By Paul Siegel

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Note that the equivalence theory above is A Tutorial on Satisfiability Modulo Theories 27 addeqlit(x = y, F, D) := F, D , if (skip) F ∗ (x) ≡ F ∗ (y) ∗ ∗ ⊥, if F (u) ≡ F (v) for some u = v ∈ D (union) addeqlit(x = y, F, D) := F , D , otherwise where x = F ∗ (x) ≡ F ∗ (y) = y , F = union(F )(x, y) (contrad ) addeqlit(x = y, F, D) := ⊥, if F ∗ (x) ≡ F ∗ (y) addeqlit(x = y, F, D) := F, D , if F ∗ (x) ≡ F ∗ (x ), F ∗ (y) ≡ F ∗ (y ), (skipdiseq) for x = y ∈ D addeqlit(x = y, F, D) := F, {x = y} ∪ D , otherwise.

The negation of a literal p is ¬p, and the negation of ¬p is just p. A formula is a clause if it is the iterated disjunction of literals of the form l1 ∨ . . ∨ ln for literals li , where 1 ≤ i ≤ n. A formula is in conjunctive normal form (CNF) if it is the iterated conjunction of clauses Γ1 ∧ . . ∧ Γm for clauses Γi , where 1 ≤ i ≤ m. 2 First-Order Logic In defining a first-order signature, we assume countable sets of variables X, function symbols F , and predicates P. A first-order logic signature Σ is a partial map from F ∪ P to the natural numbers corresponding to the arity of the symbol.

M [[an ]]) ∈ M (p) M |= ¬ψ ⇐⇒ M |= ψ M |= ψ0 ∨ ψ1 ⇐⇒ M |= ψ0 or M |= ψ1 M |= ψ0 ∧ ψ1 ⇐⇒ M |= ψ0 and M |= ψ1 M |= (∀x : ψ) ⇐⇒ M {x → a} |= ψ, for all a ∈ |M | M |= (∃x : ψ) ⇐⇒ M {x → a} |= ψ, for some a ∈ |M | A first-order Σ-formula ψ is satisfiable if there is a Σ-structure M such that M |= ψ, and it is valid if in all Σ-structures M , M |= ψ. A Σ-sentence is either satisfiable or its negation is valid. We focus on the satisfiability problem for quantifier-free first-order formulas. 3 SAT Solving The principles of modern SAT solving have their origin in the 1960 procedure of Davis and Putnam [DP60], as simplified in 1962 by Davis, Logemann, and Loveland [DLL62].

### Understanding Digital Computers by Paul Siegel

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