Lecture Slides for MAT-60556 PART III: First order logic: Formulas, - PowerPoint PPT Presentation

Lecture Slides for MAT-60556 PART III: First order logic: Formulas, Models, Tableaux Henri Hansen September 20, 2013 1

Relations and predicates • Theories with axioms and theorems in mathematics are defined on sets such as the set of integers Z • We need to be able to write and manipulate for- mulas that contain relations on the elements from arbitrary sets • R ⊆ D n is called an n -ary relation over the domain D . We also call relations predicates . • R can be represented by a boolean function P R : D n �→ { T, F } 2

First Order Logic Symbols • Let P , A , and V be countable sets of predicate, con- stant , and variable symbols . Each predicate p n ∈ P is associated with arity . • “ ∀ ” is the universal quantifier, and “ ∃ ” is the exis- tential quantifier • Variables and constants are called terms . Terms are elements of some (possibly unknown) set • The arguments of a predicate are terms. 3

FOL Syntax • An atomic formula is an n -ary predicate followed by a list of arguments. A formula is a tree defined recursively as follows: 1. A formula is a leaf that is labeled by an atomic formula 2. A formula is a node labeled with a logical oper- ator (as in propositional logic), and its children are formulas 3. A formula is labeled by ∀ x or ∃ x for some variable x , and it has a single child that is a formula 4

Scope of variables • A formula such as ∀ xA is a quantified formula . x is the quantified variable, and its scope is the formula A . x is not required to actually appear in A • Let A be a formula. An occurrence of x in A is a free variable of A iff x is not within the scope of a quantified variable x . A variable that is not free, is bound • A formula with no free variables is closed and a formula with free variables is open 5

Interpretation and Assignment • Let A be a formula where { p 1 , . . . , p m } are the pred- icates, and { a 1 , . . . , a k } are the constants that ap- pear. An interpretation for A , I A is a triple ( D, { R 1 , . . . , R m } , { d 1 , . . . , d k } ) where D is a non-empty set called domain , R i ⊆ D n i is a relation assigned to the predicate p i and d i ∈ D is the element assigned to the constant a i . • An assignment σ I A : V �→ D is a function that maps each free variable to an element of D 6

Validity and satisfiability • The truth value of a closed formula depends only on the interpretation (not assignment) • Given a closed formula A in first-order logic 1. A is true in the intrepretation I iff ν I ( A ) = T , denoted I | = A 2. A is valid if it is true for all interpretations, de- noted | = A 3. A is satisfiable if there is some interpretation I such that I | = A 7

Logical equivalence • Given two formulas A 1 and A 2 we write A 1 ≡ A 2 if they are true for the same interpretations • Given U = { A 1 , . . . , a n } of closed formulas, 1. A ≡ B iff | = A ↔ B 2. U | = A iff | = ( A 1 ∧ · · · ∧ A n ) → A 3. ∀ xA ( x ) ≡ ¬∃ x ¬ A ( x ) 4. ∃ xA ( x ) ≡ ¬∀ x ¬ A ( x ) 8

Semantic Tableaux • let A be a quantified formula, either ∀ xA 1 ( x ) or ∃ xA 1 ( x ) and let a be a constant. An instantiation of A is the formula A 1 ( a ), where all the free occur- rences of x have been replaced by a • A literal is a closed atomic formula p ( a 1 , . . . , a k ), where all arguments are constants, or the negation of such a formula. • A semantic tableau is a tree T where each node is labeled by a pair ( U ( n ) , C ( n )) where U is a set of formulas and C is a set of constants that appear in the formulas of U ( n ) 9

Semantic Tableaux (contd.) • A branch of a semantic tableau can be infinite, finite and marked open, or finite and marked closed • The root of the tableau for the formula φ is marked ( { φ } , { a 0 1 , . . . , a 0 k } ), where the constants are the constans appearing in φ • If U ( l ) contains a complementary pair of literals, then it should be marked closed, or × 10

Rules for Tableaux • The same α - and β - rules as used with tableaux for propositional formulas applies for first order logic • In addition, we have γ and δ rules • γ - formula is either ∀ xA ( x ) or ¬∃ xA ( x ), and the result is A ( a ) or ¬ A ( a ) resp. where a is an arbitrary constant • δ - formula is either ∃ xA ( x ) or ¬∀ xA ( x ), and the result is A ( a ) or ¬ A ( a ) 11

Semantic Tableau algorithm for the formula φ . • Start with {{ φ } , { a 0 , . . . , a n }} at the root. • Choose an unmarked leaf { U, C } and apply the first applicable rule from the following list and start over. 1. If the set U contains a complementary pair of literals, close the branch. 2. Choose a formula A ∈ U that is either an α − , β − or δ -formula; apply the rule 3. Apply the γ -rule. If the resulting formula is the same as before applying it, mark the branch open. 12

Handling of formulas • If A is an α -formula, create a new node as the child of l , and label it with (( U ( l ) − { A } ) ∪ { α 1 , α 2 } , C ( l )) • If A is a β - formula, create two new nodes as chil- dren of l and label them with 1. (( U ( l ) − { A } ) ∪ { β 1 } , C ( l )) 2. (( U ( l ) − { A } ) ∪ { β 2 } , C ( l )) • If A is a δ - formula, create a new node as the child of l , and label it with (( U ( l ) −{ A } ) ∪{ δ ( a ′ ) } , C ( l ) ∪{ a ′ } ), where a ′ does not appear in U ( l ) 13

Handling of formulas (contd.) • Given the set { γ 1 , . . . , γ m } ⊆ U , of the γ -formulas of U , create a new node as a child of l , and label it m ∪ ∪ with formulas ( U ∪ { γ i ( c ) } , C ) i =1 c ∈ C • In other words: a γ -formula γ i is added by γ i ( c ) for all the constants c in the formula. 14

Soundness of Tableaux • Let φ be a formula of first order logic, and T be a tableau for φ . If T closes, then φ is unsatisfiable – Proof is by induction on the height of a node; α and β -rules are like in the propositional case. – γ -rules add A ( a ) to a child, when ∀ xA ( x ) ap- pears, and if A ( a ) is unsat, then ∀ xA ( x ) is unsat – δ -rules replace ∃ xA ( x ) with A ( a ) for a new a . Assume the former is satisfiable. Then it has a model. As it has a model, then there is some constant a for which A ( a ) is satisfiable, and thus A ( a ) has a model too. 15

Hintikka sets • U is a Hintikka set if for all formulas A ∈ U ∈ U or A c / 1. If A is a literal, then either A / ∈ U 2. If A is an α -formula, then α 1 ∈ U and α 2 ∈ U 3. If A is a β -formula, then β 1 ∈ U or β 2 ∈ U 4. If A is a γ -formula, then γ ( c ) ∈ U for all constants in the formulas of U 5. If A is a δ -formula, then δ ( c ) ∈ U for some con- stant c 16

Completeness: • Theorem: If U is a Hintikka set, then there is a model for U • Theorem: If A is a valid formula, the semantic tableau for ¬ A closes – If the tableaux for ¬ A does not close, this means that we can construct a Hintikka set from an open branch, which implies there is a model for ¬ A , and thus, A is not valid • In summary: Tableaux can establish the validity of A by finding whether ¬ A has a model, but the tableau itself does not directly yield a model 17

Deductive systems • As with propositional logic, deductive systems can be used to prove formulas of first-order logic. They do so by taking axioms and rules of inference as before, but new rules and axioms are needed for quantifiers • We leave familiarization with deductive systems (Gentzen and Hilbert) as an exercise 18

Normal forms in FOL • Consider the formula ∀ x ∀ y ∀ z ( p ( x, y ) ∧ p ( y, z )) → p ( x, z ), under the interpretation { Z , { < } , {}} • The formula states that the less-than- relation is transitive in the domain of the integers • Suppose we wish to express something like ∀ x ∀ y ∀ z ( x < y ) → x + z < y + z • The diffeerence is that in addition to the predicate “ < ”, it uses the function “+”. We first extend first-order logic with functions in order to discuss such statements 19

FOL with functions • Let F be a (countable) set of function symbols , where each symbol has an arity denoted by a su- perscript. • A term is defined recursively as follows – A variable, constant, or 0-ary function is a term – if f n is an n -ary function symbol and { t 1 , . . . , t n } is a set of terms, then f ( t 1 , . . . , t n ) is a term • In the extended logic, an atomic formula is a n -ary predicate followed by a list of terms, i.e., p ( t 1 , . . . , t n ) 20

FOL interpretation with functions • Let U be a set of formulas, and p 1 , . . . , p k the pred- icate symbols, f 1 , . . . , f l the function symbols and a 1 , . . . , a m the constant symbols of U . An interpre- tation for U is a 4-tuple 1 , . . . , F n l { D, { R 1 , . . . , R k } , { F n 1 l } , { d 1 , . . . , d m }} consisting of the domain D , relations R i on D as- sociated with each predicate symbol p i , functions F i on D associated with f i and elements d i of D associated with a i • The semantics of the interpretation are the same as for FOL without functions 21