Semantic Tableaux and Resolution in Propositional Logic

Semantic Tableaux and Resolution are two important methods used in propositional logic to determine the validity or satisfiability of logical formulas. Let’s explore each method in more detail:

1. Semantic Tableaux (also known as Truth Trees): Semantic Tableaux is a proof technique used to determine the satisfiability or validity of a logical formula in propositional logic. It involves systematically constructing a tree-like structure called a truth tree, where the branches of the tree represent different possible truth value assignments to the propositional variables.

The procedure starts with the negation of the formula we want to analyze and builds the truth tree by applying a set of expansion rules. These rules manipulate the logical connectives and propagate truth values through the tree. If every branch of the truth tree leads to a contradiction (where contradictory formulas appear on a branch), the original formula is unsatisfiable. If every branch closes (all branches reach a contradiction), the original formula is valid. Otherwise, if at least one branch remains open (does not close or reach a contradiction), the formula is satisfiable.

Semantic Tableaux provide a systematic method for analyzing complex logical formulas by exhaustively exploring the possible truth value assignments. It is a sound and complete method, meaning that it can correctly identify valid and unsatisfiable formulas.

2. Resolution: Resolution is a proof technique used in propositional logic to establish the validity of logical formulas. It is based on the principle of refutation, which aims to show that the negation of a formula leads to a contradiction.

The resolution method is applied to a logical formula in conjunctive normal form (CNF), which is a conjunction (AND) of clauses, where each clause is a disjunction (OR) of literals (propositional variables or their negations). The resolution process involves repeatedly applying a resolution rule, which combines two clauses by resolving a complementary pair of literals.

The resolution rule states that if two clauses contain complementary literals (i.e., a literal and its negation), they can be resolved by eliminating the complementary literals and forming a new clause with the remaining literals. This process is repeated until no more resolutions can be made or a contradiction (an empty clause) is derived.

If, during the resolution process, a contradiction (empty clause) is derived, it means that the original formula is unsatisfiable, and hence, it is valid. On the other hand, if the resolution process reaches a point where no more resolutions can be made, the formula is satisfiable.

Resolution is a sound and complete method for propositional logic. It allows for the automated reasoning and proof of logical formulas by systematically applying the resolution rule.

Both Semantic Tableaux and Resolution are powerful tools in propositional logic for analyzing the validity and satisfiability of logical formulas. They provide systematic procedures for evaluating complex logical expressions and are widely used in automated theorem proving, model checking, and other areas of logic and computer science.