ˘ R (cid:17) ( φ , . . . , φ n ) , . . . , (cid:16) b R ◦ ˘ R (cid:17) ( φ , . . . , φ n ) (cid:17) (20)= (cid:16) b R ◦ ˘ R (cid:17) (cid:16) b R ( φ i ) , . . . , b R ( φ i ) (cid:17) (21)= (cid:16) b R ◦ b R (cid:17) ( φ i ) (22)= ˘ R ( φ , . . . , φ n ) . (23)Note that (23) is implied by that for any formula φ i , (cid:16) b R ◦ b R (cid:17) ( φ i ) = φ i ,which is defined in Definition 3.3 . – Else if (19) is satisfied, there is (cid:16) ˘ R ◦ b R (cid:17) (cid:16)(cid:16) ˘ R ◦ b R (cid:17) ( φ , . . . , φ ) , . . . , (cid:16) ˘ R ◦ b R (cid:17) ( φ n , . . . , φ n ) (cid:17) (24)= (cid:16) ˘ R ◦ b R (cid:17) (cid:16) b R ( φ ) , . . . , b R ( φ n ) (cid:17) (25)= h ˘ R ◦ (cid:16) b R ◦ b R (cid:17)i ( φ , . . . , φ n ) (26)= ˘ R ( φ , . . . , φ n ) . (27)10ased on what have been proved, we know if R is semi-expressive for { b R, ˘ R } ,then R can represent ˘ R . • For b R , we know that no matter (18) or (19) is satisfied, there is ∀ φ i , (cid:16) b R ◦ ˘ R (cid:17) ( φ i , . . . φ i ) = (cid:16) ˘ R ◦ b R (cid:17) ( φ i , . . . φ i ) = b R ( φ i ) (28)Thus, if R is semi-expressive for { b R, ˘ R } , then R can represent b R .To summarize, we can finally draw a conclusion that Theorem 3.10 is true.Up to now, we have prepared all the necessary definitions and theorems aboutthe abstract operators. Later in the next subsection, we will use them to propose asufficient and necessary condition of functional completeness.
Here we propose our theorem of the sufficient and necessary condition of func-tional completeness.
Theorem 3.12.
For a operator set R , it is functionally complete if and only if it issemi-expressive for { b R ≥ , ˘ R ≥ } .Proof. To prove the sufficient and necessary condition proposed in
Theorem 3.12 ,our proof consists of two parts. • At first, based on
Theorem 3.10 , we know that if R is semi-expressive for { b R ≥ , ˘ R ≥ } , then R can represent b R ≥ and ˘ R ≥ . • At second, we prove the sufficiency, which means that every operator set thatis semi-expressive for { b R ≥ , ˘ R ≥ } will be functionally complete. Given that R can represent b R ≥ and ˘ R ≥ , we can have following analyses. – For ˘ R ≥ , we know if v (cid:16) ˘ R ≥ ( φ , . . . , φ n ) (cid:17) = v ( φ i ), and v ( φ i ) ≥ { v ( φ j ) | j = i } , then it’s trivial that v ( φ i ) = v (cid:16) ((( φ ∨ φ ) . . . ) ∨ φ n ) (cid:17) . Thus, if R represents ˘ R ≥ , then R can represent ∨ . – For b R ≥ , since for each formula φ i v ( ¬ φ i ) = 1 − v ( φ i ), we can prove if R represents b R ≥ , then R can represent ¬ directly based on Theorem 3.9 .11ased on what have be proved, we know a operator set R that is semi-expressivefor { b R ≥ , ˘ R ≥ } can represent {¬ , ∨} , which is a minimum functionally completeoperator set. It’s known if R can represent a functionally complete operator set,then it can represent {¬ , ∨ , ∧ , → , ↔} . Thus, the sufficiency has been proved. • At third, we prove the necessity, which means that every functionally completeoperator set must be semi-expressive for { b R ≥ , ˘ R ≥ } . Following the idea used in Theorem 3.10 , we know that ∀ φ i , (cid:16) b R ≥ ◦ ˘ R ≥ (cid:17) ( φ i , . . . φ i ) = (cid:16) ˘ R ≥ ◦ b R ≥ (cid:17) ( φ i , . . . φ i ) = b R ≥ ( φ i ) (29)if there doesn’t exist any set of operators R , . . . , R k in R satisfies (18) nor (19) , then for any formula φ , v (cid:16) b R ≥ ( φ ) (cid:17) can never be defined by R . Basedon Theorem 3.9 , it’s easy to prove that ¬ can’t be represented by R . Thus,the necessity is proved.To sum up, we have proved that a operator set R is functionally complete if andonly if it is semi-expressive for { b R ≥ , ˘ R ≥ } .Note that since ≥ and ≤ are inverse relations of each other, we can replace { b R ≥ , ˘ R ≥ } in Theorem 3.12 as { b R ≤ , ˘ R ≤ } and it still holds.The Theorem 3.12 and its proof are simple since we have proposed all necessaryfoundations of them in the previous analysis. An important thing about
Theorem3.12 is that the method can verify the functional completeness of any n -elementuntested operator set since there is no further limitation for it. Finally, we propose two applications of
Theorem 3.12 , where we use it to verifythe functional completeness of several truth function operator sets.To provide a better understanding for the functional completeness verificationbased on
Theorem 3.12 , we propose several trivial examples at first.
Lemma 3.13.
Define four operators R ( φ , . . . , φ n ) = ¬ ((( φ ∨ φ ) . . . ) ∨ φ n ) , (30) R ( φ , . . . , φ n ) = ¬ ((( φ ∧ φ ) . . . ) ∧ φ n ) , (31) R ( φ , . . . , φ n ) = ((( ¬ φ ) ∨ ( ¬ φ )) . . . ) ∨ ( ¬ φ n ) , (32) R ( φ , . . . , φ n ) = ((( ¬ φ ) ∧ ( ¬ φ )) . . . ) ∧ ( ¬ φ n ) , (33) then { R } , { R } , { R } and { R } are all functionally complete. roof. It’s trivial that (30) and (31) have the same form with (18) while (32) and (33) have the same form with (19) , so
Lemma 3.13 can be directly proved basedon
Theorem 3.12 .Then, we propose three elementary but non-trivial examples.
Lemma 3.14.
Define three operators R ( φ , . . . , φ n ) = ¬ ((( φ ⋆ φ ) . . . ) ⋆ n − φ n ) , (34) R ( φ , . . . , φ n ) = ((( ¬ φ ) ⋆ ( ¬ φ )) . . . ) ⋆ n − ( ¬ φ n ) , (35) R ( φ , . . . , φ n ) = ((( (cid:7) φ ) ⋆ ( (cid:7) φ )) . . . ) ⋆ n − ( (cid:7) n φ n ) , (36) where each ⋆ k is randomly selected from {∨ , ∧} and each (cid:7) k is randomly selectedfrom {¬ , ¬¬} . Then • { R } and { R } are all functionally complete; • { R } is functionally incomplete if there exists at least one x ( x ≤ n ) thatsatisfies (cid:7) x is ¬¬ .Proof. We respectively prove (34) , (35) and (36) . • For (34) , given that each ⋆ k is randomly selected from {∨ , ∧} , our idea todemonstrate the proof is given as following: – with the truth values of given formulas { v ( φ ) , . . . , v ( φ n ) } , we can findthe greatest element v ( φ i ) and least element v ( φ j ) of this set with thetotal order relation ≥ . – Then for a given operator R , we can confirm whether ⋆ n − is ∨ or ∧ ,and ∗ if ⋆ n − is ∨ , then we know v (cid:16)(cid:16) b R ≥ ◦ ˘ R ≥ (cid:17) ( φ , . . . , φ n ) (cid:17) = v (cid:16) ( ¬ ((( . . . ⋆ . . . ) . . . ) ⋆ n − φ i )) (cid:17) , (37)where φ i is put at the end of the sequence so that ⋆ n − can act onit. As for other formulas { φ x | x = i } , there is no limitation for theirlocation so they are represented by “ . . . ” (this omission is also used inthe later analysis). We can see that in (37) , { R } is semi-expressivefor { b R ≥ , ˘ R ≥ } . Based on Theorem 3.12 , it’s functionally complete.13 if ⋆ n − is ∧ , then we know v (cid:16)(cid:16) b R ≤ ◦ ˘ R ≤ (cid:17) ( φ , . . . , φ n ) (cid:17) = v (cid:16) ( ¬ ((( . . . ⋆ . . . ) . . . ) ⋆ n − φ j )) (cid:17) , (38)thus { R } is semi-expressive for { b R ≤ , ˘ R ≤ } , which makes { R } be func-tionally complete. – To sum up, using the idea described above, we can conclude that { R } described in (34) is functionally complete. • For (35) , the same idea used to proof (34) can also be used. It’s trivial that – if ⋆ n − is ∨ , then v (cid:16)(cid:16) ˘ R ≥ ◦ b R ≥ (cid:17) ( φ , . . . , φ n ) (cid:17) = v (cid:16) (( . . . ⋆ . . . ) . . . ) ⋆ n − ( ¬ φ j ) (cid:17) , (39)which implies that { R } is functionally complete when it’s semi-expressivefor { b R ≥ , ˘ R ≥ } . – if ⋆ n − is ∧ , then v (cid:16)(cid:16) ˘ R ≤ ◦ b R ≤ (cid:17) ( φ , . . . , φ n ) (cid:17) = v (cid:16) (( . . . ⋆ . . . ) . . . ) ⋆ n − ( ¬ φ i ) (cid:17) , (40)which implies that { R } is semi-expressive for { b R ≤ , ˘ R ≤ } and functionallycomplete.Thus, { R } described in (35) is functionally complete. • For (36) , we know v (cid:16) R ( φ , . . . , φ n ) (cid:17) = v (cid:16)(cid:16) ˘ R ≥ ◦ b R ≥ (cid:17) ( φ , . . . , φ n ) (cid:17) ⇔ (41) v (cid:16) ((( (cid:7) . . . ) ⋆ ( (cid:7) . . . )) . . . ) ⋆ n − ( (cid:7) n φ j ) (cid:17) = v ( ¬ φ j ) , (42)or v (cid:16) R ( φ , . . . , φ n ) (cid:17) = v (cid:16)(cid:16) ˘ R ≤ ◦ b R ≤ (cid:17) ( φ , . . . , φ n ) (cid:17) ⇔ (43) v (cid:16) ((( (cid:7) . . . ) ⋆ ( (cid:7) . . . )) . . . ) ⋆ n − ( (cid:7) n φ i ) (cid:17) = v ( ¬ φ i ) , (44)However, if there exists at least one x ( x ≤ n ) that satisfies (cid:7) x is ¬¬ , then14 if every (cid:7) x in (36) is ¬¬ , then for each formula φ x , there is (cid:7) x φ x = φ x . Thus, it’s easy to know { R } can never represent b R ≤ nor b R ≥ . Inother words, ¬ can not be represented by { R } , which makes { R } befunctionally incomplete; – if (cid:7) n in (36) is ¬ , then ∗ assume ⋆ n − is ∨ , given that v ( ¬ φ j ) ≥ { v ( ¬ φ k ) | k = j } , thenit’s trivial that (42) is guaranteed (no matter how each ⋆ k is se-lected) if and only if v ( ¬ φ j ) ≥ { v ( φ k ) | k = j } . For the formula set { φ , . . . , φ n } does not meet this condition, (42) can be false. Thus, { R } is functionally incomplete; ∗ assume ⋆ n − is ∧ , it’s trivial that (44) is guaranteed (no matter howeach ⋆ k is selected) if and only if v ( ¬ φ i ) ≤ { v ( φ k ) | k = i } . Forother situations, (44) can be false, which makes { R } be functionallyincomplete.Finally, we can conclude that { R } described in (36) is functionally incompletewith the given condition.The examples used in Lemma 3.14 are proposed in general forms. In fact, thewell known Peirce operator ↓ (there is φ ↓ ψ = ¬ ( φ ∨ ψ ), {↓} is functionally complete[16]) is a special case of (34) . For those non-trivial examples, the method proposedby us can provide an unified process to verify the functional completeness.
4. Conclusion and Discussion
Let’s have a brief review for what have been proposed in our theory. Our workaims at providing a general verification method for the functional completeness (orfunctional incompleteness). To realize this ambition, we define two abstract operators b R and ˘ R in Definition 3.2 and
Definition 3.3 to construct a new theory, where wepropose the sufficient and necessary condition for a truth function operator set to befunctionally complete in
Theorem 3.12 . In our work, we demonstrate the theory onseveral examples, which suggests that our method can offer an efficient verification.Compared with the object-specific verification in previous [11, 12, 13, 14], our theorycan be applied more generally.In our theory, what interests us most is the properties of those two abstractoperators proposed in
Subsection 3.2 . Although we mainly use them to studythe functional completeness, they might also be inspiring for the researches of other15opics in logic since they can offer abstract representation for specific families oftruth function operators. Apart of that, the conception of semi-expressiveness forabstract operators proposed in
Subsection 3.3 might provide a new perspectiveto understand the underlying connection between representing a composite operator R ◦ . . . ◦ R n and representing the operators { R , . . . , R n } respectively. It inspiresus to address a new question: what is the sufficient and necessary condition for anycomposite operator R ◦ . . . ◦ R n to be able to represent { R , . . . , R n } ? In the feature,we will continue to explore the relevant mechanisms of them.Moreover, we suggest that our theory has the potential to be applied in thedata science. Recently, it has received increasing attention to abstract and analyselogic structures from various data sets [17, 18, 19]. There might be lots of poten-tial truth function operators detected in the experiments. To verify their functionalcompleteness (this determines how many logic relations the data contains), a purelyconstructive verification proof can be inefficient. Based on our method, their func-tional completeness can be verified in a more unified process, which makes it possibleto design the computer-aided proof.In a word, we suggest that the theory proposed in our work has the potential tobe further explored. References [1] J. R. Shoenfield, Mathematical logic, CRC Press, 2018.[2] M. M. Radzki, On axiom systems of s lupecki for the functionally complete three-valued logic, Axiomathes 27 (4) (2017) 403–415.[3] N. Belnap, S. McCALL, et al., Every functionally complete m-valued logic hasa post-complete axiomatization., Notre Dame Journal of Formal Logic 11 (1)(1970) 106.[4] Y. Venema, et al., Expressiveness and completeness of an interval tense logic.,Notre Dame journal of formal logic 31 (4) (1990) 529–547.[5] Z.-R. Wang, Y.-T. Su, Y. Li, Y.-X. Zhou, T.-J. Chu, K.-C. Chang, T.-C. Chang,T.-M. Tsai, S. M. Sze, X.-S. Miao, Functionally complete boolean logic in 1t1rresistive random access memory, IEEE Electron Device Letters 38 (2) (2016)179–182.[6] E. Lehtonen, M. Laiho, Stateful implication logic with memristors, in: 2009IEEE/ACM International Symposium on Nanoscale Architectures, IEEE, 2009,pp. 33–36. 167] V. Varshavsky, V. Marakhovsky, I. Levin, N. Kravchenko, Functionally completeelement for fuzzy control hardware implementation, in: The 2004 47th MidwestSymposium on Circuits and Systems, 2004. MWSCAS’04., Vol. 3, IEEE, 2004,pp. iii–263.[8] W. Williamson III, B. K. Gilbert, Resonant tunneling diode structures for func-tionally complete low power logic, uS Patent 5,698,997 (Dec. 16 1997).[9] J. D. Yetter, Functionally complete family of self-timed dynamic logic circuits,uS Patent 5,208,490 (May 4 1993).[10] V. Tokmen, A functionally-complete ternary system, Electronics Letters 14 (3)(1978) 69–71.[11] R. L. Graham, On ηη