Logic Programming

Theory Lecture 2:

(In)completeness for Propositional Prolog

Alex Simpson

School of Informatics

1st October 2012

Recap (Lecture 1): Propositional logic program

A definite clause is a formula of one of the two shapes below

q (a Prolog fact q . )

p1 ∧ · · · ∧ pk → q (a Prolog rule q :- p1, . . . , pk .)

where p1, . . . , pk , q are all atoms (that is, atomic statements).

A logic program is a list F1, . . . , Fn of definite clauses

A goal is a list g1, . . . , gm of atoms.

The job of the system is to ascertain whether the logicalconsequence below holds.

F1, . . . , Fn |= g1 ∧ · · · ∧ gm .

Recap (Lecture 1): Inference system

The definite clauses F1, . . . , Fn in the program are taken as axioms.

We use a single inference rule to derive new established goals fromgoals already established

g1, . . . , gl−1, h1, . . . , hk , gl+1, . . . , gm h1 ∧ · · · ∧ hk → gl

g1, . . . , gl−1, gl , gl+1, . . . , gm

where 1 ≤ l ≤ m and k ≥ 0.

A derivation (or proof) of a goal g1, . . . , gm is a tree of applicationsof the rule, in which every leaf is one of the definite clausesF1, . . . , Fn comprising the program, and the root of the tree is thegoal g1, . . . , gm.

Example logic program

hotterSun → warmerClimate

carbonIncrease → warmerClimate

warmerClimate → iceMelts

iceMelts → albedoDecrease

albedoDecrease → warmerClimate

carbonIncrease

A derivation of the goal iceMelts

carbonIncrease carbonIncrease→ warmerClimate

warmerClimate warmerClimate→ iceMelts

iceMelts

Top-down proof search

A top-down search for a derivation maintains a list g1, . . . , gn ofatoms, the current goal. The procedure is:

I For l with 1 ≤ l ≤ n, find an axiom (program clause) of form

h1 ∧ · · · ∧ hk → gl

Replace gl with h1, . . . , hk . So the current goal becomes

g1, . . . , gl−1, h1, . . . , hk , gl+1, . . . , gn

This includes, with k = 0, the case in which gl is itself anaxiom, in which case it simply gets removed from the list.

The choice of l and choice of axiom means that the search spacebranches, producing a search tree.

The goal of the proof search procedure is to find a branch in thesearch tree ending in a leaf labelled with the empty list [].

Prolog search tree

Prolog search specializes the above by always considering theindividual goal atoms in goal order.

This means that given a current goal

g1, . . . , gn

Prolog always looks for an axiom (program clause) of form

h1 ∧ · · · ∧ hk → g1

That is l on the previous slide is always chosen to be 1.

This policy is built into the Prolog search tree for a query. In aProlog search tree, the branching reflects only the choice involvedin selecting which axiom to use.

Prolog search tree for goal iceMelts

iM

wC

hS cI

yes

aD

iM

wC

hS cI

yes

aD

...

Prolog proof search

Prolog proof search is depth first:

I The search always moves to an unvisited (i.e., not previouslyvisited) child (i.e., immediately below) node of the currentnode, whenever such a node exists.

I If there is no such node, the search backtracks to the mostrecently visited node from which such an unvisited child nodeis available.

Prolog proof search follows program order.

I Child nodes are visited in the order that the axioms (Prologrules) that determine the child node appear in the program(i.e., from left to right in the trees as we are drawing them).

Prolog search for goal iceMelts

iM (1)

wC (2)

hS (3) cI (4)

yes (5)

aD

iM

wC

hS cI

yes

aD

...

Reordering program can improve efficiency . . .

carbonIncrease → warmerClimate

hotterSun → warmerClimate

warmerClimate → iceMelts

iceMelts → albedoDecrease

albedoDecrease → warmerClimate

carbonIncrease

Resulting Prolog search

iM (1)

wC (2)

cI (3)

yes (4)

hS aD

iM

wC

cI

yes

hS aD

...

. . . or reduce efficiency . . .

hotterSun → warmerClimate

albedoDecrease → warmerClimate

carbonIncrease → warmerClimate

warmerClimate → iceMelts

iceMelts → albedoDecrease

carbonIncrease

. . . drastically!

iM (1)

wC (2)

hS (3) aD (4)

iM (5)

wC (6)

hS (7) aD (8)

... (∞)

cI

yes

cI

yes

In Sicstus Prolog

Program:

warmerClimate :- hotterSun.warmerClimate :- albedoDecrease.warmerClimate :- carbonIncrease.iceMelts :- warmerClimate.albedoDecrease :- iceMelts.carbonIncrease.hotterSun :- false.

Query:

| ?- iceMelts.! Resource error: insufficient memory

Incompleteness of Prolog proof search

The correct answer to the query iceMelts is yes, becauseiceMelts is a logical consequence of the program (irrespective ofhow we order the axioms in the program).

Our original derivation of iceMelts is a valid derivation(irrespective of how we order the axioms in the program).

However, whether Prolog proof search is successful or not dependson the order in which the axioms in the program are given.

The Prolog proof search procedure is said to be incomplete: itdoes not always find a derivation, even if a derivation exists.

Notation

I Goal g1, . . . , gm is a logical consequence of F1, F2, . . . , Fn

F1, . . . , Fn |= g1 ∧ · · · ∧ gm .

I Goal g1, . . . , gm is derivable from F1, F2, . . . , Fn:

F1, . . . , Fn ` g1, . . . , gm .

i.e., there is some derivation of g1, . . . , gm from axiomsF1, F2, . . . , Fn

I Goal g1, . . . , gm is Prolog derivable from F1, F2, . . . , Fn:

F1, . . . , Fn `Prolog g1, . . . , gm ,

i.e., Prolog proof search is successful.

Fundamental properties

Correctness of Prolog proof search

(straightforward)

F1, . . . , Fn `Prolog g1, . . . , gm implies F1, . . . , Fn ` g1, . . . , gm .

Soundness of inference system

(will prove first)

F1, . . . , Fn ` g1, . . . , gm implies F1, . . . , Fn |= g1 ∧ · · · ∧ gm .

Completeness of inference system

(will prove second)

F1, . . . , Fn |= g1 ∧ · · · ∧ gm implies F1, . . . , Fn ` g1, . . . , gm .

Incompleteness of Prolog proof search

(already shown)

F1, . . . , Fn |= g1∧· · ·∧gm doesn’t imply F1, . . . , Fn `Prolog g1, . . . , gm .

Fundamental properties

Correctness of Prolog proof search (straightforward)

F1, . . . , Fn `Prolog g1, . . . , gm implies F1, . . . , Fn ` g1, . . . , gm .

Soundness of inference system

(will prove first)

F1, . . . , Fn ` g1, . . . , gm implies F1, . . . , Fn |= g1 ∧ · · · ∧ gm .

Completeness of inference system

(will prove second)

F1, . . . , Fn |= g1 ∧ · · · ∧ gm implies F1, . . . , Fn ` g1, . . . , gm .

Incompleteness of Prolog proof search

(already shown)

F1, . . . , Fn |= g1∧· · ·∧gm doesn’t imply F1, . . . , Fn `Prolog g1, . . . , gm .

Fundamental properties

Correctness of Prolog proof search (straightforward)

F1, . . . , Fn `Prolog g1, . . . , gm implies F1, . . . , Fn ` g1, . . . , gm .

Soundness of inference system (will prove first)

F1, . . . , Fn ` g1, . . . , gm implies F1, . . . , Fn |= g1 ∧ · · · ∧ gm .

Completeness of inference system

(will prove second)

F1, . . . , Fn |= g1 ∧ · · · ∧ gm implies F1, . . . , Fn ` g1, . . . , gm .

Incompleteness of Prolog proof search

(already shown)

F1, . . . , Fn |= g1∧· · ·∧gm doesn’t imply F1, . . . , Fn `Prolog g1, . . . , gm .

Fundamental properties

Correctness of Prolog proof search (straightforward)

F1, . . . , Fn `Prolog g1, . . . , gm implies F1, . . . , Fn ` g1, . . . , gm .

Soundness of inference system (will prove first)

F1, . . . , Fn ` g1, . . . , gm implies F1, . . . , Fn |= g1 ∧ · · · ∧ gm .

Completeness of inference system (will prove second)

F1, . . . , Fn |= g1 ∧ · · · ∧ gm implies F1, . . . , Fn ` g1, . . . , gm .

Incompleteness of Prolog proof search

(already shown)

F1, . . . , Fn |= g1∧· · ·∧gm doesn’t imply F1, . . . , Fn `Prolog g1, . . . , gm .

Fundamental properties

Correctness of Prolog proof search (straightforward)

F1, . . . , Fn `Prolog g1, . . . , gm implies F1, . . . , Fn ` g1, . . . , gm .

Soundness of inference system (will prove first)

F1, . . . , Fn ` g1, . . . , gm implies F1, . . . , Fn |= g1 ∧ · · · ∧ gm .

Completeness of inference system (will prove second)

F1, . . . , Fn |= g1 ∧ · · · ∧ gm implies F1, . . . , Fn ` g1, . . . , gm .

Incompleteness of Prolog proof search (already shown)

F1, . . . , Fn |= g1∧· · ·∧gm doesn’t imply F1, . . . , Fn `Prolog g1, . . . , gm .

Proof of soundness

We need to prove:

F1, . . . , Fn ` g1, . . . , gm implies F1, . . . , Fn |= g1 ∧ · · · ∧ gm .

Suppose that F1, . . . , Fn ` g1, . . . , gm.

Let I be an interpretation for which

I |= F1 and . . . and I |= Fn

We need to show that I |= g1 ∧ · · · ∧ gm.

To this end, let Π be a derivation tree for g1, . . . , gm.

We show on next 2 slides that, for every established goalg ′1, . . . , g ′m′ . that occurs in Π, it holds that I |= g ′1 ∧ · · · ∧ g ′m′ .

Thus, in particular, I |= g1 ∧ · · · ∧ gm, as required.

Proof of soundness (continued)

We show: for every established goal g ′1, . . . , g ′m′ . that occurs in Π,it holds that I |= g ′1 ∧ · · · ∧ g ′m′ .

We work from the leaves (axioms) of Π downwards, propagatingthe property that I |= g ′1 ∧ · · · ∧ g ′m′ as we go.

(Formally, this is a proof by induction on the structure/height ofthe derivation.)

There are two cases:

1. If the goal g ′1, . . . , g ′m′ is an axiom then m′ = 1 and g ′1 is oneof F1, . . . , Fn.

So I |= q is immediate (due to the assumptions made aboutI on previous slide).

Proof of soundness (continued)

2. Otherwise the derivation of g ′1, . . . , g ′m′ within Π must look like

···g ′1, . . . , g ′l−1, h1, . . . , hk , g ′l+1, . . . , g ′m′ h1 ∧ · · · ∧ hk → g ′l

g ′1, . . . , g ′m′

where h1 ∧ · · · ∧ hk → g ′l is an axiom.

By the property we are propagating through the tree,

I |= g ′1 ∧ · · · ∧ g ′l−1 ∧ h1 ∧ · · · ∧ hk ∧ g ′l+1 ∧ · · · ∧ g ′m′ .

Because h1 ∧ · · · ∧ hk → g ′l is an axiom,

I |= h1 ∧ · · · ∧ hk → g ′l .

Thus indeed I |= g ′1 ∧ · · · ∧ g ′m′ .

Illustration of the propagation process

(1) I |=

cI

(1) I |=

cI→ wC

(2) I |=

wC

(1) I |=

wC→ iM

(3) I |=

iceMelts

Illustration of the propagation process

(1) I |= cI (1) I |= cI→ wC

(2) I |=

wC (1) I |= wC→ iM

(3) I |=

iceMelts

Illustration of the propagation process

(1) I |= cI (1) I |= cI→ wC

(2) I |= wC (1) I |= wC→ iM

(3) I |=

iceMelts

Illustration of the propagation process

(1) I |= cI (1) I |= cI→ wC

(2) I |= wC (1) I |= wC→ iM

(3) I |= iceMelts

Proof of completeness

We need to prove:

F1, . . . , Fn |= q1 ∧ · · · ∧ qm implies F1, . . . , Fn ` q1, . . . , qm .

Suppose that F1, . . . , Fn |= q1 ∧ · · · ∧ qm.

We need to show that F1, . . . , Fn ` q1, . . . , qm.

To this end, define the following interpretation I:

I(q) =

{true if F1, . . . , Fn ` q

false otherwise

We show on next 2 slides that

I |= F1 and . . . and I |= Fn

Proof of completness (continued)

I(q) =

{true if F1, . . . , Fn ` q

false otherwise

We show that I |= Fi for every Fi in F1, . . . , Fn.

There are 2 cases to consider.

1. If Fi is an atom q then trivially F1, . . . , Fn ` q. SoI(q) = true. I.e., I |= q.

Proof of completness (continued)

I(q) =

{true if F1, . . . , Fn ` q

false otherwise

2. In the case Fi is an implication p1 ∧ · · · ∧ pk → q, we need toshow I |= p1 ∧ · · · ∧ pk → q.

Suppose I |= p1 and . . . and I |= pk . That is,F1, . . . , Fn ` p1 and . . . and F1, . . . , Fn ` pk .

Combine the derivations of p1, . . . , pk to get one of q:

···p1 . . .

···pk

p1, . . . , pk p1 ∧ · · · ∧ pk → q

q

So F1, . . . , Fn ` q. Thus I |= q.

Proof of completeness (completed)

I(q) =

{true if F1, . . . , Fn ` q

false otherwise

We have assumed that F1, . . . , Fn |= q1 ∧ · · · ∧ qm,and we have shown that I |= F1 and . . . and I |= Fn.

Therefore I |= q1 ∧ · · · ∧ qm. Whence, by definition of I:

F1, . . . , Fn ` q1 and . . . and F1, . . . , Fn ` qm

We now combine the derivations:

···q1 . . .

···qm

q1, . . . , qm

Showing that F1, . . . , Fn ` q1, . . . , qm as required.

Discussion

We have a complete inference system for propositional definiteclause logic, but Prolog’s proof search procedure is incomplete.

Incompleteness can be remedied in different ways, e.g.:

I Add loop checking to the Prolog search procedure to checkwhen the search reconsiders an atom already encountered.

This provides a decision procedure for propositionaldefinite-clause logic: the search will say yes if a derivationexists, and no if no derivation exists.

I Use breadth-first search instead of depth-first search.

This gives a complete proof procedure (a derivation will befound whenever one exists) without loop checking, but not adecision procedure. Loop checking still needs to be added inorder to detect when no derivation is available.

Discussion

We have a complete inference system for propositional definiteclause logic, but Prolog’s proof search procedure is incomplete.

Incompleteness can be remedied in different ways, e.g.:

I Add loop checking to the Prolog search procedure to checkwhen the search reconsiders an atom already encountered.

This provides a decision procedure for propositionaldefinite-clause logic: the search will say yes if a derivationexists, and no if no derivation exists.

I Use breadth-first search instead of depth-first search.

This gives a complete proof procedure (a derivation will befound whenever one exists) without loop checking, but not adecision procedure. Loop checking still needs to be added inorder to detect when no derivation is available.

Discussion

We have a complete inference system for propositional definiteclause logic, but Prolog’s proof search procedure is incomplete.

Incompleteness can be remedied in different ways, e.g.:

I Add loop checking to the Prolog search procedure to checkwhen the search reconsiders an atom already encountered.

This provides a decision procedure for propositionaldefinite-clause logic: the search will say yes if a derivationexists, and no if no derivation exists.

I Use breadth-first search instead of depth-first search.

This gives a complete proof procedure (a derivation will befound whenever one exists) without loop checking, but not adecision procedure. Loop checking still needs to be added inorder to detect when no derivation is available.

Loop checking

iM (1)

wC (2)

hS (3) aD (4)

iM (5)

wC

hS aD

...

cI

yes

cI (6)

yes (7)

Breadth-first search

iM (1)

wC (2)

hS (3) aD (4)

iM (6)

wC

hS aD

...

cI

yes

cI (5)

yes (7)

Discussion (continued)

There are sound pragmatic reasons, however, that Prolog does notimplement such modified search algorithms.

I Loop checking makes proof search less efficient, and it isanyway not a useful modification once we move to thepredicate (first-order) logic used by Prolog (wherenon-terminating behaviour can occur without the same goalever repeating itself).

I Breadth-first search is in general inefficient, since it alwaysexplores all branches, irrespective of their usefulness.

The benefit of program-order depth-first search is that theprogramer can order the clauses in the program to maximizethe efficiency of Prolog’s proof search algorithm.

Conclusions

Prolog proof search is an example of good engineering design basedon an interplay between theoretical and practical considerations.

I It has a strong theoretical foundation, in being based on acomplete inference system for definite-clause logic.

I However, its incomplete proof search procedure is a goodimplementation choice, allowing efficient proof search in thecontext of predicate logic, and permitting the programmer totailor programs with regard to efficiency issues.

Main points today

search tree for top-down proof search

Prolog search strategy as program-order depth-first search

incompleteness of Prolog proof search strategy

correctness of Prolog proof search strategy

soundness of inference system (and proof)

completeness of inference system (and proof)