bisimulations

PHIL 452 · Modal Logic

Gabriel Uzquiano

bisimulation

Bisimulations are to modal propositional logic what isomorphisms are to first-order logic:

• two isomorphic models verify exactly the same first-order formulas.
• two bisimilar possible worlds models verify the same modal formulas

definition

A bisimulation between \((W, R, V)\) and \((W', R', V')\) is a relation \(E \subseteq W \times W'\) where for all \(w \in W\), for all \(w'\in W'\), if \(wEw'\), then:

1. \(w \in V(p)\) iff \(w' \in V'(p)\), for all propositional atoms \(p\) (atoms);
2. if \(wRu\), then for some \(u' \in W'\), \(w'R'u'\) and \(uEu'\) (forth);
3. if \(w'R'u'\), then for some \(u \in W\), \(wRu\) and \(uEu'\) (back).

example: a bisimulation

• \(W = \{w, u, v\}\); \(R = \{(w, u), (w, v)\}\); \(V(p) = \{u, v\}\)

• \(W' = \{w', u'\}\); \(R' = \{(w', u')\}\); \(V'(p) = \{u'\}\)

• \(E = \{(w, w'), (u, u'), (v, u')\}\)

\(E\) is a bisimulation between \((W, R, V)\) and \((W', R', V')\):

• atoms:

  • \(w\) and \(w'\) verify no propositional atoms.
  • \(u\), \(v\) and \(u'\) verify just \(p\).

• forth:

  • since \(wRu\) and \(wRv\), we have \(w'R'u'\) with \(uEu'\) and \(vEu'\).
  • \(u\) and \(v\) see nothing.

• back:

  • since \(w'R'u'\), we have \(wRu\) with \(uEu'\).
  • \(u'\) sees nothing.

invariance lemma

Proposition 1 If there is a bisimulation \(E\) between two models \((W, R, V)\) and \((W', R', V')\), then for all \(w\in W\), for all \(w'\in W'\), if \(wEw'\), then for each modal formula \(\varphi\) of \(\mathcal{L}^\Box\): \[\begin{array}{lll} (W, R, V), w \Vdash \varphi & \text{iff} & (W', R', V'), w' \Vdash \varphi. \end{array}\]

Proof.

By induction on the complexity of formulas. We use the condition:

• For all \(w\in W\), for all \(w'\in W'\), if \(wEw'\): \[ \begin{array}{lll} (W, R, V), w \Vdash \varphi & \text{iff} & (W', R', V'), w' \Vdash \varphi. \end{array} \]

base and Boolean cases

Assume \(wEw'\).

• base case, \(\varphi = p\): \[ \begin{array}{llll} M, w \Vdash p & \text{iff} & w \in V(p) & \\ & \text{iff} & w' \in V'(p) & \textsf{atoms}\\ & \text{iff} & M', w' \Vdash p & \end{array} \]

• inductive step for \(\neg\): \[ \begin{array}{llll} M, w \Vdash \neg\varphi & \text{iff} & M, w \nVdash \varphi & \\ & \text{iff} & M', w' \nVdash \varphi & \textsf{inductive hypothesis}\\ & \text{iff} & M', w' \Vdash \neg\varphi & \end{array} \]

• inductive step for \(\to\): \[ \begin{array}{llll} M, w \Vdash \varphi \to \psi & \text{iff} & M, w \nVdash \varphi \ \text{or} \ M, w \Vdash \psi & \\ & \text{iff} & M', w' \nVdash \varphi \ \text{or} \ M', w' \Vdash \psi & \textsf{inductive hypothesis}\\ & \text{iff} & M', w' \Vdash \varphi \to \psi & \end{array} \]

inductive step for \(\Box\): (\(\Rightarrow\))

Assume \(wEw'\) and \(M, w \Vdash \Box\varphi\). Let \(w'R'u'\).

• By \(\textsf{back}\), for some \(u \in W\), \(wRu\) and \(uEu'\).
• Since \(M, w \Vdash \Box\varphi\) and \(wRu\): \(M, u \Vdash \varphi\).
• Since \(uEu'\), by inductive hypothesis: \(M', u' \Vdash \varphi\).
• \(u'\) was arbitrary, so \(M', w' \Vdash \Box\varphi\).

inductive step for \(\Box\): (\(\Leftarrow\))

Suppose \(wEw'\) and \(M', w' \Vdash \Box\varphi\). Let \(wRu\).

• By \(\textsf{forth}\), for some \(u' \in W'\), \(w'R'u'\) and \(uEu'\).
• Since \(M', w' \Vdash \Box\varphi\) and \(w'R'u'\): \(M', u' \Vdash \varphi\).
• Since \(uEu'\), by inductive hypothesis: \(M, u \Vdash \varphi\).
• \(u\) was arbitrary, so \(M, w \Vdash \Box\varphi\).

Every model is bisimilar to some irreflexive model

Given a model \(M = (W, R, V)\), we let \(M' = (W', R', V')\) where

• \(W' = W \times \{0, 1\}\); We write \(w_i\) for \((w, i)\).
• \(w_i R' u_i\)  iff  \(wRu\) and \(i \neq j\);
• \(w_i \in V'(p)\)  iff  \(w \in V(p)\).

\(R'\) is irreflexive, since \(i \neq i\) is never the case.

\(E = \{(w, w_i) : w \in W,\ i \in \{0, 1\}\}\) is a bisimulation between the two models:

• atoms:
  • trivial by the definition of \(V'\).
• forth:
  • if \(wEw_i\) and \(wRu\), then \(w_i R' u_{1-i}\) and \(uEu_{1-i}\).
• back:
  • if \(wE w_i\) and \(w_i R' u_j\), then \(wRu\) and \(uE u_j\).

The simplest case: a reflexive point and its double, a two-cycle.

Irreflexivity is not modally definable

Proposition 2 No formula \(\varphi\) is valid in all and only irreflexive frames.

Proof. Suppose \(\varphi\) is valid in all and only the irreflexive frames.

• The frame \((\{w\}, \{(w, w)\})\) is not irreflexive. So, by assumption, for some \(V\), \((\{w\}, \{(w, w)\}, V), w \nVdash \varphi\).
• Its two-cycle counterpart is irreflexive. So, by assumption, \(\varphi\) is valid in that frame, which means that \((W', R', V'), w_0 \Vdash \varphi\).
• But notice that \(wE w_0\) in the given bisimulation \(E\).
• So, by Invariance, \((\{w\}, \{(w, w)\}, V), w \Vdash \varphi\). Contradiction.

The recipe, and more corollaries

To show that a structural condition \(C\) is not modally definable, find a frame \(G\) in which \(C\) fails such that

• for each model \(M\) based on \(G\) and each world \(w\) in \(M\), there is a model \(N\) based on some frame \(F\) that satisfies \(C\) and a world \(u\) in \(N\), and a bisimulation \(E\) between \(M\) and \(N\) with \(wEv\).

condition \(C\)	\(G\) lacks \(C\)	\(F\) has \(C\)
irreflexive	reflexive point	two-cycle
asymmetric	reflexive point	\((\mathbb{N}, S)\)
intransitive	reflexive point	\((\mathbb{N}, S)\)
infinite	reflexive point	\((\mathbb{N}, S)\)
antisymmetric	two-cycle	\((\mathbb{N}, S)\)

In the infinite chain \((\mathbb{N}, S)\):

• \(S = \{(n,m): m = n+1\}\)

Given a model \((\{a, b\}, \{(a, b), (b, a)\}, V)\), consider a model based on \((\mathbb{N}, S)\) where for each atom \(p\):

\[ V'(p) = \{n \in \mathbb{N}: (a \in V(p) \ \text{and} \ n \ \text{even}) \ \text{or} \ (n \in V(p) \ \text{and} \ n \ \text{odd}) \} \] \(E = \{(a, n): n \ \text{even}\} \cup \{(b, n): n \ \text{odd}\}\)