Predicative proof theory of PDL and basic applications
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Propositional dynamic logic (PDL) is presented in Schütte-style mode as one-sided semiformal tree-like sequent calculus Seq_ω^pdl with standard cut rule and the omega-rule with principal formulas [ P^∗] A. The omega-rule-free derivations in Seq_ω^pdl are finite (trees) and sequents deducible by these finite derivations are valid in PDL. Moreover the cut-elimination theorem for Seq_ω^pdl is provable in Peano Arithmetic (PA)extended by transfinite induction up to Veblen's ordinal φ_ω( 0) . Hence (by the cutfree subformula property) such predicative extension of PA proves that any given [ P^∗] -free sequent is valid in PDL iff it is deducible in Seq_ω^pdl 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 S=〈 P^∗〉A∨ Z for star-free A and Z is valid in PDL iff there is a k≥ 0 and a cut- and omega-rule-free derivation of sequent A,〈 P〉 ^1A,...,〈 P〉 ^kA,B where 〈 P〉^iA is an abbreviation for itimes〈 P〉...〈 P〉A. This eventually leads to PSPACE-decidability of PDL-validity of S, provided that P is atomic and A is in a suitable basic conjunctive normal form. Furthermore we consider star-free formulas A in dual basic disjunctive normal form, and corresponding expansions S=〈 P^∗〉A∨ Z whose PDL-validity problem is known to be EXPTIME-complete. We show that cutfree-derivability in Seq_ω^pdl (hence PDL-validity) of such S is equivalent to plain validity of a suitable "transparent" quantified boolean formula S. Hence EXPTIME = PSPACE holds true iff the validity problem for any S 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 φ_ω( 0) -- actually in the corresponding primitive recursive weakening, PRA_φ_ω( 0).
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