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An overview of various methods of giving semantics to programming languages, with a focus on transition semantics. The lecture covers operational semantics, axiomatic semantics, denotational semantics, and alternative methods. It includes examples and explanations of each method, making it useful for students in computer science or related fields.
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Outline Transition SemanticsLanguage Semantics
Mark Hills [email protected]
University of Illinois at Urbana-Champaign
July 13, 2006
(^1) Based on slides by Mattox Beckman, as updated by Vikram Adve, Gul Agha, and Elsa Gunter
Outline Transition Semantics^ Language Semantics
Language Semantics
Transition Semantics
Outline Transition Semantics^ Language Semantics^ An Overview of Semantics^ Varieties of Language Semantics
Time flies like an arrow. Fruit flies like a banana.
So far, we have syntax, but we don’t know what any of it means! Language semantics provide a way for us to assign meaning to constructs in a programming language.
Outline Transition Semantics^ Language Semantics^ An Overview of Semantics^ Varieties of Language Semantics
Semantics can be grouped into two general categories: static semantics and dynamic semantics. ◮ (^) Static semantics provide meaning based solely on the form of a language construct, or the syntax – not based on execution. These are usually restricted to type checking and inference. ◮ (^) Dynamic semantics provide meaning based on models of evaluation for the language. These tell us what it means to execute a program, for instance.
Outline Transition Semantics^ Language Semantics^ An Overview of Semantics^ Varieties of Language Semantics
◮ (^) First, start with a definition of our “machine” which we run programs on ◮ (^) The machine state, including the program, is often called the configuration ◮ (^) Next, describe how to execute programs in a given language by describing how to execute individual statements and parts of statements – rules follow the structure of the program ◮ (^) A program’s “meaning” is defined by how it changes the configuration ◮ (^) Useful as a basis for implementations
Outline Transition Semantics^ Language Semantics^ An Overview of Semantics^ Varieties of Language Semantics
Axiomatic semantics are also called Floyd-Hoare logic ◮ (^) based on logic – first-order predicate calculus ◮ (^) semantics represented as a logical system build from axioms and inference rules ◮ (^) mainly suited to simple imperative languages ◮ (^) used to prove a post-condition from a pre-condition: given something holds in the starting state, we can show something else holds in the end state
{Precondition} Program {Postcondition}
Outline Transition Semantics^ Language Semantics^ An Overview of Semantics^ Varieties of Language Semantics
There are many other methods which have been devised to give semantics to languages. ◮ (^) “Compiler-based” semantics – the meaning is whatever the compiler says it is ◮ (^) Abstract Machines – similar to operational, an abstract machine with instructions, etc is devised and used to give semantics ◮ (^) Term Rewriting – semantics are formulated as term rewriting systems, with evaluation given by the rewriting relation ◮ (^) Rewriting Logic – an extension of equational logic that provides for concurrency
Outline Transition SemanticsLanguage Semantics
Introduction to Transition Semantics A Sample LanguageConfigurations Semantic Rules ExampleExtending the Language
Transition semantics is a form of operational semantics. ◮ (^) Configurations include the code and machine state: (C , m) ◮ (^) Semantics specified as transitions between configurations, altering the machine state ◮ (^) Rules of the form: 〈C , m〉 → 〈C ′, m′〉 ◮ (^) C , C ′^ represents the code yet to be executed ◮ (^) m, m′^ represents the state (store, memory, etc), often a finite map from names to values ◮ (^) May not need m – simple calculator languages with only numbers and operations don’t, for instance Key point: each transition indicates exactly one step of computation
Outline Transition SemanticsLanguage Semantics
Introduction to Transition Semantics A Sample Language Configurations Semantic Rules ExampleExtending the Language
Using a variant of BNF, we can specify the syntax for IMP as:
Aexp a ::= n | X | a 0 + a 1 | a 0 − a 1 | a 0 × a 1 Bexp b ::= true | false | a 0 = a 1 | a 0 ≤ a 1 | ¬b | b 0 ∧ b 1 | b 0 ∨ b 1 Com c ::= skip | X := a | c 0 ; c 1 | if b then c 0 else c 1 | while b do c
Important Point: We assume reasonable input programs that parse and we don’t care about precedence, associativity, etc – we assume all that has been figured out for us
Outline Transition SemanticsLanguage Semantics
Introduction to Transition Semantics A Sample Language Configurations Semantic Rules ExampleExtending the Language
Our configurations will contain two elements: ◮ (^) The code (a, b, or c) ◮ (^) The state We can define the set of states Σ as functions σ : Loc → N, or functions from locations to integer values.
Outline Transition SemanticsLanguage Semantics
Introduction to Transition Semantics A Sample LanguageConfigurations Semantic Rules ExampleExtending the Language
〈a 0 , σ〉 → 〈a′ 0 , σ〉 〈a 0 + a 1 , σ〉 → 〈a′ 0 + a 1 , σ〉
〈a 1 , σ〉 → 〈a′ 1 , σ〉 〈n 0 + a 1 , σ〉 → 〈n 0 + a′ 1 , σ〉
〈n 0 + n 1 , σ〉 → n, where n = n 0 +int n 1
Outline Transition SemanticsLanguage Semantics
Introduction to Transition Semantics A Sample LanguageConfigurations Semantic Rules ExampleExtending the Language
〈a 0 , σ〉 → 〈a′ 0 , σ〉 〈a 0 − a 1 , σ〉 → 〈a′ 0 − a 1 , σ〉
〈a 1 , σ〉 → 〈a′ 1 , σ〉 〈n 0 − a 1 , σ〉 → 〈n 0 − a′ 1 , σ〉
〈n 0 − n 1 , σ〉 → n, where n = n 0 −int n 1
Outline Transition SemanticsLanguage Semantics
Introduction to Transition Semantics A Sample LanguageConfigurations Semantic Rules ExampleExtending the Language
Our semantic relation for boolean expressions will be of the form:
〈b, σ〉 → t
We assume boolean expressions have no side-effects. ◮ (^) We have simple axioms for boolean constants, including true: 〈true, σ〉 → true ◮ (^) and false: 〈false, σ〉 → false
Outline Transition SemanticsLanguage Semantics
Introduction to Transition Semantics A Sample LanguageConfigurations Semantic Rules ExampleExtending the Language
〈a 0 , σ〉 → 〈a′ 0 , σ〉 〈a 0 = a 1 , σ〉 → 〈a′ 0 = a 1 , σ〉
〈a 1 , σ〉 → 〈a′ 1 , σ〉 〈n 0 = a 1 , σ〉 → 〈n 0 = a′ 1 , σ〉
〈n 0 = n 1 , σ〉 → b, where b = (n 0 =int n 1 )
The rules for ≤ are similar.
