Institute of Informatics, Warsaw University
A Complete Deductive System for
the µ-Calculus
Igor Walukiewicz∗
Doctoral Thesis
Written under the supervision of Professor Jerzy Tiuryn
Warsaw June, 1993
(*) This work was partially supported by Polish KBN grant No. 2 1192 91 01
To Parents
Contents
1 Overview 1
1.1 The µ-calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Other Logics of Programs . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Propositional Dynamic Logic . . . . . . . . . . . . . . 7
1.2.2 Total correctness ..................... 10
1.2.3 Temporal Logics ..................... 11
1.3 Automata on infinite objects . . . . . . . . . . . . . . . . . . 13
1.4 Games ............................... 16
1.5 Completeness problem ...................... 17
1.6 Proposed system ......................... 22
1.7 Synopsis .............................. 24
2 Characterization 27
2.1 Tableaux ............................. 28
2.2 Games ............................... 32
2.3 Characterization ......................... 33
2.4 Complexity ............................ 38
3 Axiomatization 41
3.1 The system ............................ 41
3.2 Discussion ............................. 47
4 Proof of Completeness 51
4.1 Automaton ............................ 55
4.2 Constructions on the graph ................... 66
4.3 Coding states into formulas ................... 69
4.4 Completeness proof ........................ 79
5 Conclusions 87
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