The Logic Paradigm. Joseph Spring. 7COM1023 Programming Paradigms

Transcription

1 The Logic Paradigm Joseph Spring 7COM1023 Programming Paradigms 1

2 Discussion The Logic Paradigm Propositional and Predicate Logic See also notes and slides on PP website Horn Clauses Definition, Examples and Theorem Resolution and Unification Handout: Logic Programming in Prolog 2

3 The Logic Paradigm Logic paradigm also known as Declarative or Rule Based Emerged in 1970 s Different to other paradigms in that programmer has to declare the goals of the computation as opposed to an algorithm through which the goals can be achieved Goals are expressed as a set of assertions/rules for the outcomes and constraints ~ hence label: rule based programming 3

4 The Logic Paradigm Two main areas for applications: Artificial Intelligence Prolog is popular Also Mycin for Expert systems Also CLIPS Database retrieval SQL (Structured Query Language) popular We focus on SWI-Prolog and applications to: Natural Language Processing Problem Solving 4

5 The Logic Paradigm Two noteworthy features of logic programming: Nondeterminism Nondeterministic program may find several solutions to a problem in contrast to other paradigms which find one solution Backtracking Enables non determinism Built in to Prolog interpreter Implicit in all Prolog programs Using other languages to write backtracking requies programmer to define backtracking mechanism 5

6 Logic Programs Express specifications for solutions through mathematical logic Approach evolved out of needs of researchers in Natural language processing Automatic theorem proving Conventional programming languages not well suited to the researchers needs Writing down the specification of theorem or grammar (e.g. BNF)as formal logical expression provides effective vehicle for theorem proving and natural language analysis in experimental laboratory setting 6

7 Provide the formal foundation for logic paradigm Propositional Logic Propositions are either True or False I am 21 years old I am not 21 years old See Background Slides and Notes on PP webpage Predicate Logic Predicate provide information about relationships (manages(x,y) )or properties (bossy(x)) that exist for objects in the domain of interest See Background Slides and Notes on PP webpage 7

8 Horn Clause A particular variation on predicate logic Underlies the syntax of Prolog Definition A Horn Clause has a head h, which is a predicate and a body which is a list of predicates pp p n Horn clauses are written in the following style: h pp pn Which is interpreted to mean h is true only if are simultaneously true. pp pn Compare to the syntactic turnstyle in background notes and slides for predicate logic 8

9 Horn Clause Example Suppose we want to capture the idea that it is snowing in city C only if there is precipitation in C and it is freezing in city C The following Horn Clause captures this as snowing( C) precipitation( C), freezing( C) Theorem. (i) Any Horn Clause may be written as a predicate (ii) Not every predicate can be expressed as a Horn clause 9

10 Proof ( i) h abcd,,,,..., gis equivalent to the predicate a b c d... g h (see predicates in extension in predicate notes / slides) and this is equivalent to the predicate ( a b c d... g) h which is equivalent to the predicate a b c d... g h ( ii) Consider the statement: 'Every literate person reads or writes. This may be expressed as the predicate x( literate( x) reads( x) writes( x)). This may be expressed in the form literate( x) reads( x) writes( x). It follows that for all x we have literate( x) reads( x) writes( x) which does not have a single term on the right. 10

11 Theorem. (i) Any Horn Clause may be written as a predicate (ii) Not every predicate can be expressed as a Horn clause Proof ( i) h abcd,,,,..., gis equivalent to the predicate a b c d... g h (see predicates in extension in predicate notes / slides) and this is equivalent to the predicate ( a b c d... g) h which is equivalent to the predicate a b c d... g h ( ii) Consider the statement: 'Every literate person reads or writes. This may be expressed as the predicate x( literate( x) reads( x) writes( x)). This may be expressed in the form literate( x) reads( x) writes( x). it follows that for all x we have literate( x) reads( x) writes( x) which does not have a single term on the right. 11

12 Resolution and Unification Definition (Resolution) When applied to Horn clauses resolution infers that if h is the head of a Horn clause and it matches with one of the terms of another Horn clause, then that term can be replaced by h Example If h terms and t t, h, t then we can resolve 1 2 the second clause as t t, terms, t

13 Resolution and Unification So with respect to the languages example in practical 8 speaks( mary, english) and talkswith( X, Y ) speaks( X, L), speaks( Y, L), X Y We note that the first statement is a Horn clause with an empty list of terms, and so this is unconditionally true. Resolution allows us to deduce talkswith( mary, Y ) speaks( mary, english), speaks( Y, english), Mary Y 13

14 Resolution and Unification Definition The assignment of variables to values during resolution is referred to as instantiation Definition Unification is a pattern matching process that determines the instantiations possible when making a series of simultaneous resolutions 14

15 Summary The Logic Paradigm Propositional and Predicate Logic See also notes and slides on PP website Horn Clauses Definition, Examples and Theorem Resolution and Unification Handout: Logic Programming in Prolog 15

16 References 1) Tucker A. B. and Noonan R. E., Programming Languages: Principles and Paradigms, McGraw Hill,