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Variants of PDL with intersection of programs
Philippe Balbiani
To cite this version:
Variants of P DL with intersection of programs
Philippe Balbiani
Institut de recherche en informatique de Toulouse
Dynamic logics are logics arising from the combination of relation algebras with the modality of necessity. Together with their variations and close relatives, they constitute a part of the field of modal logic that is concerned in actions and programs. The propositional versions of dynamic logics are collectively called P DL. In P DL, a modal connective [α] is associated with each program α of a programming language formed from atomic programs by induction using oper-ations on binary reloper-ations. The chief operoper-ations on binary reloper-ations considered since the first development of P DL have been the operations ;, ∪, and⋆. The
op-eration ; is the sequential composition opop-eration, with programs like α; β being read “do α, then do β”. ∪ is the operation of nondeterministic choice. Programs of the form α ∪ β mean “nondeterministically choose one of α or β and execute it”. As for ⋆, the operation of iteration, the intended meaning of a program
like α⋆ is “execute α some nondeterministically chosen finite number of times”.
The semantics of P DL comes from the semantics of modal logic. A frame for the language sketchily described above should be a Kripke structure of the form (W, {Rα: α is a program}) with Rα a binary relation on W for each program
α. Intuitively, we can think of the binary relations Rα as sets of input / output
pairs of states. Thus: Rα; β should be equal to the relational composition of Rα
and Rβ, the input / output relation associated to α ∪ β should correspond to
the union of the relations Rα, Rβ, and α⋆ should be interpreted by the reflexive
transitive closure of Rα. From the beginning, a great deal of attention has been
focused on the complexity of P DL. Papers [5, 14] are but a sample of the work that has been done: the exponential-time lower bound for P DL being estab-lished by Fischer and Ladner [5] and deterministic exponential-time algorithms being given by Pratt [14]. But perhaps the key results on P DL come from Segerberg [15] who formulated a Hilbert-style deductive system for P DL and from Gabbay [6] and Parikh [12] who showed independently its completeness.
Subsequently, a variety of different extensions of P DL have been proposed for increasing the expressive power of the logic: P DL with nonregular pro-grams [8], P DL with converse operation [12], P DL with predicates for well-foundedness [16], etc. We shall concentrate in this paper on the extension of P DL studied in Danecki [3] and Harel et al. [9], P DL with intersection of programs, because this extension is linked to a number of P DL-like systems investigated in Fari˜nas del Cerro and Or"lowska [4], Gargov and Passy [7], and Mirkowska [11] under the various names DAL, BML, and P AL. P DL with
intersection is called P DL∩. The intersection operation ∩ is a program
oper-ation that allows programs to be run in parallel: programs like α ∩ β being read “do α and β in parallel” for each α, β. The essential feature of P DL∩ is
that Rα∩ Rβ, the input / output relation associated to α ∩ β in Kripke frames,
is not modally definable in the language of P DL. As a result, less is known about the complexity and the proof theory of P DL∩. Regarding the
complex-ity of P DL∩, an exponential-space lower bound for P DL∩ has been established
only this year [10] whereas it is still unknown at present whether the doubly exponential-time algorithm given by Danecki [3] is near-optimal. As regards the proof theory of P DL∩, a sound and complete Hilbert-style deductive
sys-tem for an extension of P DL∩ with data constants, i.e. special atomic formulas
interpreted in Kripke frames by singletons, has been presented by Passy and Tinchev [13] but technical difficulties have made impracticable the project of finding a sound and complete axiomatization for P DL∩in its ordinary language.
We shall present such an axiom system for P DL∩. Based on a step-by-step
con-struction, its completeness proof brings in the new concept of maximal program initiated by [2] and furthered by [1]. Like maximal theories, which are special sets of formulas, maximal programs are special sets of programs with properties to be thoroughly studied in this paper.
References
[1] Balbiani, P.: Eliminating unorthodox derivation rules in an axiom system for iteration-free P DL with intersection. Fundamenta Informaticæ56 (2003) 211–242.
[2] Balbiani, P., Vakarelov, D.: Iteration-free P DL with intersection: a com-plete axiomatization. Fundamenta Informaticæ45 (2001) 173–194.
[3] Danecki, R.: Non-deterministic propositional dynamic logic is decidable. In Skowron, A. (Editor): Computation Theory. Springer-Verlag, Lecture Notes in Computer Science 208 (1985) 34–53.
[4] Fari˜nas del Cerro, L., Orlowska, E.: DAL — a logic for data analysis. The-oretical Computer Science 36 (1985) 251–264.
[5] Fisher, M., Ladner, R.: Propositional dynamic logic of regular programs. Journal of Computer and System Sciences 18 (1979) 194–211.
[6] Gabbay, D.: Axiomatizations of logics of programs. Unpublished (1977). [7] Gargov, G., Passy, S.: A note on Boolean modal logic. In Petkov, P. (Editor):
Mathematical Logic. Plenum Press (1990) 299–309.
[8] Harel, D., Pnueli, A., Stavi, J.: Propositional dynamic logic of nonregular programs. Journal of Computer and System Sciences 26 (1983) 222–243.
[9] Harel, D., Pnueli, A., Vardi, M.: Two dimensional temporal logic and P DL with intersection. Unpublished (1982).
[10] Lange, M.: A lower complexity bound for propositional dynamic logic with intersection. This volume.
[11] Mirkowska, G.: P AL — propositional algorithmic logic. Fundamenta In-formaticæ4 (1981) 675–760.
[12] Parikh, R.: The completeness of propositional dynamic logic. In Winkowski, J. (Editor): Mathematical Foundations of Computer Science 1978. Springer-Verlag, Lecture Notes in Computer Science 64 (1978) 403– 415.
[13] Passy, S., Tinchev, T.: An essay in combinatory dynamic logic. Information and Computation 93 (1991) 263–332.
[14] Pratt, V.: A near-optimal method for reasoning about actions. Journal of Computer and System Sciences 20 (1980) 231–254.
[15] Segerberg, K.: A completeness theorem in the modal logic of programs. In Traczyk, T. (Editor): Universal Algebra and Applications. Polish Scientific Publishers, Banach Center Publications 9 (1982) 31–36.
[16] Streett, R.: Propositional dynamic logic of of looping and converse is ele-mentarily decidable. Information and Control 54 (1982) 121–141.
