Symbolic Visibly Pushdown Automata

Abstract

Abstract. Nested words model data with both linear and hierarchical structure such as XML documents and program traces. A nested word is a sequence of positions together with a matching relation that connects open tags (calls) with the corresponding close tags (returns). Visibly Pushdown Automata are a restricted class of pushdown automata that process nested words, and have many appealing theoretical properties such as closure under Boolean operations and decidable equivalence. However, like any classical automata models, they are limited to finite alphabets. This limitation is restrictive for practical applications to both XML processing and program trace analysis, where values for individual symbols are usually drawn from an unbounded domain. With this motivation, we introduce Symbolic Visibly Pushdown Automata (SVPA) as an executable model for nested words over infinite alphabets. In this model, transitions are labeled with predicates over the input alphabet, analogous to symbolic automata processing strings over infinite alphabets. A key novelty of SVPAs is the use of binary predicates to model relations between open and close tags in a nested word. We show how SVPAs still enjoy the decidability and closure properties of Visibly Pushdown Automata. We use SVPAs to model XML validation policies and program properties that are not naturally expressible with previous formalisms and provide experimental results for our implementation.

Control-flow recovery from partial failure reports

“…Symbolic visibly-pushdown automata (s-VPAs) describe languages of nested words over large or infinite alphabets [11]. Nested words are linear encodings of words with hierarchical structure, such as traces of procedural programs.…”

“…To avoid explicit transition on all possible symbols, s-VPA transitions carry predicates describing sets of symbols from Σ. The set of predicates must be closed under Boolean operators, and checking satisfiability of a predicate must be decidable [11].1 We let P X represent the set of predicates in our algebra with free variables X. For example, if our alphabet is the set of numbers {1, .…”

“…In figures and descriptions, we largely follow the style of D'Antoni and Alur [11], and label transitions "I: ϕ x ," "C: ϕ x /p," and "R: ϕ x,y /p" for Internal, Call, and Return respectively. Here, ϕ x denotes a unary predicate on the input symbol x (for internal and call transitions), while ϕ x,y denotes a binary predicate (for return transitions) with y as the current 1 In this paper, we only focus on the algebra where predicates are unions of intervals over integers and we allow equalities between variables.…”

Abstraction Refinement and Antichains for Trace Inclusion of Infinite State Systems

“…Data words are also studied in the context of symbolic visibly pushdown automata (SVPA) [10]. Language inclusion is decidable for SVPAs with transition guards from a decidable theory because SVPAs are closed under complement and the emptiness can be reduced to a finite number of queries expressible in the underlying theory of guards.…”

“…[ \ The algorithm iterates the loop on lines (10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) until Sim and PrevSim define the same relation. Since, in general the data constraints Sim i j , at each iteration step, might form an infinitely decreasing sequence, we use the matrix Cnt of integer counters, initially set to some input value K ą 0 (line 4).…”