O determinizaci zásobníkových automatů a konverzi regulárních stromových výrazů na determinizovatelné zásobníkové automaty

Abstract

Trees are one of the fundamental and well-studied data structures in Computer Science. Arbology uses pushdown automata (PDAs), which read linearised notations of trees, as its suitable model of computation. Regular tree expressions (RTEs) are a formalism for describing regular tree languages (RTLs), whose standard model of computation is usually a finite tree automaton. It has been proven that the class of regular tree languages is a proper subclass of tree languages whose linear notations can be accepted by deterministic (string) PDAs. In this dissertation thesis, two new algorithms for converting RTEs to equivalent real-time height-deterministic PDAs (rHPDAs) that accept the trees in their linear notation are presented. Visibly pushdown automata (VPDAs), introduced by Alur and Madhusudan in 2004 [6], are PDA whose pushdown operations are determined by the input symbol, where the input alphabet is partitioned into three parts for the push, pop, and local pushdown operations. It is well known that nondeterministic VPDAs can be determinised. Height-deterministic pushdown automata (HPDAs), a natural generalisation of VPDAs where for any given input string the stack heights during any (nondeterministic) computation are a priori fixed, were introduced by Dirk Nowotka and Jiˇr´ı Srba in 2007 [36]. This can be seen as a generalisation of the property of VPDAs where the input alphabet is partitioned into three parts for a push, pop and local pushdown operations which implies this condition. The subgroup of rHPDAs was proven to be determinisable. In this dissertation thesis, new algorithms for the determinisation of VPDAs and rHPDAs are presented. The algorithms are based upon theoretical proofs of determinisation and result in reasonably small deterministic PDAs for most practical examples. This is achieved in a way that only necessary and accessible states and pushdown symbols are computed and constructed during the determinisation.

Trees are one of the fundamental and well-studied data structures in Computer Science. Arbology uses pushdown automata (PDAs), which read linearised notations of trees, as its suitable model of computation. Regular tree expressions (RTEs) are a formalism for describing regular tree languages (RTLs), whose standard model of computation is usually a finite tree automaton. It has been proven that the class of regular tree languages is a proper subclass of tree languages whose linear notations can be accepted by deterministic (string) PDAs. In this dissertation thesis, two new algorithms for converting RTEs to equivalent real-time height-deterministic PDAs (rHPDAs) that accept the trees in their linear notation are presented. Visibly pushdown automata (VPDAs), introduced by Alur and Madhusudan in 2004 [6], are PDA whose pushdown operations are determined by the input symbol, where the input alphabet is partitioned into three parts for the push, pop, and local pushdown operations. It is well known that nondeterministic VPDAs can be determinised. Height-deterministic pushdown automata (HPDAs), a natural generalisation of VPDAs where for any given input string the stack heights during any (nondeterministic) computation are a priori fixed, were introduced by Dirk Nowotka and Jiˇr´ı Srba in 2007 [36]. This can be seen as a generalisation of the property of VPDAs where the input alphabet is partitioned into three parts for a push, pop and local pushdown operations which implies this condition. The subgroup of rHPDAs was proven to be determinisable. In this dissertation thesis, new algorithms for the determinisation of VPDAs and rHPDAs are presented. The algorithms are based upon theoretical proofs of determinisation and result in reasonably small deterministic PDAs for most practical examples. This is achieved in a way that only necessary and accessible states and pushdown symbols are computed and constructed during the determinisation.