Symbolic Abstraction

Symbolic abstraction is a technique that computes the best abstract domain element that overapproximates a given formula in first-order logic. It serves as a bridge between concrete program analysis and efficient abstract reasoning, enabling scalable verification and analysis of complex systems.

The key idea is to transform a concrete logical formula into an abstract representation that captures the essential behavior while being more amenable to automated analysis. This abstraction preserves important properties (like satisfiability) while discarding unnecessary details that would make analysis computationally expensive.

Overview

Arlib’s symbolic abstraction module (arlib/symabs) provides a comprehensive suite of abstraction techniques for program analysis and verification. The module is organized around several key approaches:

  1. Abstract Interpretation-based Symbolic Abstraction (ai_symabs)

  2. Optimization Modulo Theory-based Symbolic Abstraction (omt_symabs)

  3. Predicate Abstraction (predicate_abstraction)

  4. Model Counting-based Abstract Interpretation (mcai)

  5. Congruence Closure-based Abstraction (congruence)

  6. Range Set SAT-based Abstraction (rangeset_sat)

Each approach offers different trade-offs between precision, scalability, and applicability to different problem domains.

Core Concepts

Abstract Domain

An abstract domain defines the lattice structure and operations for abstract elements:

class AbstractDomain:
    def join(self, elements):  # Least upper bound
    def meet(self, elements):  # Greatest lower bound
    def gamma_hat(self, alpha):  # Concretization function
    def model(self, formula):   # Find concrete model

The gamma_hat function translates an abstract element back to a concrete formula, while the join operation computes the least upper bound in the abstract lattice.

Concretization and Abstraction

  • Concretization (γ̂): Maps abstract elements to concrete formulas

  • Abstraction: Maps concrete formulas to abstract elements

  • Galois Connection: The pair (α, γ̂) forms a Galois connection if α(γ̂(α̂)) = α̂

Precision Metrics

The quality of an abstraction is measured by:

  • Precision: How closely the abstract formula approximates the concrete one

  • Scalability: Computational cost of abstraction computation

  • Expressiveness: Ability to capture important program properties

Abstract Interpretation (ai_symabs)

The ai_symabs submodule implements classic abstract interpretation domains based on Thakur’s PhD thesis. It provides a framework for computing symbolic abstractions using various abstract domains.

Supported Abstract Domains

Interval Domain

The interval domain represents variable ranges using lower and upper bounds:

from arlib.symabs.ai_symabs.domains.interval.domain import IntervalDomain

# Create interval domain for variables ['x', 'y']
domain = IntervalDomain(['x', 'y'])

# Abstract state represents: x ∈ [0, 10], y ∈ [-5, 5]
# This concretizes to: 0 ≤ x ∧ x ≤ 10 ∧ -5 ≤ y ∧ y ≤ 5

Sign Domain

The sign domain tracks the sign of variables (negative, zero, positive):

from arlib.symabs.ai_symabs.domains.sign.domain import SignDomain

# Sign domain for variable 'x'
# Possible abstract values: -, 0, +, T (top)

Reduced Product Domain

Combines multiple domains for better precision:

from arlib.symabs.ai_symabs.domains.reduced_product.domain import ReducedProductDomain

# Combine interval and sign domains
combined = ReducedProductDomain([interval_domain, sign_domain])

Abstraction Algorithms

Bilateral Algorithm

The bilateral algorithm efficiently computes abstractions by working with both the positive and negative forms of the formula:

from arlib.symabs.ai_symabs.domains.algorithms.bilateral import bilateral

# Compute abstraction using bilateral algorithm
abstract_state = bilateral(concrete_formula, domain)

RSY Algorithm

Recursive abstraction refinement algorithm that iteratively improves precision:

from arlib.symabs.ai_symabs.domains.algorithms.rsy import rsy

# Compute abstraction with RSY algorithm
abstract_state = rsy(concrete_formula, domain)

OMT-based Symbolic Abstraction (omt_symabs)

The omt_symabs submodule uses Optimization Modulo Theory (OMT) solvers to compute optimal abstractions. This approach leverages modern OMT engines like Z3’s optimizer to find the best abstract representation.

Key Features

  • Optimal Abstractions: Computes the most precise abstraction within the domain

  • Multiple Domains: Supports intervals, zones, and octagons

  • Theory Support: Handles both linear arithmetic (LIA/LRA) and bit-vectors

Linear Arithmetic Abstraction

from arlib.symabs.omt_symabs.lira_symbolic_abstraction import LIRASymbolicAbstraction

# Create LIRA symbolic abstraction
symabs = LIRASymbolicAbstraction()

# Initialize from SMT formula
symabs.init_from_formula(formula)

# Compute interval abstraction
interval_abs = symabs.compute_interval_abstraction()

# Compute zone abstraction
zone_abs = symabs.compute_zone_abstraction()

Bit-Vector Abstraction

from arlib.symabs.omt_symabs.bv_symbolic_abstraction import BVSymbolicAbstraction

# Create bit-vector symbolic abstraction
bv_symabs = BVSymbolicAbstraction()

# Initialize with bit-vector formula
bv_symabs.init_from_formula(bv_formula)

# Compute abstraction
abstract_state = bv_symabs.compute_abstraction()

Predicate Abstraction (predicate_abstraction)

Predicate abstraction computes the strongest Boolean combination of given predicates that is entailed by the concrete formula. This approach is particularly effective for software verification tasks.

Core Algorithm

The predicate abstraction algorithm works by:

  1. Finding satisfying models of the concrete formula

  2. Evaluating predicates in each model

  3. Constructing the strongest Boolean formula consistent with the evaluations

from arlib.symabs.predicate_abstraction.predicate_abstraction import compute_predicate_abstraction

# Define predicates
predicates = [x > 0, y > 0, x + y < 10]

# Concrete formula: x > 5 ∧ y > 3 ∧ x + y < 8
concrete_formula = z3.And(x > 5, y > 3, x + y < 8)

# Compute predicate abstraction
abstract_formula = compute_predicate_abstraction(concrete_formula, predicates)
# Result: x > 0 ∧ y > 0 ∧ x + y < 10 (strongest consequence)

Additional Components

Model Counting-based Abstract Interpretation (mcai)

The mcai submodule combines model counting with abstract interpretation to provide precision metrics and analysis for different abstract domains. This is particularly useful for:

  • Evaluating abstraction quality

  • Comparing different abstract domains

  • Bit-vector formula analysis

Congruence Closure-based Abstraction (congruence)

The congruence submodule implements congruence closure techniques for abstraction, particularly useful for equality reasoning and uninterpreted functions.

Range Set SAT-based Abstraction (rangeset_sat)

The rangeset_sat submodule provides SAT-based abstraction techniques for range and set operations over bit-vectors, offering efficient analysis for hardware verification.

Usage Examples

Basic Symbolic Abstraction

import z3
from arlib.symabs.ai_symabs.domains.interval.domain import IntervalDomain
from arlib.symabs.ai_symabs.domains.algorithms.bilateral import bilateral

# Define concrete formula: 0 ≤ x ∧ x ≤ 10 ∧ x + y = 15
x, y = z3.Ints('x y')
formula = z3.And(x >= 0, x <= 10, x + y == 15)

# Create interval domain
domain = IntervalDomain(['x', 'y'])

# Compute abstraction
abstract_state = bilateral(formula, domain)

# Get concrete representation
concrete_repr = domain.gamma_hat(abstract_state)
print(f"Abstract state concretizes to: {concrete_repr}")

OMT-based Abstraction

from arlib.symabs.omt_symabs.lira_symbolic_abstraction import LIRASymbolicAbstraction

# Create symbolic abstraction engine
symabs = LIRASymbolicAbstraction()

# Load formula from file
symabs.init_from_file('example.smt2')

# Set OMT engine type
symabs.set_omt_engine_type('z3opt')

# Compute optimal interval abstraction
interval_abs = symabs.compute_interval_abstraction()
print(f"Interval abstraction: {interval_abs}")

Predicate Abstraction Example

import z3
from arlib.symabs.predicate_abstraction.predicate_abstraction import compute_predicate_abstraction

# Define variables and predicates
x, y = z3.Ints('x y')
predicates = [x >= 0, y >= 0, x + y <= 10]

# Concrete formula representing a program path
concrete = z3.And(x >= 5, y >= 3, x + y <= 8)

# Compute abstraction
abstract = compute_predicate_abstraction(concrete, predicates)
print(f"Predicate abstraction: {abstract}")

Applications

Symbolic abstraction techniques find applications in various domains:

Program Verification

  • Software Model Checking: Abstract program states to enable scalable verification

  • Shape Analysis: Abstract heap structures and pointer relationships

  • Numerical Program Analysis: Abstract numerical computations for overflow/underflow detection

Security Analysis

  • Information Flow Analysis: Abstract security levels and taint propagation

  • Cryptographic Protocol Verification: Abstract cryptographic operations

  • Buffer Overflow Detection: Abstract memory operations and bounds checking

Hardware Verification

  • Bit-Vector Abstraction: Abstract bit-level operations for hardware verification

  • Range Analysis: Abstract value ranges for timing and power analysis

  • Equivalence Checking: Abstract circuit behaviors for equivalence verification

Optimization and Synthesis

  • Compiler Optimizations: Abstract program semantics for optimization

  • Program Synthesis: Abstract specifications for synthesis tasks

  • Test Generation: Abstract program paths for test case generation

Performance Considerations

Choosing the Right Approach

When to use ai_symabs:

  • Fast analysis with reasonable precision

  • Classic abstract domains (intervals, signs, octagons)

  • Large-scale program analysis

  • Resource-constrained environments

When to use omt_symabs:

  • Need for optimal precision

  • Linear arithmetic formulas

  • Small to medium-sized problems

  • When exact bounds are critical

When to use predicate_abstraction:

  • Software verification tasks

  • User-defined predicates available

  • Boolean program analysis

  • Counterexample-guided abstraction refinement (CEGAR)

Scalability Trade-offs

  • Precision vs. Scalability: OMT provides highest precision but scales poorly

  • Domain Selection: Interval domains offer good precision/scalability balance

  • Algorithm Choice: Bilateral algorithm often provides best performance

  • Parallelization: Many abstraction algorithms can be parallelized

Future Directions

Research Areas

  1. Theory Development: - New abstract domains for emerging computing paradigms - Combination methods for heterogeneous domains - Completeness and decidability results for abstraction algorithms

  2. Algorithm Improvement: - Machine learning-guided abstraction refinement - Parallel and distributed abstraction algorithms - Incremental abstraction techniques for dynamic analysis

  3. Integration with Modern Verification: - Deep learning integration for abstraction learning - Quantum computing abstraction techniques - Blockchain and smart contract verification

  4. Industrial Applications: - Large-scale software verification - Autonomous systems analysis - Cyber-physical systems verification

References

Core Theory

  • Thakur, A. V. (2014). Symbolic Abstraction: Algorithms and Applications. Ph.D. dissertation, University of Wisconsin-Madison. https://www.cs.wisc.edu/~aws/pubs/thesis.pdf

  • Cousot, P., & Cousot, R. (1977). Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. POPL’77.

  • Graf, S., & Saïdi, H. (1997). Construction of abstract state graphs with PVS. CAV’97.

Abstract Interpretation

  • Cousot, P., Cousot, R., & Logozzo, F. (2013). A parametric segmentation functor for fully automatic and scalable array content analysis. POPL’13.

  • Miné, A. (2006). The octagon abstract domain. Higher-Order and Symbolic Computation.

  • Singh, G., Püschel, M., & Vechev, M. (2015). Fast numerical program analysis with reinforcement learning. CAV’15.

OMT-based Abstraction

  • Sebastiani, R., & Trentin, P. (2015). OPTIMathSAT: a tool for optimization modulo theories. CAV’15.

  • Borralleras, C., Larraz, D., Oliveras, A., Rodríguez-Carbonell, E., & Rubio, A. (2019). The iSAT solver. TACAS’19.

Predicate Abstraction

  • Graf, S., & Saïdi, H. (1997). Construction of abstract state graphs with PVS. CAV’97.

  • Ball, T., Majumdar, R., Millstein, T., & Rajamani, S. K. (2001). Automatic predicate abstraction of C programs. PLDI’01.

  • Henzinger, T. A., Jhala, R., Majumdar, R., & McMillan, K. L. (2004). Abstractions from proofs. POPL’04.

Applications and Case Studies

  • Chen, Y., Wang, Y., Zhu, Z., Song, L., & Zhang, L. (2021). Program Analysis via Efficient Symbolic Abstraction. OOPSLA’21.

  • Albarghouthi, A., Li, Y., & Zhang, L. (2019). Automating Abstract Interpretation. VMCAI’16.

  • Gulwani, S., & Tiwari, A. (2007). Combining abstract interpretation with model checking. ESEC/FSE’07.