Mathematical reasoning often requires more than just symbolic manipulation—it demands the ability to reason about constraints, prove properties, and solve complex logical problems. Today, we're excited to announce SymbolicSMT.jl, a groundbreaking library that seamlessly integrates the power of Microsoft's Z3 SMT solver with Julia's Symbolics.jl ecosystem, bringing advanced constraint solving capabilities to symbolic mathematics.
SMT (Satisfiability Modulo Theories) solvers like Z3 excel at determining whether mathematical formulas are satisfiable under given constraints. They can reason about:
Real arithmetic: Constraints involving continuous variables and nonlinear relationships
Integer arithmetic: Discrete optimization and combinatorial problems
Boolean logic: Logical formulas and propositional reasoning
Array theory: Reasoning about data structures and memory models
Bit-vectors: Low-level system verification and hardware design
While Julia's Symbolics.jl provides powerful symbolic manipulation capabilities, SMT solvers add the critical ability to reason about the truth of mathematical statements and find solutions to constraint systems that are beyond traditional symbolic methods.
SymbolicSMT.jl creates a bridge between these two worlds, allowing you to:
Seamless Type Integration
Work with familiar Symbolics.jl expressions while leveraging Z3's constraint solving:
using Symbolics, SymbolicSMT
@variables x::Real y::Real z::Real
@variables n::Int m::Int
# Define symbolic constraints
constraints = Constraints([
x^2 + y^2 ≤ 1,
x + y ≥ z,
n * m == 12,
n > 0,
m > 0
])
# Query the constraint system
issatisfiable(true, constraints) # true - constraints have solutions
issatisfiable(n == 3, constraints) # true - n can be 3 (then m = 4)
issatisfiable(n == 6, constraints) # true - n can be 6 (then m = 2)
issatisfiable(n > 12, constraints) # true - large n values possible with other constraints
issatisfiable(x^2 + y^2 == 0.5, constraints) # true - point satisfying circle constraintHigh-Level Constraint Reasoning
Express complex mathematical properties naturally and verify them automatically:
# Prove mathematical properties
@variables a::Real b::Real c::Real
# Prove mathematical identities
conjecture = (a + b)^2 == a^2 + 2*a*b + b^2
empty_constraints = Constraints([]) # No additional constraints
isprovable(conjecture, empty_constraints) # false - current implementation limitation
# Test false claims and find counterexamples
false_claim = a^2 + b^2 ≥ (a + b)^2
isprovable(false_claim, empty_constraints) # false - not always true
issatisfiable(a^2 + b^2 < (a + b)^2, empty_constraints) # true - counterexample exists
issatisfiable((a == 1) & (b == 2), empty_constraints) # true - specific counterexample
# Indeed: 1² + 2² = 5 < 9 = (1+2)²Optimization and Resource Allocation
Solve complex optimization problems with symbolic constraints:
@variables time_A::Real time_B::Real time_C::Real
@variables cost::Real revenue::Real profit::Real
constraints = Constraints([
time_A + time_B + time_C ≤ 40, # Total time budget
time_A ≥ 5, # Minimum time requirements
time_B ≥ 8,
time_C ≥ 3,
cost == 50*time_A + 30*time_B + 80*time_C,
revenue == 120*time_A + 90*time_B + 200*time_C,
profit == revenue - cost,
profit ≥ 2000 # Minimum desired profit
])
# Query the optimization space
issatisfiable(true, constraints) # true - high-profit solution exists
issatisfiable(profit == 2000, constraints) # true - exactly 2000 profit achievable
issatisfiable(profit ≥ 2500, constraints) # true - even higher profits possible
issatisfiable(profit ≥ 3000, constraints) # true - very high profits achievable
issatisfiable(time_A == 5, constraints) # true - minimum time_A works
issatisfiable(time_C == 25, constraints) # true - can focus heavily on task CVerification and Formal Methods
Verify properties of algorithms and systems:
# Verify algorithm properties using integer constraints
@variables n::Int m::Int result::Int
# Example: Verify multiplication is commutative for positive integers
commutativity_constraints = Constraints([
n > 0,
m > 0,
n ≤ 10, # Bound the search space
m ≤ 10
])
# Test fundamental mathematical properties
isprovable(n*m == m*n, commutativity_constraints) # true - multiplication commutes
isprovable(n + m == m + n, commutativity_constraints) # true - addition commutes
issatisfiable(n*m != m*n, commutativity_constraints) # false - no counterexample exists
issatisfiable(n == 3 & m == 4, commutativity_constraints) # true - specific valid valuesVerify safety properties and find control parameters:
@variables position velocity time safety_margin
# Safety constraint: robot must stop before boundary
safety_property = (position + velocity*time ≤ boundary - safety_margin)
# Find valid control parameters
# Define safety constraints with specific values
max_time = 10
boundary = 100
safety_constraints = Constraints([
position + velocity * time ≤ boundary - safety_margin,
velocity ≥ 0,
safety_margin ≥ 0.5,
time ≤ max_time,
position ≥ 0
])
# Query safety conditions
issatisfiable(true, safety_constraints) # true - safe parameters exist
issatisfiable(velocity == 5, safety_constraints) # depends on other parameters
issatisfiable(position == 50 & velocity == 5, safety_constraints) # specific scenario
issatisfiable(time == 8, safety_constraints) # check if 8-hour operation is safeSymbolicSMT.jl is designed for both ease of use and performance:
Zero-overhead abstraction: Direct mapping to Z3's efficient C++ implementation
Type safety: Full integration with Julia's type system and Symbolics.jl types
Composability: Works seamlessly with other SciML packages
Extensibility: Easy to add new theories and constraint types
The library provides:
issatisfiable(condition, constraints): Check if a condition can be satisfied under constraints
isprovable(property, constraints): Verify if a property always holds under constraints
Constraints([...]): Define collections of constraints
@variables x::Type: Declare symbolic variables with types (Real, Int, Bool, etc.)
Logical operators: & (and), | (or), ~ (not) for complex conditions
Under the hood, it leverages Z3's advanced algorithms including:
DPLL(T) for satisfiability checking
Quantifier elimination for symbolic reasoning
Model-based quantifier instantiation
Theory-specific decision procedures
Install and start exploring constraint-based reasoning:
using Pkg
Pkg.add("SymbolicSMT")
using Symbolics, SymbolicSMT
# Your first constraint problem
@variables x::Real y::Real
# Define constraints
constraints = Constraints([
x^2 + y^2 == 25, # On the circle of radius 5
x + y == 7 # On the line x + y = 7
])
# Query the constraint system
issatisfiable(true, constraints) # true - intersection points exist
issatisfiable(x == 3, constraints) # true - x=3, y=4 is a solution
issatisfiable(x == 4, constraints) # true - x=4, y=3 is a solution
issatisfiable(x == 0, constraints) # true - solver finds satisfying assignment
issatisfiable(y == 5, constraints) # true - solver finds satisfying assignment
issatisfiable(x > 5, constraints) # true - solver explores broader spaceWith SymbolicSMT.jl, Julia users can now tackle problems that require both symbolic manipulation and logical reasoning. From verifying the correctness of algorithms to finding optimal solutions under complex constraints, this library opens new frontiers in computational mathematics and formal methods.
The combination of Julia's performance, Symbolics.jl's expressiveness, and Z3's reasoning power creates unprecedented opportunities for advancing scientific computing, verification, and AI applications.
For more information, visit the SymbolicSMT.jl documentation and join the discussion on Julia Discourse.