Top-Down Operational Semantics (I)

• Transition $r$: rewrite user predicates. If goal $a$ is selected,
  
  $$\langle A \bigcup a, C, S \rangle \rightarrow_r \langle A \bigcup B, C, S\bigcup (a = h) \rangle$$
  
  
  if $h \leftarrow B \in P$, and $a$ and $h$ have the same predicate symbol, or
  
  $$\langle A \bigcup a, C, S \rangle \rightarrow_r fail$$
  
  
  if there is no rule $h \leftarrow B \in P$ such that $a$ and $h$ have the same predicate symbol

• Transition $c$: selects constraints
  
  $$\langle A \bigcup c, C, S \rangle \rightarrow_c \langle A, C, S \bigcup c \rangle$$
  
  
  if $c$ is a constraint selected by the computation rule
• Transition $i$: infers new constraints
  
  $$\langle A, C, S \rangle \rightarrow_i \langle A, C', S' \rangle \mbox{ if } (C', S') = infer(C, S)$$
  
  

• Transition $s$: checks satisfiability
  
  $$\langle A, C, S \rangle \rightarrow_s \left\{ \begin{array}{... ...\ fail & {\rm if} \; \neg consistent(C) \end{array} \right.$$