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A proof-theoretic approach to abstract interpretation

This paper establishes a proof-theoretic framework for abstract interpretation by systematically constructing logical systems whose algebraic structures correspond to given abstract lattices, thereby unifying program analysis with proof theory and algebraic logic through soundness and completeness results.

Original authors: Vijay D'Silva, Alessandra Palmigiano, Apostolos Tzimoulis, Caterina Urban

Published 2026-05-27

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CC BY 4.0

Original authors: Vijay D'Silva, Alessandra Palmigiano, Apostolos Tzimoulis, Caterina Urban

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to describe a massive, chaotic city (the concrete world) to a friend who only speaks a simplified, symbolic language (the abstract world). The city has infinite streets, buildings, and people moving in complex patterns. Your friend can't handle that much detail, so you need a way to summarize the city's behavior without lying about it. This is the core problem of Abstract Interpretation: creating a safe, simplified map of a complex reality.

This paper proposes a new way to build the "grammar" or logic for that simplified map. Instead of just guessing what rules the map should follow, the authors suggest a mechanical recipe to generate a perfect logic system that matches the map exactly.

Here is the breakdown of their ideas using everyday analogies:

1. The Translator and the Map

Think of the complex city as a giant set of all possible scenarios. The "Abstract Lattice" is a finite, manageable checklist of properties (e.g., "Is the traffic light red?" "Is the bridge open?").

To connect the city to the checklist, you need two translators:

The Up-Translator (Abstraction): Takes a messy real-world situation and says, "This fits into category A."
The Down-Translator (Concretization): Takes a category from the checklist and says, "This represents all the real-world situations that fit here."

The authors' goal is to create a Logic (a set of rules for reasoning) where the "dictionary" of that logic is perfectly identical to the checklist. If the checklist says "A implies B," the logic should prove "A implies B" without fail.

2. The Recipe for a Custom Logic

The paper offers a step-by-step "recipe" to build this logic for any finite checklist:

Pick the Tools: Look at the checklist. Which tools (like "AND," "OR," "NOT") work correctly when you translate back and forth between the city and the checklist? Only keep those tools.
Name the Items: Give every item on the checklist a name (like a label on a box).
Write the Rules:
- If the checklist says "Box A is a subset of Box B," write a rule in the logic: "If you have A, you have B."
- If the checklist says "Combining Box A and Box B makes Box C," write a rule: "A AND B equals C."
The Result: The authors prove that if you follow this recipe, the resulting logic system is sound (it never lies about the city) and complete (it can prove everything that is true about the checklist).

The "Naive" Warning: The authors admit this recipe is a bit like using a sledgehammer to crack a nut. It works for any checklist, but it might create too many rules, some of which are redundant. It's a "brute force" method that guarantees correctness but isn't the most efficient way to do it.

3. The "Cartesian" vs. "Non-Cartesian" Puzzle

The paper then looks at a specific problem: What happens when you have two variables, like $x$ and $y$ ?

The Cartesian Approach (The Grid): Imagine a grid where you check $x$ and $y$ separately. It's like checking the temperature in the kitchen and the temperature in the bedroom independently. This is easy to handle because the rules for the whole grid are just the rules for the kitchen plus the rules for the bedroom.
The Non-Cartesian Approach (The Shape): Sometimes, $x$ and $y$ are linked in a weird shape. For example, "The sum of $x$ and $y$ must be less than 10." This creates a diagonal cut across the grid. You can't just look at $x$ and $y$ separately; you have to look at the shape they make together.

The authors observe that dealing with these "weird shapes" (non-Cartesian abstractions) is actually easier for their logic-building recipe than trying to force them into a simple grid. They suggest a strategy: Build the theory for the complex, linked shapes first, and then see how the simple grid case fits into that.

4. The Octagon Example

To test their theory, they looked at a specific type of shape called an "octagon" (predicates like $x + y \geq 5$ ).

They found that while you can easily say "NOT ( $x+y \geq 5$ )", you cannot easily say "( $x+y \geq 5$ ) AND ( $x-y \geq 5$ )" using their specific set of rules, because the intersection of those two shapes doesn't fit the simple "line" format of their checklist.
This revealed a limitation: If you only allow "NOT" and no "AND," your logic is very weak.
The Fix: They proposed allowing "AND" and "OR" as meta-rules (rules about the rules) rather than strict parts of the checklist. This lets them handle complex contradictions (like proving a situation is impossible) without breaking their system.

Summary

In simple terms, this paper is a blueprint for building a custom language that perfectly matches a simplified model of a computer program.

The Problem: We need to verify complex software, but we can't check every single possibility. We use simplified models.
The Solution: The authors provide a mechanical way to generate the exact set of logical rules needed to reason about that simplified model.
The Insight: Sometimes, treating linked variables as a single complex shape (non-Cartesian) is mathematically cleaner than trying to force them into separate, independent buckets (Cartesian).

The paper doesn't claim to solve all software bugs or predict future medical outcomes; it strictly provides the mathematical machinery to ensure that the "simplified maps" we use for verification have a consistent and reliable set of logical rules.

Technical Summary: A Proof-Theoretic Approach to Abstract Interpretation

Problem Statement
Abstract interpretation is a foundational theory for formal program verification, designed to generate sound approximations of program semantics to solve undecidable or computationally intractable problems. Traditionally, this theory relates a complex concrete model (e.g., the powerset $P(\Sigma)$ of program executions) to a finite abstract lattice $A$ via an adjoint pair of maps: a concretization map $\gamma$ and an abstraction map $\alpha$ . While the connection between abstract interpretation and logic has been recognized since the theory's inception, it has only recently been systematized. Previous work (e.g., [5], [3]) introduced the notion of an "internal logic" for an abstraction and identified logics underlying specific abstractions. However, a general, procedural method for generating the internal logic of an arbitrary abstraction, along with a corresponding proof system, remains an open challenge.

Methodology
The authors propose a general procedure to generate a logic $L_A$ corresponding to a finite abstraction $A = (A, \sqsubseteq, Op_A)$ with a concretization map $\gamma: A \to (P(\text{Struct}), \subseteq, Op_c)$ . The methodology constructs the logic such that its Lindenbaum-Tarski algebra is isomorphic to the abstract algebra $A$ . The construction proceeds through the following steps:

Connectives: The logical connectives of $L_A$ are defined as those operations preserved by the concretization map $\gamma$ . If $A$ is a finite distributive lattice, it is a bi-Heyting algebra, making implication and co-implication available candidates.
Predicates: For every element $a \in A$ , a unary predicate symbol $a(x)$ is added to the language.
Introduction Rules: For every connective preserved by $\gamma$ , appropriate introduction rules are added to the proof system.
Axiomatic Rules:
- For binary connectives $\star$ where $a \star b = c$ , the rule $a(x) \star b(x) \dashv\vdash c(x)$ is added.
- For unary connectives $\star$ where $\star a = b$ , the rule $\star a(x) \dashv\vdash b(x)$ is added.
- For order relations where $a \leq b$ , the rule $a(x) \vdash b(x)$ is added.

The authors acknowledge that this procedure is "naive" and computationally inefficient due to potential redundancy in axioms, but emphasize its applicability to any finite lattice and concretization map.

Key Results and Theoretical Findings
The paper establishes three primary lemmas regarding the constructed logic $L_A$ :

Soundness (Lemma 1): The logic $L_A$ is sound with respect to the concretization $\gamma$ . This is proven by induction on proof depth, relying on the fact that $\gamma$ preserves the connectives and that the axioms reflect the relationships in the concrete algebra.
Isomorphism (Lemma 2): The Lindenbaum-Tarski algebra $L$ of $L_A$ is isomorphic to the abstract algebra $A$ . The mapping $e: A \to L$ defined by $e(a) = [a(x)]$ (the equivalence class of the predicate) is shown to be a surjective, injective order-embedding and a homomorphism.
Completeness (Lemma 3): If $\gamma$ is an order-embedding, $L_A$ is complete with respect to the concretization. This result strengthens previous findings (specifically [5, Proposition 5]) by relaxing the condition from $\gamma$ being an injective right adjoint to simply being an order-embedding.

Case Studies and Extensions
The paper applies this framework to specific scenarios:

Cartesian Products: The methodology generalizes to $n$ variables. When predicates are formed by conjunctions of single-variable predicates ( $\phi_i(x) = \bigwedge \psi^i_k(x_k)$ ), the resulting lattice is a Cartesian product of single-variable lattices. The concretization map corresponds to the assignment of sets to $n$ -dimensional rectangles.
Logic of Octagons: The authors analyze predicates of the form $\pm x \pm y \geq c$ . They observe that while $\gamma$ preserves negation, it does not preserve conjunction or disjunction (as the intersection of such predicates often results in geometric shapes not representable by a single predicate in the lattice). Consequently, a logic relying solely on classical negation is weak. To address this, the authors propose using conjunction and disjunction as meta-connectives to handle sequents, while noting that the lattice structure of octagons consists of meet- and join-irreducible elements (excluding top and bottom), allowing for an involutive negation.

Significance and Claims
The paper positions itself as a collection of preliminary results and observations from ongoing work. Its primary significance lies in:

Generalization: Extending the results of [3] to provide a general procedure for generating the internal logic of any finite abstraction.
Systematic Construction: Offering a concrete, albeit naive, algorithm to generate a proof system where the algebraic structure of the logic is guaranteed to be isomorphic to the abstract domain.
Refinement of Theory: Strengthening the theoretical link between order-theoretic properties of concretization maps and the completeness of the associated logic.

The authors explicitly state that the current procedure is inefficient and ignores computational limitations, framing the work as a foundational step toward characterizing what properties "percolate" from non-Cartesian abstractions to Cartesian cases, rather than presenting a ready-to-use verification tool.