Programming Language Theory 101: Inference Rules and Extensions

Inference rules are fundamental concepts in formal logic & reasoning, widely used in mathematics, computer science & especially in programming language theory.

They provide a structure way of:

• To represent logic deductions
• Define a behavior of programming construct.

Key points:

1. Structure: They consist of

• premises (above a line)
• conclusion (below the line)

Premise1  Premise2  ...  PremiseN
———————————————————————————————————
         Conclusion

2. Interpretation:

if all premises are true, the conclusion must be true.

3. Usages in programming language:

• Define how expressions are evaluated
• Specify type checking rules.
• Formalize language semantics.
• They allow rigorous reasoning about program behavior.

Understanding inference rules is crucial for:

• Formal verification of programs.
• Designing type systems.
• Providing properties like type safety or termination.

Examples:

Modus Ponens, a classic logical inference rule: 

P    P → Q
———————————
    Q

P is true, if P implies Q, Q is true

Typing rule for function application (in programming language theory): 

Γ ⊢ e1 : T1 → T2    Γ ⊢ e2 : T1
——————————————————————————————————
      Γ ⊢ (e1 e2) : T2

e1 has a type T1 → T2

e2 has a type T1

then the application (e1 e2) has a type T2

T1 → T2 is a single type, not two separate types. It's the type of a function

T1 → T2 is read as a function from T1 to T2

For example:

• If T1 is Int and T2 is String, then T1 → T2 would be Int → String
• This would be the type of a function that takes an Int as input and returns a String

So, e1 having type T1 → T2 means:

• e1 is a function
• It takes an argument of type T1
• It returns a result of type T2

Γ ⊢ e1 : T1 → T2    Γ ⊢ e2 : T1
——————————————————————————————————
      Γ ⊢ (e1 e2) : T2

This rules states that if we have a function e1 of type (t1 -> t2), and an expression e2 of type t1, then applying e1 to e2 results in the expression of type t2.
plus 0 nv2 = nv2
plus (succ nv1) nv2 = succ (plus nv1 nv2)

Base case:

---------------
plus 0 nv2 = nv2

this rules has no premises, so there is nothing above the line. It states that adding 0 to any number gives that number. 

Recursive case: 

plus nv1 nv2 = result 
----------------------------------
plus (succ nv1) nv2 = succ result

t is defined as :

1. true - A boolean true value
2. false - A boolean false value
3. if3 t t t - An if-then-else construct that takes three t expressions
4. 0 - The number zero
5. succ t - The successor of a number (adding 1)
6. pred t - The predecessor of a number (substracting 1)
7. iszero t - A check if number if zero
8. bad - A new construct representing a bad state or error
9. try t expect t - A new error-handling construct.

This grammar defines the syntax for a simple language that can express boolean & arithmetic operations along with some error handling capabilities.

The bad & try except constructs are extensions to a basic arithmetic language to explore language design and semantics.

Each of these constructs would typically have associated

1. Operational semantics(rules for how they behave)
2. Typing rules (rules for type-checking)

The grammar defines a tiny language that can do basic math & logic, plus handle errors.

The "bad" is like an error state. The "try-except" is like error handling in regular programming languages

For a normal software developer, understanding this is useful because:

1. It shows how programming language works under the hood. Even though you might not build a language, knowing how they work can make you a better programmer.
2. Error handling is crucial in real-world programming. This simple model shows the core idea behind try-catch block you use in language like Java, Python or JS.
3. Introduces formal way of thinking code behaviour. This kind of thinking helps when designing robust systems or debugging complex issues.
4. Modern programming concepts, like functional programming or strong typing have roots in these kinds of formal language models.

These concepts are part of theoretical foundations of programming languages. They are used to rigorously define how languages work, prove properties about them & ensure they behave consistently.

While they are not typically used in everyday coding, they inform the design of languages, compilers & advanced development tools.

1. Formal Language Grammar: Precise & mathematical way of defining a programming language's syntax.
2. Operational semantics: Rules that formally describe how programs in a language execute.
3. Type Systems: Formal rules for categorizing & checking the types of values in a program.
4. Lambda calculus: A minimal foundational model of computation that underlies functional programing.
5. Error Handling Mechanism: Formal ways of representing & dealing with error states in a language.
6. Reduction Strategies: Different ways of evaluating expressions in a programming language.
7. Formal Reasoning about Programs: Using mathematical logic to prove properties of programs.
8. Abstract Syntax Trees(ASTs): Representing programs as tree structures for analysis & manipulation.

Who needs to understand these concepts:

1. Programming language designers: These people create new language or update existing ones. They need a deep understanding of these concepts to design language features & ensure the language is consistent and powerful.
2. Compiler & Interpreter Developers: Professional who build the tools that translate human readable code into machine instructions. They use these formal definitions to ensure their compilers correctly implement language features.
3. Static Analysis Tool Developers: People who create tools that analyze code without running it (like advanced linters or type checkers). They use these formal models to reason about code properties.
4. Academic Computer Scientist: Researchers studying programming language, type theory or formal methods use these concepts to advance our understanding of computation.
5. Advanced functional programmers: Languages like Haskell or OCaml have type systems closely related to these formal models.

Operational Semantics Rules:

How can we extend this operational semantic rules, by adding two new constructs in this operation

1. For "bad": when a expression reduce to bad
2. For "try t1 except t2": if t1 is bad then t2, if t1 is normal value then only t1

To accommodate these, you would typically:

1. Extend the syntax construct to include these new constructs
2. Add new evaluation rules that specially deal with "bad" & "try...except"
3. May be modify existing rules if they need to interact with these new constructs.

The goal is to precisely define how these new language features behave in all possible scenarios within the existing arithmetic language.

1. Rules for 'bad' construct:

pred bad → bad
succ bad → bad
iszero bad → bad
if3 bad t2 t3 → bad

2. Rules for stuck states reducing to 'bad':

pred true  → bad
pred false → bad
succ true  → bad
succ false → bad
iszero true → bad
iszero false → bad

3. Rules for "try ... except" construct:

if t1 reduces, the whole expression reduces 

t1 → t1'
─────────────────────────────
try t1 except t2 → try t1' except t2

try t1 except t2

1. If t1 succeeds, doesn't result in bad, we use its result.
2. If t1 fails, result in bad, we use t2 as a fallback or recovery option.
3. for example:
4. try (pred 0) except 42 would evaluate to 0, because pred 0 evaluates normally.
5. try (pred true) except 42 would evaluate to 42, because (pred true) would result in bad, so we use expect part.

t ::= true | false | if3 t t t | 0 | succ t | pred t | iszero t

This grammar defines the syntax for a simple language that can express:

• Boolean values (true and false)
• A conditional statement (if3)
• Natural numbers (starting from 0 and built up with succ)
• Basic arithmetic operations (successor and predecessor)
• A zero check

given:

New terms
| bad | try t except t

New Value
 v ::= ... | bad

(a) How do we extend the operational semantic rules in slide 28 to accommodate for the two

1. Try-Eval:

Try-Eval is about the ongoing evaluation of t1.

If t1 can be further evaluated to t1'
t1 → t1'
──────────────────────────── (Try-Eval) 
try t1 except t2 → try t1' except t2
 

this rule applies when t1 can be further evaluated. It says that if t1 can become t1', then whole try-except expression takes a step evaluating t1 while keeping t2 unchanged.

2. Try-Value:

Try value is about what happens when t1 successfully evaluates to a non bad value.

v ≠ bad
────────────────────────────   (Try-Value)
try v except t2 → v

This rules applies when t2 have been fully evaluated to value v that is not bad, in this case, the entire try-except expression evaluates to v, ignoring t2.

3. Try-Bad:

It's about what happens when t1 evaluates to the bad state.

────────────────────────────   (Try-Bad)
try bad except t2 → t2

This rule applies when t1 has evaluated to bad, in this case the entire try except expression evaluates to t2.

Together these rules define the complete behaviour of the try-except construct:

1. Keep evaluating t1 (try-eval)
2. if t1 become a non bad , return that value (try-value)
3. if t1 becomes bad, evaluate t2 instead (try-bad)

How do we extend the operational semantic rules so that for

if3 t1 t2 t3 we first evaluate t2, and then t3, and finally t1?

t1 → t1'
─────────────────────── (If3-Eval)
if3 t1 t2 t3 → if3 t1' t2 t3

───────────────────────── (If3-True)
if3 true t2 t3 → t2

───────────────────────── (If3-False)
if3 false t2 t3 → t3

this evaluates t1 condition first, then selects either t2 or t3 based on whether t1 evaluates to true or false.

Modified rules for evaluating t2 & t3 before t1.

To enforce evaluation of t2, then t3, & finally t1, we introduce new evaluation steps. We ensure that the system progresses through t2 & t3 before considering evaluation of t1.

1. Evaluating t2 first: if t2 is not a value we evaluate t2 first

 t2 → t2'
─────────────────────── (If3-Eval-T2)
if3 t1 t2 t3 → if3 t1 t2' t3

2. Evaluating t3 next: Once t2 has been fully evaluated i.e t2 is now a value, we move to evaluating t3.

 t3 → t3'
─────────────────────── (If3-Eval-T3)
if3 t1 v2 t3 → if3 t1 v2 t3'

3. Finally evaluating t1: Once both t2 & t3 have been evaluated to values, we evaluate t1:

 t1 → t1'
─────────────────────── (If3-Eval-T1)
if3 t1 v2 v3 → if3 t1' v2 v3

4.- Conditional selection:

After t1 evaluates to either true or false, we can perform the final selection based on t1:

──────────────────────────────── (If3-True)
if3 true v2 v3 → v2

──────────────────────────────── (If3-False)
if3 false v2 v3 → v3

Finally summary of extended rules:

1. If3-Eval-T2: We evaluate t2 until it becomes a value
2. If3-Eval-T3: After t2 is fully evaluated, we evaluate t3.
3. If3-Eval-T1: After both t2 & t3 are values, we evaluate t1.
4. After evaluating t1, select between v2 & v3 based on wheater t1 is true or false.

How do we extend the typing rules for these two constructs?

To extend the typing rules for the new construct bad & try...except, we need to consider

1. Their behaviour
2. How they interact with existing type system.

Here, since we have added 2 new construct, what typing rules should we have?

Typing rules help us understand what kind of data we're dealing with in our program.

For example: is something a number or a true/false value?

For Bad feature:

we want to say that bad can be treated as any type of data. This is because an error could happen anywhere in our program.

For the try...except feature:

We want to make sure that whether we get an error or not, the end result is the same type of data.

Γ (Gamma):

Represent the typing context or environment. Essentially a collection of all the information we know about the types of variables in our program at a given point.

for example:

1. If we have a variable x that we know is a number (Nat), Γ include this information. Nat is a natural number.
2. If have another variable y that we know is a boolean (Bool) , Γ would include this too.
3. Γ = { x: Nat, y : Bool}

In the typing rule:

Γ ⊢ t1 : T 
⊢, means proves or entrails

What we know about the types of variable (Γ), we can determine that t1 has type T.

The importance of gamma Γ is that it allows us to type check expressions that might contain variables.

1. Rule for Bad:

──────────── (T-Bad)
Γ ⊢ bad : T

This rules means that in any context gamma, bad can have any type T.

2. Rule for 'try-except':

For this we want to ensure that both the main expression & the exception handler have the same type.

Γ ⊢ t1 : T  Γ ⊢ t2 : T
───────────────────────── (T-Try)
Γ ⊢ try t1 except t2 : T

How to read this typing rule?=

1. Premises:
2. Γ ⊢ t1 : T, reads as in the context of gamma, t1 has type T.
3. Γ ⊢ t2 : T, in the context of gamma, t2 has type T.
4. Conclusion:
5. Γ ⊢ try t1 except t2 : T, reads as in the context of gamma, try t1 except t2 has type T.

Putting it together we reads as:

If in the context of gamma(Γ), t1 has type T, & in the same context gamma(Γ) t2 also has type T, then in the context gamma (Γ), the expression try t1 except t2 has type T.

In simpler terms:

If the main expression (t1) & the exception handler (t2) both have the same type T, then the entire try-except expression also has type T.

This ensures that both branch t1 & t2 return values of the same type, maintaining consistency & type safety.

Consider the termination theorem. With our extension in context with two new features, is our semantics still terminating? What adjustment has to be made to prove termination?

To determine if the semantics are still terminating with the extensions added, see what adjustments may be needed to prove the termination, we need to consider how the new constructs i.e bad & try...except affect the evaluation process.

Original Termination Theorem:

The theorem stated that for any well typed term, the evaluation process will eventually reach a value or a stuck state after a finite number of steps.

Uniqueness of Normal Forms:

This theorem states that if a term t can be evaluated to two normal from u & u', then u = u'.

Termination of Evaluation:

This theorem states that for any term t, there exists some normal form u such that t can be evaluated to u, This is the core of termination property we need to preserve with our extensions.

Analysis of New Constructs:

1. Bad construct:
2. Bad itself is a normal form, so it doesn't affect termination theorem directly.
3. It is introduced as a value representing an error state.
4. It does not introduce any reduction steps by itself.
5. It provides a way to explicitly represent a stuck states that previously had no reduction Thus strengthens the termination property.
6. Try...Except construct:
7. This introduced new evaluation rule.
8. Evaluating t1 in try t1 except t2
9. Handling a case where t1 evaluates to bad
10. Handling the case where t1 evaluates to a non bad value.
11. It does not violate the termination therorem.
12. We need to ensure that evaluation of try t1 except t2 always leads to a normal form.

Impact on Termination:

This semantics will still be terminating if:

1. The evaluation of t1 in try t1 expect t2 is guaranteed to terminate either to
2. a value
3. or including a bad or stuck state.
4. Once t1 is fully evaluated, the construct immediately terminates or moves to evaluate t2 in case t1 evaluated to bad.
5. The evaluation of t2 if reached, is also guaranteed to terminated.

To prove termination these new constructs, we need to adjust:

1. Extend the definition of normal form to include bad.
2. Modify termination theorem to account for try...except:
3. For any term t, there is some normal form u such that t -> *u, where u is either:
4. t evaluates to a value (including bad) in a finite number of steps
5. t reaches a stuck state in a finite number of steps
6. t evaluates to a term of the form 'try bad except t2' in a finite number of steps.

For the notes and task discussed in this lecture, credit goes to CSCE-550, Principles of Program Language class. And notes i took are study materials in the class. Credit goes to Dr. Chen.

#CSCE550 #Principles #ProgramLanguage #550 #inferenceRules #languageSemantics