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Questions and Answers
What is the main importance of the concept of completeness?
What is the main importance of the concept of completeness?
What is the condition for a logical calculus to be considered complete?
What is the condition for a logical calculus to be considered complete?
Why does the resolution rule terminate for propositional logic?
Why does the resolution rule terminate for propositional logic?
What is the satisfiability problem (SAT) in propositional logic?
What is the satisfiability problem (SAT) in propositional logic?
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What is the Hilbert calculus used for in propositional logic?
What is the Hilbert calculus used for in propositional logic?
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What is the significance of the P = NP-problem?
What is the significance of the P = NP-problem?
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What is the main characteristic of NP-complete problems?
What is the main characteristic of NP-complete problems?
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What is the relationship between a proposition being a tautology and being non-satisfiable?
What is the relationship between a proposition being a tautology and being non-satisfiable?
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What is the significance of completeness in propositional logic?
What is the significance of completeness in propositional logic?
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What is the significance of soundness in a logical calculus?
What is the significance of soundness in a logical calculus?
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What is true about a logical calculus that is semantically complete?
What is true about a logical calculus that is semantically complete?
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What is the resolution rule used for in propositional logic?
What is the resolution rule used for in propositional logic?
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What is the limitation of the theorem on completeness of logical calculus for propositional logic?
What is the limitation of the theorem on completeness of logical calculus for propositional logic?
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What is the characteristic of the runtime of algorithms for NP-complete problems?
What is the characteristic of the runtime of algorithms for NP-complete problems?
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What is the relationship between the number of propositional variables and the number of possible clauses in the resolution rule?
What is the relationship between the number of propositional variables and the number of possible clauses in the resolution rule?
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What is the significance of propositional logic in programming?
What is the significance of propositional logic in programming?
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What is the property of soundness in a logical calculus?
What is the property of soundness in a logical calculus?
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What is the main difference between the resolution rule and the Hilbert calculus?
What is the main difference between the resolution rule and the Hilbert calculus?
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What is the significance of the number of propositional variables in the resolution rule?
What is the significance of the number of propositional variables in the resolution rule?
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What is the relationship between the completeness of a logical calculus and the soundness of a logical calculus?
What is the relationship between the completeness of a logical calculus and the soundness of a logical calculus?
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Study Notes
Propositional Logic
- Propositional logic is a subfield of mathematical logic that studies how to utilize the structure of known propositions to derive new and valid facts and propositions.
- It provides the basis for the formulation of conditions that are often used in programming.
- Propositional logic investigates the truth values of propositions, which are defined as phrases to which a truth value "true" or "false" can be attached.
Basic Concepts
- Propositional variables are statements that can be assigned a truth value, such as "it is raining" or "London is the capital of the United Kingdom."
- Propositional logic studies how to reason about the truth of propositions, including assessing how the truth of one proposition relates to the truth of another.
Boolean Operators
- Boolean operators are used to combine propositional variables, including:
- NOT (¬): inverts the truth value of a given propositional variable
- AND (∧): corresponds to "and" in colloquial language, connecting two propositional variables
- OR (∨): corresponds to "or" in everyday language, connecting two propositional variables
- IF-THEN (→): represents a logical implication, connecting two propositional variables
- IF-AND-ONLY-IF (): represents an equivalence or biconditional, connecting two propositional variables
- Truth tables are used to define the behavior of these operators.
Complete Basis
- A set of Boolean operators is a complete basis if every other Boolean operator can be expressed with elements of the set.
- OR and NOT form a complete basis, as do NAND and other operators.
Propositional Logic as a Language
- Propositional logic can be understood as a formal language, following a set of rules defined by its syntax and semantics.
- The language is defined via a set of rules, including the use of propositional variables, Boolean operators, and parentheses.
Calculation Rules and Normal Forms
- Laws in propositional logic, such as the idempotent law, commutative law, associative law, absorption law, distributive law, and De Morgan's law, allow for the transformation of formulas.
- These laws can be used to simplify and manipulate propositional logic formulas.
Applications
- Propositional logic plays a key role in computer science and appears in various situations, including:
- Formulation of conditions and constraints in programming (e.g., if-then-else statements)
- Formulation of complex formulas in spreadsheets
- Setting up logic circuits in hardware design
- Artificial intelligence
- Logic programming (e.g., in Prolog)
- Program verification, formal specifications, and formal semantics### Propositional Logic
- De Morgan's Laws: allow to transform a negated bracket or to switch between AND and OR connectives.
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Normal Forms: standardized ways of describing a fact in propositional logic.
- Disjunctive Normal Form (DNF): a proposition is represented as a disjunction of conjunctions.
- Conjunctive Normal Form (CNF): a proposition is represented as a conjunction of disjunctions.
- Reduced Normal Forms: require that every literal appears only once in a conjunction of a DNF, or in every disjunction of a CNF.
Proofs
- Deduction Laws: used to deduce new formulas from existing ones.
- Axioms: formulas that are considered true and are used to derive new formulas.
- Formal System: an abstract structure used to infer or prove formulas from axioms according to a set of rules.
- Hilbert Calculus: a logical calculus that consists of axioms and transformation laws.
- Truth Tables: can be used to prove or disprove whether a proposition is a tautology.
Interpretation and Satisfiability
- Interpretation: a function that assigns a truth value to each propositional variable.
- Model: a formula that is true under a given interpretation.
- Satisfiability: a formula is satisfiable if at least one interpretation exists that makes the formula true.
- Tautology: a formula that is true for every possible interpretation.
- Falsifiable: a formula that is false for at least one interpretation.
- Non-satisfiable: a formula that is false for every possible interpretation.
Three-Valued Logic
- Undefined or Unknown Values: an extension of two-valued logic to include a third truth value.
- Short Evaluation: a conjunction or disjunction is evaluated as soon as the outcome is determined.
- Strict Interpretation: the conjunction and disjunction are evaluated in the classical manner.
Proof by Contradiction and Resolution
- Proof by Contradiction: a method of proof that assumes the opposite of what is to be proven, and then derives a contradiction.
- Resolution: a method of proof that is based on the complement rule and can be automated.
- Clause: a propositional formula that is a disjunction of literals.
- Resolution Rule: a rule for combining two clauses to form a new one.
Soundness and Completeness
- Soundness: a property of a logical system that ensures that every formula that can be deduced is a tautology.
- Completeness: a property of a logical system that ensures that it is always sufficient to prove or disprove whether a proposition is a tautology.
- Theorem: the logical systems of propositional logic discussed are sound, including the calculus of truth tables, the Hilbert calculus, and the resolution rule.### Completeness of a Logical Calculus
- A logical calculus is (semantically) complete if every tautology can be proved within the system.
- A calculus is complete if ⊨ φ always implies ⊢ φ.
- In propositional logic, the following logical calculi are complete:
- The calculus of truth tables
- The Hilbert calculus with axioms and laws of propositional logic
- The resolution rule described in the previous algorithm
Propostional Logic
- Propostional logic studies the truth values of propositions.
- A proposition is a statement that can be assigned a truth value (true or false).
- Propostional variables are the simplest well-formed formulas.
- Boolean operators are used to combine propositional variables:
- NOT (¬): inverts the truth value of a propositional variable
- AND (∧): true if and only if both propositional variables are true
- OR (∨): true if at least one propositional variable is true
- IF-THEN (→): true unless the antecedent is true and the consequent is false
- IF-AND-ONLY-IF (): true if and only if both propositional variables have the same truth value
Boolean Operators
- The list of Boolean operators includes:
- XOR (exclusive OR)
- NAND (not-and)
- NOR (not-or)
- A set of Boolean operators is a complete basis if every other Boolean operator can be expressed with elements of that set.
- Theorem: OR and NOT form a complete basis, and NAND forms a complete basis.
Propositional Logic as a Language
- Propositional logic can be understood as a formal language.
- A language is defined via its syntax (which expressions are allowed?) and its semantics (what do certain expressions mean in the language?).
- The language of propositional logic follows a specific set of rules, making it a context-free language (CFL).
- In programming languages, Boolean operators are available for use in logical expressions.
Additional Concepts
- Fuzzy logic: a framework for working with imprecise or unclear terms.
- Temporal logic: deals with propositional variables whose truth values are time-dependent.
- Predicate logic (first-order logic): deals with statements that contain variables that must be interpreted.
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Description
Learn the fundamentals of propositional logic, a crucial subfield of mathematical logic that deals with deriving new facts from known propositions, with applications in programming.
