The Computational Complexity of Propositional Cirquent Calculus
Introduced in 2006 by Japaridze, cirquent calculus is a refinement of sequent calculus. The advent of cirquent calculus arose from the need for a deductive system with a more explicit ability to reason about resources. Unlike the more traditional proof-theoretic approaches that manipulate tree-like objects (formulas, sequents, etc.), cirquent calculus is based on circuit-style structures called cirquents, in which different âpeerâ (sibling, cousin, etc.) substructures may share components. It is this resource sharing mechanism to which cirquent calculus owes its novelty (and its virtues). From its inception, cirquent calculus has been paired with an abstract resource semantics. This semantics allows for reasoning about the interaction between a resource provider and a resource user, where resources are understood in the their most general and intuitive sense. Interpreting resources in a more restricted computational sense has made cirquent calculus instrumental in axiomatizing various fundamental fragments of Computability Logic, a formal theory of (interactive) computability. The so-called “classical” rules of cirquent calculus, in the absence of the particularly troublesome contraction rule, produce a sound and complete system CL5 for Computability Logic. In this paper, we investigate the computational complexity of CL5, showing it is -complete. We also show that CL5 without the duplication rule has polynomial size proofs and is NP-complete.
Key words and phrases:cirquent calculus, computability logic, resource semantics, proof theory, substructural logics
Introduced in 2006 by Japaridze , cirquent calculus is a refinement of classical sequent calculus. The advent of cirquent calculus arose from the need for a deductive system with a more explicit ability to reason about resources. Unlike the more traditional proof-theoretic approaches that manipulate tree-like objects (formulas, sequents, etc.), cirquent calculus is based on circuit-style structures called cirquents, in which different âpeerâ (sibling, cousin, etc.) substructures may share components. It is this resource sharing mechanism to which cirquent calculus owes its novelty (and its virtues). Cirquents come in a variety of forms. One way to characterize the sort of cirquents studied in this paper in familiar terms is to say that a cirquent is a multiset of sequents (called âgroupsâ) where each formula — more precisely, each occurrence of a formula — may simultaneously belong to more than one sequent. This explains the origin of the word âcirquentâ, which is a hybrid of âcircuitâ and âsequentâ. From its inception, cirquent calculus has been paired with an abstract resource semantics. This semantics allows for reasoning about the interaction between a resource provider and a resource user, where resources are understood in the their most general and intuitive sense.
Interpreting resources in a more restricted computational sense has made cirquent calculus instrumental in axiomatizing various fundamental fragments of Computability Logic, a formal theory of (interactive) computability. The so-called “classical” rules of cirquent calculus, in the absence of the particularly troublesome contraction rule, produce a sound and complete system CL5 for Computability Logic. Born in , Computability Logic (CoL) is an ambitious research program aimed at developing a formal theory of interactive computability. To this end, formulas of CoL represent computational problems modeled at games. The notion of “truth” for such formulas becomes synonymous with the notion of “computability” for the computational problems they represent.
While CL5 has shown to be sound and complete with respect Japridze’s abstract resource semantics, it has also shown to validate a strictly larger class of formulas than affine logic (sequent calculus without the contraction rule), the latter being well studied and sound as a logic of resources. However, due to no shortcomings in effort, a complete resource aware semantics has not been found for affine logic. This had lead Japridze to conclude that “CL5 rather than affine logic adequately materializes the resource philosophy traditionally associated with the latter”. Indeed, the “semantics before syntax” philosophy upon which CL5 was conceived, together with its completeness result provide compelling evidence in this direction.
In parallel with the development of cirquent calculus, recent work [3, 4, 5, 6, 10] has introduced sound and complete axiomizations for several fragments of CoL, the most fundamental of these being CL4, which contains the propositional connectives (negation), (parallel disjunction), (parallel conjunction), (choice disjunction) and (choice conjunction) as well as the “choice” quantifiers and and the “blind” quantifiers and . For a full discussion of these operators and the semantics of CoL, see . While the semantics of CoL differs drastically from that of classical logic, the set of valid formulas of CL4 is identical to that of classical logic when restricted to its operators and the so-called “elementary” sort of atoms. Thus, the presence of the quantifiers and within CL4’s language guarantees its undecidability. In , however, the ,-free fragment of CL4 was shown to be decidable in polynomial space. A PSPACE-completeness proof was later given in .
The deductive apparatus of CL4 is far removed from traditional Gentzen or Hilbert style axiomizations. Further, the system does not extend very naturally to other fragments of Computability Logic. A rectification to this shortcoming came with the advent of cirquent calculus. As mentioned previously, the fragment of cirquent calculus not containing the contraction rule, produces a sound and complete system for the fragment of Computability Logic known as CL5. This system contains the operators , and and in this paper we show it to be -complete.
We further investigate the complexity of the logical system described by the rules of CL5, but with the absence of the duplication rule. We call this logic CL5 and show that every theorem in CL5 has a polynomial size proof. This effectively places the problem of deciding CL5 provability in . A reduction from the vertex cover problem to CL5 provability is then given, solidifying the logic as NP-complete. In some sense, CL5 is the CoL “counterpart” to multiplicative linear logic, the former being a proper extension of the latter. Here, CL5 lies strictly between multiplicative affine logic (linear logic with weakening) and CL5. Among the virtues of multiplicative affine logic is that, unlike the coNP-complete classical logic, it is in NP. CL5 thus presents a nice and natural extension of multiplicative affine logic that gets “closer” to classical logic while still remaining in NP.
With the tight relationship between cirquent calculus and Computability Logic, one should not overlook the potential impact of cirquent calculus on other areas of logic and proof theory. In , Xu constructs a cirquent calculus based system for the propositional fragment of independence friendly (IF) logic, allowing one to account for independence from propositional connectives in the same vein that traditional IF logic accounts for independence from quantifiers. In  a cirquent calculus based proof system was developed which, among other benifits, yields polynomial size proofs for all instances of the pigeonhole principle. To date, this is the first proof system to achieve such a result without leveraging cut, extension or substitution.
This paper is organized as follows. In section The Computational Complexity of Propositional Cirquent Calculus, we provide an introduction to the rules of cirquent calculus and describe the systems CL5 and CL5. Section The Computational Complexity of Propositional Cirquent Calculus studies the system CL5 and proves its NP-completeness. In the final section, we show the full CL5 to be -complete by a reduction from the TQBF- problem.
section1[Core cirquent calculus]Core cirquent calculus
The present section provides the technical definitions and information necessary for understanding the results of this paper. For a full discussion of the concepts of cirquent calculus, see [7, 9]. As noted previously, a formula of the language of cirquent calculus is syntactically identical to that of classical propositional logic, where formulas are built from the connectives , and as well as non-logical atoms (also known as propositional letters). The only difference between the two being that the language of cirquent calculus does not contain logical atoms such as or ; the latter can simply be understood as abbreviations of and , respectively, for some (whatever) atom . Our language further mandates that can only be applied to atoms. As always, a literal is an atom with or without the prefix .
A -ary pool, for , is a sequence Pl = of formulas. The formulas in a pool are not required to be unique. We refer to a particular occurrence of a formula as an oformula. A -ary structure, for , is a finite sequence St = for , where each , called a group of St, is a subset of . It is permitted that for and hence we use the term ogroup to refer to a particular occurrence of a group. We are now ready to give our definition for a cirquent.
 A -ary () cirquent is a pair = (St, Pl), where St, called the structure of , is a -ary structure, and PL, called the pool of , is a -ary pool.
For example, let Pl = and St = . Here, has 4 oformulas and 3 ogroups and is typically represented by the diagram below.
Following , we will adopt the notion that a diagram simply “is” (rather than just “represents”) a cirquent. When an ogroup is connected with an arc to an oformula , we say that contains .
Before defining the rules for our inference system, let us first give some additional terminology necessary for the definition of our rules. For a given cirquent, two oformulas and are called adjacent if is positioned immediately to the right of . In such a case, it is said that immediately precedes and immediately follows . Merging two adjacent ogroups and in a given cirquent means replacing in the two ogroups and by a single ogroup . The rightmost cirquent below represents the cirquent that results from merging the first and second ogroups in the leftmost cirquent.
Merging two adjacent oformulas and into a new oformula is the result of replacing an by and redirecting to it all of the arcs that were pointing to or . The rightmost cirquent below represents the cirquent obtained from merging, in the leftmost cirquent, (the first) and into a single oformula .
With these definitions in mind, we are now equipped to present the so-called “core” cirquent calculus rules. The rules we give here are only those relevant to our language and are by no means exhaustive. In its full generality, CoL encompasses numerous other logical operators all accompanied by a deep and meaningful semantics. As predicted earlier in , recent works have produced cirquent calculus axiomazations for logics containing some of the more powerful operators of Computability Logic. See, for example, [10, 11].
subsection2[Axioms (A)]Axioms (A) Axioms are rules with an empty set of premises. In cirquent calcus, they come in two forms: the empty cirquent axiom and the identity axiom. Both varieties of the axiom rule are illustrated below. It is important to note that the identity axiom is actually a scheme of infinitely many axioms, as stands for an arbitrary formula.
As will be our convention throughout, the letter placed to the right of the horizontal line represents the rule by which the conclusion was obtained.
subsection2[Mix (M)]Mix (M)
The mix rule takes two premises. Its conclusion is obtained by placing each of the two premise cirquents side by side in a single cirquent, as illustrated below.
The remaining rules described in subsections The Computational Complexity of Propositional Cirquent Calculus to The Computational Complexity of Propositional Cirquent Calculus all take a single premise cirquent.
subsection2[Exchange (E)]Exchange (E)
The exchange rule comes in two varieties, oformula exchange and ogroup exchange. The conclusion of the oformula exchange rule is obtained by swapping the positions of two adjacent oformulas in the premise cirquent. The ogroup exchange rule is similar in that it allows two adjacent ogroups in a premise cirquent to exchange positions in the conclusion. In both varieties of the rule, the arcs from each ogroup to its oformulas should be preserved. Below is an example of oformula (resp. ogroup) exchange in which the oformulas F and G (resp. ogroups 1 and 2) are swapped.
subsection2[Weakening (W)]Weakening (W) The weakening rule has two forms; ogroup weakening and pool weakening. In the case of ogroup weakening, the conclusion is obtained from the premise by adding a new arc between a pre-existing ogroup and oformula pair. In a application of pool weakening, the conclusion is obtained by inserting a new oformula at any position in the pool of the premise.
subsection2[Duplication (D)]Duplication (D) The duplication rule can be applied in one of two ways, called downward duplication and upward duplication. The conclusion of the downward duplication rule is obtained from its premise by replacing an ogroup with two adjacent ogroups that each have arcs to exactly the same oformulas as the original ogroup. An application of upward duplication works in the opposite direction in that the premise cirquent is obtained by replacing an ogroup in the conclusion by two adjacent ogroups that are both identical to the original ogroup.
subsection2[Contraction (C)]Contraction (C) The contraction rule takes a premise cirquent which contains two adjacent and identical oformulas. The conclusion of an application of this rule is obtained from the premise by merging two adjacent and identical oformulas and into a single oformula .
subsection2[-introduction ()]-introduction () The conclusion of this rule is obtained from the premise by merging two adjacent oformulas and into a single oformula such that all arcs pointing to either or now point to . Two examples of an application of this rule are given below.
subsection2[-introduction ()]-introduction ()
This rule takes a premise cirquent that contains two adjacent oformulas and such that no ogroup contains both and and every ogroup that contains (resp. ) is immediately followed (resp. preceded) by an ogroup containing (resp. ). The conclusion in an application of this rule is obtained from its premise by merging each ogroup that contains with the ogroup containing that immediately follows it. The oformulas and should then merge into . We again give two examples below.
subsection2[The systems CCC, CL5 and CL5]The systems CCC, CL5 and CL5
The cirquent calculus system built from all eight of the above rules has been aptly named “Classical Cirquent Calculus”, or CCC. By removing the contraction rule from CCC, we get the system CL5. We further let CL5 denote the system that results from additionally removing the duplication rule from CL5.
Let be one of the cirquent calculus systems CCC, CL5 or CL5. A proof of a cirquent in is a tree of cirquents whose root is where each node follows from its children by one of the rules of . When we say a formula is provable in , we mean the cirquent containing a single oformula with one ogroup and arc is provable in . In this paper, for simplicity, we are only interested in proving formulas, even though all of our results almost straightforwardly extend from formulas (as special cases of cirquents) to all cirquents. Correspondingly, we agree that, unless suggested otherwise by the context, “provability” means “provability of formulas”. It was shown in  that the provable formulas of CCC coincide exactly with those of classical propositional logic. Interestingly enough, the provable formulas of CL5 can also be described in a very natural way. Because we will rely on this result and its associated concepts in several of our later proofs, we shall give the relevant details here.
A substitution for a formula is a function that maps every atom in to some formula . If is an atom for every in , then we say that is an atomic-level substitution. This notion can be extended to all formulas by requiring that , and . Let and be formulas. is said to be an instance of iff there exists a substitution such that . If is an atomic level substitution, then is an atomic-level instance of . A formula is called binary iff no atom has more than two occurrences in it. A binary formula is said to be normal iff, whenever an atom occurs twice in the formula, one occurrence is positive and the other is negative. The following theorem is a combination of Theorem 12 and Lemma 9 of .
(Japaridze) A formula is provable in CL5 iff it is an instance of a binary tautology iff it is an atomic-level instance of a normal binary tautology.
At this point, a semantical characterization of the provable formulas of CL5 has yet to be given. In light of the results of this paper, such a finding would perhaps be very interesting.
section1[CL5 without duplication is NP-complete]CL5 without duplication is NP-complete
In this section, we show that deciding provability for a formula of the logic CL5 is NP-complete. This result is achieved by a combination of two theorems, the first of which states that every formula provable in CL5 has a proof whose size is polynomial in the size of . Towards this goal, we begin by presenting a set of technical lemmas.
Every formula provable in CL5 is an atomic-level instance of a normal binary tautology.
This follows immediately from the “” direction of Theorem 0.2, given that the theorems of CL5 form a subset of the theorems of CL5.
Let be a formula provable in CL5 with positive occurrences of atoms. No CL5 proof of can contain a cirquent with more than ogroups.
Let and be as in the condition of the lemma and let be a proof tree for . Assume for a contradiction that has a cirquent that contains more than ogroups. Now let be the subtree of rooted at . A careful examination of the rules of CL5 will show that, no matter which series of rules are applied in , the total number of ogroups in all the leaves of must be greater than or equal to the number of ogroups in . This is because the premise cirquent(s) in the application of a CL5 rule must, together, contain at least as many ogroups as the conclusion. Every leaf in must be derived by either the identity axiom or the empty cirquent axiom and thus cannot contain more than 1 ogroup. This means has at least leaves with a single ogroup, each of which must follow by the identity axiom. To be a consequence of the identity axiom, a cirquent must contain exactly two oformulas and . But the total number of positive occurrences of atoms in oformulas in the leaves of cannot exceed . This is because, by our assumption, the conclusion of contains a single oformula with positive literals. Further, the oformulas of the premise cirquent(s) in the application of a CL5 rule cannot contain more positive occurrences of atoms than the oformulas of the conclusion. This means at most applications of the identity axiom are possible, contradiction.
The length of a formula is defined as the total number of occurrences of literals and connectives in . The size of a group is the total number of oformulas it contains. The size of a cirquent is then the sum of the lengths of the oformulas in its pool plus the sum of the sizes of each ogroup in its structure. Naturally, the size of a proof is the sum of the sizes of the cirquents it contains.
Let be a formula provable in CL5 and let be the length of . There exists a CL5 proof of using applications of rules.
Let be a formula and let be a CL5 proof tree for . Given , we show that a new proof can be obtained from such that proves using applications of rules.
We begin by noting that any proof containing applications of the empty cirquent axiom can be transformed into one in which no applications of the rule are made111The only purpose of this axiom in  is to ensure the provability of the empty cirquent itself; everything else is provable without using the empty cirquent axiom.. Indeed, observe that the empty cirquent can only be a (“dummy”) premise of mix. In such a case, the conclusion of mix is simply the same as its other premise. Therefore, the empty cirquent can be deleted and the application of mix can be skipped. This can be done for all empty cirquents contained in the proof until none remain. Let be the proof that results from removing all applications of the empty cirquent axiom in the preceding manner.
Observe now that must contain exactly one application of conjunction introduction or disjunction introduction for each occurrence of or in , respectively. This is because every application of either rule introduces, in its conclusion, a single or connective that cannot be later removed by the application of any CL5 rule. The number of positive occurrences of atoms in also bounds the number of applications of the identity axiom. This is because, in a bottom up view of a proof, no CL5 rule allows formulas to be removed or merged. With the number of applications of conjunction introduction, disjunction introduction and the identity axiom all bounded by in , we can also bound the number of applications of the mix rule. To see this, first note that because contains no applications of the empty cirquent axiom, every leaf node must be derived by the identity axiom. That is, there are paths from root to leaf in . Viewing the proof tree in a top down fashion, mix is the only CL5 rule which allows the proof tree to “branch”, given it is the only rule with more than one premise. Thus, each application of mix, which must take two premises, increases the number of paths from root to leaf by 1. Because the number of such paths is bounded by , so too must be the number of applications of the mix rule.
Each application of pool weakening introduces, in its conclusion, an oformula that must be present (perhaps only as a proper subformula of some oformula) in the conclusion of . This is because no CL5 rule can remove, in its conclusion, an oformula contained in its premise cirquent. Thus we have an immediate bound of for the number of applications of pool weakening in . Observe that this also implies a bound of on the number of oformulas contained in any cirquent. To bound the number of possible applications of ogroup weakening in , notice that no cirquent in can contain more than arcs. This is because, by Lemma 0.4, bounds the maximum number of ogroups in any cirquent and, as noted previously, bounds the number of oformulas in any cirquent. Further, conjunction introduction and disjunction introduction are the only CL5 rules whose conclusion can contain fewer arcs than its premise. The number of applications of these rules in does not exceed and each application can remove no more than the maximum arcs contained in any premise cirquent. Thus, no more than total arcs can be removed from premise to conclusion for all applications of conjunction and disjunction introduction in . Each application of ogroup weakening creates a single arc in its conclusion that was not present in its premise. As the conclusion of contains a single arc, no more than arcs can be created from premise to conclusion in by applications of ogroup weakening.
Every leaf of must be derived by the identity axiom, which is applied at most times in . That is, there are at most paths from root to leaf in . Each such path for contains at most applications of CL5 rules other than oformula or ogroup exchange. Thus, any can contain at most sequences for where each (for ) is a sequence of cirquents such that every cirquent follows from by exchange (of either sort) for all . In , each can be of arbitrary length. We show, however, that a new proof can be obtained from such that each uses no more than applications of oformula and ogroup exchange. This is because, the last cirquent in any can always be obtained from using no more than applications of oformula and ogroup exchange. By lemma 0.4, the first cirquent can contain at most ogroups. Moving all of these ogroups to an arbitrary position in can be done with applications of ogroup exchange. Additionally, moving each of possible oformulas in to an arbitrary position in also requires no more than applications of oformula exchange. This means that every sequence in can be replaced with a new sequence such that uses no more than applications of oformula or ogroup exchange. Let be the CL5 proof tree that results from replacing each in by . must again contain paths from root to leaf. Each such path now has a bound of applications of oformula or ogroup exchange, resulting in a total bound of applications of exchange in all branches of .
It should be clear that if proves , then will also prove . The number of applications of non-exchange rules does not change from to and remains . Further, uses no more than applications of exchange. Totaling the number of applications of every rule in , we obtain a bound of .
Let be a formula provable in CL5 and let be the size of . There exists a CL5 proof of whose size is polynomial in .
By lemma 0.5, there exists a CL5 proof of that uses a polynomial number of rule applications. Further, by lemma 0.4, the size of any cirquent in must be polynomial in . Thus, the maximum size of each cirquent multiplied by the maximum number of cirquents in yields a polynomial bound on the size of .
The result of Theorem 0.6 effectively places the provability problem for CL5 formulas in NP. We further this result by additionally showing that the problem is NP-complete. Before giving the proof, we solidify some standard concepts that will be used within it. A graph is an ordered pair made up of a set of vertices () and edges (), with each edge being an unordered pair of vertices. The degree of a vertex in , denoted deg(v), is defined as the number of edges incident to . Given a graph , a vertex cover is such that every edge of is incident to at least one vertex in . In complexity theory, the vertex cover problem can be stated as a decision problems as follows. Given a graph and number , does have a vertex cover, i.e. a vertex cover using at most vertices? It is well known that this problem is NP-complete.
Deciding provability for the logic CL5 is NP-complete.
It follows from Theorem 0.6 that CL5 provability is in NP. To see it is NP-hard, we give a polynomial time reduction to it from the vertex cover problem. Fix some arbitrary graph and some . The reduction, borrowed from , follows.
Above, is a new atom that differs from all other atoms in the formula. In , and are the vertices on the endpoints of edge for . It is also important to note that each for represents both an atom in as well as the label of a vertex in .
Obviously the mapping is computable in polynomial time. It now remains to show that a graph has a vertex cover if and only if the formula is provable in CL5.
“” Our reduction is identical to that of  (Section 5.2) for multiplicative affine (direct) logic. As a direct consequence of Theorem 3 from , CL5 proves every formula provable in multiplicative affine logic222From Theorem 3 of , we have affine logic = CL5, where CL5 is the system CL5 with the limitation that cirquents contained in proofs do not have groups that share oformulas. Because upward (resp. downward) duplication is only applicable when the premise (resp. conclusion) of the rule is a cirquent in which groups share oformulas, all cirquents provable in CL5 are also provable in CL5. , so the result follows immediately.
“” Assume is provable in CL5. We need to show that there exists a vertex cover of . By Theorem 0.2, there exists a normal binary tautology and an atomic-level substitution such that . Let be the subformula of such that . Let be the set of the atoms of , and let be the set of those members of that have (not only positive but also) negative occurrences of . For each , let be the conjunct in the component of that contains . By our construction, each takes the form
for . Define a vertex cover for as the set of all vertices in labeled where is contained in for some .
To see that is indeed a vertex cover for , note first that is an atomic-level substitution, and hence each positive literal must be mapped to a unique positive occurrence of in . By the definition of , however, there are exactly positive occurrences of in . This means contains no more than atoms and subsequently the vertex cover derived from contains no more than vertices.
Because is an atomic-level instance of , both formulas take exactly the same form, the only difference being in the atoms they are built from. Let us now define a model for . For each let . For any atom with the property that , let . Further, for each atom of where for some , let . All remaining atoms of should be interpreted as .
Note that every atom of the earlier defined is interpreted as , meaning that . Next let be the subformula of such that . Each disjunct of takes the form for some positive integer . If then and , making the entire subformula evaluate to under . If then and . Note, however, that when then where and . This means and , again making the subformula evaluate to under . Because each disjunct of evaluates to under , we have .
Finally, let be the subformula of such that . Because is a tautology where and , we have . Each conjunct of takes the form such that and are the endpoints of an edge in . Further, by our construction, every edge in is represented by such a conjunct in . Observer that, by our definition of , we have for an atom of only when . Thus, is true only when every edge in has an endpoint in . That is, is a vertex cover of .
section1[CL5 is -complete]CL5 is -complete
This section contains our main result, namely, the -completeness of the decision problem for provability of formulas in CL5.
Deciding provability for CL5 is in .
The following is a algorithm that, if view of Theorem 0.2, decides provability of a formula in CL5. On input , existentially guess a binary formula such that is an instance of . Then, universally guess a truth assignment for . If is true under , accept. Otherwise, reject.
Let TQBF- be the problem of deciding truth for a quantified Boolean formula of the form , where and are sequences of variables and is a quantifier-free Boolean formula all of whose variables are among or . As shown in Theorem 4.1 of , this problem is -complete. We say an atom is isolated in a formula if there is only a single (positive or negative) occurrence of in . Otherwise it is non-isolated.
Deciding provability for the logic CL5 is -complete.
By lemma 0.8, the problem of deciding provability for CL5 is in . To show it is -hard, we construct a polynomial time mapping reduction from TQBF- to CL5-provability. We can safely restrict our attention to instances of TQBF- where the Boolean portion of the formula is in disjunctive normal form, as the complexity of the problem is not reduced with these limitations. Our reduction follows.
Let be a formula that takes the form with and being sets of variables and being a boolean formula in disjunctive normal form all of whose variables are among . The following steps are used to construct the corresponding CL5 formula .
For each , let and be the number of positive occurrences of the literals and in , respectively. Define
where , and are all fresh333Here, fresh variables are those not occurring elsewhere in . pairwise distinct variables, uniquely chosen for . Here, if (resp. ) is 0, the second (resp. third) conjunct should be omitted. Now let and let be the formula . Here is a formula derived from such that, for each , every positive occurrence of in is replaced by a unique literal from and every occurrence of in is replaced by a unique literal from .
Consider any . Let be the number of positive occurrences of in , and be the number of negative occurrences. For each pair with and , we choose a fresh and unique variable . Now, define to be the result of replacing in , for each , every positive occurrence of the literal by and every (positive) occurrence of the literal by where (resp. ) is unique for each replacement of an occurrence of (resp. ).
“” Assume is true. Then there exists some truth assignment such that is true under any truth assignment that extends to . By our assumption, is in disjunctive normal form and hence takes the form
where each is either a positive or negative literal. Now let and represent the antecedent and consequent of the outermost implication in such that . Our construction guarantees that takes the form
where each is a disjunction of literals (such a “disjunction” may have only a single “disjunct”). That is, is obtained from by replacing each oliteral by , where is a disjunction of literals. We will henceforth use to represent a unique position in and to represent the corresponding position in .
We want to show that is provable in CL5. By Theorem 0.2, it suffices to show that is an instance of a binary tautology. We construct a binary tautology of which is an instance. Namely, we let be the formula obtained from as follows. For each , if (resp. ) replace the third (resp. second) occurrence of by an atom such that does not occur elsewhere in and is unique for each . It should be easy to see that is a quantifier free binary formula and is an instance of . We need only show that is a tautology. Again notice that takes form where is the same as in .
Given , we have where each matches one of the following forms444It is possible that the third (resp. second) conjunct in formula 1 (resp. 2) is absent..
For any in where , if then , and hence . If , then all of the literals in the consequent of the implication in with antecedent literal must be true under , otherwise we will have and again . Thus, we need only guarantee under truth assignments such that, for each conjunct in , and every literal in the consequent of the implication containing antecedent literal are true under .
Let be the formula that results from replacing in every positive occurrence of or , where , by its truth value under . For example, if , replace by and by . It should be easy to see that is a tautology. For any position that contains (resp. ) in and in , the position in must contain a single literal (resp. ) such that (resp. ) if . This is because, by our construction, if (resp. ), every for (resp. for ) must occur in the consequent of the implication in with non-isolated antecedent . For reasons already discussed, such a literal must be true under any truth assignment where . Define as the formula such that, for each occurrence of or in position of , the literal in position of is replaced by the same value ( or ) as . Note that only substitutes the logical atom in position of when contains a positive or negative occurrence of an atom in . Since our goal is to show that is true under truth assignments that make true, we need only show that is a tautology.
For a contradiction, assume is not a tautology. Then there exists some truth assignment defined on the variables of such that . We define a truth assignment for as follows. If a variable is such that, for some with , , we let ; otherwise we let . It is not hard to see that, if a subformula of in a position is false under , then the subformula of in the corresponding position is false under (but not necessarily vice versa). This, in view of the monotonicity of and , obviously implies that , because and have the same -structures. Now we are dealing with a contradiction, because the tautological cannot be false.
“” Assume is provable in CL5. By Theorem 0.2, for some normal binary tautology there exists an atomic level substitution such that . Let , where and represent the antecedent and consequent of the outermost implication in . We want to define a truth assignment such that is true for any truth assignment that extends to . In what follows, we define such a partial truth assignment for while concurrently defining a partial truth assignment for .
Procedure 1 - Consider a and let be the atom of such that the first occurrence of in originates from A (i.e., A gets replaced by ) when transitioning from to .
Case 1: The second and third occurrences of in originate from and , respectively (for some in ). Define , and . The consequent555Here and later in similar contexts, the “consequent of W” should be understood as the consequent of the implication whose antecedent is W. Similarly for “antecedent of W”. of the second occurrence of in should take the form for some positive literals in and the consequent of should take the form for some negative literals . Let all of and be true under .
Case 2: The second and third occurrences of in originate from and , respectively (for some in ). Define , and . The consequent of the second occurrence of in should take the form for some negative literals and the consequent of should take the form for some positive literals . Let all of and be false under .
Case 3: There are only two occurrences of in , both of which originate from . If the consequent of the second occurrence of in takes the form for some positive literals then define and . Further, let all of be true under . If the consequent of the second occurrence of in takes the form for some negative literals then define and . Further, let all of be false under .
Case 4: If none of the cases are satisfied, then the second and third occurrences of in must originate from and , respectively (where and ). Define , and . Further, let every atom in the consequent of evaluate to under and every atom in the consequent of evaluate to under .
As we remember, in disjunctive normal form, taking the form
where each is a literal. It is also the case that takes the form
where each is a disjunction of literals (possibly with just a single “disjunct”). Thus, we will use our previous convention in which represents a unique position in and represents the corresponding position in .
Our construction guarantees that for each position in containing some positive (resp. negative) literal (resp. ) such that , contains a positive (resp. negative) literal (resp. ). By procedure 1 we have . As in the previous direction, let be the formula that results from replacing, in , for every , all positive occurrences of the literals and by their truth values under . Let be the formula such that, for each position in containing a logical atom , the literal in the corresponding position of is replaced by . The interpretation defined as part of procedure 1 is such that . Because is a tautology, any extension of defined on all atoms of must make true. We also know that for each position in that contains a logical atom (resp. ), the literal in position of evaluates to (resp. ) under . This means that is true regardless of how its non-logical atoms are interpreted and is a tautology. Our goal is to show that is a tautology as well.
Pick some arbitrary truth assignment for . We show . For each position in containing a positive literal , the corresponding position in contains a disjunction of positive literals () such that , , for some . Similarly, for each position in containing a negative literal , the corresponding position in contains a disjunction of negative literals () such that , , , for some . We now define a truth assignment for . If , let be such that all of the corresponding literals , for every evaluate to false under . If , let be such that all corresponding literals , for every evaluate to false under . With some thought, one can see that whenever has a false (under ) literal in a position , has a false (under ) disjunction of literals in the corresponding position . So, if , then