SLD-resolution

1
SLD-resolution

• Introduction
• Most general unifiers
• SLD-resolution
• Soundness
• Completeness

2
Proof of A refutation of ?A

• ? true (any valid formula)
• ? false (any unsatisfiable formula)
• Fact 1
• P A iff P ? ? A ?
• Fact 2
• if P ? ? A - ? then P ? ? A ?
• supposing the inference rules are sound
• if and only if if they are sound and complete
• ? idea of proof by refutation
• start from P and ?A
• apply inference rules until ? is produced
• if so, we know that P A

3
Refutation proofs with clauses

• Definite clauses
• F can be read as F ? ?
• ? G1, ..., Gm ? ? ? G1, ..., Gm
• empty clause ? ? ? ? ? ?
• Refutation proofs
• start from P and ? ? G1,...,Gm
• try to derive the empty clause ?
• example pp. 34-35 a sort of generalized Modus
  Ponens

4
How to automate refutations?

• G ? G1, ...,Gm
• select some Gi p(t1, ..., tn)
• look for clauses defining p/n
• H ? B1, ..., Bk (we may have several such)
• make Gi and H syntactically identical
• reason Modus Ponens requires it
• ? find substitution (unifier) ? such that H?
  Gi?
• if such ? exists, apply it to (B1,...,Bk) and the
  rest of G
• new goal (? G1, ..., B1, ..., Bk, ... Gm)?
• continue until G is empty (if possible)

5
What did we just compute?

• G (? ?(G1 ? ... ? Gm)) ? ?(? G1 ? ... ? Gm)
• if G lead to contradiction
• then we have the conclusion that ? G1 ? ... ? Gm
  holds
• Derivation any chain of inference steps
• Refutation of G finite derivation of falsity
  starting from G
• consists of k inferences with k unifications ?i
  (1 ? i ? k)
• ? ((? G1 ? ... ? Gm) ?1... ?k) was shown to be
  unsatisfiable
• fact p(X) is unsatisfiable iff some instance of
  p(X) is
• Thus P (G1 ? ... ? Gm) ?1 ... ?k
• ?1 ... ?k is the counter-example or the answer to
  our question
• if some variables are left unbound we may assume
  the universal closure

6
Troubles in the computation

• Selections deterministic or not?
• goal literals
• program clauses
• unifiers
• Computationssuccess/fail/infinite?
• dead ends (no matching clauses)
• infinite number of solutions (several unifiers)
• loops (p ? p)

7
Most general unifiers

• Algorithmic solution to the matching problem
• deterministic efficient
• Vocabulary
• predicate symbols functors constructors
• atoms, terms structures
• principal symbol, direct substructures
• size of a structure ( of symbols)

8
Unifier (definition)

• Let s1, s2 be structures and ? a substitution
• if s1? s2? then ? is a unifier of s1 and s2
• Notes
• if s contains X then X and s have no unifier
• proof size(s) gt size(X) ? size(s?) gt size(X?)
  whatever ? is
• unifier ? may leave (some) variables free
• composition of ? with any substitution ? yields
  another unifier
• each of them is more specific than ?

9
Generality of unifiers

• let ? and ? be unifiers of s and t
• ? ?? ? is more general than ?
• any s? is an instance of s?
• Note generality is not antisymmetric
• but the following holds
• if ? is more general than ? and vice versa then
• s? and s? are identical up to the renaming of
  variables

10
Most general unifier(s), mgu

• substitution ? is a mgu of s t if
• ? is a unifier of s t
• ? is more general than any other unifier of s t
  
• Thm mgus are unique up to renaming!
• mgus are the substitutions we are looking for in
  the matching
• problem how to find them effectively

11
(yet) Some (more) terminology

• Set of equations X1t1,,Xntn
• is in solved form iff
• X1,,Xn are distinct variables, and
• none of X1,,Xn appear in t1,,tn
• Proposition
• If E X1t1,,Xntn is in solved form
• then ? X1/t1,,Xn/tn is a mgu of E
• Definition E1 and E2 are equivalent
• both have the same set of unifiers
• consequently same solutions in Herbr.
  interpretations

12
Unification algorithm

• Several implementations exist
• simple O(min(size(s), size(t))) presented
• with occurs check quadratic time
• Basic ideas
• if s is a variable, bind it to t (and vice versa)
  
• if both are structures (or constants)
• principal symbols must be the same
• direct substructures must unify (use recursion)
• note bindings may spread elsewhere in terms
• same variable may have several occurrences

13
Our algorithm (p. 40)

• To unify s t
• try to transform s t into an equivalent one
  in solved form
• if this succeeds, the result gives also a mgu of
  s t
• if this fails, no unifier exists
• note constants are treated as 0-ary functors
• Theorem Algorithm returns
• the equivalent solved form of s t
• or failure if no such exists

14
Resolution rule

• Inference rule for definite clauses goals
• From
• (?? (G1,...,Gi,...,Gm)) and
• (? H ? B1,...,Bk)
• such that
• Gi and H have a mgu ?
• derive
• (?? (G1,...,B1,...,Bk,...,Gm)?)

15
Procedural view

• FROM
• query G and clause C
• SUCH THAT
• subgoal Gi of G and the head H of C unify
• DERIVE
• a new query G with
• Gi replaced with body of C and
• mgu of H and Gi applied to the result

16
Resolution technicalities (i)

• Both premises universally closed
• scopes of the quantifiers are disjoint
• Conclusion is universally closed
• variables in the premises must be disjoint
• What to do?
• rename variables in clauses before inference
• e.g. attach the of the inference step to all
  variables
• allows multiple uses of the same clause
• essential with recursive clauses

17
Resolution technicalities (ii)

• Selection of subgoals
• abstraction assume some selection function ?
• a.k.a. computation rule
• SLD-resolution
• Linear resolution for Definite clauses with
  Selection function
• linear resolution
• P1 is always the query created in the previous
  step
• P2 is always a program clause

18
Using SLD-resolution

• G0 ? A1,...,Am
• ?(G0) Ai
• apply resolution to
• Ai and
• a (renamed copy of a) program clause C
• construct new goal G1, then G2, ...
• computations end (at some goal Gi) when
• ?(Gi) does not unify with any clause head or
• Gi false (empty goal)
• infinite computations possible, too

19
Formalizing resolution proofs

• SLD-derivation
• From Gi (a copy of) Ci, determine G(i1)
• Note Gi, ? Ci determine ?(i1)
• reason mgus are unique (up to var. names)
• ? no need to state ? separately in derivations
• including ?s improves the readability, though
• G(i1) is said to be derived from Gi Ci
• Gi Ci resolve into G(i1)
• special notation (jagged arrow, p. 44)
• Each derivation G0,...,Gn yields a computed
  substitution ?1?n

20
Successful derivations

• Finite and end to the empty goal
• SLD-refutation of G
• answer substitution computed substitution of a
  refutation
• Theorem (independence of the computation rule)
• If G has a SLD-refutation with some rule ?1 then
• G has one with the same answer with any other
  rule ?2, and
• the refutation ?2 is obtained by permuting the
  clauses used in the refutation ?1 proof
• intuitive reason
• all subgoals are selected sooner or later
• same clauses are used to refute each of them

21
Classifying derivations

• Successful
• lead to false
• Failed
• lead to some goal Gi such that
• ?(Gi) does no unify with any clause head
• note it doesnt matter if there are some other
  subgoals than the selected one which unify with
  some clause head, we are stuck with the selected
  one anyway
• Infinite
• Complete derivation any of the above
• taken to the extreme, continued as long as
  possible

22
SLD-trees...

• Consider alternative clauses for a selected
  subgoal
• ?(Gi) may unify with several clauses
• ? Gi may have several (and different) complete
  derivations with rule ?
• SLD-trees formalize the set of all such
  derivations
• nodes goals (G0 at the root)
• edges are labeled with the clause used in
  inference
• descendant the resulting new goal
• Each path represents some derivation
• some may fail, some succeed, some may be infinite
• Definition 3.18.

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...SLD-trees

• SLD-trees are (usually) distinct for different
  selection functions
• Finite tree with ?1 infinite with ?2 possible!
• Independence of computation rule tells that
• refutations in tree T1 (using ?1)
• are also found in T2 (using ?2)
• and are permutations of the paths in T1
• Note
• there may exist other failing or infinite paths
  for which the above does not hold
• but they dont interest us that much anyway
• it is enough that the successful derivations are
  found

24
Soundness of SLD-resolution

• Final conclusion for a give goal G
• try to find a refutation of ? G
• if found, apply the answer substitution to G
• this step is actually an additional inference
  rule
• ? SLD-resolution gives answers
• ? p(X), , empty goal, ? X/tom,
• reasoning is sound, if P p(tom)
• Answer computed substitution restricted on
  query variables
• negative answer is possible, too
• Soundness (or correctness)
• Shown w.r.t. computed answer substitutions
• Reason user is not (usually) interested in
  individual resolution steps, but the final answer

25
Soundness theorem (Clark -79)

• Let
• P be a definite program
• ? a selection function
• ? a ?-computed answer for goal ? A1,,Am
• Then
• P (? (A1 ? ? Am) ?)
• Note
• Requires occurs-check in unification
• see example 3.21

26
Completeness theorem...

• Does SLD-resolution compute all logical
  consequences?
• strictly speaking NOT, but practically YES, since
• every correct answer is an instance of some
  computed answer
• Let
• P, ? A1,,Am, and ? be as before
• If
• P ? (A1 ? ? Am)?
• Then
• there exists a refutation of ? A1,,Am with
  answer ?
• such that
• (A1 ? ? Am)? is an instance of (A1 ? ? Am)?
• See Example 3.23

27
...Completeness theorem

• Most general unifiers are used in inference steps
• ? refutations give the most general answers
• necessity for completeness theorem to hold
• Thm. confirms only the existence of a refutation
• does not tell how to find one
• in practice, we have to systematically search in
  the SLD-tree for the successful derivations

28
Search in SLD-trees

• Prolog systems
• Textual ordering of clauses ? ordering of edges
• Depth-first search in this order
• Backtrack from leaf nodes
• Solutions are reported at empty leaves
• Property the above is complete for finite
  SLD-trees
• Infinite trees ? infinite paths ? infinite search
• Breadth-first?
• theoretically better
• makes implementation (memory management) much
  more complicated
• too slow for practical applications

29
Proof trees

• Structural representations for SLD-derivations
• like derivation trees in CF grammars
• each clause represents an elementary tree
• derivation tree combination of elementary trees
• joint nodes labeled with
• equations for clause head corresponding body
  atom above
• proof tree
• a complete derivation tree, i.e. all leaf nodes
  are constants true

30
Proof trees

• Alternative view
• proof tree collection of equations (joint
  nodes)
• consistent tree equation set has a solved form
• not all derivation trees are consistent
• Note works also in other interpretations
• i.e. no matter what the equality stands for
• Extension use (atomic) goals as the root
• solved form an answer for the goal
• Simplification of proof trees
• apply the substitution induced by the solved form
  to the tree
• collapse identical equations into one node
• simplified tree shows (clearly) the consistence

31
Searching for a consistent proof tree

• Two (possibly) interleaved processes
• combination of elementary trees
• given this subgoal, try out this clause
• simplification of nodes
• not necessary to simplify the whole tree at once
• In which order to simplify?
• construct the whole tree check then
• check while building
• check the whole set of equations when new ones
  are added
• simplify the tree as new equations are introduced
• Prolog approach
• build tree in depth-first manner
• simplify immediately

32
Derivations and proof trees

• Many derivations may map to the same proof tree
• Clear due to the independence of computation rule
• computation rule tells the order in which
  equations are solved
• the final result stays the same