--- dg-publish: true --- 18 Setembro 2023 - #MFES # Conteúdo 1. [[MFES/T - Aula 2#1. Intro|Intro]] 1. [[MFES/T - Aula 2#1.1 SAT|SAT]] 2. [[MFES/T - Aula 2#1.2 Proposicional Logic (PL)|Lógica Proposicional]] 3. [[MFES/T - Aula 2#SAT Solvers|SAT Solvers]] # 1. Intro *Formal modeling* - formally represent the system and its properties in the syntactic conventions that the tool understands and can process. Formal Logic = logical language (logical symbols + non-logical symbols) + semantics +proof system ### 1.1 SAT The Boolean satisfiability (SAT) problem: Find an assignment to the propositional variables of the formula such that the formula evaluates to TRUE, or prove that no such assignment exists. - SAT is an NP-complete decision problem. - SAT was the first problem to be shown NP-complete. - There are no known polynomial time algorithms for SAT. Usually SAT solvers deal with formulas in conjunctive normal form (CNF) - **literal**: propositional variable or its negation A, ¬A, B, ¬B, C, ¬C - **clause**: disjuntion of literals. (A _ ¬B _ C) - **conjunctive normal form**: conjuction of clauses. (A _ ¬B _ C) ^ (B _ ¬A) ^ ¬C > [!info]+ Cook's theorem(1971) > SAT is NP-complete ## 1.2 Proposicional Logic (PL) >[!note] Nota >Esta secção basicamente só contém revisão de conceitos. Aconselha-se a ver a coisa rapidamente, porque é só a formalidade de lógica escrita por extenso. Let $A$ be an assignment and let $F$ be a formula. If $A(F) = 1$, then we say **$F$ holds under assignment**, or **$A$ models $F$.** We write A $\models F$ iff $A(F)=1$, and $A \not \models F$ iff $A(F) = 0$. An assignment is a function $A$ : $V_{prop} \implies {0,1}$ , that assigns to every propositional variable a truth value. An assignment $A$ naturally extends to all formulas, $A$ : **Form** $\implies {0,1}$. The truth value of a formula is computed using **truth tables**: | F | $A$ | $B$ | $\neg A$ | $A \land B$ | $A \lor B$ | $A \implies B$ | $A \iff B$ | $\bot$ | $\top$ | | --------- | --- | --- | -------- | ----------- | ---------- | -------------- | ---------- | ------ | ------ | | $A_1 (F)$ | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | | $A_2 (F)$ | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | | $A_3 (F)$ | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | | $A_4 (F)$ | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | A formula $F$ is: 1. **valid** iff it holds under every assignment. We write $\models F$. A valid formula is called a *tautology*. 2. **satisfiable** iff it folds (true) under some assignment. 3. **unsatisfiable** iff it holds under no assignment. An unsatisfiable formula is called a *contradiction*. 4. **refutable** iff it is not valid. > [!tip]+ Proposition > $F$ is **valid** iff $\neg F$ is **unsatisfiable**. - $F \models G$ iff for every assignment $A$, if $A \models F$ then $A \models G$. We say $G$ is a **consequence** of $F$. - $F \equiv G$ iff $F \models G$ and $G \models F$. We say $F$ and $G$ are **equivalent**. - Let $\Gamma = { F_1, F_2, F_3,... }$ be a set of formulas. - $A \models \Gamma$ iff $A \models F_i$ for each formula $F_i$ in $\Gamma$. We say $A$ models $\Gamma$. - $\Gamma \models G$ iff $A \models \Gamma$ implies $A \models G$ for every assignment $A$. We say $G$ is a **consequence** of $\Gamma$. > [!tip]+ Proposition > - $F \models G$ iff $\models F \implies G$. > - $\Gamma \models G$ and $\Gamma$ finite iff $\models \land \Gamma \implies G$. > - $\Gamma$ is *consistent* or *satisfiable* iff there is an assignment that models $\Gamma$. - We say that $\Gamma$ is inconsistent or unsatisfiable iff there is not consistent and denote this by $\Gamma \models \bot$. > [!tip]+ Proposition > - {$F, \neg F$} $\models \bot$ > - If $\Gamma \models \bot$ and $\Gamma \subseteq \Gamma '$, then $\Gamma ' \models \bot$ > - $\Gamma \models F$ iff $\Gamma, \neg F \models \bot$ - Formula $G$ is a subformula of formula F if it occurs syntactically within F - Formula G is a strict subformula of F if G is a subformula of $F$ and $G \neg = F$ **Basic Equivalences:** 1. $\neg \neg A \equiv A$ 2. $A \lor A \equiv A$ 3. $A \land A \equiv A$ 4. $A \land \neg A \equiv \bot$ 5. $A \lor \neg A \equiv \top$ 6. $A \lor B \equiv B \lor A$ 7. $A \land B \equiv B \land A$ 8. $A \land \top \equiv A$ 9. $A \lor \top \equiv \top$ 10. $A \land \bot \equiv \bot$ 11. $A \lor \bot \equiv A$ 12. $A \land (B \lor A) \equiv A$ 13. $A \land (B \lor C) \equiv (A \land B) \lor (A \land C)$ 14. $A \lor (B \land C) \equiv (A \lor B) \land (A \lor C)$ 15. $\neg (A \lor B) \equiv \neg A \land \neg B$ 16. $\neg (A \land B) \equiv \neg A \lor \neg B$ 17. $A \implies B \equiv \neg A \lor B$ 18. $A \iff B \equiv (A \implies B) \land (B \implies A)$ # 2. SAT Solvers - There are several techniques and algorithms for SAT solving. - Usually SAT solvers receive as input a formula in a specific syntatical format. - SAT solvers deal with formulas in **conjunctive normal form (CNF)**. - Most current state-of-the-art SAT solvers are based on the **Davis-Putnam-Logemann-Loveland (DPLL) framework**. ## 2.1 DPLL Framework The idea is to **incrementally construct an assignment compatible with a CNF**, propagating the implications of the decisions made that are easy to detect and simplifying the clauses. A CNF is satisfied by an assignment if all its clauses are satisfied. And a clause is satisfied if at least one of its literals is satisfied.