Adding Time to Pushdown Automata

In this tutorial, we illustrate through examples how we can combine two classical models, namely those of pushdown automata (PDA) and timed automata, in order to obtain timed pushdown automata (TPDA). Furthermore, we describe how the reachability problem for TPDAs can be reduced to the reachability problem for PDAs.

💡 Research Summary

The paper introduces Timed Pushdown Automata (TPDA), a formal model that integrates the unbounded stack of pushdown automata (PDA) with the real‑valued clocks of timed automata (TA). After a brief recap of PDA (used for context‑free languages) and TA (used for real‑time systems), the authors define TPDA as a 5‑tuple (Q, Σ, Γ, X, δ). A transition is a tuple (q, a, g, r, q′) where a is the input symbol (or ε), g is a guard that may involve clock constraints and the top‑of‑stack symbol, r is a set of clocks to be reset, and q′ is the target state. When a transition fires, the guard must be satisfied by the current clock valuation and the stack’s top symbol; then the specified clocks are reset and a push or pop operation is performed. This definition allows one to express properties such as “a recursive call must return within 5 time units” directly in the automaton’s semantics.

The central technical contribution is a reduction of the reachability problem for TPDA to the classic reachability problem for PDA. Because clocks range over an infinite continuous domain, direct exploration is impossible. The authors therefore apply a “clock‑region” or “interval abstraction” technique: each clock is partitioned into a finite set of intervals (e.g.,