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Satisfiability Checking Satisfiability Checking First-Order Logic Prof. Dr. Erika Ábrahám RWTH Aachen University Informatik 2 LuFG Theory of Hybrid Systems WS 14/15 Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 14/15 1 / 38 First-order logic We have seen that natural languages are not well-suited for correct reasoning. Propositional logic is useful but sometimes not expressive enough for modeling. First-order (FO) logic is a framework with the syntactical ingredients: 1 Theory symbols: constants, variables, function symbols 2 Lifting from theory to the logical level: predicate symbols 3 Logical symbols: Logical connectives and quantifiers 3 is fixed Fixing 1 and 2 gives different FO instances Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 14/15 2 / 38 Constants, variables, function symbols, terms Theory symbols: constants, variables, function symbols Example: Constants:0 ; 1 Variables: x; y; z;::: Function symbol binary + Terms (theory expressions) are inductively defined by the following rules: 1 All constants and variables are terms. 2 If t1;:::; tn (n > 0) are terms and f an n-ary function symbol then f (t1; :::; tn) is a term. Only strings obtained by finitely many applications of these rules are terms. Example terms:0, x,0 + 1, x + 1, +(x; +(y; 1))( x + (y + 1)) Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 14/15 3 / 38 Predicates, constraints Predicates lift terms from the theory to the logical level. Example predicate symbols: binary ≥; >; =; <; ≤ (Theory) constraints are inductively defined by the following rule: 1 If P is an n-ary predicate symbol and t1;:::; tn are terms then P(t1;:::; tn) is a constraint. Only strings obtained by finitely many applications of this rule are constraints. Example constraints: x<x + 1, (x + 1) + y=((x + y) + 1) Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 14/15 4 / 38 Logical connectives and quantifiers, formulas Logical connectives: unary :, binary ^; _; !; $;::: Universal quantifier 8 (“for all”), existential quantifier 9 (“exists”) (Well-formed) formulas are inductively defined by the following rules: 1 If c is a constraint then c is a formula (called atomic formula). 2 If ' is a formula then (:') is a formula. 3 If ' and are formulas then (' ^ ) is a formula. 4 Similar rules apply to other binary logical connectives. 5 If ' is a formula and x is a variable, then (8x:') and (9x:') are formulas. Only expressions which can be obtained by finitely many applications of these rules are formulas. Example formulas: x < x + 1 (atomic formula) :(x < 0) x < x + 1 ^ (x + 1) + y =( x + y) + 1 8x:9y: y = x + 1 Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 14/15 5 / 38 Example Assume the argumentation: 1 All men are mortal. 2 Socrates is a man. 3 Therefore, Socrates is mortal. We can formalize it by defining Constants: Socrates Variables: x Predicate symbols: unary isMen, isMortal Formalization: 1 8x: isMen(x) ! isMortal(x) 2 isMen(Sokrates) 3 isMortal(Sokrates) Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 14/15 6 / 38 Some remarks and notation Constants can also be seen as function symbols of arity0. Sometimes equality (=) is included as a logical symbol. E.g., the logical connectives negation (:) and conjunction (^) and the existential quantifier (9) would be sufficient, the remaining syntax (_; !; $;:::; 8) are syntactic sugar. We omit parenthesis whenever we may restore them through operator precedence: binds stronger : ^ _ ! $ 9 8 Thus, we write: ::a for (:(:a)); 9a: 9b: (a ^ b ! P(a; b)) for 9a: 9b: ((a ^ b) ! P(a; b)) Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 14/15 7 / 38 Free and bound variables The free and bound variables of a formula are defined inductively: If ' is an atomic formula then a variable x is free in ' iff x occurs in '. Moreover, there are no bound variables in any atomic formula. A variable x is free in (:') iff x is free in '. Moreover, x is bound in (:') iff x is bound in '. x is free in (' ^ ) iff x is free in either ' or . Moreover, x is bound in (' ^ ) iff x is bound in either ' or . The same rule applies to any other binary connective in place of ^. x is free in (9y:') iff x is free in ' and x is a symbol different from y. Moreover, x is bound in (9y:') iff x is y or x is bound in '. The same rule holds with 8 in place of 9. Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 14/15 8 / 38 Free and bound variables Examples: In P(z) _ 8x: 8y: (P(x) ! Q(z)), x and y are bound variables, z is a free variable, and w is neither bound nor free. In Q(z) _ 8z:P(z), z is both bound and free. Being free or bound is for specific occurrences of variables in a formula. In Q(z) _ 8z:P(z), the first occurrence of z is free while the second is bound. Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 14/15 9 / 38 Signature Σ, Σ-formula, Σ-sentence A signature Σ fixes the set of non-logical symbols. A Σ-formula is a formula with non-logical symbols from Σ. A Σ-sentence is a Σ-formula without free variables. In the previous example: Σ = (Sokrates; isMen(·); isMortal(·)) with Sokrates a constant and isMen and isMortal unary predicate symbols. The formulas 1 8x: isMen(x) ! isMortal(x) 2 isMen(Sokrates) 3 isMortal(Sokrates) are Σ-sentences (the only variable x is bound). Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 14/15 10 / 38 Further examples Σ = f0; 1; +;> g 0; 1 are constant symbols + is a binary function symbol > is a binary predicate symbol Examples of Σ-sentences: 9x: 8y: x > y 8x: 9y: x > y 8x: x + 1 > x 8x: :(x + 0 > x _ x > x + 0) Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 14/15 11 / 38 Further examples Σ = f0; 1; +; ∗;<; isPrimeg 0; 1 constant symbols +; ∗ binary function symbols < binary predicate symbol isPrime unary predicate symbol An example Σ-sentence: 8n: (1 < n ! (9p: isPrime(p) ^ n < p < 2 ∗ n)) Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 14/15 12 / 38 Example Let Σ = f0; 1; +; =g where0 ; 1 are constants, + is a binary function symbol and = a binary predicate symbol. Let ' = 9x: x + 0 = 1a Σ-formula. Q: Is ' true? A: So far these are only symbols, strings. No meaning yet. Q: What do we need to fix for the semantics? A: We need a domain for the variables. Let’s say N0. Q: Is ' true in N0? A: Depends on the interpretation of ’+’ and ’=’! Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 14/15 13 / 38 Structures, satisfiability, validity A Σ-structure is given by: a domain D, an interpretation I of the non-logical symbols in Σ that maps each constant symbol to a domain element, each (free) variable to a domain element, each function symbol of arity n to a function of type Dn ! D, and each predicate symbol of arity n to a predicate of type Dn ! f0; 1g. A Σ-formula is satisfiable if there exists a Σ-structure that satisfies it. A Σ-formula is valid if it is satisfied by all Σ-structures. Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 14/15 14 / 38 Semantics Semantics of terms and formulas under a structure (D; I ): constants: [[c]](D;I ) = I (c) variables: [[x]](D;I ) = I (x) functions: [[f (t1;:::; tn)]](D;I ) = I (f )([[t1]](D;I );:::; [[tn]](D;I )) predicates: [[p(t1;:::; tn)]](D;I ) = I (p)([[t1]](D;I );:::; [[tn]](D;I )) logical structure: 1 if [[']](D;I ) = 0 [[:']](D;I ) = 0 if [[']](D;I ) = 1 1 if [[']](D;I ) = 1 and [[ ]](D;I ) = 1 [[' ^ ]](D;I ) = 0 if [[']](D;I ) = 0 or [[ ]](D;I ) = 0 1 if there exists v2D such that [[']](D;I [x7!v]) = 1 [[9x:']](D;I ) = 0 if for all v2D we have [[']](D;I [x7!v]) = 0 Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 14/15 15 / 38 Example Σ = f0; 1; +; =g ' = 9x: x + 0 = 1a Σ-formula Q: Is ' satisfiable? A: Yes. Consider the structure S: Domain: N0 Interpretation: 0 and1 are mapped to0 and1 in N0 + means addition = means equality S satisfies '. S is said to be a model of '. Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 14/15 16 / 38 Example (cont.) Σ = f0; 1; +; =g ' = 9x: x + 0 = 1a Σ-formula Q: Is ' valid? A: No. Consider the structure S0: Domain: N0 Interpretation: 0 and1 are mapped to0 and1 in N0 + means multiplication = means equality S0 does not satisfy '. Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 14/15 17 / 38 Theories T , T -safisfiability and T -validity A Σ-theory T is defined by a set of Σ-sentences. A Σ-formula ' is T -satisfiable if there exists a structure that satisfies both the sentences of T and '. A Σ-formula ' is T -valid if all structures that satisfy the sentences defining T also satisfy '. The number of sentences that are necessary for defining a theory may be large or infinite. Instead, it is common to define a theory through a set of axioms. The theory is defined by these axioms and everything that can be inferred from them by a sound inference system. Satisfiability Checking — Prof. Dr. Erika Ábrahám (RWTH Aachen University) WS 14/15 18 / 38 Examples Σ = f0; 1; +; =g ' = 9x: x + 0 = 1a Σ-formula.
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