Dynamic logic (modal logic)

In <a href="/facts/Logic/8JnpJLtt">logic</a>, <a href="/facts/Philosophy/zAH5PVD6">philosophy</a>, and <a href="/facts/Theoretical_computer_science/jSkGGuvy">theoretical computer science</a>, dynamic logic is an extension of <a href="/facts/Modal_logic/I7uXvYYT">modal logic</a> capable of encoding properties of <a href="/facts/Computer_program/e38jH9N9">computer programs</a>.
A simple example of a statement in dynamic logic is

The ground is dry
        
        →
        [
        
          It rains
        
        ]
        
          The ground is wet
        
        ,
      
    
    {\displaystyle {\text{The ground is dry}}\to [{\text{It rains}}]{\text{The ground is wet}},}

which states that if the ground is currently dry and it rains, then afterwards the ground will be wet.
The syntax of dynamic logic contains a language of propositions (like "the ground is dry") and a language of actions (like "it rains"). The core modal constructs are 
 
 
 
 [
 a
 ]
 p
 
 
 {\displaystyle [a]p}
 
, which states that after performing action a the proposition p should hold, and 
 
 
 
 ⟨
 a
 ⟩
 p
 
 
 {\displaystyle \langle a\rangle p}
 
, which states that after performing action a it is possible that p holds.
The action language supports operations 
 
 
 
 a
 
 ;
 
 b
 
 
 {\displaystyle a\mathbin {;} b}
 
 (doing one action followed by another), 
 
 
 
 a
 ∪
 b
 
 
 {\displaystyle a\cup b}
 
 (doing one action or another), and iteration 
 
 
 
 a
 
 ∗
 
 
 
 {\displaystyle a{*}}
 
 (doing one action zero or more times). The proposition language supports <a href="/facts/Boolean_algebra/VkCToAmg">Boolean operations</a> (and, or, and not). The action logic is expressive enough to encode programs. For an arbitrary program 
 
 
 
 P
 
 
 {\displaystyle P}
 
, <a href="/facts/Precondition/dWbBdG2e">precondition</a> 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
, and <a href="/facts/Postcondition/4tF1Havd">postcondition</a> 
 
 
 
 
 φ
 ′
 
 
 
 {\displaystyle \varphi '}
 
, the dynamic logic statement 
 
 
 
 φ
 →
 [
 P
 ]
 
 φ
 ′
 
 
 
 {\displaystyle \varphi \to [P]\varphi '}
 
 encodes the correctness of the program, making dynamic logic more general than <a href="/facts/Hoare_logic/UoMg08wo">Hoare logic</a>.
Beyond its use in <a href="/facts/Formal_verification/BYD4GnJR">formal verification</a> of programs, dynamic logic has been applied to describe complex behaviors arising in <a href="/facts/Linguistics/oQuamvf2">linguistics</a>, <a href="/facts/Philosophy/zAH5PVD6">philosophy</a>, <a href="/facts/Artificial_intelligence/lJGauQwX">AI</a>, and other fields.

Dynamic logic (modal logic) open-in-new

Dynamic logic (modal logic)