Computational Logic

Logic Programming:
Model and Fixpoint Semantics

Towards the Model and Fixpoint Semantics

• We have seen previously the operational semantics (SLD-resolution).
• We now present the (declarative) Model Semantics:
  • We define our semantic domain (Herbrand interpretations).
  • We introduce the Minimal Herbrand Model.
• And the (also declarative) Fixpoint Semantics.
  • We recall some basic fixpoint theory.
  • Present the $T_P$ operator and the classic fixpoint semantics.

Declarative Semantics - Herbrand Base and Universe

• Given a first-order language $L$, with a non-empty set of variables, constants, function symbols, relation symbols, connectives, quantifiers, etc. and given a syntactic object $A$,
  
  $$ground(A) = \{ A\theta \vert \exists \theta \in Subst, var(A\theta) = \emptyset \}$$
  
  
  i.e. the set of all ``ground instances'' of $A$.

• Given $L$, $U_L$ (Herbrand universe) is the set of all ground terms of $L$.

• $B_L$ (Herbrand Base) is the set of all ground atoms of $L$.

• Similarly, for the language $L_P$ associated with a given program $P$ we define $U_P$, and $B_P$.

Declarative Semantics - Herbrand Base and Universe (example)

• Program:

  
  $P = \{$ $p(f(X)) \leftarrow p(X).$ 
  
  $p(a).$ 
  
  $q(a).$ 
  
  $q(b).$ }
  

• Herbrand universe:
  $U_P = \{a,b,f(a),f(b),f(f(a)),f(f(b)),\ldots\}$
  

• Herbrand base:
  $B_P = \{p(a),p(b),q(a),q(b),p(f(a)),p(f(b)),q(f(a)),\ldots\}$

Herbrand Interpretations and Models

• A Herbrand Interpretation is a subset of $B_L$, i.e. the set of all Herbrand interpretations $I_L = \wp(B_L)$.

  (Note that $I_L$ forms a complete lattice under $\subseteq$ - important for fixpoint operations to be introduced later).

• In previous example: $P = \{ p(f(X)) \leftarrow p(X). p(a). q(a). q(b). \}$
  $U_P = \{a,b,f(a),f(b),f(f(a)),f(f(b)),\ldots\}$
  $B_P = \{p(a),p(b),q(a),q(b),p(f(a)),p(f(b)),q(f(a)),\ldots\}$
  $I_P =$ all subsets of $B_P$

• A Herbrand Model is a Herbrand interpretation which contains all logical consequences of the program.

• The Minimal Herbrand Model $H_P$ is the smallest Herbrand interpretation which contains all logical consequences of the program. (Theorem: it is unique.)

• Example:
  $H_P = \{q(a),q(b),p(a),p(f(a)),p(f(f(a))),\ldots\}$

Declarative Semantics, Completeness, Correctness

• Declarative semantics of a logic program $P$:
  the set of ground facts which are logical consequences of the program (i.e., $H_P$).
  (I.e., the Minimal Herbrand model (or ``least model'') of $P$).

• Intended meaning of a logic program $P$:
  the set $I$ of ground facts that the user expects to be logical consequences of the program.

• A logic program is correct if $H_P$ $\subseteq$ $I$.

• A logic program is complete if $I$ $\subseteq$ $H_P$.

• Example:
  father(john,peter).
  father(john,mary).
  mother(mary,mike).
  grandfather(X,Y) $\leftarrow$ father(X,Z), father(Z,Y).
  
  with the usual intended meaning is correct but incomplete.

Towards a Fixpoint Semantics for LP - Fixpoint Basics

• A fixpoint for an operator $T : X \rightarrow X$ is an element of $x \in X$ such that $x = T(x)$.

• If $X$ is a poset, $T$ is monotonic if $\forall x,y \in X, x \leq y \Rightarrow T(x) \leq T(y)$

• If $X$ is a complete lattice and $T$ is monotonic the set of fixpoints of $T$ is also a complete lattice [Tarski]

• The least element of the lattice is the least fixpoint of $T$, denoted $l\!f\!p (T)$

• Powers of a monotonic operator (successive applications):

  

  We abbreviate $T\uparrow \alpha(\bot)$ as $T\uparrow \alpha$

• There is some $\omega$ such that $T\uparrow \omega = l\!f\!p\ T$. The sequence $T\uparrow 0, T\uparrow 1,... ,l\!f\!p T$ is the Kleene sequence for $T$

• In a finite lattice the Kleene sequence for a monotonic operator $T$ is finite

Towards a Fixpoint Semantics for LP - Fixpoint Basics (Contd.)

• A subset $Y$ of a poset $X$ is an (ascending) chain iff $\forall y,y' \in Y, y \leq y' \vee y' \leq y$

• A complete lattice $X$ is ascending chain finite (or Noetherian) if all ascending chains are finite

• In an ascending chain finite lattice the Kleene sequence for a monotonic operator $T$ is finite

Lattice Structures

A Fixpoint Semantics for Logic Programs

• Semantic domain: $I_L = \wp(B_L)$.

• I.e., the elements of the semantic domain and interpretations
  (subsets of the Herbrand base).

• Semantic operator (defined on programs):
  the immediate consequences operator, $T_P$:
  • $T_P$ is a mapping: defined by:
    
    $T_P(I) = \{ A \in B_P \vert \exists C \in ground(P), C = A \leftarrow L_1,... ,L_n \mbox{and } L_1, \ldots L_n \in I \}$
  (in particular, if $(A \leftarrow) \in P$, then every element of $ground(A)$ is in $T_P(I)$, $\forall$ $I$).

• $T_P$ is monotonic, so:
  • it has a least fixpoint $I^*$ so that $T_P(I^*) = I^*$,
  • this fixpoint can be obtained by applying $T_P$ iteratively starting from the bottom element of the lattice (the empty interpretation).

A Fixpoint Semantics for Logic Programs: Example 1 (finite)

$P = \{$ $p(X,a) \leftarrow q(X).$ 

$p(X,Y) \leftarrow q(X),r(Y).$ 

$q(a).     r(b).$ 

$q(b).     r(c).$ }

$U_P = \{ a,b,c \}$

$B_P = \{$ $p(a,a), p(a,b), p(a,c), p(b,a), p(b,b), p(b,c), p(c,a), p(c,b), p(c,c),$

$q(a), q(b), q(c),$

$r(a), r(b), r(c) \}$

$I_P =$ all subsets of $B_P$
$H_P = \{ q(a), q(b), r(b), r(c), p(a,a), p(b,a), p(a,b), p(b,b), p(a,c), p(b,c) \}$

$T_P \uparrow 0 = \{ q(a), q(b), r(b), r(c) \}$
$T_P \uparrow 1 = \{ q(a), q(b), r(b), r(c) \} \cup \{ p(a,a), p(b,a), p(a,b), p(b,b), p(a,c), p(b,c) \}$
$T_P \uparrow 2 = T_P \uparrow 1 = lfp(T_P) = H_P$

A Fixpoint Semantics for Logic Programs: Example 2 (infinite)

$P = \{$ $p(f(X)) \leftarrow p(X).$ 

$p(a).$ 

$q(a).$ 

$q(b).$ }

$U_P = \{a,b,f(a),f(b),f(f(a)),f(f(b)),\ldots\}$
$B_P = \{p(a),p(b),q(a),q(b),p(f(a)),p(f(b)),q(f(a)),\ldots\}$
$I_P =$ all subsets of $B_P$
$H_P = \{q(a),q(b),p(a) \} \cup \{ p(f^n(a)) \vert n \in \cal{N} \}$
where we define $f^n(a)$ to be $f$ nested $n$ times and then applied to $a$.
(i.e., $q(a),q(b),p(a),p(f(a)),p(f(f(a))),p(f(f(f(a)))), ...$)

$T_P\uparrow 0 = \{p(a),q(a),q(b)\}$
$T_P\uparrow 1 = \{p(a),q(a),q(b),p(f(a))\}$
$T_P\uparrow 2 = \{p(a),q(a),q(b),p(f(a)),p(f(f(a)))\}$
$\ldots$
$T_P\uparrow \omega = H_P$

A Fixpoint Semantics for Logic Programs: Example 3 (infinite)

• Example:

  
  $P = \{$ $nat(0).$ 
  
  $nat(s(X)) \leftarrow nat(X).$ 
  
  
  
  $sum(0,X,X).$ 
  
  $sum(s(X),Y,s(Z)) \leftarrow sum(X,Y,Z).$ }
  

  $U_P = \{ 0 \} \cup \{ s(x) \vert x \in U_P \}$
  
  (i.e., $\{ 0, s(0), s(s(0)), s(s(s(0))), ... \}$).
  
  
  $B_P = \{ nat(x) \vert x \in U_P \} \cup \{ sum(x,y,z) \vert x,y,z \in U_P \}$
  
  (i.e., $\{ nat(0), nat(s(0)), nat(s(s(0))), ... \} \cup \{ sum(0,0,0), sum(s(0),0,0), sum(0,s(0),0), sum(0,0,s(0)), ... \}$).

A Fixpoint Semantics for Logic Programs: Example 3 (infinite, cont.)

Constructing the least fixpoint of the $T_P$ operator:

$T_P \uparrow 0$ = $\{ nat(0) \} \cup \{ sum(0,x,x)  \vert  x \in U_P \}$


$T_P \uparrow 1$ = $T_P \uparrow 0$ $\cup  \{ nat(s(0)) \}$ 

$\cup  \{ sum(s(0),y,s(y))  \vert  y \in U_P \}$


$T_P \uparrow 2$ = $T_P \uparrow 1$ $\cup  \{ nat(s(s(0))) \}$ 

$\cup  \{ sum(s(s(0)),y,s(s(y)))  \vert  y \in U_P \}$


$T_P \uparrow 3$ = $T_P \uparrow 2$ $\cup  \{ nat(s(s(s(0)))) \}$ 

$\cup  \{ sum(s(s(s(0))),y,s(s(s(y))))  \vert  y \in U_P \}$


...


$T_P \uparrow \omega$ = $\{ nat(x)  \vert  x \in U_P \}  \cup$ 

$\{ sum(s^n(0),y,s^n(y))  \vert  y \in U_P \wedge n \in \cal{N} \}$

where we define $s^x(y)$ to be $s$ nested $x$ times and then applied to $y$.

Semantics - Equivalences

• (Characterization Theorem) [Van Emden and Kowalski]
  A program $P$ has a Herbrand model $H_P$ such that :
  • $H_P$ is the least Herbrand Model of $P$.
  • $H_P$ is the least fixpoint of $T_P$ ($l\!f\!p T_P$).
  • $H_P = T_P\uparrow \omega$.
  
  I.e., least model semantics ($H_P$) $\equiv$ fixpoint semantics ($l\!f\!p T_P$)

• In addition, there is also an equivalence with the operational semantics (SLD-resolution):
  • SLD-resolution answers ``yes'' to $a \in B_P$ $\iff a \in H_P$.

  
  

• Because it gives us a way to directly build $H_P$ (for finite models), the least fixpoint semantics can in some cases also be an operational semantics (e.g., for datalog in deductive databases).