1) What is a semantic tableau and what is the main advantage of using it to dete

Question

1) What is a semantic tableau and what is the main advantage of using it to determine if a formula is satisfiable?
- Theorem 2.60 in our textbook states: A set of literals is satisfiable if and only if it does not contain a complementary pair of literals. Discuss and illustrate with examples the meaning of this theorem.
(note: a literal is a propositional variable or a negation of a propositional variable)
2) Discuss the algorithm for the construction of a semantic tableau for a formula in propositional logic. What is the main idea of the algorithm?
- Show and discuss examples of the use of semantic tableaux.

3)Consider the following sentence: Let A and B be arbitrary formulas. If A and B are logically equivalent then A <-> B is a tautology. Is that correct? Why? How about the converse?
4)The Lecture Notes given for this module provides a list of logical equivalences and call them “algebraic laws” of the propositional logic. Why call them like that?
5) How can we demonstrate or prove the laws of the propositional logic? Please illustrate with one example.
6) What do you understand by “simplification of a propositional formula”? Show an exampl

Explanation / Answer

Semantic scene:

Definition : An exacting is a particle or the nullification of an iota. A molecule is a positive exacting and the refutation of an iota is a negative strict. For any molecule p, {p,¬p} is a reciprocal match of literals.

For any recipe A, {A,¬A} is an integral combine of equations. An is the supplement of ¬A and ¬A is the supplement of A.

Hypothesis: An arrangement of literals is satisfiable if and just in the event that it doesn't contain an integral combine of literals.

Confirmation Let L be an arrangement of literals that does not contain a corresponding pair. Characterize the translation I by:

I(p) = T if p L,

I(p) = F if ¬p L.

The understanding is all around characterized—there is just a single esteem alloted to every molecule in L—since there is no reciprocal combine of literals in L. Every strict in L assesses to T so L is satisfiable. On the other hand, if {p,¬p} L, at that point for any translation I for the particles in L, either vI (p) = F or vI (¬p) = F, so L isn't satisfiable.

Calculation (Construction of a semantic scene):

Information: A recipe of propositional rationale.

Yield: A semantic scene T for the greater part of whose leaves are stamped.

At first, T is a tree comprising of a solitary root hub named with the singleton set {}. This hub isn't stamped.

Rehash the accompanying advance to the extent that this would be possible: Choose an unmarked leaf l named with an arrangement of recipes U(l) and apply one of the accompanying guidelines.

• U (l) is an arrangement of literals. Stamp the leaf shut × on the off chance that it contains a correlative match of literals. If not, stamp the leaf open .

• U (l) isn't an arrangement of literals. Pick a recipe in U (l) which isn't an exacting. Characterize the recipe as a -equation An or as a -recipe B and perform one of the accompanying strides as indicated by the arrangement:

– A is an -formula. make a fresh node l_ as a child of l and label l_ by:

U(l_) = (U(l) {A}) {A1,A2}.

(In that case that A is A1, there is a refusal A2.)

– B is a -formula. make two fresh nodes l_ and l__ as children of l. Label l_ by:

U(l_) = (U(l) {B}) {B1},

and label l__ by:

U(l__) = (U(l) {B}) {B2}.

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