Advanced Topics in LP and FP

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1 Lecture 1: Prolog and

2 Summary of this lecture 1 Introduction to Prolog 2 3 Truth value evaluation 4

3 Prolog Logic programming language Introduction to Prolog Introduced in the 1970s Program = collection of logical sentences in the First-Order Predicate Logic (FOPL) Execution environment = theorem prover which indicates If a fact is true or false. In which circumstances is a fact true or false. Prolog resource available on Wikibooks:

4 Prolog Features Introduction to Prolog Solid theoretical foundation of the reasoning process The only way of program execution is with a reasoning engine limited ways to control program execution Automated search of values of the free variables (if this is necessary) Can be used to perform symbolic proofs and deductions.

5 What is logic? Introduction to Prolog Symbolic formalism for the representation of facts and reasoning rules Based on the notion of truth value, that is, true or false Allows the construction of explanations and proofs deduction, induction, resolution, and so on.

6 Logic Programming Introduction to Prolog Programs are made of logical sentences (Horn clauses, if we use Prolog) The execution environment can use these sentences to prove theorems and to deduce facts. In order to understand how programs work in a logic programming language, we must understand What are sentences, what is their meaning, and how can we represent them; The functionality of the theoretical processes used by the execution environment.

7 What is propositional logic? Framework for Describing the properties of objects in a language with a clearly defined semantics Deducing new properties, based on properties we already know. A proposition is a language expression which corresponds to a statement, that can be true or false. Example. The weather is nice. A proposition can be viewed in two ways: Syntactic: As a sequence (or string) of symbols used to write it down Semantic: the proper meaning of the sentence, in an interpretation

8 Syntax Introduction to Prolog Two kinds of propositions: simple: they express atomic facts: The weather is nice. compound: they express relations between more elementary propositions: The phone rings and the dog barks. Simple propositions: p, q, r,... Negations: α Conjunctions: (α β) Disjunctions: (α β) Implications: (α β) Equivalences: (α β)

9 Semantics Introduction to Prolog Purpose: development of processing mechanisms for propositions, which can be applied regardless of their truth value in concrete situations Emphasis is put on the relation between the truth value of a compound proposition and the truth values between the component propositions. The concept of interpretation is used to explain the meaning (or truth value) of propositions.

10 Semantics Interpretation of simple sentences Definition (Interpretation) Associates a truth vale to every simple proposition of a language.

11 Semantics Interpretation of simple sentences Definition (Interpretation) Associates a truth vale to every simple proposition of a language. Example Intepretation I p I = false q I = true r I = false Interpretation J p J = true q J = true r J = true

12 Semantics Interpretation of simple sentences Definition (Interpretation) Associates a truth vale to every simple proposition of a language. Example Intepretation I p I = false q I = true r I = false Interpretation J p J = true q J = true r J = true How can we know if p is true or false?

13 Semantics Interpretation of simple sentences Definition (Interpretation) Associates a truth vale to every simple proposition of a language. Example Intepretation I p I = false q I = true r I = false Interpretation J p J = true q J = true r J = true How can we know if p is true or false? We can answer this question only if we fix the interpretation p is just a name given to a concrete sentence.

14 Semantics Interpretation of compound sentences The interpretation α I of a compound sentence α is defined by recursion on its syntactic structure { true if α Negation: ( α) I := I = false false otherwise { true if α Conjunction: (α β) I := I = true and β I = true, false otherwise. { false if α Disjunction: (α β) I := I = false and β I = false, true otherwise. { false if α Implication: (α β) I := I = true and β I = false, true otherwise. { true if α Equivalence: (α β) I := I = β I, false otherwise.

15 Evaluation How to compute the truth value of an arbitrary proposition? Apply recursively the semantic rules defined on the previous slide. Example Consider the interpretation I where p I = false, q I = true, r I = false. The truth value of the sentence φ = (p q) (q r) is computed as follows: φ I = (false true) (true false) = false false = false.

16 Satisfiability Introduction to Prolog Definition A proposition is satisfiable if it is true in at least one interpretation. That interpretation satisfies the proposition. We can check satisfiability with the truth table method. Example p q r (p q) (q r) true true true true true true false true true false true true true false false true false true true true false true false false false false true false false false false false

17 Validity Introduction to Prolog Definition A proposition is valid if it is true in all interpretations. Valid propositions are also known as tautologies. Example (Validity) The proposition p p is valid, regardless of the truth value of p, thus it is valid.

18 Validity Introduction to Prolog Definition A proposition is valid if it is true in all interpretations. Valid propositions are also known as tautologies. Example (Validity) The proposition p p is valid, regardless of the truth value of p, thus it is valid. We can use the truth table method to check the validity of a proposition.

19 Unsatisfiability Introduction to Prolog Definition A proposition is unsatisfiable if it is false in all interpretations. An unsatisfiable proposition is also known as a contradiction. Example (Unsatisfiability) The proposition p p is false regardless of the truth value of p, thus it is unsatisfiable.

20 Unsatisfiability Introduction to Prolog Definition A proposition is unsatisfiable if it is false in all interpretations. An unsatisfiable proposition is also known as a contradiction. Example (Unsatisfiability) The proposition p p is false regardless of the truth value of p, thus it is unsatisfiable. We can use the truth table method to check the unsatisfiability of a proposition.

21 Derivability Definition Introduction to Prolog We write = φ and say that proposition φ is a consequence of a set of propositions if φ I = true whenever I is an interpretation such that ξ I for all ξ. Example {p} = p q {p, q} = p q {p} = p q {p, p q} = q

22 Derivability How to prove the derivability of φ from? Example With the truth table method: check if all rows in the table satisfy the following condition: if the truth values of all ξ are true then the truth value of φ is true. Prove that {p, p q} = q. p q p q true true true true false false false true true false false true The only interpretation in which both p and p q are true is the first row in the truth table, and q is true in that row. Thus, {p, p q} = q.

23 Derivability Equivalent statements Introduction to Prolog The following statements are equivalent: 1 {φ 1,..., φ n } = φ 2 φ 1... φ n φ is valid 3 φ 1... φ n φ is unsatisfiable

24 Inference Motivation Introduction to Prolog Remark: Exponential growth of the number of interpretations of a compound proposition w.r.t. the number of simple propositions semantic methods, such as the truth table method, have little practical value. An alternative to the semantic method are the syntactic methods, which manipulate only the symbolic representations of propositions. Inference mechanical derivation possibility to check logical derivability by symbolic computation. The usage of inference methods allow us to construct a computation machine

25 Inference Definition Introduction to Prolog By inference we understand the mechanical derivation of conclusions from a set of premises. Definition (Inference rule) An inference rule inf is a procedure capable to derive conclusions from a set of premises. We write inf φ if φ is a conclusion derived by the inference rule inf from the set of premises.

26 Inference rules Parameterised templates for reasoning General form: premise 1... premise n conclusion Example α β Modus Ponens (MP): β α β β Modus Tollens: α α

27 Desirable properties of inference rules An inference rule inf is sound if = φ whenever inf φ This means that all conclusions determined by inf from are logical consequences of the premises from. complete if inf φ whenever = φ This means that all logical consequences from can be obtained as conclusions of inf from.

28 A sound and consistent inference rule = very powerful inference rule Can be used to build a sound an complete theorem prover The search space for proofs is smaller than in several other proof systems is designed to work with propositions in clausal form: proposition = set of clausal forms (interpreted as a conjunction) clause = set of literals (interpreted as a disjunction) literal = atom or negated atom atom = simple proposition

29 Clausal forms Definitions Introduction to Prolog Definition (Literal) p or p where p is a simple proposition. Definition (Clausal expression) Literal or disjunction of literals. E.g., p q r. Definition (Clause) Set of literals from a clausal expressions. E.g., {p, q, r} Definition (Clausal form - CNF) The representation of a proposition as a set of clauses, implicitly connected by conjunction.

30 The clausal form (CNF) Example The CNF of the proposition p ( q r) ( p r) is {{p}, { q, r}, { p, r}}.

31 Bringing a proposition to CNF Every proposition can be brought to a CNF: 1 Eliminate implication: (α β) α β 2 Push all negations in parentheses, until they reach the atomic propositions: (α β) α β, (α β) α β, ( α) α 3 Apply the distributivity rules: α (β γ) (α β) (α γ) α (β γ) (α β) (α γ) 4 Transform the expressions into clauses: φ 1... φ n {φ 1,..., φ n } φ 1... φ n {φ 1 },..., {φ n }

32 Bringing a proposition to CNF Example 1: Let s bring p (q r) to CNF: 1 Eliminate implications p ( q r) 2 Form clauses {{p}, { q, r}}

33 Bringing a proposition to CNF Example 1: Let s bring p (q r) to CNF: 1 Eliminate implications p ( q r) 2 Form clauses {{p}, { q, r}} Example 2: Let s bring (p (q r)) to CNF: 1 Eliminate implications (p ( q r)) 2 Push negations inside p (q r) 3 Apply distributivity rules ( p q) ( p r) 4 Form clauses {{ p, q}, { p, r}}

34 Main reasoning principle: resolution step {p q} { p r} Ideea: {q, r} p can be cancelled because: p false p true r true p true q true p p is always true at least one of q and r is true (q r) is true. The general form of a resolution step is: {p 1,..., r,..., p m } {q 1,..., r,..., q n } {p 1,..., p m, q 1,..., q n }

35 Special cases Introduction to Prolog The empty clause signals a contradiction: { p} {p} {} = If more than one resolvent is possible, we should choose only one: {p, q} { p, q} {p, q} { p, q} or {p, p} {q, q}

36 Other inference rules special cases of resolution Modus ponens p q q p { p, q} {p} {q} Modus tollens p q q p Transitivity of implication p q q r p r { p, q} { q} { p} { p, q} { q, r} { p, r}

37 proofs Introduction to Prolog Proof un unsatisfiability derivability of the empty clause. Derivability of φ from premises φ 1,..., φ n φ 1... φ n φ is unsatisfiable Proof of validity of φ proof of unsatisfiability of φ...

38 as a proof algorithm 1 Assume there are given premises φ 1,..., φ n and conclusion φ 2 Transform φ 1,..., φ n, φ and φ in CNF set of clauses φ 1,..., φ n, φ 3 Proceed by choosing two clauses to which the resolution step can apply If the result is the empty clause, then SUCCESS. otherwise goto step 3.

39 Using resolution as a proof algorithm Example Let s prove that {p q, q r} p r by showing that the set {p q, q r, (p r)} contains a contradiction. 1. { p, q} premise 2. { q, r} premise 3. {p} negated conclusion (assumed to be true) 4. { r} negated conclusion (assumed to be true) 5. {q} resolution 1,3 6. {r} resolution 2,5 7. {} resolution 4,6. empty clause, thus SUCCESS.

40 Soundness and completeness Introduction to Prolog Theorem The inference rule of resolution is sound and complete: = φ if and only if rez φ The running of the resolution algorithm is guaranteed to terminate, because A finite number of clauses implies a finite number of conclusions that can be drawn from them.

41 Summary Introduction to Prolog This lecture was about Foundations of propositional logic Syntax and semantics Inference rules, resolution, normal forms

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