Syntax, Semantics and Intended Meaning The 1-slide course

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Syntax, Semantics and Intended MeaningThe
1-slide course

• Syntax
• Symbols a set of rules for their valid
  composition into aggregate structures
• Semantics
• The relationship between syntactic structures of
  a symbol system and what they are intended to
  denote -- their intended meaning
• Intended Meaning
• A set of truths, T, about some world
• Source is external to the symbol system
• Represented in the symbol system by syntactic
  structures (S1,Sn)
• The system is good/useful/fruitful/meaningful
  if
• All truths in T may be represented by some Si
  (complete)
• Every Si entailed by the system represents some
  truth in T (sound)

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LP Syntactic Structures

• Terms
• Constant, Variable, Functor, Term, Compound Term
• Instances
• Substitution, Instance
• Facts and Queries
• Facts and the rule of instantiation
• Queries and the rule of generalization
• Rules and Logic Programs
• Rules, Unification, Logic Programs

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TermThe single data structure

• Constant
• letter, number, string of letters or numbers
  beginning with a lower case letter.
• Intended to represent entities, individuals,
  existent etc. dog, cat, alfred, r2d2, selmer,
  the number 2.
• Variable
• Capital letter, or string of letters or numbers
  beginning with a capital letter
• Intended to represent an yet to be
  identified/deduced constant.
• Operationally Write Once, Ready Many. Once
  found truth can not be unfound.
• Compound term
• Functor(A1,,An)
• Functor is any constant
• Ai is a term.
• Intended to represent relations or predicates
  about things father(selmer,r2d2).

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SubstitutionA formal definition

• A finite set of pairs of the form xi tj, where
• xi is a variable
• ti is a term
• xi lt gt xj for every i lt gt j
• variables are bound only once in the substitution
  
• xi does not occur in tj, for any i and j.
• no recursive references in a binding
• Substitutions can be applied to terms
• foo(X,Y) Xselmer,Yr2d2 ? foo(selmer, r2d2).

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InstancesA relationship between terms

• One term may be an instance of another term.
• A is an instance of B if
• There exists a substitution q such that ABq
• Think of a substitution as a set of variable
  bindings
• Examples
• 1 is an instance of X.
• The substitution is X1
• foo(bar,1) is an instance of foo(V,1)
• the substitution is VBar
• father(big_Tony,franky_three_shots) is an
  instance of father(X,Y)
• What is the substitution?
• A ground instance is an instance with NO
  variables
• foo(V,Y)Vbar) foo(bar,Y) is NOT ground.
• foo(V,Y)Vbar, Y1) foo(bar,1) is ground.

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Queries

• A compound term preceded by -
• - functor(t1,,tn)
• Variables are existentially quantified
• Example
• LP - father(big_Tony,X)
• FOL ?x father(big_Tony, x)
• English
• Is Big Tony anyone's father?
• Does there exist a thing, X, such that Big Tony
  is the father of X?

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Rule of Generalization

• An existential query P is a logical consequent of
  an instance of P, Pq, for any substitution q.
• Example
• - father(big_Tony, X) is a logical consequent of
  
• father(big_Tony, two_Ton_Tommy)
• Different ways of saying the same thing
• ?X p(X) is true if p(anything here) is true.
• The existential generalization is true if an
  instance of it is true.

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A Fact

• Compound Term
• Functor(t1,,tn)
• Variables are universally quantified
• father(big_Tony, X) means
• English big Tony is everyones father
• FOL ?(X) father(big_Tony,X)

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Rule of Instantiation

• From a universally quantified statement P (i.e.,
  a fact) deduce an instance of it, Pq, for any
  substitution q.
• From father(big_Tony,X) deduce
• father(big_Tony, two_Ton_Tommy),
• father(big_Tony, selmer),
• father(big_Tony, franky_Three_shots),
• father(big_Tony, r2d2)

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A Rule

• Complex query composed of simpler queries
• Has a head (a term) and a body (a sequence of
  terms)
• H ? B1,,Bn
• A fact is a rule where n0
• Example
• LP grandmother(X,Z) ? mother(X,Y), mother(Y,Z).
• FOL ?(X,Y,Z) mother(X,Y) mother(Y,Z) ?
  grandmother(X,Z)
• English A person who is the mother of another
  person who is herself the mother of a third
  person is the grandmother of the third person
• Which representation do you prefer?

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Rule of Universal Modus Ponens

• From
• A ? B1,,Bn ? (B1,,Bn)
• where for all i between 1 and n, Bi is an
  instance of Bi
• A is an instance of A
• Deduce
• A
• From
• grandfather(X,Z) ? father(X,Y), father(Y,Z)
• father(big_Tony, little_Tony), father(little_Tony,
  tiny_Tony)
• Deduce
• grandfather(big_Tony, tiny_Tony)

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A Logic Program

• A logic program is a finite set of rules.
• Example Logic Program The Family Tree
• father(big_Tony, little_Tony).
• father(little_Tony, tiny_Tony).
• boss(quick_Nick, franky_three_shots).
• boss(buddy_the_button, mikey_the_mop).
• boss(big_tony,Y) ? boss(Y,Z).
• grandfather(X,Z) ? father(X,Y), father(Y,Z).
• Example Queries
• - boss(big_Tony, quick_Nick)
• yes
• - boss(big_Tony, r2d2)
• no.why? Whats the intended meaning of the
  boss(big_tony_Y) rule above?

Facts are rules where the body is empty.
Rules with bodies
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More on Semantics

• Declarative and Operational

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Declarative SemanticsWhat it means is what it is.

• How a the syntactic structure of a program
  relates to its intended meaning
• The meaning of a logic program is the set of all
  the statements it entails
• The meaning of a logic program P is the minimal
  model for P

• where the minimal model for P is the intersection
  of all models of P,
• where a model of P is an interpretation, I, for P
  where, for every ground instance of a clause A ?
  B in P, A is in I if B is in I
• where an interpretation is a subset of the
  Herbrand Base
• where the Herbrand Base is the set of all ground
  goals that can be formed (syntactically) form the
  predicates in P and the terms in the Herbrand
  Universe
• where the Herbrand Universe is the set of all
  ground terms that can be formed from the
  constants and function symbols appearing in P.

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Operational SemanticsWhat it means is what a
procedure produces.

• The meaning of a logic program, P, is the set of
  ground goals that are instances of queries
  produced by P according to a LP computation.
• A computation of a program P is a procedure.
• The procedure starts with a query G, progresses
  via goal reduction and it either succeeds, fails
  or never terminates.
• If it succeeds then the instance of G produced is
  in the meaning of P.
• If it fails then no instance of G is in the
  meaning of P.
• If it never terminates, then we dont know if any
  instance of G is in the meaning of P.
• How does goal reduction produces statements
  (i.e., answers)?
• How it works is the operational semantics of
  logic programming.
• The meaning is the output of the procedure. The
  operational semantics is a formal description of
  the procedure for producing the output from the
  input.

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Classic Programming Languages

• What are the declarative semantics?
• Scott-Strachey approach to programming language
  theory
• Denotational Semantics (book by Joseph Stoy)
• Lambda Calculus (dates back to Lisp in the
  1950s)
• Concept Given a program, P and the environment,
  E (contents of the registers of the machine or of
  variable/value bindings), there is a formal
  theory for logically deducing the next
  environment -- the environment that is entailed
  by P E.
• What are the operational semantics?
• How the compiler processes the program to
  determine the assignment of variables to values.

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Goal Reduction

• The computational core of LP

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Elements of Goal Reduction

• Unification Pattern Matching
• Resolution Inference
• Chronological Backtracking Search

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Unification

• A constructive pattern matching procedure
• Determines the most general unifier (MGU) of two
  terms.
• Using the MGU constructs the most general common
  instance of two terms
• Used to compute the resolution inference
• Resolution is used to iteratively reduce a goal
  generating its proof from the program if possible
  (i.e., goal reduction)

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Most General Unifier(MGU)

• A common instance of A and B is an instance of
  both A and B.
• foo(1,2) is a common instance of foo(1,Y) and of
  foo(X,2)
• A unifier of two terms is a substitution that
  renders the terms identical
• X1,Y2 applied to foo(1,Y) and to foo(X,2)
  produces identical terms
• The MGU of two terms is a unifier that produces
  the most general common instance of the two
  terms.
• Which are instances of both foo(X,Y), foo(1,Y)?
  What is the MGU?
• bar(1, 2)
• foo(A, 1)
• foo(A, B)
• foo(1, 2)
• foo(1, B)

None, None XA, Y1, None XA, YB,
None X1,Y2, Y2 X1,YB, YB
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Unification Procedure

• Computes the MGU of two terms if it exists else
  fails.
• unify(X,1) X,1
• unify(a(X,1), a(1,Y)) X1, Y1)
• Produces the most general common instance
• If two terms unify it is NOT necessarily the case
  that one is an instance of the other
• a(X,1) is not an instance of a(1,Y)
• a(1,Y) is not an instance of a(X,1)

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Unification Examples

• Fact Schema Supports(Video_Card, Display_Type,
  Number_of_Colors)
• Example 1 Neither A nor B entails the other
• Facts (A and B) supports(V,b14,C),
  supports(nvidia,D,256)
• MCI supports(nvidia, b14, 256)
• MGU Vnvidia, Db14, C256)
• Example 2 B is an instance of A
• Facts (A and B) supports(V,b14,C),
  supports(rage,b14,16)
• MCI supports(rage, b14, 16)
• MGU Vrage, C16
• Example 3 A and B are not unifiable
• Facts (A and B) supports(V,b14,16),
  supports(nvidia,D,256)
• MCI None
• MGU None

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Resolution

• Enables a reduction of two rules into one
• Eliminates a common term from the consequent of
  one rule and from the antecedent of another
• Efficient application depends on the constructive
  unification procedure
• General Form
• Rule 1 A ? (B M)
• Rule 2 (C N) ? D
• Resolution Inference (A C)q ? (B D) q
• where q is the MGU of M and N.

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Resolution Example

• Rule 1
• affinity(Y,Z) - likes(Y,Z), even(Z).
• Rule 2
• basic_affinity(Y) - affinity(Y,2)
• Inference
• basic_affinity(Y) - likes(Y,2), even(2).
• The inference allowed the
• elimination of affinity(Y,Z) and
• reduction to a single rule through finding
• the MGU of affinity(Y,Z) and affinity(Y,2) and
• applying it to the body of rule 1.
• What fact would make - basic_affinity(r2d2) true?

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Chronological Backtracking

• A search procedure for finding unifying terms
• Conducted over an explicit set of clauses
• Upon failure (a dead-end) previous variable
  bindings are chronologically unwound and the
  search begins again until all possible bindings
  are exhausted
• Note
• Fruitful bindings may be unrelated to chronology
• Can therefore be terrible inefficient
• Must be aware of how this works to write
  efficient LPs

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Backtracking Example

• - plugsIn(X,Y)
• outPort(o5)
• inPort(in1)
• compatible(o5,in1)
• FAIL and Backtrack
• inPort(in2)
• compatible(o5,in2)
• FAIL and Backtrack
• inPort(in3)
• compatible(o5,in3)
• Success(1). plugsIn(o5,in3).
• inPort(Y)
• FAIL and Backtrack
• outPort(X)
• FAIL

• plugsIn(X,Y) -
• outPort(X),
• inPort(Y),
• compatible(Y,Y).
• inPort(in1).
• inPort(in2).
• inport(in3).
• outPort(o5).
• compatible(o5,in3).

Search order matters. Put compatible first in the
plugsIn rule?
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Practical Introduction to Prolog

• Basics

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Programming Excursions

• Exploring Unification
• Basic Constructs
• Arithmetic, Recursion and Lists
• Database Programming
• Trees and Parsing

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Practical Introduction to Prolog

• Negation and the Cut