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Computational Logic
Automated Deduction Fundamentals

Elements of First-Order Predicate Logic

First Order Language:

Important: Notation Convention Used

(A bit different from standard notational conventions in logic, but good for compatibility with LP systems)

Variables: start with a capital letter or a ``_'' (X, Y, _a, _1)

Atoms, functors, predicate symbols: start with a lower case letter or are enclosed in ' ' (f, g, a, 1, x, y, z, 'X', '_1')

Terms and Atoms

We define by induction two classes of strings of symbols over a given alphabet.

The class of terms:

The class of atoms (different from LP!):

The class of Well Formed Formulas (WFFs):

Literal: positive or negative (non-negated or negated) atom.


Examples of Terms


Correct: spot, f(john), f(X), +(1,2,3), +(X,Y,L), f(f(spot)), h(f(h(1,2)),L)
Incorrect: spot(X), +(1,2), g, f(f(h))

Examples of Literals

Given the elements above and:
Correct: q, r, dog(spot), p(X,f(john))...

Incorrect: q(X), barks(f), dog(barks(X))

Examples (Contd.)

Examples of WFFs
Given the elements above

Correct: q, q $\rightarrow$ r, r $\leftarrow$ q, dog(X) $\leftarrow$ barks(X), dog(X) , p(X,Y), $\exists$ X (dog(X) $\wedge$ barks(X) $\wedge$ $\neg$ q), $\exists$ Y (dog(Y) $\rightarrow$ bark(Y))

Incorrect: q $\vee$, $\exists$ p

More about WFFs

Towards Efficient Automated Deduction

Towards Efficient Automated Deduction (Contd.)

Clausal Form

Deduction Mechanism

Classical Clausal Form: Conjunctive Normal Form


Substitutions (Contd.)

Composition of Substitutions


Unification Algorithm

Unification Algorithm revisited

Unification Algorithm revisited (Contd.)

Resolution with Variables

Resolution with Variables (Contd.)

Basic Properties

Proof Tree

Proof Tree (Contd.)

Characteristics of the Proof Tree

General Strategies

General Strategies (Contd.)

General Strategies (Contd.) (Contd.)

Linear Strategies

Characteristics of these Strategies

If $\Box$ can be derived from $K$ by using resolution with variables, it can also be derived by linear resolution

Let $K$ be $K\lq  \cup \{C_0\}$ where $K\lq $ is a satisfiable set of clauses, i.e. $\Box$ cannot be derived from $K\lq $ by using resolution with variables. If $\Box$ can be derived from $K$ by using resolution with variables it can also be derived by linear resolution with root $C_0$.

From (1), if the strategy is breadth first, it is complete.

From (2), if we want to prove that $B$ is derived form $K\lq $ then we can apply linear resolution to $K = K\lq  \cup \{\neg B\}$.

Depth first with backtracking is not complete:

Input Strategies

  • Those which only explore input derivations
  • A derivation $K,F_0\cdots F_m$ is input if
    • $F_0$ is obtained by resolution or replacement using $C_0$
    • $F_i, i < 0$ is obtained by resolution or replacement using at least a clause in $K$
  • Example:


Input Strategies

Ordered Strategies

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Last modification: Wed Nov 22 23:29:13 CET 2006 <>[CLIP] [FIM] [UNM]