Modal Logic – an Appetizer

Modal Logic – An Appetizer

2 What is ‘Modal Logic’? ‘Modal logic’ refers to a (huge!) family of extensions of classical logic. Like classical logic, modal logics come in levels/orders. We will focus on propositional modal logics, here. This is no accident: Propositional modal logics allow one to express features that would need quantiﬁers in classical logic! Syntax: additional unary connective(s): modal operator (‘box’) and the dual operator ♦ (‘diamond’): ♦ =def ¬¬ the (inductive) deﬁnition of formulas is as usual Straightforward extension to to multi-modal logics

E.g., K1,..., Kn, EG , CG for n-agent epistemic logic other multi-modal logics:

I temporal logics: operators to refer time points: hF iA, [P]A, A until B I dynamic logic: each program is a modal operator!

We will only consider logics based on here. 3 Intended semantics Modal operators refer to a mode of assertion:

Intended Meaning of F Meaning of ♦F (= ¬¬F )

necessarily F possibly F (an agent) knows that F (an agent) deems F consistent with what is known to her/him (an agent) believes that F (an agent) deems F possible it always holds that F at some time F F should be the case F is permitted F is provable F is not refutable F holds after every state in F holds after some state in which a (nondeterministic) pro- which a (nondeterministic) program terminates gram terminates

4 Possible world semantics Example: Reasoning about databases

Formulas F are built up from atoms P(t1,..., tn), where ground atoms are entries in a database. Such classical formulas correspond to queries, but may also be used to formulate integrity constraints, i.e., formulas that have to hold in every instance of a database. To express that F is an integrity constraint (and not just a formula holding in the current instance of the database, or just a valid formula) we have to extend classical logic (CL): F ... F is an integrity constraint To extend (Tarski’s) semantics for CL to the evaluation of statements like F , we have to refer to all possible states of a database. Similar examples: (physically) necessary −→ (physically) possible states of the world always −→ system states at arbitrary time points etc. [check previous slide!] 5 (Formal) possible world semantics: Kripke semantics Note: We only treat propositional modal logics, here. A Kripke interpretation (model) is a tuple M = hW , R, V i where: non-empty set W of (possible) worlds (states, points) an accessibility relation R ⊆ W × W (variable) assignment V :(PV × W ) 7→ {1, 0} (alternatively: V 0 : PV 7→ 2W or V 00 : W 7→ 2PV ) Formulas without are evaluated as usual (in each world), e.g.:  0 if v (F , w) = 1  M vM(F ⊃ G, w) = and vM(G, w) = 0  1 otherwise

New: ( 1 if ∀u wRu ⇒ v (F , u) = 1 v ( F , w) = M M 0 otherwise

7 Kripke semantics (ctd.) Since ♦ = ¬¬ we obtain: ( 1 if ∃u wRu and v (A, u) = 1 v ( A, w) = M M ♦ 0 otherwise

Note: The deﬁning conditions for ♦ (vM(♦A, w)) can be derived from ♦ = ¬¬ and from the consitions for F and for ¬F .

Alternative notation for vM(A, w) = 1: (M, w) |= A Read:‘ A holds at state/point/world w (in M)’ or:‘ A is satisﬁed at state/point/world w (in M)’ or:‘ A is true at state/point/world w (in M)’ A is true in interpretation (model) M = hW , R, V i if vM(A, w) = 1 for all w ∈ W . Notation: M |= A The pair hW , Ri of an interpretation M = hW , R, V i is called the (Kripke) frame on which M is based. 8 Four levels of ‘truth’ in Kripke semantics

1 Truth at a world: (M, w) |= A 2 Truth in a model: M |= A. 3 Validity (truth) in a frame: A is valid in frame F = hW , Ri if A is true in all interpretations based on F. Notation: F |= A 4 Validity (truth) in a class of frames: If Φ is a class (set) of frames we say that A is true in Φ if F |= A for all F ∈ Φ. Note: A modal logic does not refer to a particular frame, but to a whole class of frames determined by some property of the accessibility relations.

9 Evaluation in a Kripke interpretation — Example M = h{0, 1, 2, 3, 4}, R, V i: ¨ ? - - - 0 1 3 4 q p q p, q

? © 2 p, q Exercise : Evaluate the following formulas in diﬀerent worlds in M: 1. p ⊃ q, q ⊃ p, p ∨ q, p ∨ ¬p 2. ♦p, p, ♦q, q, ♦¬q, ¬q 3. (p ∨ q), ♦(p ∨ q), ♦(p ⊃ q), ♦(p ∧ ¬p), (p ∨ ¬p) 4. p ∨ ¬p, p ⊃ ♦p, ¬♦p ⊃ p, q ⊃ q, p ∨ ♦p 5. ♦p, ♦q, ♦♦p, p ⊃ p, q ⊃ q, p ⊃ p 6. ♦(p ∧ q) ⊃ ♦p, ♦¬q ⊃ ¬q, (p ∧ q) ⊃ (p ∧ q) 10 Validity in frames Consider the frame F = h{w, u}, Ri:

¨ ¨ ? ? - w u The following formulas (schemata) are valid in F: A ⊃ ♦A — not valid without R(w, w) or R(u, u). ♦(A ⊃ A) — not valid without, e.g., R(w, w) and R(w, u). Exercise : Find some further examples of modal formulas with one schematic variable that are valid in F, above, such that removal of some accessibilities leads to invalidity.

11 A syntactic deﬁnition of ‘logic’ A propositional logic L is a set of formulas that is 1 closed under substitutions( PV 7→ FORMULAS) 2 closed under modus ponens (MP): FF ⊃ G MP G read: if F ∈ L and F ⊃ G ∈ L then also G ∈ L Note: CL (classical logic) is a logic, in this sense (like all logics mentioned in this course) every Hilbert-type system (with MP) induces a logic adding or removing axiom schemata and derivation rules to a Hilbert-type system results in new logics (in general) Exercise : Show that the intersection of two logics is also a logic. What about unions of logics?

12 From models and frames to logics Note: In general, the set of formulas that are true in a particular Kripke interpretation(model) does not form a logic. In contrast: The set of formulas that are valid in a particular Kripke frame do always form a logic that extends classical logic. Observe: The following formulas (in fact: schemata of formulas) are valid in every Kripke frame: ♦(A ∧ B) ⊃ ♦A ♦¬A ⊃ ¬A (A ∧ B) ⊃ (A ∧ B) (A ⊃ B) ⊃ (A ⊃ B) Exercise :

Which of the above implicative formulas can/cannot be inverted? 13 From models and frames to logics (ctd.) Fact: for any Kripke interpretation M: if M |= A then M |= A It follows, that also for all frames F: if F |= A then F |= A Deﬁnition: A normal modal logic is a logic that contains all CL-tautologies and (K) (A ⊃ B) ⊃ (A ⊃ B) F and is closed under the following necessitation rule: F Be careful! Although the validity of F implies the validity of F the formula F ⊃ F is not valid in general. In other words: the deduction theorem does not hold for modal logics! Exercise : Find a counter-example to F ⊃ F . 14 From models and frames to logics Consequence of the previous deﬁnition: A Hilbert-style system for the smallest normal modal logic — called K, in honour of Saul Kripke — can be presented as follows: appropriate axioms for CL (e.g.: all CL-tautologies) modal axiom (K): (A ⊃ B) ⊃ (A ⊃ B) derivation rules: MP and necessitation Theorem: (Soundness and completeness of K) K is the set of all formulas that are valid in all frames. More exactly, we have: Soundness: K ⊆ {A | for all F : F |= A} Completeness: {A | for all F : F |= A} ⊆ K

15 Frame properties Observe: For all reflexive frames F we have F |= A ⊃ A. Terminology:‘ X frame’ is short for ‘frame hW , Ri, where the accessibility relation R satisfies the property X’ Why? [−→ blackboard] Fact: also the inverse holds: F |= A ⊃ A implies that F is reflexive. Why? [indirect proof −→ sketch on blackboard] Definition: B characterizes frame property X means: F |= B iff F is X.

16 Important frame properties E1 reﬂexive: ∀s sRs E2 symmetric: ∀s∀t sRt ⇒ tRs E3 serial: ∀s∃t sRt E4 transitive: ∀s∀t∀u (sRt & tRu) ⇒ sRu E5 euclidian: ∀s∀t∀u (sRt & sRu) ⇒ tRu E6 partially functional: ∀s∀t∀u (sRt & sRu) ⇒ t=u E7 functional: ∀s∃!†t sRt E8 (weakly) dense: ∀s∀t sRt ⇒ ∃u (sRu & uRt) E9 (weakly) connected: ∀s∀t∀u (sRt & sRu) ⇒ (tRu or t=u or uRt) E10 (weakly) directed: ∀s∀t∀u (sRt & sRu) ⇒ ∃v (tRv & uRv) † ∃! ... ‘there exists exactly one’: E7: ∀s(∃t sRt & ∀u(sRu ⇒ u=t))

17 Characterizing formulas

A1 (T) A ⊃ A A2 (B) A ⊃ ♦A A3 (D) A ⊃ ♦A A4 (4) A ⊃ A A5 (5) ♦A ⊃ ♦A A6 ♦A ⊃ A A7 (♦A ⊃ A) ∧ (A ⊃ ♦A) A8 A ⊃ A A9 (L) ((A ∧ A) ⊃ B) ∨ ((B ∧ B) ⊃ A) A10 (Geach) ♦A ⊃ ♦A Theorem: The formula Ai characterizes frame property Ei for all 1 ≤ i ≤ 10. Note: Applying modal logics – e.g., reasoning about knowledge and belief of agents scenarios or characterizing dynamic behavior of systems – often starts by locating relevant frame properties and assembling an appropriate logic by determining corresponding characterizing formulas. 18 Can all frame properties be characterized? Answer: No! E.g., irreﬂexivity, non-symmetry, etc., cannot be characterized by modal formulas. This can be shown using the concept of bounded morphisms. Exercise (voluntary):

1 Find (at least one) appropriate internet resource for ‘bisimulation’ as well as for ‘bounded morphism’ (also called ‘p-morphism’). 2 Summarize the central deﬁnition and fact(s) precisely. 3 Give non-trivial examples of bisimilar models. 4 Apply a bounded morphism to show that asymmetry is not characterizable.

19 Do all modal formulas characterize frame properties? Answer (even if restricted ‘ﬁrst-order properties’): No! E.g., ♦A ⊃ ♦A does not correspond to a ﬁrst-order property of the accessibility relation in Kripke frames. However: There are important general positive results. E.g., for all i, j, k, l ≥ 0:

k l m n F |= ♦ A ⊃ ♦ A if and only if ∀s∀t∀u(sRk t & sRmu) ⇒ ∃v(tRl v & uRnv) uRi w . . . there is path from u to w using i R-steps uR0w ... u = w Remark: The study of the relation between frame properties and modal formulas is called correspondence theory.