[Paper Review] The Tree Width of Separation Logic with Recursive Definitions

Radu Iosif, Adam Rogalewicz+1 more|arXiv (Cornell University)|Jan 22, 2013

Advanced Database Systems and Queries5 references4 citations

TL;DR

This paper introduces a decidable fragment of Separation Logic with Recursive Definitions, called SLRD_btw, capable of expressing complex mutable data structures like trees with parent pointers and linked leaves. By translating formulas into Monadic Second-Order logic over graphs of bounded tree width, the authors establish decidability of satisfiability and entailment through reduction to tree automata emptiness, providing a general solution beyond restricted data structures.

ABSTRACT

Separation Logic is a widely used formalism for describing dynamically allocated linked data structures, such as lists, trees, etc. The decidability status of various fragments of the logic constitutes a long standing open problem. Current results report on techniques to decide satisfiability and validity of entailments for Separation Logic(s) over lists (possibly with data). In this paper we establish a more general decidability result. We prove that any Separation Logic formula using rather general recursively defined predicates is decidable for satisfiability, and moreover, entailments between such formulae are decidable for validity. These predicates are general enough to define (doubly-) linked lists, trees, and structures more general than trees, such as trees whose leaves are chained in a list. The decidability proofs are by reduction to decidability of Monadic Second Order Logic on graphs with bounded tree width.

Motivation & Objective

• To address the long-standing open problem of decidability for expressive fragments of Separation Logic with recursive definitions.
• To extend decidability beyond singly-linked lists and trees to more complex data structures such as doubly-linked lists and trees with parent pointers.
• To define a general logic fragment that captures unbounded, dynamically allocated data structures while preserving decidability.
• To provide a uniform framework for automated verification of heap-manipulating programs using bounded tree-width properties.
• To establish a reduction-based approach from SLRD to decidable MSO logic over graphs with bounded tree width.

Proposed method

• Define the fragment SLRD_btw, which ensures all models have bounded tree width by restricting recursive predicate definitions.
• Construct a translation from SLRD_btw formulas into equivalent Monadic Second-Order (MSO) logic formulas over graph structures.
• Model the heap as a directed graph with nodes for memory cells and edges for pointer and data relationships.
• Encode recursive predicate definitions using tree automata and MSO constraints to capture parameter assignments and structural invariants.
• Use the bounded tree width of models to reduce satisfiability and entailment problems to the emptiness problem of tree automata.
• Prove correctness of the translation via a bisimulation-like correspondence between SLRD_btw and MSO models, ensuring semantic equivalence.

Experimental results

Research questions

• RQ1 Can a general fragment of Separation Logic with recursive definitions be defined such that satisfiability and entailment are decidable for unbounded data structures?
• RQ2 Does the bounded tree width property of models enable decidability for expressive recursive predicates beyond lists and trees?
• RQ3 Can the translation from SLRD_btw to MSO logic preserve semantic equivalence while ensuring decidability?
• RQ4 Is it possible to reduce the satisfiability and entailment problems in SLRD_btw to the emptiness problem of tree automata?
• RQ5 What is the expressive power of SLRD_btw in capturing complex data structures such as tll (trees with linked leaves) or doubly-linked lists?

Key findings

• The fragment SLRD_btw is shown to be decidable for both satisfiability and entailment, extending decidability beyond previously known restricted fragments.
• The logic can express complex data structures such as trees with parent pointers and linked leaves, which are not expressible in standard decidable fragments.
• The translation from SLRD_btw to MSO logic over graphs of bounded tree width is correct and preserves semantic equivalence.
• The decidability result is achieved via reduction to the emptiness problem of tree automata, leveraging the bounded tree width of models.
• The approach provides a uniform framework for verifying programs manipulating complex heap-allocated data structures using automated reasoning.
• The complexity of the decision problems is conjectured to be elementary, with tight upper bounds to be established in future work.