Solving Non-linear Arithmetic Dejan Jovanovic and Leonadro de Moura Presented by Boris Trayvas

Outline Background - Introduction - Conflict-Driven Clause Learning - Definitions An Abstract Decision Procedure - States and Well Formed States - Search, Clause & Conflict Rules Producing Explanations - More Definitions - Intro to Cylindrical Algebraic Decomposition - Projection Based Explanations Experimental Results

Introduction I Everyone loves polynomials! Tarski (1951) showed a decision procedure for elimination of quantifiers over the reals, which shows the existence of an (impractical) decision procedure for solving polynomial constraints. Collins (1975) first showed a procedure using cylindrical algebraic decomposition (CAD).

Introduction II

Conflict-Driven Clause Learning The Algorithm: Black boxes: - PickBranchingVariable() – What variable to “guess” next? - UnitPropagation() – What can I learn from the new assignment? - ConflictAnalysis() – What clause should I add to prevent the conflict?

CDCL Example

Definitions I

Definitions II

Definitions III

Definitions IV

Polynomial Constraint Problem

An Abstract Decision Procedure Although the algorithm is a CDCL-style algorithm, we will present it as an abstract transition system. This system can be generalized to apply to theories other than non-linear arithmetic.

Trail and Trail Elements

Trail Examples I

The Implied Assignment

Stage Increasing

Trail Examples II

Value

Well-Formed State

The Transition System

Search Rules I

Search Rules – Select-Clause The Select-Clause rule selects a clause from the current stage for which the value is undefined and moves to the clause processing mode to satisfy it.

Search Rules - Conflict The Conflict rule detects a clause from the current stage that is inconsistent and tries to resolve the conflict it by moving to the conflict analysis mode.

Search Rules Lift-Level

Search Rules - SAT

Search Rules - Forget The last rule is the Forget rule, that allows us to eliminate a clause that was added while analyzing conflicts.

Search Rules II Note that once we move to the next stage, all the clauses of the previous stages are satisfied, and they will not be selected by the Select-Clause or Conflict rules.

Clause Processing Rules

Processing Rules – B-Propagate

Processing Rules – R-Propagate

Processing Rules – Decide-Literal If neither of the propagate rules works, we assume that some literal is true for a compatible literal.

Conflict Analysis Rules

Conflict Rules – Resolve- Propagation

Conflict Rules – Resolve Decision

Conflict Rules - Consume

Conflict Rules – Drop-Stage

Conflict Rules - Unsat Finally, if we backtrack all the way to the beginning with a conflict, we will return unsat.

Termination Theorem

The Explain Procedure

Definitions V

A much needed example

Intro to CAD

Projection-Based Explanations

Experimental Results The algorithm (nlsat) did overall better than other algorithms. Each other algorithm was generally good at one or two problem sets and not so good on the others, but this algorithm was pretty good on all the problem sets.

Fin