[Paper Review] A Symbolic Decision Procedure for Symbolic Alternating Finite Automata

Loris D’Antoni, Zachary Kincaid+1 more|arXiv (Cornell University)|Oct 6, 2016

Formal Methods in Verification20 references5 citations

TL;DR

This paper introduces Symbolic Alternating Finite Automata (s-AFA), a succinct and decidable model for representing sets of finite sequences over large or infinite alphabets. It proposes a symbolic equivalence checking algorithm based on bisimulation up to congruence, leveraging SAT/SMT solvers to efficiently explore state spaces, and demonstrates significant performance gains over existing methods in LTL satisfiability checking and regular expression equivalence verification.

ABSTRACT

We introduce Symbolic Alternating Finite Automata (s-AFA) as an expressive, succinct, and decidable model for describing sets of finite sequences over arbitrary alphabets. Boolean operations over s-AFAs have linear complexity, which is in sharp contrast with the quadratic cost of intersection and union for non-alternating symbolic automata. Due to this succinctness, emptiness and equivalence checking are PSpace-hard. We introduce an algorithm for checking the equivalence of two s-AFAs based on bisimulation up to congruence. This algorithm allows us to exploit the power of SAT and SMT solvers to efficiently search the state space of the s-AFAs. We evaluate our decision procedure on two verification and security applications: 1) checking satisfiability of linear temporal logic formulas over finite traces, and 2) checking equivalence of Boolean combinations of regular expressions. Our experiments show that our technique often outperforms existing techniques and it can be beneficial in both such applications.

Motivation & Objective

• Address the limitations of traditional finite automata in handling large or infinite alphabets, especially in security and program analysis applications.
• Overcome the exponential blowup in state complexity during Boolean operations—particularly intersection and complementation—common in non-alternating symbolic automata.
• Develop a scalable decision procedure for equivalence and emptiness checking in symbolic alternating automata, which are PSpace-complete but highly succinct.
• Enable practical verification by integrating SAT/SMT solvers into a symbolic bisimulation algorithm that exploits structural congruence to reduce search space.
• Demonstrate the utility of s-AFA in real-world applications such as finite-trace LTL satisfiability and Boolean combinations of regular expressions.

Proposed method

• Define Symbolic Alternating Finite Automata (s-AFA) by extending symbolic finite automata with alternation, allowing transitions to be Boolean formulas over target states.
• Use a symbolic representation where transitions are labeled with first-order predicates over a Boolean algebra, enabling compact representation over large or infinite alphabets.
• Develop a bisimulation up to congruence algorithm tailored for s-AFA, exploiting lattice structure on configurations (disjunctions of states) to compute equivalence relations efficiently.
• Integrate SAT solvers to compute congruence closure and perform representative enumeration of alphabet elements, mimicking AllSat solving for symbolic alphabet exploration.
• Optimize the algorithm by maintaining a relation R that generates a bisimulation up to congruence, avoiding full bisimulation computation and reducing complexity.
• Apply the algorithm to two applications: finite-trace LTL satisfiability and regular expression equivalence, using the s-AFA model to represent formulas and regexes symbolically.

Experimental results

Research questions

• RQ1 Can symbolic alternating finite automata (s-AFA) provide a more succinct and efficient representation for sets of finite sequences over large or infinite alphabets compared to non-alternating symbolic automata?
• RQ2 Can a bisimulation-based equivalence checking algorithm for s-AFA be designed that scales efficiently by exploiting congruence closure and SAT/SMT solving?
• RQ3 Does the proposed symbolic decision procedure outperform existing techniques in practical verification tasks such as LTL satisfiability and regular expression equivalence?
• RQ4 To what extent can the bisimulation up to congruence technique be generalized from non-deterministic to alternating automata over symbolic alphabets?
• RQ5 Can representative enumeration of alphabet elements be effectively used in symbolic automata to enable scalable reasoning without explicit enumeration?

Key findings

• s-AFA support Boolean operations with linear complexity, in contrast to the quadratic cost of intersection and union in non-alternating symbolic automata.
• Emptiness and equivalence checking for s-AFA are PSpace-complete, reflecting the trade-off between succinctness and computational complexity.
• The proposed bisimulation up to congruence algorithm efficiently checks equivalence by leveraging SAT solvers to compute congruence closure and representative character enumeration.
• In experiments, the s-AFA-based decision procedure outperformed existing techniques in both finite-trace LTL satisfiability and regular expression equivalence checking.
• The algorithm's performance advantage stems from avoiding explicit state explosion through symbolic manipulation and congruence-based optimization.
• The framework generalizes the bisimulation up to congruence technique from NFAs to alternating automata and extends it to symbolic and infinite alphabets via AllSat-style enumeration.