p&s 3.5, 3.7: the logic and operation of prolog revisited ...ddekok/pp/lectures/bjoern.pdf ·...

P&S 3.5, 3.7: Thelogic and operationof Prolog revisited,

definite clausegrammar

Bjorn Rudzewitz

Logic andOperationRevisited

Substitution

Unification

Semantics of Prolog

Definite ClauseGrammars

Parse Trees

Tree Operations inProlog

Removal of Nodes

Lexicon Simplification

References

P&S 3.5, 3.7: The logic and operation ofProlog revisited, definite clause grammar

Bjorn Rudzewitz

[email protected]

University of Tubingen

June 1, 2016

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

Logic and Operation RevisitedSubstitutionUnificationSemantics of Prolog

Definite Clause GrammarsParse TreesTree Operations in PrologRemoval of NodesLexicon Simplification

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

Introduction

I How does Prolog search/proof ?I search criteria:

1. efficient2. predictable

I search operations: substitution, unification, resolution

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

Substitution

I Substitute variables X by terms T with substitutionfunction σ:

σ : {x1, · · · xk} → {t1, · · · , tk},{x1 = t1, · · · , xk = tk}

I application of σ to expression e: [e]σ

I if [e1]σ = e2, thenI e1 subsumes e2

if not e2 subsumes e1, e1 is more generalI e2 is an instance of e1

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

Subsumption in Practice

Figure: Subsumption predicates in SWI-Prolog source: http:

//www.swi-prolog.org/pldoc/doc_for?object=terms%3Asubsumes_chk/2

(last accessed 31/05/16)

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http://www.swi-prolog.org/pldoc/doc_for?object=terms%3Asubsumes_chk/2

http://www.swi-prolog.org/pldoc/doc_for?object=terms%3Asubsumes_chk/2



Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References


practical example !

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References


Figure: Subsumption checks in SWI-Prolog

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

Unification

I special case of substitution: unifying substitution

[e1]σ = [e2]σ

→ result of substitutions identical

I unifier σ always ’most general’:for every other unifier σ′ of e1,2, [e1]σ subsumes [e1]σ′

I i.e. the term resulting from applying σ′ is an instance ofthe term obtained by applying σ

I application of most general unifier: unification (ofe1, e2)

I result of most general unifiers: identical expressions, orat most other variable names

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

UnificationUnification Algorithm

I for 2 expressions, compute most general unifier

I various implementations

I key idea: find mismatches in expressions, findsubstitutions/bindings to solve them

I occurs check: never construct a term as a substitutionof a variable such that the term contains the variableagain

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

Unification AlgorithmOccurs check

e1: f(X)e2: f(g(X))naive unification algorithm: substitute x by g(x)e1’: f(g(X))e2: f(g(g(X))) · · ·occurs check needed for unification !

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

Unification AlgorithmOccurs check

see http://www.swi-prolog.org/pldoc/man?predicate=unify_with_

occurs_check/2

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http://www.swi-prolog.org/pldoc/man?predicate=unify_with_occurs_check/2

http://www.swi-prolog.org/pldoc/man?predicate=unify_with_occurs_check/2



Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

Semantics of Prolog

I different perspectives on goal finding:I declarative: conjunction of program and the negation of

the goal unsatisfiable→ a logical statement doesn’t hold

I procedural: negated goal and program generate emptyclause→ the result of a resolution is empty

I difference: procedural way is not necessarilyI complete: not always a solution is found due to

depth-first searchI sound: no occurs check

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

Definite Clause Grammars

I extension of Context Free Grammars: nonterminal canhave arguments (and are not just atoms such as in CFG)

I string positions encoded as last two arguments

Note: DCGs exist outside Prolog

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

Definite Clause GrammarsReminder: Built-In Support and Notation

s --> np(Number), vp(Number).

pn(singular) --> [he].

I --> operator

I RHS comma-separated (terminals and/or nonterminals)

I terminal as list (empty string: empty list)

I implicit encoding of position arguments (P0,P1),encodes lists

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

Definite Clause GrammarsShorthand Notation

The following grammars are equivalent:

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

Definite Clause Grammars

I DCG internally transformed to arity 2 predicate withdifference lists (for partial parsing)

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

Parse Trees

I additional argument for NTs to indicate span / subtree

I epsilon for empty subtree

s(s(NP,VP)) --> np(NP), vp(NP).

pn(pn(he)) --> [he].

I while parsing, a tree is built up in this argument

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

DCGs

I internally transformed to difference lists (for partialparsing)

I for the full parse, give an empty difference list as 3rdargument

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

Parse Trees

I more general parse tree representation (by Daniel):I general tree t predicate for parse treeI tree daughters in list argument

I Example:s(t(s,[NP,VP])) --> np(NP), vp(VP).

I see file dcg-t.pl

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../Code/dcg-t.pl



Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

Tree Operations

I ’normal’ Prolog statements can be specified in additionto/inside DCG rules

I notation in brackets {operation} to make difference togrammar clear

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

Node Removal: Example from Pereira andSheiber [1989]

Terry writes a program.Terry writes a program that halts.

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References


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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References


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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References


I How to get rid of the node without terminal span ?

I Solution: add a special builder predicate

I define a predicate build np:

build np(Det, N, rel(epsilon), np(Det,N))

build np(Det, N, rel(that,VP), np(Det,N,

rel(that,VP)))

I define np rule as follows:

np(NP) --> det(Det), n(N), optrel(Rel),

{build np(Det,N, Rel, NP)}

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References


I DCG lexicon so far:

n --> [house].

n --> [program].

det --> [the].

I new idea:

n --> [Word], {n(Word)}n(program).

n(house)

I instead of many rules, one rule per POS

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References


I also applicable to parse tree grammar:

n(n(Word)) --> [Word], {n(Word)}.n(Number) --> [Word], {n(Word, Number)}.n(program).

n(professors, plural).

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References


I even more radical approach: store singular and pluraltogether

n(program, programs).

n(professor, professors).

n(singular) --> [Word], {n(Word, )}.n(plural) --> [Word], {n( , Word)}.

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References


see file dcg-t-compact.pl

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Bjorn Rudzewitz


Substitution

Unification

Semantics of Prolog


Parse Trees


Removal of Nodes


References

Patrick Blackburn, Johannes Bos, Kristina Striegnitz, et al.Learn prolog now!, volume 7. College PublicationsLondon, 2006.

Fernando Pereira and Stuart Sheiber. Prolog andnatural-language analysis: Fernando cn pereira and stuartm. sheiber, csli lecture notes 10 (the university of chicagopress, il, 1987); 266 pages, 13.95(paperback),28.95(cloth)., 1989.

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p&s 3.5, 3.7: the logic and operation of prolog revisited ...ddekok/pp/lectures/bjoern.pdf ·...

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