Big Step Semantics Dr. Mattox Beckman, based in part on slides by - PowerPoint PPT Presentation

Introduction The Rules Proof Trees Conclusion Big Step Semantics Dr. Mattox Beckman, based in part on slides by Elsa Gunter. University of Illinois at Urbana-Champaign Department of Computer Science

Introduction The Rules Proof Trees Conclusion Objectives ◮ Describe the components of a big-step semantic rule. ◮ Know how to build a proof tree for an expression. ◮ Explain the correspondence between big step semantics and small step semantics.

skip Introduction The Rules Proof Trees Conclusion Grammar for Simple Imperative Programming Language The Language S ::= u := A | S 1 ; S 2 | if B then S 1 else S 2 fi | while B do S 1 od | B ::= E ∼ E | true | false E ::= u int | E ⊕ E | ◮ Let u be a possibly subscripted variable. ◮ E represents arithmetic expressions, ⊕ is an arithmetic operator. ◮ B represents boolean expressions, ∼ represents relationals.

Introduction The Rules Proof Trees Conclusion The Downarrow Notation ◮ In small step semantics we use the → to represent one step of computations. ◮ In big step semantics we use ⇓ to represent an entire evaluation. Statements < S , σ > ⇓ σ ′ Expressions < E , σ > ⇓ e v Booleans < B , σ > ⇓ b b

Introduction The Rules Proof Trees Conclusion Expressions Integers Variables if u := v ∈ σ . if i is an integer. < u , σ > ⇓ e v < i , σ > ⇓ e i Operations < e 1 , σ > ⇓ e v 1 < e 2 , σ > ⇓ e v 2 < e 1 ⊕ e 2 , σ > ⇓ e v 1 ⊕ v 2 Here ⊕ represents typical binary operations like + , − , × , etc.

Introduction The Rules Proof Trees Conclusion Booleans Booleans Variables < true , σ > ⇓ b true if u := v ∈ σ . < u , σ > ⇓ b v < false , σ > ⇓ b false Relational Operations < e 1 , σ > ⇓ e v 1 < e 2 , σ > ⇓ e v 2 < e 1 ∼ e 2 , σ > ⇓ b v 1 ∼ v 2 Here ∼ represents the typical comparison operations like <, ≤ , = , � = , >, ≥ , etc.

skip Introduction The Rules Proof Trees Conclusion Skip and Assignment < skip , σ > ⇓ σ assignment < e , σ > ⇓ e v < x := e , σ > ⇓ σ [ x := v ]

skip Introduction The Rules Proof Trees Conclusion Skip and Assignment < skip , σ > ⇓ σ assignment < e , σ > ⇓ e v < x := e , σ > ⇓ σ [ x := v ] Next is sequencing. See if you can guess what the rule looks like.

Introduction The Rules Proof Trees Conclusion Sequencing < S 2 , σ ′ > ⇓ σ ′′ < S 1 , σ > ⇓ σ ′ < S 1 ; S 2 , σ > ⇓ σ ′′ ◮ Compare this to the small step version... < S 1 , σ > → < S 2 , σ ′ > < S 1 ; S , σ > → < S 2 ; S , σ ′ >

Introduction The Rules Proof Trees Conclusion Sequencing < S 2 , σ ′ > ⇓ σ ′′ < S 1 , σ > ⇓ σ ′ < S 1 ; S 2 , σ > ⇓ σ ′′ ◮ Compare this to the small step version... < S 1 , σ > → < S 2 , σ ′ > < S 1 ; S , σ > → < S 2 ; S , σ ′ > Next is if . There are two rules for this. See if you can guess what the rules looks like.

Introduction The Rules Proof Trees Conclusion If Statements < B , σ > ⇓ b true < S 1 , σ > ⇓ σ ′ < if B then S 1 else S 2 fi , σ > ⇓ σ ′ < B , σ > ⇓ b false < S 2 , σ > ⇓ σ ′ < if B then S 1 else S 2 fi , σ > ⇓ σ ′

Introduction The Rules Proof Trees Conclusion If Statements < B , σ > ⇓ b true < S 1 , σ > ⇓ σ ′ < if B then S 1 else S 2 fi , σ > ⇓ σ ′ < B , σ > ⇓ b false < S 2 , σ > ⇓ σ ′ < if B then S 1 else S 2 fi , σ > ⇓ σ ′ Next is while . There are two rules for this. See if you can guess what the rules looks like. The second one uses induction!

Introduction The Rules Proof Trees Conclusion While statements < B , σ > ⇓ b false < while B do S od , σ > ⇓ σ

Introduction The Rules Proof Trees Conclusion While statements < B , σ > ⇓ b false < while B do S od , σ > ⇓ σ < while B do S od , σ ′ > ⇓ σ ′′ < B , σ > ⇓ b true < S , σ > ⇓ σ ′ < while B do S od , σ > ⇓ σ ′′

Introduction The Rules Proof Trees Conclusion Proof Trees ◮ To show the effect of a program, we need to build proof trees. ◮ What is 2 × y + 9 × x ? ◮ Let σ = { x := 3 , y := 4 } . Here is what we want to evaluate. What kind of expression is this? < 2 × y + 9 × x , σ > ⇓ e ???

Introduction The Rules Proof Trees Conclusion Proof Trees ◮ To show the effect of a program, we need to build proof trees. ◮ What is 2 × y + 9 × x ? ◮ Let σ = { x := 3 , y := 4 } . Move up the two subexpressions. < 2 × y , σ > ⇓ e ??? < 9 × x , σ > ⇓ e ??? < 2 × y + 9 × x , σ > ⇓ e ???

Introduction The Rules Proof Trees Conclusion Proof Trees ◮ To show the effect of a program, we need to build proof trees. ◮ What is 2 × y + 9 × x ? ◮ Let σ = { x := 3 , y := 4 } . < y , σ > ⇓ e ??? < 2 , σ > ⇓ e ??? < 2 × y , σ > ⇓ e ??? < 9 × x , σ > ⇓ e ??? < 2 × y + 9 × x , σ > ⇓ e ???

Introduction The Rules Proof Trees Conclusion Proof Trees ◮ To show the effect of a program, we need to build proof trees. ◮ What is 2 × y + 9 × x ? ◮ Let σ = { x := 3 , y := 4 } . < y , σ > ⇓ e ??? < 2 , σ > ⇓ e 2 < 2 × y , σ > ⇓ e ??? < 9 × x , σ > ⇓ e ??? < 2 × y + 9 × x , σ > ⇓ e ???

Introduction The Rules Proof Trees Conclusion Proof Trees ◮ To show the effect of a program, we need to build proof trees. ◮ What is 2 × y + 9 × x ? ◮ Let σ = { x := 3 , y := 4 } . < y , σ > ⇓ e 4 < 2 , σ > ⇓ e 2 < 2 × y , σ > ⇓ e ??? < 9 × x , σ > ⇓ e ??? < 2 × y + 9 × x , σ > ⇓ e ???

Introduction The Rules Proof Trees Conclusion Proof Trees ◮ To show the effect of a program, we need to build proof trees. ◮ What is 2 × y + 9 × x ? ◮ Let σ = { x := 3 , y := 4 } . < y , σ > ⇓ e 4 < 2 , σ > ⇓ e 2 < 2 × y , σ > ⇓ e 8 < 9 × x , σ > ⇓ e ??? < 2 × y + 9 × x , σ > ⇓ e ???

Introduction The Rules Proof Trees Conclusion Proof Trees ◮ To show the effect of a program, we need to build proof trees. ◮ What is 2 × y + 9 × x ? ◮ Let σ = { x := 3 , y := 4 } . < y , σ > ⇓ e 4 < x , σ > ⇓ e ??? < 2 , σ > ⇓ e 2 < 9 , σ > ⇓ e ??? < 2 × y , σ > ⇓ e 8 < 9 × x , σ > ⇓ e ??? < 2 × y + 9 × x , σ > ⇓ e ???

Introduction The Rules Proof Trees Conclusion Proof Trees ◮ To show the effect of a program, we need to build proof trees. ◮ What is 2 × y + 9 × x ? ◮ Let σ = { x := 3 , y := 4 } . < y , σ > ⇓ e 4 < x , σ > ⇓ e ??? < 2 , σ > ⇓ e 2 < 9 , σ > ⇓ e 9 < 2 × y , σ > ⇓ e 8 < 9 × x , σ > ⇓ e ??? < 2 × y + 9 × x , σ > ⇓ e ???

Introduction The Rules Proof Trees Conclusion Proof Trees ◮ To show the effect of a program, we need to build proof trees. ◮ What is 2 × y + 9 × x ? ◮ Let σ = { x := 3 , y := 4 } . < y , σ > ⇓ e 4 < x , σ > ⇓ e 3 < 2 , σ > ⇓ e 2 < 9 , σ > ⇓ e 9 < 2 × y , σ > ⇓ e 8 < 9 × x , σ > ⇓ e ??? < 2 × y + 9 × x , σ > ⇓ e ???

Introduction The Rules Proof Trees Conclusion Proof Trees ◮ To show the effect of a program, we need to build proof trees. ◮ What is 2 × y + 9 × x ? ◮ Let σ = { x := 3 , y := 4 } . < y , σ > ⇓ e 4 < x , σ > ⇓ e 3 < 2 , σ > ⇓ e 2 < 9 , σ > ⇓ e 9 < 2 × y , σ > ⇓ e 8 < 9 × x , σ > ⇓ e 27 < 2 × y + 9 × x , σ > ⇓ e ???

Introduction The Rules Proof Trees Conclusion Proof Trees ◮ To show the effect of a program, we need to build proof trees. ◮ What is 2 × y + 9 × x ? ◮ Let σ = { x := 3 , y := 4 } . < y , σ > ⇓ e 4 < x , σ > ⇓ e 3 < 2 , σ > ⇓ e 2 < 9 , σ > ⇓ e 9 < 2 × y , σ > ⇓ e 8 < 9 × x , σ > ⇓ e 27 < 2 × y + 9 × x , σ > ⇓ e 35

Introduction The Rules Proof Trees Conclusion Statement Proof Tree ◮ Let σ = { x := 10 , y := 20 } . ◮ Let σ ′ = { x := 10 , y := 20 , m := 20 } Here is an example that will use all three versions of ⇓ . < if x > y then m := x else m := 2 × x fi , σ > ⇓ ???

Introduction The Rules Proof Trees Conclusion Statement Proof Tree ◮ Let σ = { x := 10 , y := 20 } . ◮ Let σ ′ = { x := 10 , y := 20 , m := 20 } This is an if statement, so... < x > y , σ > ⇓ b ??? ??? < if x > y then m := x else m := 2 × x fi , σ > ⇓ ???

Introduction The Rules Proof Trees Conclusion Statement Proof Tree ◮ Let σ = { x := 10 , y := 20 } . ◮ Let σ ′ = { x := 10 , y := 20 , m := 20 } Deconstructing x > y . < x , σ > ⇓ e ??? < y , σ > ⇓ e ??? < x > y , σ > ⇓ b ??? ??? < if x > y then m := x else m := 2 × x fi , σ > ⇓ ???

Introduction The Rules Proof Trees Conclusion Statement Proof Tree ◮ Let σ = { x := 10 , y := 20 } . ◮ Let σ ′ = { x := 10 , y := 20 , m := 20 } Evaluate x and y . < x , σ > ⇓ e 10 < y , σ > ⇓ e 20 < x > y , σ > ⇓ b ??? ??? < if x > y then m := x else m := 2 × x fi , σ > ⇓ ???