When is Containment Decidable for Probabilistic Automata?

The containment problem for quantitative automata is the natural quantitative generalisation of the classical language inclusion problem for Boolean automata. We study it for probabilistic automata, where it is known to be undecidable in general. We restrict our study to the class of probabilistic automata with bounded ambiguity. There, we show decidability (subject to Schanuel's conjecture) when one of the automata is assumed to be unambiguous while the other one is allowed to be finitely ambiguous. Furthermore, we show that this is close to the most general decidable fragment of this problem by proving that it is already undecidable if one of the automata is allowed to be linearly ambiguous

Affine extensions of integer vector addition systems with states

We study the reachability problem for affine Z-VASS, which are integer vector addition systems with states in which transitions perform affine transformations on the counters. This problem is easily seen to be undecidable in general, and we therefore restrict ourselves to affine Z-VASS with the finite-monoid property (afmp-Z-VASS). The latter have the property that the monoid generated by the matrices appearing in their affine transformations is finite. The class of afmp-Z- VASS encompasses classical operations of counter machines such as resets, permutations, transfers and copies. We show that reachability in an afmp-Z-VASS reduces to reachability in a Z-VASS whose control-states grow polynomially in the size of the matrix monoid. Our construction shows that reachability relations of afmp-Z-VASS are semilinear, and in particular enables us to show that reachability in Z-VASS with transfers and Z-VASS with copies is PSPACE-complete

Dynamic data structures for timed automata acceptance

We study a variant of the classical membership problem in automata theory, which consists of deciding whether a given input word is accepted by a given automaton. We do so through the lenses of parameterized dynamic data structures: we assume that the automaton is fixed and its size is the parameter, while the input word is revealed as in a stream, one symbol at a time following the natural order on positions. The goal is to design a dynamic data structure that can be efficiently updated upon revealing the next symbol, while maintaining the answer to the query on whether the word consisting of symbols revealed so far is accepted by the automaton. We provide complexity bounds for this dynamic acceptance problem for timed automata that process symbols interleaved with time spans. The main contribution is a dynamic data structure that maintains acceptance of a fixed one-clock timed automaton A with amortized update time 2O(|A|) per input symbol

When are Emptiness and Containment Decidable for Probabilistic Automata?

The emptiness and containment problems for probabilistic automata are natural quantitative generalisations of the classical language emptiness and inclusion problems for Boolean automata. It is well known that both problems are undecidable. In this paper we provide a more refined view of these problems in terms of the degree of ambiguity of probabilistic automata. We show that a gap version of the emptiness problem (that is known to be undecidable in general) becomes decidable for automata of polynomial ambiguity. We complement this positive result by showing that the emptiness problem remains undecidable even when restricted to automata of linear ambiguity. We then turn to finitely ambiguous automata. Here we give a conditional decidability proof for containment in case one of the automata is assumed to be unambiguous while the other one is allowed to be finitely ambiguous. Part of our proof relies on the decidability of the theory of real exponentiation, which has been shown, subject to Schanuel’s Conjecture, by Macintyre and Wilkie

Complexity of two-variable logic on finite trees

Verification of properties expressed in the two-variable fragment of first-order logic FO2 has been investigated in a number of contexts. The satisfiability problem for FO2 over arbitrary structures is known to be NEXPTIME-complete, with satisfiable formulas having exponential-sized models. Over words, where FO2 is known to have the same expressiveness as unary temporal logic, satisfiability is again NEXPTIME-complete. Over finite labelled ordered trees, FO2 has the same expressiveness as navigational XPath, a popular query language for XML documents. Prior work on XPath and FO2 gives a 2EXPTIME bound for satisfiability of FO2 over trees. This work contains a comprehensive analysis of the complexity of FO2 on trees, and on the size and depth of models. We show that different techniques are required depending on the vocabulary used, whether the trees are ranked or unranked, and the encoding of labels on trees. We also look at a natural restriction of FO2, its guarded version, GF2. Our results depend on an analysis of types in models of FO2 formulas, including techniques for controlling the number of distinct subtrees, the depth, and the size of a witness to satisfiability for FO2 sentences over finite trees

Complexity of two-variable logic on finite trees

Verification of properties expressed in the two-variable fragment of first-order logic FO2 has been investigated in a number of contexts. The satisfiability problem for FO2 over arbitrary structures is known to be NEXPTIME-complete, with satisfiable formulas having exponential-sized models. Over words, where FO2 is known to have the same expressiveness as unary temporal logic, satisfiability is again NEXPTIME-complete. Over finite labelled ordered trees, FO2 has the same expressiveness as navigational XPath, a popular query language for XML documents. Prior work on XPath and FO2 gives a 2EXPTIME bound for satisfiability of FO2 over trees. This work contains a comprehensive analysis of the complexity of FO2 on trees, and on the size and depth of models. We show that different techniques are required depending on the vocabulary used, whether the trees are ranked or unranked, and the encoding of labels on trees. We also look at a natural restriction of FO2, its guarded version, GF2. Our results depend on an analysis of types in models of FO2 formulas, including techniques for controlling the number of distinct subtrees, the depth, and the size of a witness to satisfiability for FO2 sentences over finite trees