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propositional

# T3 each internal node with two successors is labeled

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Unformatted text preview: ariable in p. T2. Each internal node with a single successor is labeled by a subformula ¬q of p and has q as a successor. T3. Each internal node with two successors is labeled by a subformula aXb of p with X in {∧,∨,⊕,→, } and has a as a left successor and b as a right successor. 14 T1. Each leaf is an occurrence of a propositional variable in p. T2. Each internal node with a single successor is labeled by a subformula ¬q of p and has q as a successor. T3. Each internal node with two successors is labeled by a subformula aXb of p with X in {∧, ∨, ⊕, →, ↔} and has a as a left successor and b as a right successor. Example Example 4. The formation tree of the formula ((a ∧ b) ∨ ¬c) is given by ((a ∧ b) ∨ ¬c) ✚ ✚ (a ∧ b) ¬c ✡❏ ✡❏ ab c Remark. In a course on compiler construction, you will learn how to write a parser for languages such as the one that we have speciﬁed for propositional logic. You can check out lex and yacc if you want to write a parser for propositional logic in C or C++ now. In Haskell, you can use for example the parser generator Happy. 4 15 Assigning Meanings to Formulas We know that each formula corresponds to a unique binary tree. We can evaluate the formula by - giving each propositional variable an interpretation. - defining the meaning of each logical connective - propagate the truth values from the leafs to the root in a unique way, so that we get an unambiguous evaluation of each formula. 16 4 Semantics The Semantics of Propositional Logic Let B = {t, f } denote the set of truth values, where t and f represent true and false, respectively. We associate to the logical connective ¬ the function M¬ : B → B given by P M¬ (P ) Let4B={t,f}.Semantics of Propositional Logic function Mx: B->B The Assign to each connective x a f t t f Let B = {t, f } denote semantics values, that determines thethe set of truth of x. where t and f represent true and falseP ) is true if We associate is false. This justiﬁes the name negation Thus, M¬ (, respectively. and only if P to the logical connective ¬ the function for M¬ : B → B given by this connective. The graph of the function M¬ given above is called the P M¬ Similarly, we associate to a connective truth table of the negation connective.(P ) f t X in the set {∧, ∨, ⊕, →, ↔} a binary function MX : B × B → B. The truth t f tables of these connectives are as follows: Thus, M¬ (P ) is true if and only if P is false. This justiﬁes the name negation for P Q M∧ (P, Q) M∨ (P, Qthe M⊕ (P, QM¬ M→ (P, Q) is called Q) this connective. The graph of ) function ) given above M↔ (P, the truth table of the negation connective. Similarly, we associate to a connective ff f f f t t X in the set {∧, ∨, ⊕, →, ↔} a binary function MX : B ×tB → B. The truth ft f t t f tables of these connectives are t follows: t as tf f f f t P t Q M∧tP, Q) M∨ (P, Q) M⊕ (P, Q) M→ (P, Q) M↔ (P, Q) t f t t ( ff f f f t t You should tvery carefully inspect this table! It is t critical that you memorize f f t t f and fully understand the meaning of each connective. tf f t t f f The t...
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