1 Extended abstract
Propositional dynamic logic (PDL) is presented in Schütte-style mode as one-sided semiformal tree-like sequent calculus Seq with standard cut rule and the omega-rule with principal formulas . The omega-rule-free derivations in Seq are finite (trees) and sequents deducible by these finite derivations are valid in PDL. Moreover the cut-elimination theorem for Seq is provable in Peano Arithmetic (PA) extended by transfinite induction up to Veblen’s ordinal . Hence (by the cutfree subformula property) such predicative extension of PA proves that any given -free sequent is valid in PDL iff it is deducible in Seq by a finite cut- and omega-rule-free derivation, while PDL-validity of arbitrary star-free sequents is decidable in polynomial space. The former also implies a Herbrand-style conclusion that e.g. a given formula for star-free and is valid in PDL iff there is a and a cut- and omega-rule-free derivation of sequent where is an abbreviation for . This eventually leads to PSPACE-decidability of PDL-validity of , provided that is atomic and is in a suitable basic conjunctive normal form. Furthermore we consider star-free formulas in dual basic disjunctive normal form, and corresponding expansions whose PDL-validity problem is known to be EXPTIME-complete. We show that cutfree-derivability in Seq (hence PDL-validity) of such is equivalent to plain validity of a suitable “transparent” quantified boolean formula . Hence EXPTIME = PSPACE holds true iff the validity problem for any involved is solvable by a polynomial-space deterministic TM. This may reduce the former problem to a more transparent complexity problem in quantified boolean logic. The whole proof can be formalized in PA extended by transfinite induction along – actually in the corresponding primitive recursive weakening, .
2 Introduction and survey of results
Propositional dynamic logic (PDL) was derived by M. J. Fischer and R. Ladner ,  from dynamic logic where it plays the role that classical propositional logic plays in classical predicate logic. Conceptually, it describes the properties of the interaction between programs (as modal operators) and propositions that are independent of the domain of computation. The semantics of PDL is based on Kripke frames and comes from that of modal logic. Corresponding sound and complete Hilbert-style formalism was proposed by K. Segerberg  (see also , ). Gentzen-style treatment is more involved. This is because the syntax of PDL includes starred programs which make finitary sequential formalism similar to that of (say) Peano Arithmetic with induction (PA) that allows no full cut-elimination. In the case of PA, however, there is a well-known Schütte-style solution in the form of infinitary (also called semiformal) sequent calculus with Carnap-style omega-rule that allows full cut elimination, provably in PA extended by transfinite induction up to Gentzen’s ordinal (cf. , ). By the same token, in the case of PDL, we introduce Schütte-style semiformal one-sided sequent calculus Seq whose inferences include the omega-rule with principal formulas and prove cut-elimination theorem using transfinite induction up to Veblen’s predicative ordinal (that exceeds , see , ). The omega-rule-free derivations in Seq are finite and sequents deducible by these finite derivations are valid in PDL. Hence by the cutfree subformula property we conclude that any given -free sequent is valid in PDL iff it is deducible in Seq by a finite cut- and omega-rule free derivation, which by standard methods enables better structural analysis of the validity of -free sequent involved. 111cf. e.g. Gentzen-style conclusion that any given false equation (in particular ) is not valid in PA, since obviously it has no cutfree derivation. The latter is related to computational complexity of decision problems in PDL. Namely, the satisfiability (and hence the validity) problem in PDL is known to be EXPTIME-complete (cf. , ). Actually the EXPTIME-completeness holds for PDL-validity of special -free basic disjunctive normal expansions (abbr.: ), whose negations express that satisfying Kripke frames encode accepting computations of polynomial-space alternating TM. Thus the conjecture EXPTIME = PSPACE holds true iff PDL-validity of is decidable in polynomial space. We show that cutfree-derivability in Seq (and hence PDL-validity) of any given , , is equivalent to the validity of a suitable “transparent” quantified boolean formula . Having this we conclude that EXPTIME = PSPACE holds true iff boolean validity of any involved is decidable by a polynomial-space deterministic TM. Hence EXPTIME = PSPACE holds true iff is equivalid with another quantified boolean formula whose size is polynomial in the size of , for every . This may reduce the former problem to a more transparent complexity problem in quantified boolean logic, which will be investigated more deeply elsewhere. The whole proof can be formalized in PA extended by transfinite induction along – actually in the corresponding primitive recursive weakening, .
3 More detailed exposition
3.1 Hilbert-style proof system Pdl
Programs PRO (abbr.: , , , , possibly indexed):
include P-variables , , …(abbr.: , , , possibly indexed),
are closed under modal connectives and and star operation .
Formulas FOR (abbr.: , , , , , , , , etc., possibly indexed):
include F-variables , , … (abbr.: , , , possibly indexed),
are closed under implication , negation and modal operation
, where . 222Boolean constants are definable as usual e.g. by and .
3.2 Semiformal sequent calculus SEq
The language of Seq includes seq-formulas and sequents. Seq-formulas are built up from literals and by propositional connectives and and modal operations and for arbitrary . Seq-negation is defined recursively as follows, for any seq-formula .
In the sequel we use abbreviations and . For any , let For any () and let . By , and we abbreviate for arbitrary , and , respectively. Formulas from are represented as seq-formulas recursively by , and, conversely, by , , . Sequents (abbr.: , , , , possibly indexed) are viewed as multisets (possibly empty) of seq-formulas. A sequent is called valid iff so is the corresponding disjunction . Plain complexity of a given formula and/or program in is its ordinary length (= total number of occurrences of literals and connectives , , , , ).
Ordinal complexity of formulas, programs and sequents in is defined recursively as follows, where is the symmetric sum of ordinals and .
Seq includes the following axiom and inference rules , , , , , , , , , in classical one-sided sequent formalism in the language . In we allow . 444We assume that all rules exposed have nonempty premises. 555 has infinitely many premises. Ii is called the -rule.
For the sake of brevity we’ll drop “seq-” when referring to seq-formulas of Seq. is called derivable in Seq if there exists a (tree-like, possibly infinite) Seq derivation with the root sequent (abbr.: ). We assume that Seq derivations are well-founded. The simplest way to implement this assumption is to supply nodes in with ordinals such that ordinals of premises are always smaller that the ones of the corresponding conclusions. Having this we let and call it the height of .
In Seq, formulas occurring in and/or are called side formulas, whereas other (distinguished) ones are called principal formulas, of axioms or inference rules exposed. These axioms and inferences, in turn, are called principal with respect to their principal formulas. Principal formulas of are also called the corresponding cut formulas. We’ll sometimes specify as to indicate principal program involved.
Theorem 4 (soundness and completeness)
Seq is sound and complete with respect to PDL. Moreover any PDL-valid sequent (in particular formula) is derivable in Seq using ordinals .
Proof. The soundness says that any sequent that is derivable in Seq is valid in Kripke-style semantics of PDL. It is proved by transfinite induction on of well-founded involved. 666Plain (finite) induction is sufficient for -free derivations. Actually it suffices to verify that every inference rule of Seq preserves Kripke validity, which is easy (we omit the details; see also Remark 5 below).
The completeness is proved by deducing in Seq the axioms and inferences , , , of PDL.
is deducible by standard method via extended axiom whose finite cutfree derivation is constructed by recursion on plain complexity of (in particular we pass by from to ).
and are trivial, while , are derivable as follows.
etc. via , and .
Obviously these derivations don’t use and require ordinal assignments . Seq inferences and are obviously derivable by and , respectively. These increase ordinals by one, which makes an arbitrary Hilbert-style PDL deduction interpretable as a Seq derivation of the height , as required.
The validity of also follows from that of , , and plain generalzation , e.g. like this:
3.3 Cut elimination procedure
3.3.1 Auxiliary sequent calculus SEq
Seq is a modification of Seq that includes the following upgraded inferences , , , .
Obviously these upgrades are still sound in PDL and cut-free derivable in Seq. Hence Seq and Seq are proof theoretically equivalent.
3.3.2 Derivable refinements
The following inferences are derivable in Seq minus. Moreover, for any inversion involved we have . In we assume that , and . Note that is a particular case of .
Proof. Induction on proof height and/or formula complexity. Cases , are standard. Note that with principal is trivial, e.g.
Case is an obvious iteration of .
Cases , , are standard (and trivial) boolean inversions.
Case ( analogous). We omit trivial case of principal inversion of and show only the crucial cases of principal (in simple form):
Case ( analogous). As above, we omit trivial case of principal inversion of and show the crucial cases of principal (in simple form):
Case is analogous to , via trivial inversion of :
Note that , , , , don’t increase derivation heights.
3.3.3 Cut elimination proper
Theorem 8 (Predicative cut elimination)
The following is provable in PA extended by transfinite induction up to Veblen-Feferman ordinal . Any sequent derivable in Seq is derivable in Seq minus . Hence any PDL-valid sequent (formula) is derivable in the cut-free fraction of Seq, and hence also in Seq minus .
Proof. Our cut elimination operator satisfying is defined for any derivation in Seq by simultaneous transfinite recursion on and ordinal cut-degree .
Namely, for any inference rule with
we respectively let
Otherwise, if with
then we stipulate
with respect to a suitable cut reduction operation such that
which makes , , definable by induction on and .
Now is defined for any
by following double induction on ordinal complexity of and , provided that .
1. Case and for . This case is standard. Namely, is principal left-hand side cut formula only if for . But then infers by derivable weakening . That is, graphically speaking, is bottom up constructed by (1) substituting for every side formula predecessor of the cut formula while ascending up to its disappearance due to or else principal appearance in followed by (2) adding to every side formula predecessor of the cut formula while ascending .
2. Case and . Use derivable inversions :
3. Case and . Analogous reduction to ’s on and by derivable inversions , , .
4. Case and . Immediate reduction to on by derivable inversions , .
5. Case and where and , while . The reduction is either trivial, if , or else defined hereditarily with respect to left-hand side non-principal subcases like
with, when we let
or analogous non-principal subcases ,
as well as the following principal subcases 5 , 5 , 5 .
5 . and
with . Let
5 . and
with. Then let
5 . and
with. Then we let
where is defined by induction on – either trivially, if , or hereditarily, in the non-principal subcases, while in the principal subcases