Structural Properties of Dynamic Reasoning

We characterize the structural properties of dynamic inference in general update models, and show that these are exactly the ones of public update in epistemic logic. 1 DYNAMIC INFERENCE IN ABSTRACTO Logical dynamics is about actions that change information, such as information update in communication. At an abstract level, we view propositions A as partial functions TA taking input states meeting the preconditions of update with A to output states: TA More generally, this process can be modeled by means of transition models M = (S, {TA | A∈Prop}) consisting of the relevant information states S with a family of transition relations TA over these, one for each proposition A in some abstract index set Prop. Such update propositions suggest the following notion of dynamic inference: the action of the successive premises enforces the conclusion. 2 Meaning: the Dynamic Turn Definition A sequence of propositions P1, ..., Pk dynamically implies conclusion C in transition model M, if any sequence of premise updates starting from any state in M ends in a state which is a fixed point for the conclusion: whenever s1 Tp1 s2 ... Tpk sk+1, then sk+1 C sk+1 Alternatively, we say the sequent P1, ..., Pk ⇒ C is true in the model – written as: M |= P1, ..., Pk ⇒ C. In what follows, we will use symbols P, Q, R to stand for finite sequences of propositions, and A, B, C for single propositions. Over transition models, dynamic inferential sequents of this form lack the standard structural rules of classical consequence. This situation is analyzed in van Benthem (1996, Chapter 7). Some simple counter-examples establish the following result. Fact None of the following structural rules hold for dynamic inference: Monotonicity, Contraction, Permutation, Reflexivity, or Cut. One can show this formally, but the main idea is simply this. Any cooking recipe may be disturbed by inserting arbitrary instructions, deleting repeats of an instruction, interchanging instructions, etc. Even the Cut Rule fails, at least in its general form: if P ⇒ A and R, A, Q ⇒ C, then R, P, Q ⇒ C But as more often in non-classical logics, some 'substitute rules' turn out to hold. Fact Partial update functions validate the following rules for dynamic inference: if P ⇒ C, then A, P ⇒ C Left-Monotonicity if P ⇒ A and P, A, Q ⇒ C, then P, Q ⇒ C Left-Cut if P ⇒ A and P, Q ⇒ C, then P, A, Q ⇒ C Cautious Monotonicity Proof For instance, consider Left-Cut. If we move from state s to t via P, and then from t to u via Q, the first premise P ⇒ A tells us that t A t, and so the sequence s, t, t, u fits the action of P, A, Q, whence u C u by the second premise. ! 2 AN ABSTRACT COMPLETENESS THEOREM The above structural rules are characteristic for dynamic inference with partial update functions. The proper setting for this is the following representation result. Take any set Prop of propositions, seen as abstract objects – with a binary relation ⇒ between finite sequences of propositions and propositions – again written with finite sequents: Structural Properties of Dynamic Reasoning 3 Theorem The following are equivalent for any structure (Prop, ⇒) : (a) ⇒ satisfies {Left-Monotonicity, Left-Cut, Cautious Monotonicity}, viewed as abstract conditions on relations of type sequence-to-object, (b) there is a transition model (S, {TA| A∈Prop}) with partial maps TA whose relation of dynamic inference as defined above coincides with the given relation ⇒ among the abstract propositions A. Proof The direction from (b) to (a) is the preceding Fact. Now from (a) to (b). For any given abstract structure (Prop, ⇒), we define a transition model M as follows. States are finite sequences X, Y, ... of propositions. For each proposition A, we then define the following partial function over these states: TA = {(X, X) | X ⇒ A} ∪ { (X,< X, A>) | not X ⇒ A} We must check that the following equivalence holds: M |= P1, ..., Pk ⇒ C iff P1, ..., Pk ⇒ C is true in (Prop, ⇒) 'If'. Suppose that s1 Tp1 s2 ... Tpk sk . By the definition of the transformations TA, each step in this sequence of states either adds a proposition at the end, or 'pauses'. Here is a typical illustration of what may happen: X Tp1 (not X ⇒ P1) Tp2 ( ⇒ P2) Tp3 (not ⇒ P3) We must show that the final state is a fixed point for TC : i.e., ⇒ C First we have (in this particular case) that ⇒ C in Prop, and hence by LeftMonotonicity in that structure also ⇒ C: Then following the above three transition steps, we can suppress one proposition thanks to the truth of ⇒ P2, by using Left-Cut: ⇒ C This argument is really completely general. 'Pauses' involve valid sequents that can be used to cut out items in the left sequence P1, ..., Pk at the right places. 'Only if'. This direction essentially involves the remaining structural rule. Again, one example demonstrates the general procedure. Suppose dynamically imply C in the above 4 Meaning: the Dynamic Turn transition structure M. Start with the empty sequence – . We choose three particular transitions for the premises. If – ⇒ P1 in Prop, our first transition is –, – ; otherwise, we go to an extended sequence ; etc. Suppose that, for our three propositions, this yields the following sequence of transformations: –, , (where P1 ⇒ P2!) , Now by the assumption of this case, the final state is a fixed point for TC, i.e., P1, P3 ⇒ C is true in Prop But then by the fact that P1 ⇒ P2 plus Cautious Monotonicity: P1, P2, P3 ⇒ C is true in Prop Again the general trick is clear. We can insert propositions wherever required. ! This representation also yields a completeness theorem for sequents interpreted as above on models M. For this purpose, we need to define valid consequence among sequents on transition models, for which we introduce a new arrow: "from set of sequents Σ to sequent σ" Σ ! σ Definition We have valid consequence between a set Σ and a sequent σ if σ is true in all transition models where all sequents from Σ are true. Here is the more general thrust of the preceding theorem. Corollary A sequent σ is a valid consequence of a set of sequents Σ iff σ is derivable from sequents in Σ using the three mentioned structural rules. Proof From right to left, this follows from the soundness of the structural rules. Going from left to right requires a small modification of the above construction. Suppose that σ is not derivable from the set Σ. Take the structure (Prop, ⇒) with the relation ⇒ holding only for sequents derivable from the set Σ using the three given structural rules. Now represent this structure just as above. The result is a transition model M where all sequents in Σ are true, while σ is false. ! There are also other notions of dynamic inference, placing other requirements on the update action associated with the conclusion. Analogous abstract characterizations for their structural properties may be found in van Benthem (1996). Structural Properties of Dynamic Reasoning 5 3 CONCRETE MODELS: PUBLIC UPDATE IN EPISTEMIC LOGIC Epistemic logic and information models One of the most concrete update systems has public announcements transforming multi-S5 models for epistemic logic. Here is an illustration. Two players draw from a set of red and white cards. Each can see the colour of their own card, the other cannot. Also, it is known that no two white cards have been drawn. Here is the epistemic model, which may be viewed as an information state for the group {1, 2}. Its three 'worlds' are tuples like rw, standing for the physical situation in which "1 has a red card, and 2 has a white one". The bold-face tuple rr represents what actually happened, as seen by an outside observer: both players drew a red card.