Predicate Logic: Syntax Alice Gao Lecture 12 CS 245 Logic and - - PowerPoint PPT Presentation

Predicate Logic: Syntax

Alice Gao

Lecture 12

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Outline

Learning goals Symbols Terms Formulas Parse Trees Revisiting the learning goals

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Learning goals

By the end of this lecture, you should be able to

▶ Defjne the set of terms inductively. ▶ Defjne the set of formulas inductively. ▶ Determine whether a variable in a formula is free or bound. ▶ Prove properties of terms and formulas by structural

induction.

▶ Draw the parse tree of a formula.

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The Language of Predicate Logic

▶ Domain: a non-empty set of objects. ▶ Individuals: concrete objects in the domain. ▶ Variables: placeholders for concrete objects in the domain. ▶ Functions: takes objects in the domain as arguments and

returns an object of the domain.

▶ Relations: takes objects in the domain as arguments and

returns true or false. They describe properties of objects or relationships between objects.

▶ Quantifjers: for how many objects in the domain is the

statement true?

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Outline

Learning goals Symbols Terms Formulas Parse Trees Revisiting the learning goals

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Predicate Language 𝑀

Eight classes of symbols:

▶ Individual symbols: 𝑏, 𝑐, 𝑑. ▶ Relation symbols: 𝐺, 𝐻, 𝐼.

A special equality symbol ≈

▶ Function symbols: 𝑔, 𝑕, ℎ. ▶ Free variable symbols: 𝑣, 𝑤, 𝑥. ▶ Bound variable symbols: 𝑦, 𝑧, 𝑨. ▶ Connective symbols: ¬, ∧, ∨, →, ↔. ▶ Quantifjer symbol: ∀, ∃. ▶ Punctuation symbols: ‘(’, ‘)’, and ‘,’

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Free and Bound Variables

In a formula ∀𝑦 𝐵(𝑦) or ∃𝑦 𝐵(𝑦), the scope of a quantifjer is the formula 𝐵(𝑦). A quantifjer binds its variable within its scope. An occurrence of a variable in a formula

▶ is bound if it lies in the scope of some quantifjer of the same

variable.

▶ is free, otherwise.

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Outline

Learning goals Symbols Terms Formulas Parse Trees Revisiting the learning goals

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Two Kinds of Expressions

Two kinds of expressions:

▶ A term refers to an object in the domain. ▶ A formula evaluates to 1 or 0.

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Terms

The set of terms Term(𝑀) is defjned below:

• 1. An individual symbol 𝑏 standing alone is a term.
• 2. A free variable symbol 𝑣 standing alone is a term.
• 3. If 𝑢1, … , 𝑢𝑜 are terms and 𝑔 is an 𝑜-ary function symbol,

then 𝑔(𝑢1, … , 𝑢𝑜) is a term.

• 4. Nothing else is a term.

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Examples of Terms

Terms:

▶ 𝑏, 𝑐, 𝑑, 𝑣, 𝑤, 𝑥 ▶ 𝑔(𝑐), 𝑕(𝑏, 𝑔(𝑐)), 𝑕(𝑣, 𝑐), 𝑔(𝑕(𝑔(𝑣), 𝑐))

A term with no free variable symbols is called a closed term. Which one(s) of the above are closed terms?

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CQ: Which expressions are terms?

Which of the following expressions is a term? If there are multiple correct answers, choose your favourite one. (A) 𝑥 (B) 𝑕(𝑏, 𝑣) (C) 𝐺(𝑔(𝑣, 𝑤), 𝑏) (D) 𝑔(𝑣, 𝑕(𝑤, 𝑥), 𝑏) (E) 𝑕(𝑣, 𝑔(𝑤, 𝑥), 𝑏) Individual symbols: 𝑏 Relation symbols: 𝐺 is a binary relation symbol. Function symbols: 𝑔 is a binary function symbol and 𝑕 is a 3-ary function symbol. Free variable symbols: 𝑣, 𝑤, 𝑥.

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Defjning the set of terms inductively

The set of terms can be inductively defjned as follows:

▶ The domain set 𝑌:

The set of fjnite sequences of symbols of 𝑀

▶ The core set 𝐷:

The set of all individual symbols and free variable symbols

▶ The set of operations 𝑄:

The set of all function symbols

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Structural induction on terms

Theorem: Every term has a property 𝑄. Proof by structural induction:

▶ Base cases:

The term is an individual symbol. The term is a free variable symbol.

▶ Inductive cases:

The term is 𝑔(𝑢1, … , 𝑢𝑜) where 𝑔 is an n-ary function and 𝑢1, ..., 𝑢𝑜 are terms. Induction hypotheses: Assume that 𝑢1, … , 𝑢𝑜 all have the property 𝑄. We need to show that 𝑔(𝑢1, … , 𝑢𝑜) has the property 𝑄.

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Outline

Learning goals Symbols Terms Formulas Parse Trees Revisiting the learning goals

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Atomic Formulas

The set of atomic formulas Atom(𝑀) is defjned below:

▶ If 𝐺 is an n-ary relation symbol and 𝑢1, … , 𝑢𝑜 (𝑜 ≥ 1) are

terms, then 𝐺(𝑢1, … , 𝑢𝑜) is an atomic formula.

▶ If 𝑢1, 𝑢2 are terms, then ≈ (𝑢1, 𝑢2) is an atomic formula. ▶ Nothing else is an atomic formula.

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Examples of Atomic Formulas

Terms:

▶ 𝑏, 𝑐, 𝑑, 𝑣, 𝑤, 𝑥 ▶ 𝑔(𝑐), 𝑕(𝑏, 𝑔(𝑐)), 𝑕(𝑣, 𝑐), 𝑔(𝑕(𝑔(𝑣), 𝑐))

Atomic formulas:

▶ 𝐺(𝑏, 𝑣, 𝑔(𝑐), 𝑔(𝑥), 𝑕(𝑤, 𝑔(𝑏))) ▶ ≈ (𝑐, 𝑥)

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Well-Formed Formulas

The set of well-formed formulas Form(𝑀) is defjned below:

• 1. An atomic formula is a well-formed formula.
• 2. If 𝐵 is a well-formed formula, then (¬𝐵) is a well-formed

formula.

• 3. If 𝐵 and 𝐶 are well-formed formulas and ⋆ is one of ∧, ∨, →,

and ↔, then (𝐵 ⋆ 𝐶) is a well-formed formula.

• 4. If 𝐵(𝑣) is a well-formed formula and 𝑦 does not occur in

𝐵(𝑣), then ∀𝑦 𝐵(𝑦) and ∃𝑦 𝐵(𝑦) are well-formed formulas.

• 5. Nothing else is a well-formed formula.

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Explaining Case 4 of Formulas

If 𝐵(𝑣) is a well-formed formula and 𝑦 does not occur in 𝐵(𝑣), then ∀𝑦 𝐵(𝑦) and ∃𝑦 𝐵(𝑦) are well-formed formulas.

▶ 𝐵(𝑣) is a well-formed formula where 𝑣 is a free variable in the

• formula. We want to quantify 𝑣.

▶ In order to do so, we need to choose a symbol for a bound

variable, e.g. 𝑦. We need to make sure that our choice of the bound variable symbol does not already occur in 𝐵(𝑣).

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Examples for Case 4

▶ We are allowed to generate the formula ∀𝑧𝐺(𝑧, 𝑧).

Start with 𝐺(𝑣, 𝑣). If we quantify 𝑣 by replacing it with 𝑧, we get ∀𝑧𝐺(𝑧, 𝑧).

▶ We are not allowed to generate the formula ∃𝑧∀𝑧𝐺(𝑧, 𝑧).

Start with ∀𝑧𝐺(𝑧, 𝑧). If we want to add the ∃ quantifjer, we will need to choose a bound variable symbol that is not 𝑧 because 𝑧 already appears in ∀𝑧 𝐺(𝑧, 𝑧). So, there is no way for us to generate ∃𝑧∀𝑧𝐺(𝑧, 𝑧).

▶ We are allowed to generate the formula ∃𝑦𝐻(𝑦) ∨ ∀𝑦𝐼(𝑦).

Start with 𝐻(𝑣) and 𝐼(𝑤) separately. We can quantify 𝑣 by replacing it with 𝑦 since 𝑦 does not appear in 𝐻(𝑣)). We get ∃𝑦𝐻(𝑦). We can quantify 𝑤 by replacing it with 𝑦 since 𝑦 does not appear in 𝐼(𝑤). We get ∀𝑦𝐼(𝑦). Connecting the two formulas using ∨, we get ∃𝑦𝐻(𝑦) ∨ ∀𝑦𝐼(𝑦).

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Examples of Formulas

Well-Formed Formulas:

▶ 𝐺(𝑏, 𝑐), ∀𝑧 𝐺(𝑏, 𝑧), ∃𝑦∀𝑧 𝐺(𝑦, 𝑧) ▶ 𝐺(𝑣, 𝑤), ∃𝑧 𝐺(𝑣, 𝑧)

A formula with no free variable symbols is called a closed formula

• r a sentence.

Which formulas above are closed formulas?

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Determine whether a formula is well-formed

Which of the following is a well-formed formula? (A) 𝑔(𝑣) → 𝐺(𝑣, 𝑤) (B) ∀𝑦 𝐺(𝑛, 𝑔(𝑦)) (C) 𝐺(𝑣, 𝑤) → 𝐻(𝐻(𝑣)) (D) 𝐻(𝑛, 𝑔(𝑛)) (E) 𝐺(𝑛, 𝑔(𝐻(𝑣, 𝑤))) Individual symbols: 𝑛. Free Variable Symbols: 𝑣, 𝑤. Bound Variable symbols: 𝑦. Relation symbols: 𝐺 and 𝐻 are binary relation symbols. Function symbols: 𝑔 is a unary function.

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Defjning the set of formulas inductively

The set of formulas can be inductively defjned as follows:

▶ The domain set 𝑌:

The set of fjnite sequences of symbols of 𝑀

▶ The core set 𝐷:

The set of all atomic formulas.

▶ The set of operations 𝑄:

𝑔1(𝑦) = (¬𝑦) 𝑔2(𝑦, 𝑧) = (𝑦 ∗ 𝑧) where ∗ is one of ∧, ∨, →, and ↔. 𝑔3(𝐵(𝑣)) = ∀𝑦 𝐵(𝑦), 𝑔4(𝐵(𝑣)) = ∃𝑦 𝐵(𝑦) where 𝑦 does not

• ccur in 𝐵(𝑣).

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Structural induction on formulas

Theorem: Every formula has a property 𝑄. Proof by structural induction:

▶ Base cases:

The formula is an atomic formula.

▶ Inductive cases:

The formula is (¬𝐵) where 𝐵 is a formula. The formula is (𝐵 ∗ 𝐶) where 𝐵 and 𝐶 are formulas and ∗ is a binary connective. The formula is ∀𝑦 𝐵(𝑦) and ∃𝑦 𝐵(𝑦) where 𝐵(𝑣) is a formula and 𝑦 does not occur in 𝐵(𝑣).

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Comparing the Defjnitions of Well-Formed Formulas

Let’s compare the set of predicate formulas to the set of propositional formulas. Questions to think about:

▶ Which parts of the two defjnitions are the same?

The cases for negation and binary connectives are the same.

▶ Which parts of the two defjnitions are difgerent?

Atomic formulas are difgerent.

▶ Atomic propositional formulas are propositional variables. ▶ Atomic predicate formulas are relations applied to terms.

Predicate formulas have one additional case for quantifjers.

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Outline

Learning goals Symbols Terms Formulas Parse Trees Revisiting the learning goals

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Parse Trees of Predicate Formulas

▶ The leaves are atomic formulas. ▶ Every quantifjer has exactly one child

(namely the formula which is its scope). Example: ∀𝑦(𝐺(𝑐) → ∃𝑧(∀𝑨 𝐻(𝑧, 𝑨) ∨ 𝐼(𝑔(𝑣), 𝑦, 𝑧)))

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Parse tree

Example: ∀𝑦(𝐺(𝑐) → ∃𝑧(∀𝑨 𝐻(𝑧, 𝑨) ∨ 𝐼(𝑔(𝑣), 𝑦, 𝑧))) Parse tree: ∀𝑦(𝐺(𝑐) → ∃𝑧(∀𝑨 𝐻(𝑧, 𝑨) ∨ 𝐼(𝑔(𝑣), 𝑦, 𝑧))) 𝐺(𝑐) → ∃𝑧(∀𝑨 𝐻(𝑧, 𝑨) ∨ 𝐼(𝑔(𝑣), 𝑤, 𝑧)) 𝐺(𝑐) ∃𝑧(∀𝑨 𝐻(𝑧, 𝑨) ∨ 𝐼(𝑔(𝑣), 𝑤, 𝑧)) ∀𝑨 𝐻(𝑥, 𝑨) ∨ 𝐼(𝑔(𝑣), 𝑤, 𝑥) ∀𝑨 𝐻(𝑥, 𝑨) 𝐻(𝑥, 𝑣1) 𝐼(𝑔(𝑣), 𝑤, 𝑥)

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A few notes on parse trees

• 1. While constructing the parse tree, when removing a quantifjer,

we change the bound variable symbol to one of the free variable symbols that hasn’t appeared in the parse tree. For example, When removing ∀𝑦, we changed 𝑦 to 𝑤. When removing ∃𝑧, we changed 𝑧 to 𝑥.

• 2. Th quantifjers have higher precedence than any other
• connective. Each quantifjer modifjes the formula that is

immediately after it. For example, ∀𝑨 modifjes 𝐻(𝑥, 𝑨) instead of 𝐻(𝑥, 𝑨) ∨ 𝐼(𝑔(𝑣), 𝑤, 𝑥).

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Revisiting the learning goals

By the end of this lecture, you should be able to

▶ Defjne the set of terms inductively. ▶ Defjne the set of formulas inductively. ▶ Determine whether a variable in a formula is free or bound. ▶ Prove properties of terms and formulas by structural

induction.

▶ Draw the parse tree of a formula.

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