Temporal Logic - Soundness and Completeness of L

Size: px

Start display at page:

Download "Temporal Logic - Soundness and Completeness of L"

Maximillian McDaniel
5 years ago
Views:

1 Temporal Logic - Soundness and Completeness of L CS402, Spring 2018

2 Soundness Theorem 1 (14.12) Let A be an LTL formula. If L A, then A. Proof. We need to prove the axioms and two inference rules to be valid. We have already shown soundness for propositional logic, and we have shown the soundness of MP rule for H earlier. Distribution (Axiom 1) and linearity (Axiom 5) have been shown to be sound. Among the remaining axioms, let us take Axiom 4: (A A) (A A).

3 Proof: Soundness of (A A) (A A). If the formula is not valid, there exists an interpretation ρ such that ρ (A A) A A. Since ρ A and ρ A, there exists a smallest value i > 0 such that ρ i A and ρ j A for 0 j < i. In particular, ρ i 1 A. But we also have that ρ (A A), so by definition of the operator, ρ i 1 A A. By MP we have ρ i 1 A and thus ρ i A, contradicting ρ i A.

4 Soundness of Generalisation: If A, then A Proof. We need to show that for all interpretations ρ, ρ A. This means that for all i 0, it is true that ρ i A. But A implies that for all interpretation ρ, ρ A, in particular, this must hold for ρ = ρ i.

5 Exercises Prove the soundness of other axioms of L.

6 Completeness Theorem 2 (14.13) Let A be a formula of LTL. If A, then L A. Proof. If A is valid, the construction of a semantic tableau for A will fail, either because it closes or because all the cycles are non-fulfulling a. We show by induction that for every node in the tableau, the disjunction of the negations of the formulas labeling the node is provable in L. Since the formula labeling the root is A, it follows that A, from which A follows by propositional logic. a This is a simplified statement, but we skipped over the actual algorithm; for our purpose here, this definition will do.

7 Completeness Proof. Cont. Here we present the proof for X -rules. Consider the following application of X -rule: A 1,..., A n, B 1,..., B k A 1,..., A n. Note that we assume negations are pushed inwards, using the linearity axiom. The induction structure is as follows: we hypothesise that A, therefore, going bottom-up in the tableau, we hypothesise the negation of distunctions, i.e. A 1... A n, and show that the formula corresponding (i.e. disjunction of negations) to the line above in the tableau is provable in L. 1. A 1... A n Inductive hypothesis 2. ( A q... A n) Generalisation, 1 3. ( A 1... A n) Expansion, 2 4. A 1... A n Distribution, 3 5. A 1... A n Linearity, 4 6. A 1... A n B 1... B k Prop, 5

8 Completeness Proof. Cont. Now we need to deal with cycles in tableaux. Intuitively, we extract the invariant of the cycle, and use it in our induction to represent that particular branch. Let us use an example: consider p p. First, we construct a semantic tableau for the negation of the formula: ( p p) p, p l s : p, p, p p, p l β : p, p, p p, p, p To node l s

9 Completeness Proof. Cont. Invariants are the conjunction of the formulas A i, where A i are the next formulas in the cycle. In our example, the invariant is p p (see l s ). We first show that the formula is inductive: 1. ( p p) (p p) ( p p) Expansion 2. ( p p) (p p p) Prop, 1 3. ( p p) ( p p) Prop, 2 4. ( p p) ( p p) Distr., 3 5. ( p p) ( p p) Induction, 4 So we have proved that p p is the invariant; consequently, we can use this to represent the cycle on the right; the left branch can be converted into an axiom by taking the disjunctive of the negations: p p p.

10 Completeness Proof. 1. p p p Axiom 0 2. (p p) p Prop, 1 3. p p Contraction, 2 4. ( p p) p Prop, 3 5. ( p p) p Generalisation, 4 6. ( p p) p Prop, prev. proof., 5 7. (p p p) p Expansion, 6 8. (p p p) p Duality, 7 9. p p p Prop, 8 That is, from the formula corresponding to the closed left branch and the invariant of the right cycle, we have proved the formula corresponding to l β.

11 As I promised way earlier: Gödel s Ontological Proof Symbols : necessity, : possibility. P(φ): φ is a positive property (i.e. φ is good). 1. P(φ) x[φ(x) ψ(x)] P(ψ) Ax.1 2. P( φ) P(φ) Ax.2 3. P(φ) x[φ(x)] Th.1, RAA, 1, 2 4. G(x) φ[p(φ) φ(x)] Df.1 5. P(G) Ax.3 6. xg(x) Th.2, MP, 4, 5, 3 7. φ ess x φ(x) ψ{ψ(x) y[φ(y) ψ(y)]} Df.2 8. P(φ) P(φ) Ax.4 9. G(x) G ess x Th E(x) φ[φ ess x yφ(y)] Df P(E) Ax xg(x) Th.4

12 Gödel s Ontological Proof The following natural language reading is by mjqxxxx ( com/questions/248548/gdels-ontological-proof-how-does-it-work): Axiom 1: If φ is good, and φ forces ψ (that is, it s necessarily true that anything with property φ has property ψ ), then ψ is also good. Axiom 2 : For every property φ, exactly one of φ and φ is good. (If φ is good, we may as well say that φ is bad.) Theorem 1 (Good Things Happen): If φ is good, then it s possible that something exists with property φ. Definition 1: We call a thing godlike when it has every good property. Axiom 3 : Being godlike is good. Theorem 2 (No Atheism): It s possible that something godlike exists. Definition 2 : We call property φ the essence of a thing x when (1) x has property φ, and (2) property φ forces every property of x. Axiom 4 : If φ is good, then φ is necessarily good. Theorem 3: If a thing is godlike, then being godlike is its essence. Definition 3 : We call a thing indispensable when something with its essence (if it has an essence) must exist. Axiom 5: Being indispensable is good. Theorem 4: Something godlike necessarily exists.

13 Gödel s Ontological Proof Proof. Theorem 3. First note that if x is godlike, it has all good properties (by definition) and no bad properties (by Axiom 2). So any property that a godlike thing has is good, and is therefore necessarily good (by Axiom 4), and is therefore necessarily possessed by anything godlike. Proof. Theorem 4. If something is godlike, it has every good property by definition. In particular, it s indispensable, since that s a good property (by Axiom 5); so by definition something with its essence, which is just being godlike (by Theorem 3), must exist. In other words, if something godlike exists, then it s necessary for something godlike to exist. But by Theorem 2, it s possible that something godlike exists; so it s possible that it s necessary for something godlike to exist; and so it is, in fact, necessary for something godlike to exist.

14 Done! What we have covered: Propositional Logic: semantics, tableau method, natural deduction, proof sequents (G and H) Predicate Logic: semantics, tableau method, natural deduction Modal/Temporal Logic: semantics, tableau method, proof sequent (L ) Using SAT/SMT Solvers for basic tasks From here, where can we go next? CS492 (Fall used to be CS453, currently under change) Automated Software Testing: if you want to put the basic knowledge of logic into serious business (i.e. software model checking). CS454 (Fall 2018) AI-Based Software Engineering: if you want to experience metaheuristic inference (sound? complete? you must be joking?) in the context of software engineering.

Propositional Logic: Gentzen System, G

Propositional Logic: Gentzen System, G CS402, Spring 2017 Quiz on Thursday, 6th April: 15 minutes, two questions. Sequent Calculus in G In Natural Deduction, each line in the proof consists of exactly one proposition. That is, A 1, A 2,...,