The Properties of Determinative Disjunctive Normal Forms and Systems Based on Them

We study some structural and numerical properties of varieties for determinative disjunctive normal forms, introduced here for given propositional formula. We consider for classical and non-classical propositional logics some proof systems, which are constructed on the base of determinative disjunctive normal forms. We investigate also the relation between the proof complexities in some well-known classical and non-classical proof systems (Resolution, Cut-free sequent, Gentzen refutation, Cutting planes etc.) and numerical properties of varieties for determinative disjunctive normal forms for classical and non-classical tautologies.


Introduction
Many of the proof systems of classical propositional logic use presentation of tautologies or contradictions in disjunctive normal forms (DNF) or in conjuctive normal forms (CNF).Some of the outstanding examples of propositional tautologies (contradictions) can be presented in various DNF (CNF) and proof complexities of these presentations can differ from each other.
It is well known that a Boolean function can be represented by different DNF: perfect, abridged, dead-end, shortest and minimal (see for example in Diskretnaya (1974)).It is also known that every conjunct from abridged (therefore from every dead-end, every minimal and at least one shortest) DNF corresponds to some prime implicant.If we consider a tautology in the capacity of the Boolean function, then it has a unique prime implicant, which corresponds to empty conjunct, but it is well known that there are "hard" and "simple" tautologies, therefore the representations of classical tautologies, moreover of non-classical tautologies, in some above varieties of DNF are not entirely correct.
Using some notions given in Chubaryan, An. and Chubaryan, Arm. (2007) for tautologies, in this paper for given propositional formula we introduce the notions of determinative conjunct, minimal determinative conjunct, to which corresponds determinative prime interval, determinative disjunctive normal form (dDNF), perfect dDNF, abridged dDNF, dead-end dDNF, shortest dDNF and minimal dDNF.We compare the properties of DNF and dDNF for the same formula.We show that there are some essential difference between the numerical and structural properties of the same varieties of DNF and dDNF for a given formula.Then we investigate the relations between numerical properties for the varieties of dDNF and proof complexities for classical and non-classical tautologies in the systems, based on dDNF, and in some other proof systems.This paper consists of 5 sections.The notions of dDNF and its varieties for given propositional formula and, in particular, for classical tautologies as well as the structural and numerical properties of dDNF are given in Section 2. The algorithm of constructions of dDNF for non-classical tautologies are given in Section 3. The systems for classical and non-classical propositional logics, based on dDNF, are defined in Section 4. In the same place the relations between the numerical properties of dDNF and proof complexities in described systems as well as in some other proof systems are investigated.The conclusions are given in the Section 5.
An extended abstract of some parts of this paper appeared as Chubaryan, An., Chubaryan, Arm. and Abajyan (2011).

dDNF for Given Propositional Formula
We will use the current concepts of the unit Boolean cube (E n ), interval in it, a propositional formula, a classical tautology, a proof system for classical propositional logic, and proof complexity.The particular choice of a language for presented propositional formulas is immaterial in this consideration.However, because of some technical reasons we assume that the language contains the propositional variables p i (i ≥ 1) and (or) p i j (i ≥ 1; j ≥ 1), logical connectives ¬, &, ∨, ⊃, ≡ and parentheses ( , ).Note that some parentheses can be omitted in generally accepted cases.
Following the usual terminology we call the variables and negated variables literals.The conjunct K (clause) can be represented simply as a set of literals (no conjunct contains a variable and its negation simultaneously).
In Chubaryan and Chubaryan (2007) the following notions were introduced.
We call a replacement-rule each of the following trivial identities for a propositional formula ψ: Application of a replacement-rule to some word consists in the replacing of some its subwords, having the form of the left-hand side of one of the above identities, by the corresponding right-hand side.
Let ϕ be a propositional formula, P = {p 1 , p 2 , . . ., p n } be the set of all variables of ϕ, and Definition 1 Determinative conjunct (Chubaryan & Chubaryan, 2007).Given σ = {σ 1 , . . ., σ m } ⊂ E m , the conjunct to each p i j and successively using replacement-rules we obtain the value of ϕ (1 or 0) independently of the values of the remaining variables.
Note that minimal determinative conjunct corresponds to determinative prime interval.
It is obvious that for every propositional formula ϕ perfect DNF is ϕ-determinative, but not every DNF for ϕ is dDNF.
Basing on these notions and following Diskretnaya (1974) we introduce the notions of determinative versions for above mentioned DNF and search the numerical characteristics of dDNF.
2) dDNF for formula ϕ is called dead-end determinative DNF (ddDNF) if after the removal of any its conjunct it does not be equal to ϕ.
3) dDNF for formula ϕ is called shortest determinative DNF (shdDNF) if it consist of the minimal number of conjuncts among all dDNF of ϕ. 4) dDNF for formula ϕ is called minimal determinative DNF (mdDNF) if it has the minimal number of literals among all dDNF of ϕ.
The abridged normal form is uniquely constructed from a Boolean function by means of a fairly simple algorithm.Its most important property is the fact that every minimal disjunctive normal form of a function and at least one shortest form can be obtained from the abridged disjunctive normal form by the elimination of certain elementary conjunctions.Therefore many minimization algorithms employ the abridged disjunctive normal form as the initial specification of a Boolean function (see for example in Diskretnaya, 1974).
The abridged dDNF for every formula ϕ has the same property, therefore we can use the same well known algorithms to obtain shortest and minimal dDNF from abridged dDNF, but in the case of dDNF we must verify the determinative property for each conjunct, which is obtained after every step of the algorithm.Note that determinative property of conjunct can be verified on-line.

The Properties of dDNF for Classical Tautologies
As only tautologies (classical or non-classical) are provable in propositional proof systems, further we will consider dDNF for tautologies.Here we investigate some structural and numerical properties of various types of dDNF for classical tautologies.
It is easy to prove the following Proposition 1 Each dDNF for given tautology must have at least one occurrence of some variable with negation.
It is well known that minimal DNF for monotone Boolean function has no occurrence of variable with negation.The situation with dDNF is other.
Proposition 2 There are monotone Boolean functions each dDNF for which (therefore mdDNF) must have at least one occurrence of some variable with negation.

Such monotone Boolean functions correspond to tautologies.
Proposition 3 The maximal number of conjuncts in shdDNF for tautologies in n variables is equal to 2 n .
Proof of upper bound can be obtained by induction, using Proposition 1.The lower bound follows by consideration of the tautologies Note that the maximal number of conjuncts in shortest DNF for Boolean functions in n variables is 2 n−1 (see in Diskretnaya, 1974).
Each dDNF for every formula ϕ is DNF for Boolean function, which is presented by a formula ϕ, therefore the various numerical characteristics of DNF are valid for dDNF.
If we consider only tautologies, then the upper bounds for numerical characteristics of DNF are also valid for dDNF.For the lower bounds we can construct the "bad" examples of tautologies on the base of the "bad" formulas from Diskretnaya (1974), using the following Proposition 4 If the formula ϕ is disjunction of the formulas ϕ 1 and ϕ 2 , then ϕ − 1-determinative conjunct must be ϕ 1 − 1-determinative or ϕ 2 − 1-determinative.
In particular, if we denote by L a (n) the maximal number of conjuncts in adDNF for tautologies in n variables, then we can prove Proof of upper bound is the same as in Diskretnaya (1974).For proving the lower bound we must "complete until sequence of tautologies" the sequence, considered in Diskretnaya (1974) "bad" formulas, by adding to each formula two formulas with disjunction.
So let S [i,k] and by D a (S [i,k]  n ) is denoted the abridged DNF of S [i,k]  n .As sequence of "bad" tautologies we consider the following formulas for n ≥ 3.
It is not difficult to see that abridged DNF, therefore dDNF of ϕ n has Note that the sequence of Boolean functions S [n/3],2[n/3] n is considered in Diskretnaya (1974).

Determinative Disjunctive Normal Forms for Non-classical Tautologies
The investigations of proof complexity start for the systems of Classical Propositional Logic (CPL).However, natural real conclusions have constructive character in most cases, therefore the investigation of the proofs complexities is important for systems of Intuitionistic Propositional Logic (IPL) and in some cases also for Minimal (Johansson's) Propositional Logic (MPL) (see for example in Kleene, 1952).In particular, logic programming is based on intuitionistic logic.
As the intuitionistic (minimal) validity is determined by derivability in some intuitionistic (minimal) propositional proof calculus, the above definition of dDNF for classical tautologies is not applicable here.
Some algorithm for construction of dDNF of a classical tautology ϕ on the base of resolution refutation for ϕ is described in Chubaryan (2002).Following Chubaryan, An., Chubaryan, Arm. and Sayadyan (2007), we give here the corresponding algorithm on the base of intuitionistic (minimal) resolution systems.
Let us recall the mentioned systems.
The system RI (resolution for IPL) is described by Mints in Mints (1985).
The axioms are the sequents p → p and ⊥ → p.
The rules of inference (resolution rules) are: , where p * for some propositional variable p can be p or ⊥.
The corresponding system for MPL is defined in the following way: RM is obtained from RI by dropping the rules (1) and (2) (see in Chubaryan, An., Chubaryan, Arm., & Sayadyan, 2007).
It is necessary to make some comments about the system RI (RM).Let ϕ be some formula and {p 1 , p 2 , ..., p n } is the set of its all variables (later we call this variables the main variables).Associating a new variable with every non-elementary subformula of ϕ, we can construct the system of disjuncts by employing method described in Mints (1985).The disjuncts of this system can be represented as the following sequents p → q ∨ r; (p ⊃ q * ) → r; q 1 , q 2 , . . ., q k → r * . (♦) Let s be the variable, associated with ϕ itself.Mints has shown, that the sequent → s is proved in RI from the axioms and from above set (♦) of disjuncts, constructed for ϕ, iff the sequent → ϕ is proved in the system NI (natural system for IPL (see in Mints, 1985).The same result can be proved for the systems RM and NM (corresponding natural system for MPL, which is obtained from NI by dropping the rule Γ→⊥ Γ→A ).Later for every formula ϕ each of disjuncts of the set (♦) is called the additional axiom.The axiom (additional or not) is called the main axiom if it contains at least one main variable.
Recall that there is a well-known notion of positive and negative occurrences of subformulas (or variables) in the formula or in the sequent (see for example in Chubaryan, 2002).If a variable p has negative occurrence in some subformula, which in its turn has negative occurrence in the formula, we say that the variable p has double negative occurrence in this formula.
It is not difficult to see, that occurrence of any variable in axioms (additional or not) or in inference rules of system RI is either positive, negative or double negative, and since p ∼ p is not derivable in IPL (MPL), then not only variable or variable with negation, but also variable with double negation can serve as literal for ϕ-determinative conjunction in IPL.
It is natural that for any variable p not only the literals p and p are contrary, but p and p also.It is not difficult to see that just the contrary pairs of literals of type p, p and p, p are subjected to resolution by application of inference rules of RI.
It is necessary to note that any ϕ-determinative conjunct can not include contrary literals.
The analogies of the ϕ-determinative d.n.f. for IPL and MPL (ϕ − I-determinative d.n.f. and ϕ − M-determinative d.n.f.accordingly) can be constructed using the following algorithm.
Let W be the proof of → s in RI (RM).The steps for the construction of the ϕ − I-determinative (ϕ − Mdeterminative) DNF are the followings: 1) We transform the proof W into tree-like proof W tree of → s.Let k be the number of the paths of this tree.
2) For every path i (1 ≤ i ≤ k) between two vertices, one associated with the main axiom and another with → s, we construct the conjunct K i as the set of all main variables (or their negations, or double negations), which have positive (negative or double negative) occurrence in the sequents of this path.
Then we take the conjunct Ki , which consists of negations of all literals of K i and after substitution of p type literals with p, we obtain the determinative conjunct K i .
Note that only variables with one or double negations are the literals in I-determinative conjuncts (p ⊃ ⊥ and (p ⊃ ⊥) ⊃ ⊥ type literals in M-determinative conjuncts).
Using various RI-proofs (RM-proofs) we can construct different ϕ − I-determinative DNF (ϕ − M-determinative DNF) and hence we can define the varieties of DNF as above for classical tautologies.

The Systems EC, EI and EM
Earlier some classical propositional proof system EC is introduced by the first co-author in Chubaryan (2002).
The axioms of EC aren't fixed, but for every formula ϕ each conjunct from some dDNF of ϕ can be considered as an axiom.
The classical elimination rule (C ε -rule) infers K ∪ K from clauses K ∪ {p} and K ∪ {p}, where K and K are clauses and p is a variable.
The proof in EC is a finite sequence of clauses such that every clause in the sequence is one of the axioms of EC, or is inferred from earlier clauses in the sequence by C ε -rule.
By analogy the corresponding proof system EI (EM) can be constructed for IPL (MPL).
As axiom is considered every I-determinative (M-determinative) conjunct from some I-determinative (M -determinative) DNF.
For EI (EM) we take the following inference rule where K and K are clauses and p is a variable.

Proof Complexity, Polynomial Simulation
To indicate the role of above mentioned systems and investigate the relation between the proof complexity and numerical properties of dDNF, let us recall some notions and notations.
By |ϕ| we denote the size of a formula ϕ, defined as the number of all variable entries.It is obvious that the full length of a formula, which is understood to be the number of all symbols or the number of all entries of logical signs, is bounded by some linear function in |ϕ|.
In the theory of proof complexity two main characteristics of the proof are: t − complexity, defined as the number of proof steps, and l − complexity, defined as total number of proof symbols.Let Φ be a proof system and ϕ be a tautology.We denote by t Φ ϕ (l Φ ϕ ) the minimal possible value of t − complexity (l − complexity) for all the proofs of tautology ϕ in Φ.
Let Φ 1 and Φ 2 be two different proof systems.Following Cook and Reckhow (1979) we recall.
Definition 4 Φ 2 p − t-simulates (p − l-simulates) Φ 1 if there exists a polynomial p() such that for every formula ϕ derivable both in Φ 1 and For CPL it is proved in Chubaryan (2002) that the systems EC, Cut-free sequent and Resolution are p-equivalent, in Arai (2000) it is proved that Resolution system and Gentzen refutation systems are p-equivalent also.For IPL (MPL) it is proved in Chubaryan, An., Chubaryan, Arm. and Sayadyan (2007) that the systems EI, RI and Multi-succedent cut-free sequent (EM, RM and Multi-succedent cut-free sequent) are p-equivalent.We suppose also that the analogies of "Cutting planes" and "Gentzen refutation" for IPL (MPL) can be constructed on the base of I-determinative (M-determinative) DNF.So the investigation of proof complexities in EC, EI and EM is very important.It is interesting also which is the relation between the proof complexities in E-type systems and numerical properties of dDNF.
The search of this questions is in progress, but at present we have some results.
By L(D(ϕ)) we denote the number of conjuncts in DNF D for formula ϕ.
Let d(ϕ) be the number of literals in minimal determinative conjunct of ϕ. 2) There is a sequence of tautologies γ n such that ).The statements of the point 1 are proved very easy.Really a) as we note above for every propositional formula ϕ perfect DNF is ϕ-determinative, b) every ϕ-determinative DNF must contain at least 2 d(ϕ) conjuncts, each of them must use as axiom in EC proof of ϕ in EC, c) the sum of sizes of axioms in the proof of ϕ in EC can not be smaller than the size of mdDNF for ϕ, d) follows from b).
The proof of statement 2 is given in Chubaryan (2002) on the base of formulas It is obvious that the analogous results can be proved for IPL and MPL.

Conclusion
As some proof systems for propositional logics are based on DNF and every "valid statement" can admit many varieties of DNF, the investigation of various representation of formulas (tautologies) in DNF can be very important for evaluation of proof complexities in many proof systems.In particular, if for given tautology d(ϕ) = Θ(|ϕ|), then the exponential lower bound for proof complexity of ϕ in many proof systems can be obtained very easy.The relation between the numerical properties of dDNF and proof complexities can be useful for "good" DNF construction also.Mints, G. (1985).Resolution systems for nonclassical logic.Semiotika i informatika,25,[120][121][122][123][124][125][126][127][128][129][130][131][132][133].

Notes
Note 1.As usual, given a propositional variable p and σ ∈ E 1 , by p σ we denote the function p σ = p, if σ = 1, p, if σ = 0.