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**Satisfiability Modulo Theories (An introduction)**

Magnus Madsen

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**What are SMT solvers? How are they used in practice?**

Todays Talk What are SMT solvers? How are they used in practice?

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**Knowledge of prop. logic**

Motivation Find 𝒙 and 𝒚 s.t.: 𝑥≥3∧ 𝑥≤0∨𝑦≥0 𝑥≥3∧𝑥≤0 ∨ 𝑥≥3∧𝑦≥0 𝑥=3∧𝑦=0 Knowledge of prop. logic Knowledge of integers Knowledge of integers Solution

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What is SMT? Satisfiability Modulo Theories +

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**What is a SMT instance? A logical formula built using**

negation, conjunction and disjuction e.g. 𝑎∧ 𝑏∨𝑐 e.g. 𝑎∨¬𝑏∨𝑐 ∧ ¬𝑏∨¬𝑥∨𝑦 ∧ 𝑏∨𝑏∨𝑥 theory specific operators e.g. 𝑥≤5, 𝑦≠𝑧 e.g. 𝑚⊕𝑛 ⊕𝑛=𝑚 e.g. 𝑓 𝑥 =𝑓(𝑦)∧𝑓(𝑓 𝑥 )≠𝑓(𝑓 𝑦 ) k-SAT theory of bitwise operators theory of integers theory of uninterpreted functions

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**Recall k-SAT The Boolean SATisfiability Problem:**

𝑎∨¬𝑏∨𝑐 ∧ ¬𝑏∨¬𝑥∨𝑦 ∧ 𝑏∨𝑏∨𝑥 ∧… 2SAT is solveable in polynomial time 3SAT is NP-complete (solveable in exponential time) clause literal or negated literal

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**Q: Why not encode every formula in SAT?**

A: Theory solvers have very efficient algorithms Graph Problems: Shortest-Path Minimum Spanning Tree Optimization: Max-Flow Linear Programming (just to name a few)

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**Q: But then, Why not get rid of the SAT solver?**

A: SAT solvers are very good at case analysis

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**SAT Theory Formula 𝑥≥3∧ 𝑥≤0∨𝑦≥0 SMT Solver 𝑥≥3∧𝑥≤0 𝑎∧ 𝑏∨𝑐 𝑥≥3∧𝑦≥0 𝑎∧𝑏**

YES 𝑎∧𝑐 NO NO YES 𝑥=3 𝑦=0 add clause: ¬ 𝑎∧𝑏

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**Important Properties Efficiency of both SAT and Theory solver!**

SAT Solver Incremental (supports adding new clauses) Theory Solver Ability to construct blocking clauses Ability to create so-called "theory lemmas"

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**Theories Theory of: Difference Arithemetic Linear Arithmetic Arrays**

Bit Vectors Algebraic Datatypes Uninterpreted Functions

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**SMT-LIB A modeling language for SMT instances**

A declarative language with Lisp-like syntax Defines common/shared terminology e.g. LRA = Closed linear formulas in linear real arithmetic e.g. QF_BC = Closed quantifier-free formulas over the theory of fixed-size bitvectors.

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Example 1 𝒙=𝟑∧𝒚=𝟎 Solution

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Example 2

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**Applications Dynamic Symbolic Execution Program Verification**

Extended Static Checking Model Checking Termination Analysis See Also: Tapas: Theory Combinations and Practical Applications

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**Dynamic Symbolic Execution**

combines dynamic and symbolic execution step 1: execute the program recording the branches taken and their symbolic constraints step 2: negate one constraint step 3: solve the constraints to generate new input to the program (e.g. by using a SMT solver) step 4: if a solution exists then execute the program on the new input

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Program Path ¬𝑐 1 Negate ¬𝑐 3 𝑐 2 ¬𝑐 3 Run SMT Solver 𝑐 4

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New Program Path ¬𝑐 1 𝑐 2 𝑐 3 𝑐 5

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**Example: Greatest Common Divisor**

Original program SSA unfolding int gcd(int x, int y) { while (true) { int m = x % y; if (m == 0) return y; x = y; y = m; } int result = gcd(2, 4) int gcd(int x0, int y0) { while (true) { int m0 = x0 % y0; assert(m0 != 0) if (m0 == 0) return y0; x1 = y0; y1 = m0; int m1 = x1 % y1; assert(m1 == 0) if (m1 == 0) return y1; }

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**Collecting Constraints**

Collected constraints SSA unfolding int result = gcd(2, 4) (assert (= m0 (mod x0 y0))) (assert (not (= m0 0))) (assert (= x1 y0)) (assert (= y1 m0)) (assert (= m1 (mod x1 y1))) (assert (= m1 0)) int gcd(int x0, int y0) { while (true) { int m0 = x0 % y0; assert(m0 != 0) if (m0 == 0) return y0; x1 = y0; y1 = m0; int m1 = x1 % y1; assert(m1 == 1) if (m1 == 0) return y1; } (assert (not (= m1 0)))

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**Computing a new path Solution: x = 2 and y = 3**

int gcd(int x, int y) { while (true) { int m = x % y; if (m == 0) return y; x = y; y = m; } Solution: x = 2 and y = 3 Iteration 1: x = 2 & y = 3 Iteration 2: x = 3 & y = 2 Iteration 3: x = 2 & y = 1

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**Program Verification Assertion Violation: low = 230, high = 230+1**

int binary_search(int[] arr, int low, int height, int key) { assert(low > high || 0 <= < high); while (low <= high) { // Find middle value int mid = (low + high) / 2; assert(0 <= mid < high); int val = arr[mid]; // Refine range if (key == val) return mid; if (val > key) low = mid + 1; else high = mid – 1; } return -1; Assertion Violation: low = 230, high = 230+1

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**SMT Solvers Z3 MathSAT5 CVC4 Many more Microsoft Research**

University of Trento CVC4 New York University Many more

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SMT-COMP A yearly competition between SMT solvers Z3

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**Research Directions in SMT**

Improving the efficiency of SAT/Theory solvers Improving the interplay between the SAT solver and the theory solver e.g. "online" solvers (partial truth assignment) Developing solvers for new theories Combining different theories

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**With Thanks to Evan Driscoll**

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References Satisfiability Modulo Theories: Introduction and Applications Leonardo De Moura & Nikolaj Bjørner Tapas: Theory Combinations and Practical Applications Z3 Tutorial Guide

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**Summary Satisfiability Modulo Theory (SMT):**

constraint systems involving SAT + Theory SMT solvers combine the best of: SAT solvers and theory solvers SMTs have applications in program analysis

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More Work To Be Done?

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