Lecture 2: First Order Logic

COMS E6998 Formal Verification of System Software
Fall 2018
Ronghui Gu

1. What is First Order Logic

1.1 v.s. Propositional Logic

• “Eric is younger than Ronghui”.
• How to present “Every student is younger than Ronghui”?
• The only way is to write this as an atomic fact.
• We need a richer logic to reason about “every”, “younger”, etc.

1.2 Introduction of First Order Logic

First Order Logic := Propositional Logic + :

• Terms: syntax only, e.g., “Eric”, “Ronghui”, etc.
• Predicates: property of terms:
  • $S$(Eric): Eric is a student
  • $Y$(Eric, Ronghui): Eric is younger than Ronghui
• How to talk about every student？
  • FOL uses variables and universal quantification $\forall$
  • $\forall x. S(x)$: “all terms are students”
  • $\forall x. S(x) \rightarrow$: “forl all terms who are students, …”
• How to talk about the existence of at least one Instructor
  • FOL uses variables and existential quantification $\exists$
  • $\exists x. I(x) \wedge Y(\text{Eric}, x)$: “there exists a term which is an Instructor and Eric is younger than this instructor”.

Example: Not all birds can fly

• $\neg\forall x.(B(x)\rightarrow F(x))$
• $\exists x.(B(x)\wedge \neg F(x))$

1.3 Syntax of FOL

Terms are strings:

• variable: x, y, z, …
• constant: c, nullary function
  • e.g., Eric, Ronghui
  • we pick constants from a function set $\mathcal{F}$
• application of function $f$ to terms $t_1, ..., t_n$:
  • e.g., S(Eric), I(Ronghui)
  • e.g., natural numbers: O, S(O), S(S(O))
  • we pick functions from the same function set $\mathcal{F}$

Formulas are strings:

• application of predicate $P$ to terms $t_1, ..., t_n$:
  • e.g., $Y(\text{Eric},x)$
  • we pick predicates from a set $\mathcal{P}$

Two parameters to FOL: $\mathcal{F}$ and $\mathcal{P}$

• natural numbers: $\mathcal{F}:=\{O, S, ...\}$, large enough $\mathcal{P}$
• propositional logic: $\mathcal{F}:=\emptyset$ and $\mathcal{P}:=\{p,q,r,...\}$

1.4 Substitution

Definition

• $\forall x. A / \exists x.A$ binds the variable $x$ in $A$
  • the scope of $x$ is $A$
  • $x$ is bound in $\forall x. A / \exists x.A$
  • $x$ is free in $A$
• any variable $x$ which appears in a formula $A$ and is not bound in $A$ is called free in $A$

Q: find the free and bound variables:

• $\forall x. (P(x) \rightarrow Q(y))$

Barendreght convention: To avoid confusion, every bound variable will be distinct from any other bound variable and all the free variables.

Substitution: replace free variables with actual terms.

Definition $A[t/x]$ is defined to be the formula we get by replacing all free occurrences of $x$ in $A$ with $t$. Q A:= $\forall x. (P(x) \rightarrow Q(y))$, what is $A[\text{Eric}/y]$?

2 FOL Syntax Reasoning (Inference Rules)

$A_1 ... A_n ⊢ B$

2.1 Propositional rules

All propositional logic rules are rules of FO

2.2 Equality rules

Only consider “=” $\in \mathcal{P}$

Q: prove the following rules:

Proof of (=sym)
 
1. t1 = t2                    (Pre)
2. t1 = t1 := (x = t1)[t1/x]  (ref)
3. t2 = t1 := (x = t1)[t2/x]  (=e, 1, 2)
 
Qed

Q: Assume $\mathcal{F}:=\{O, S, +\}$ and $\mathcal{P}:=\{=, <,>,\leq, \geq,...\}$, prove

Proof of
 
1. t1 = t2                                    (Pre)
2. t + t1 = t + t1 := (t + x = t + t1)[t1/x]  (ref)
3. t + t2 = t + t1 := (t + x = t + t1)[t2/x]  (=e, 1, 2)
 
Qed

Q: Prove $(x+1=1+x), [(x+1)>1 \rightarrow x>0]⊢(1+x)>1 \rightarrow x>0$

2.3 Quantification rules

Universal quantification elimination rule:

• Can only replace with free variables: $A:= \forall x. \exists y. x < y$, what is $A[y/x]$?
• Q: prove $P(t), (\forall x. P(x)\rightarrow Q(x)) ⊢ Q(t)$

Proof of
 
1. P(t)               (Pre)
2. ∀x. P(x) → Q(x)    (Pre)
3. P(t) → Q(t)        (∀e, 2)
3. Q(t)               (→e, 1, 3)
 
Qed

Universal quantification introduction rule:

• $x_0$ is a dummy variable that represents an arbitrary term
• Thus, to prove $\forall x.A$, we need to prove $A[x_0/x]$ for an arbitrary term $x_0$
• Q: prove $[\forall x.P(x)], [\forall x. P(x)\rightarrow Q(x)] ⊢ \forall x. Q(x)$

Existential introduction rule:

• pick an convenient $t$ and prove $A[t/x]$
• Q: prove $⊢ \exists x. x = 0$

Existential elimination rule:

• $x_0$ represents unknown term
• Q: prove $\forall y. \exists x.f(x) = g(y) ⊢ \exists x. g(5) = f(x)$

Proof
1. ∀y. ∃x. f(x) = g(y)  (Pre)
2. ∃x. f(x) = g(5)      (∀e, 1)
   -----------------------------------
3. | x0                 (Hyp)        |
4. | f(x0) = g(5)       (Hyp)        |
5. | g(5) = f(x0)       (=sym, 4)    |
6. | ∃x. g(5) = f(x)    (∃i, 5)      |
   -----------------------------------
7. ∃x. g(5) = f(x)      (∃e, 2, 3~6)
 
Qed

Q: Assume $\mathcal{F}:=\{O, S\}$ and $\mathcal{P}:=\{=, <,>,\leq, \geq,...\}$, and assume

express and prove in FOL over $\mathcal{F}$ and $\mathcal{P}$:
“Any natural number is smaller than some number”

Proof
   ------------------------------
1. | x0            (hypothesis) |
2. | x0 < S x0     (<S, 1)      |
3. | ∃y. x0 < y    (∃i, 2)      |
   ------------------------------
4. ∀x. ∃y. x < y   (∀i, 1~3)
 
Qed

Q: Prove

Proof
1. ∀x. Q(x) → R(x)    (Pre)
2. ∃y. P(y) ∧ Q(y)    (Pre)
   ---------------------------------
3. | x0               (Hyp)        |
4. | P(x0) ∧ Q(x0)    (Hyp)        |
5. | P(x0)            (∧e1, 4)     |
6. | Q(x0) → R(x0)    (∀e, 1, 3)   |
7. | Q(x0)            (∧e2, 4)     |
8. | R(x0)            (→e, 6, 7)   |
9. | P(x0) ∧ R(x0)    (∧i, 5, 8)   |
10.| ∃x. P(x) ∧ R(x)  (∃i, 9)      |
   ---------------------------------
11. ∃x. P(x) ∧ R(x)   (∃e, 2. 3~10)
 
Qed

Q: $¬∀x.A ⊣⊢ ∃x.¬A$
Q: $¬∃x.A ⊣⊢ ∀x.¬A$

3. Classical vs. Intuitionistic logic

Consider the formula

To prove this we need use rule $∃i$ and provide a term $t$ for which we can show

Classically this is a tautology:

• If $∀y.P(y)$ holds then we can pick arbitrarily term $t_0$ and show

using rule $→ i$ (also see truth table of implication).

• If $¬∀y.P(y)$ then there exists a term $t_1$ such that $¬P(t_1)$, which we can use to show

$\quad \Box$

However notice how we didn’t have to provide any concrete term $t_1$ in the second case to complete the proof of $∃x.(P(x) → ∀y.P(y))$ Let’s see this again:

• If $¬∀y.P(y)$ then there exists a term $t_1$ such that $¬P(t_1)$, which we can use to show $P(t1) → ∀y.P(y)$

  • Assume logic’s terms $\mathcal{F}$ contain natural numbers and predicates $\mathcal{P}$ standard predicates over them.
  • if $P(x)$ = even(x) then $t_1$ can be 3
  • if $P(x)$ = $x \leq 1000$ then $t_1$ can be 1001
  • …

• We can’t provide a $t_1$ if we don’t know $P$ and the parameters $\mathcal{F}$, $\mathcal{P}$, but there is always one.

• Classical logic is OK with that!

• Intuitionistic logic is not! It requires us to give a concrete $t_1$, before we know what $P$ is.

Theorem There are irrational $a$ and $b$ for which $a^b$ is rational.
Proof We know that 2 is irrational (known theorem of arithmetic which we will not prove here).
Suppose $\sqrt{2}^\sqrt{2}$ is rational. Then pick $a = b = \sqrt{2}$ and we’re done.
Otherwise $\sqrt{2}^\sqrt{2}$ is irrational. Pick $a = 2$ and $b = \sqrt{2}^\sqrt{2}$.

which is rational. $\quad \Box$

Correct but we never identified two definitely irrational numbers a and b. In fact knowing whether $\sqrt{2}^\sqrt{2}$ is rational is a difficult problem!

Riemann hypothesis Michael Atiyah, Proof by Contradiction

4. Semantics of FOL

Let $Γ$ be a list of formulas $φ_1,...,φ_n$

• $Γ ⊢ ψ$ means that there is a proof of $ψ$ from premises $Γ$ using the natural deduction rules.
• In propositional logic we also defined $Γ \models ψ$ to mean that any model (valuation) which makes $Γ$ true, also makes $ψ$ true.
  • we used finite truth-tables as the semantics of formulas
• In propositional logic we showed soundness and completeness:
  • $Γ⊢ψ \texttt{ iff } Γ\models ψ$
• How can we define a semantic entailment $Γ \models ψ$ in FOL?
  • What is the semantics of formulas?
  • What sort of models can we consider for quantifiers $∀x.φ$ and $∃x.φ$?

4.1 Usefulness of semantics

Syntactic entailment is useful to show existence of proofs.
Semantic entailment is useful to show absence of proofs. How can we show that $Γ \not ⊢ ψ$ ?

• we need to consider all possible (infinite) proofs

How can we show that $Γ \not\modelsψ$?

• we need to find one model that makes $Γ$ true and $ψ$ false.

4.2 Model

What should be the semantics of $∀x.P(x, y)$?
It depends on the semantics of the parameters of FOL: the set of terms $\mathcal{F}$ and predicates $\mathcal{P}$.

Definition A model $\mathcal{M}$ of $(\mathcal{F} , \mathcal{P})$ consists of the following:

• A non-empty set $\mathcal{V}$: the universe of concrete values.
• for each nullary function $c ∈ \mathcal{F}$, a concrete element $c^\mathcal{M} ∈ \mathcal{V}$
• for each $f ∈ \mathcal{F}$ with arity $n > 0$, a concrete mathematical function $f^\mathcal{M} : \mathcal{V}^n → \mathcal{V}$, taking $n$-tuple of $\mathcal{V}$-values to $\mathcal{V}$-values
• for each $P∈\mathcal{P}$ with arity $n>0$, a concrete mathematical function $P^M: \mathcal{V}^n \rightarrow \{T, F\}$, taking $n$-tuple of $\mathcal{V}$-values to truth values.

Example Model Natural numbers:

A model $\mathcal{M}$ may be:

1. $\mathcal{V}=\{0,1,2,\cdots\}$
2. $O^\mathcal{M} := 0$
3. $S^\mathcal{M} := fun(x) \Rightarrow (x+1)$
4. $<^\mathcal{M} := fun(x,y) \Rightarrow \text{if } x < y \text{ then T else F}$

Q: What $\mathcal{M}$ is like if $A=\{"@", "@@", "@@@",\cdots\}$

4.3 Semantics of formulas

We will give semantics to closed formulas (no free variables) using the semantics of open formulas.

The semantics of $∀x.φ$ means that for all values $v ∈ \mathcal{V}, φ[v/x]$ is true. However it’s not a valid syntax to have formulas containing semantic values from $v$. We need to use environments.

Definition
$\mathcal{E}$ is an environment if it is a function that maps syntactic variables to semantic values. (lookup tables)

Definition Given a model $\mathcal{M}$ for a pair $(\mathcal{F}, \mathcal{P})$ and given an environment $\mathcal{E}$, we define the satisfaction relation $\mathcal{M} \models_\mathcal{E} φ$ for each logical formula $φ$ over the pair $(\mathcal{F}, \mathcal{P})$ and $\mathcal{E}$ as follows.

• $\mathcal{M} \models_\mathcal{E} P(t_1,\cdots,t_n)$ holds if $P^\mathcal{M}(a_1,\cdots,a_n) = T$, where $a_1,\cdots,a_n$ are the $\mathcal{V}$-value of $t_1,\cdots,t_n$ under $\mathcal{E}$.

• $\mathcal{M} \models_\mathcal{E} φ_1 \wedge φ_2$ holds if $\mathcal{M} \models_\mathcal{E} φ_1$ holds and $\mathcal{M} \models_\mathcal{E} φ_2$ holds

• $\mathcal{M} \models_\mathcal{E} φ_1 \vee φ_2$ holds if $\mathcal{M} \models_\mathcal{E} φ_1$ or $\mathcal{M} \models_\mathcal{E} φ_2$ holds

• $\mathcal{M} \models_\mathcal{E} \neg φ$ holds if $\mathcal{M} \models_\mathcal{E} \neg φ$ does not hold

• $\mathcal{M} \models_\mathcal{E} φ_1 \rightarrow φ_2$ holds if $\mathcal{M} \models_\mathcal{E} φ_2$ holds whenever $\mathcal{M} \models_\mathcal{E} φ_1$ holds

• $\mathcal{M} \models_\mathcal{E} \forall x. φ$ holds if $\mathcal{M} \models_{\mathcal{E}\cup\{x \mapsto v\} } φ$ holds for all $v \in \mathcal{V}$

• $\mathcal{M} \models_\mathcal{E} \exists x. φ$ holds if $\mathcal{M} \models_{\mathcal{E}\cup\{x \mapsto v\} } φ$ holds for some $v \in \mathcal{V}$

5. Results of FOL

Theorem (Soundness)
For a given $(\mathcal{F} , \mathcal{P})$, if $⊢ φ$ then $\models φ$ which means for any model $\mathcal{M}$ of $(\mathcal{F} , \mathcal{P})$ and any environment $\mathcal{E}$, we have $\mathcal{M} \models_\mathcal{E} φ$.

Theorem (Strong soundness)
For a given $(\mathcal{F} , \mathcal{P})$, if $Γ ⊢ ψ$ then $Γ \models φ$ which means for any model $\mathcal{M}$ of $(\mathcal{F} , \mathcal{P})$ and any environment $\mathcal{E}$, if $\mathcal{M} \models_\mathcal{E} Γ$ then $\mathcal{M} \models_\mathcal{E} φ$.

This involves properties that are true in all models. How can we talk about properties of certain models (e.g., numbers with some standard predicates over them)?
A: Encode the necessary properties of these models in $Γ$. $Γ$ can contain the axioms we want to hold in these models.

• famous axiomatisation of natural numbers: Peano axioms

Peano Axioms
Terms: $\mathcal{F}=\{O, S\}$ Axioms:

• refl, sym, trans of equality

• $O$ is a natural number

• if $n$ is a natural number, $S(n)$ is a natural number

• $∀x.¬(S(x) = O)$

• $∀x.∀y.S(x) = S(y) \leftrightarrow x = y$

• If $K$ is a set such that

  • $O \in K$
  • for every natural number $n$, $n \in K \rightarrow S(n)\in K$

  then $K$ contains every natural number

The Peano axioms can be augmented with other operations:

• Addition
  • $a+O =a$
  • $a + S(b) = S (a + b)$
• Multiplication
  • $a \times O = O$
  • $a \times S(b) = a + (a \times b)$

Gödel’s 1st incompleteness theorem

Theorem (Incompleteness) Any set of axioms $Γ$ which is consistent (no contradictions such as 0 = 1 are derivable) and contains “enough arithmetic” cannot be complete. That is, there are true facts φ about arithmetic for which $Γ\not⊢ φ$.

Proof. Göedel gave a way to encode first-order logic itself in any axiomatisation $Γ$ containing Peano (or any other encoding of) natural numbers.
Hence for any such system he was able to write an encoding of the formula:

If $Γ ⊢ φ$ then obviously the logic is inconsistent ($Γ ⊢ φ ∧ ¬φ$).
If $Γ\not⊢ φ$ then obviously the logic is incomplete ($φ$ is true but not provable).

Gödel’s completeness theorem

Theorem (Completeness)
$\models φ$ then $⊢ φ$.

• This means that anything that is true for all models, can be syntactically proven. Examples:
  • Yes: $¬∀x.φ(x) → ∃x.¬φ(x)$
  • No: 1 + 1 = 2
  • No: The Goldbach conjecture: “Every even integer greater than 2 can be expressed as the sum of two primes”