Graph-Based Reasoning and Verification

• Graph-based verification is a method that uses structured graphs to model program semantics and dependencies, facilitating modular analysis in formal verification.
• It constructs dual graphs—a constant dependency graph and a predicate dependency graph—to represent syntactic and semantic relationships, enabling refined reduction of verification conditions.
• Graph traversal techniques, such as breadth-first search and weighted distance metrics, determine hypothesis relevance and efficiently prune verification inputs to enhance automatic proof discharge.

Graph-based reasoning and verification encompasses a spectrum of techniques that leverage graph-theoretic models and algorithms to formalize, reduce, and discharge verification tasks. Under this approach, mathematical graphs represent program states, semantic dependencies, logical structure, or data/control flows. The utility of graph-based reasoning lies in its expressive ability to encode both local and global dependencies, to enable modular analysis via selective subgraph traversal, and to systematically prune irrelevant components in large verification conditions—thereby enhancing automation, scalability, and interpretability in formal verification pipelines across software, hardware, and hybrid systems.

1. Foundational Graph Constructions in Verification

Graph-based verification capitalizes on constructing structural representations of program semantics to enable fine-grained reasoning. The methodology described in (0907.1357) demonstrates a dual-graph architecture for reducing the complexity of first-order verification conditions (VCs) in deductive verification of C programs:

• Constant Dependency Graph: An undirected graph where each vertex represents a constant—either directly appearing in the VC or introduced via “flattening” of compound terms. Edges encode the inclusion of one constant in a compound term, thus tracing variable/term dependency chains.
• Predicate Dependency Graph: A weighted, directed graph in which vertices denote predicate symbols (including special handling for comparison predicates such as =, <, ≤). Edges are drawn according to clause structure in the conjectures’ CNF form, with edge weights inversely related to clause size, representing deductive strength.

This two-graph scheme is generalizable to other symbolic domains, such as dependency graphs in hardware verification (Wu et al., 2023) or control/data flow graphs in software model checking (Leeson et al., 2022). It provides a basis for selective traversal and for encapsulating multiple axes of relevance (terms, predicates, control flow) in the verification search space.

2. Syntactic Analysis and Graph Construction Algorithms

Graph construction begins with a syntactic analysis of VCs:

• Term Flattening: Each occurrence of a compound term $f(t_1, ..., t_n)$ is replaced with a fresh constant (e.g. $f_i$), with edges relating $f_i$ to all argument constants. The process is recursively applied to inner compound arguments, yielding a tree-like expansion in the constant graph.
• Predicate Extraction and Clause CNFization: All axioms, hypotheses, and context formulas are normalized into CNF, and literals are parsed for predicate identification. For each clause, explicit rules are applied (taking literal polarity into account) to generate corresponding nodes and arcs in the predicate dependency graph:
  • For instance, if a clause contains $(¬p, q)$, an arc is drawn from $p$ to $q$.
  • Edge weights are set to $|Cl| - 1$ (clause cardinality minus one), capturing the notion that shorter clauses contribute "stronger" implications during automated reasoning.

A particularly notable refinement addresses comparison predicates: the predicate graph is “typed” (e.g., $=_{\tau}$ denotes equality over type $\tau$), and domain-specific axioms (such as those relating $\leq$, $<$, and $=$ in linear arithmetic) are encoded by adding extra arcs to preserve algebraic consistency.

This process ensures that symbolic and semantic proximity in the VC is faithfully mapped to graph connectivity, thereby supporting subsequent relevance traversals and reductions.

3. Graph Traversal for Relevance Determination

Relevance determination proceeds through controlled traversals over the constructed graphs:

• Constant Graph (Term) Traversal: Employs breadth-first search (BFS) from the set of constants in the VC’s conclusion (notated as ${}_0$), forming layers ${}_1, {}_2, ..., {}_\infty$ of reachable constants. A heuristic biases traversal towards nodes more frequently linked by multiple edges, thereby prioritizing “central” terms with likely influence on the proof.
• Predicate Graph Traversal: Initiates from all conclusion predicates (${_0}$) and computes minimal weighted distances (taking contrapositive arcs into account) to other predicates. For each threshold $i$, set ${_i}$ contains all predicate vertices reachable with cumulative path weight at most $i$—enabling stepwise inclusion of more “distant” (less immediately relevant) axioms or hypotheses.

Relevance of each hypothesis/clause is then judged against these traversed sets: a clause is retained if every constant and predicate symbol it contains appears in the corresponding reachable sets (${_j}$ for constants, ${_i}$ for predicates). Context axioms are included if any arc used in the traversals originated from them, uniting syntactic (local) and clause-theoretic (global) relevance criteria.

This traversal-based approach provides a substantial advance over symbol-matching heuristics, as it directly models the logical/deductive topology manifested in the VC and its supporting context.

4. Application: Reduction of Verification Conditions in Deductive Verification

The primary benefit of graph-based reduction manifests in the selective pruning of irrelevant or weakly connected hypotheses. The reduction process is implemented as a post-processing module (in the Why/Caduceus software verification platform):

• Initial Query: The unpruned VC is sent to an automatic theorem prover; if proof fails, the reduction strategy is activated.
• Incremental Reduction Algorithm: Iteratively increases the depth thresholds ($i$, $j$) on the predicate and constant traversals. At each step, only those hypotheses satisfying the strong inclusion criteria (all their terms/predicates are in the reachable sets) are passed to the prover.
• Context-Aware Selection: Specialized handling for quantified axioms, comparison predicates, and context axioms ensures expressivity and coverage for industrial-scale VCs.

Evaluation on the “Oslo” trusted bootloader benchmark demonstrates that, out of 771 generated VCs, 741 were initially discharged; with the graph-based reduction (and its quantitative/qualitative refinements), 762 (an additional 21) were proved automatically with SMT provers such as Simplify, Alt-Ergo, and Yices—marking a significant advance in automation for deployment-grade code bases.

Illustrative formulas show how complex function calls are flattened (e.g., via $\mathrm{shift}(...)$, $\mathrm{upd}(...)$, $\mathrm{acc}(...)$), and how the conclusion is incrementally recast to admit smaller and more relevant input subproblems.

5. Comparative Analysis with Prior Approaches

Graph-based reduction differentiates itself from traditional proof strategies:

Approach	Support Mechanism	Hypothesis Selection
Set of Support (sos)	Requires user- or system-specified clause set	Restricts inference to clauses in sos
Symbolic Heuristics	Counts or matches overlapping symbols	Ignores logical/global structure
Graph-based (this)	Two dependency graphs (term and predicate)	Traversal+relevance criteria, automatic

• Advantages of Graph-based approach:
  • Fully automatic “guess” of support sets using both syntactic and deductive structure.
  • Clause structure harnessed: CNF, clause cardinalities, clause overlap.
  • Precise inclusion of context axioms (including complex domain axioms, quantifiers, and theory-specific predicates).
  • Reduction in prover input size and improved search-space focus, directly observed in improved VC discharge rates and avoidance of divergence.

A distinctive claim (per experimental results) is that this dual-graph architecture achieves both higher efficiency (by pruning) and effectiveness (by increasing VC discharge) in benchmarks where purely syntactic or symbol-counting heuristics fail to scale.

6. Implications and Future Directions

The graph-based strategy consolidates program semantics, logical deduction, and proof relevance into unified computational structures—allowing for:

• Modular decomposition and analysis of large-scale verification conditions, with natural support for rich axiomatic and context extensions (quantifiers, theory-specific constructs).
• Tunable, incremental deepening of axiom inclusion, critical in interactive verification workflows or when dealing with multiple back-end provers.
• Potential integration with other graph-theoretic techniques including dependency slicing, symbolic execution on program graphs, or dynamic graph-based invariant inference.

A natural extension suggested by the methodology is leveraging graph representations to enhance explainability and debugging of proof failures; graph traversals can in principle expose “cut” sets of hypotheses responsible for proof bottlenecks or nontrivial logical dependencies.

In sum, graph-based reasoning and verification as applied in (0907.1357) exemplifies an effective synthesis of syntactic preprocessing, semantic graph modeling, and traversal-based relevance selection, yielding measurable advances in formal software verification at industrial scale.