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Semantics Q1 2007 S EMANTICS (Q1,’07) Week 1 Jacob Andersen PhD student andersen@daimi.au.dk

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Semantics Q1 2007 Week 1 - Outline Introduction Welcome and introduction Short-film about teaching (20')... Course Presentation [ homepage ]homepage Prerequisitional (discrete) Mathematics Relations Inference Systems Transition Systems The Language “L”

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Semantics Q1 2007 Introduction Semantics –Merriam-Webster: –Example: “I would like a cup of coffee.” “Colourless green ideas sleep furiously.” Question: Would you fly the space shuttle if you had written the guidance software??? In this course we shall explore formal ways of reasoning about the meaning / behaviour of computer programs. Main Entry: se·man·tics Pronunciation: si-'man-tiks Function: noun plural but singular or plural in construction 1 : the study of meanings

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Semantics Q1 2007 Short-film about teaching...

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Semantics Q1 2007 Course Introduction I SOLO 2, 3, 4 Verbs identifies skills / competences Actor

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Semantics Q1 2007 Course Introduction II No-one can force new skills into your head –You do not learn how to ride a bike by observing others (!!) Expected weekly time consumption: Lecture3 Reading1 Exercise class3 Hand-in2 Sum13 Fitting the pieces together Lecture3 Reading1 Reflecting / Solving exercises4 Exercise class3 Hand-in2 Sum13

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Semantics Q1 2007 Course Introduction III Course Retrospection dSem ’05:

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Semantics Q1 2007 Learning vs. Breaks #Breaks, |Breaks|,...? Conclusion: more (shorter) breaks

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Semantics Q1 2007 R ELATIONS AND I NFERENCE S YSTEMS

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Semantics Q1 2007 Relations I Example 1 : “even” relation: Written as: as a short-hand for: … and as: as a short-hand for: Example 2 : “equals” relation: Written as: as a short-hand for: … and as: as a short-hand for: Example 3 : “DFA transition” relation: Written as: as a short-hand for: … and as: as a short-hand for: | _ even Z | _ even 4 | _ even 5 4 | _ even 5 | _ even 2 3(2,3) ‘=’ ‘=’ Z Z (2,2) ‘=’ 2 = 2 ‘ ’ Q Q q q’ (q, , q’) ‘ ’ (p, , p’) ‘ ’p p’

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Semantics Q1 2007 Relations II In general, a k-ary relation, L, over the sets S 1, S 2, …, S k is a subset L S 1 S 2 … S k k is called the “arity” of L, and “L S 1 S 2 … S k ” is the “signature”.

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Semantics Q1 2007 Inference System Inference System: Inductive (recursive) specification of relations Consists of axioms and rules Example: Axiom: “0 (zero) is even”! Rule: “If n is even, then m is even (where m = n+2)” | _ even 0 | _ even n | _ even m m = n+2 | _ even Z

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Semantics Q1 2007 Terminology Meaning: Inductive: “If n is even, then m is even (provided m = n+2)”; or Deductive: “m is even, if n is even (provided m = n+2)” | _ even n | _ even m m = n+2 premise(s) conclusion side-condition(s) Question: Why have both premises and side-conditions?? Note: The “ “ is actually just another way of writing a “ ” !

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Semantics Q1 2007 Abbreviation Often, rules are abbreviated: Rule: “If n is even, then m is even (provided m = n+2)”; or “m is even, if n is even (provided m = n+2)” Abbreviated rule: “If n is even, then n+2 is even”; or “n+2 is even, if n is even” | _ even n | _ even n+2 | _ even n | _ even m m = n+2 Even so, this is what we mean Warning: Be careful !

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Semantics Q1 2007 Relation Membership (Even)? Axiom: “0 (zero) is even”! Rule: “If n is even, then n+2 is even” Is 6 even?!? The inference tree proves that: | _ even 0 | _ even 2 | _ even 4 | _ even 6 | _ even 0 | _ even n | _ even n+2 [rule 1 ] [axiom 1 ] inference tree | _ even 6

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Semantics Q1 2007 Example: “less-than-or-equal-to” Relation: Is ”1 2” ?!? Yes, because there exists an inference tree: »In fact, it has two inference trees: 0 0 n m n m+1 [rule 1 ] [axiom 1 ] ‘ ’ N N n m n+1 m+1 [rule 2 ] 0 0 0 1 1 2 [rule 2 ] [rule 1 ] [axiom 1 ] 0 0 1 1 1 2 [rule 1 ] [rule 2 ] [axiom 1 ]

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Semantics Q1 2007 Activation Exercise 1 Activation Exercise: 1. Specify the signature of the relation: ' << ' »x << y " y is-double-that-of x " 2. Specify the relation via an inference system »i.e. axioms and rules 3. Prove that indeed: »3 << 6 "6 is-double-that-of 3"

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Semantics Q1 2007 Activation Exercise 2 Activation Exercise: 1. Specify the signature of the relation: ' // ' »x // y " x is-half-that-of y " 2. Specify the relation via an inference system »i.e. axioms and rules 3. Prove that indeed: »3 // 6 "3 is-half-that-of 6" Syntactically different Semantically the same relation

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Semantics Q1 2007 Relation Definition (Interpretation) Actually, an inference system: …is a demand specification for a relation: The three relations: R = {0, 2, 4, 6, …}(aka., 2N) R’ = {0, 2, 4, 5, 6, 7, 8, …} R’’ = {…, -2, -1, 0, 1, 2, …}(aka., Z) …all satisfy the (above) specification! |_R 0|_R 0 | _ R n | _ R n+2 [rule 1 ] [axiom 1 ] | _ R Z (0 ‘| _ R ’) ( n ‘| _ R ’ n+2 ‘| _ R ’)

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Semantics Q1 2007 Inductive Interpretation A inference system: …induces a function: Definition: ‘lfp’ (least fixed point) ~ least solution: | _ R 0 | _ R n | _ R n+2 [rule 1 ] [axiom 1 ] F R : P(Z) P(Z) | _ R Z F R (R) = {0} { n+2 | n R } F(Ø) = {0}F 2 (Ø) = F({0}) = {0,2}F 3 (Ø) = F 2 ({0}) = F({0,2}) = {0,2,4} … | _ even := lfp(F R ) = F R n (Ø) n | _ R P(Z) From rel. to rel. 2N = F n (Ø) ~ “Anything that can be proved in ‘n’ steps”

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Semantics Q1 2007 T RANSITION S YSTEMS

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Semantics Q1 2007 Definition: Transition System A Transition System is (just) a structure: is a set of configurations is a binary relation (called the transition relation) –We will write instead of Other times we might use the following notations: , ’ ’( , ’) ’ ’ |_ ’ |_ ’ ’ ’,, …

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Semantics Q1 2007 The transition may be illustrated as: In this course, we will be (mostly) using: For instance: A Transition ’ ’ ’’ = system configuration = ...and have " " describe "one step of computation"

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Semantics Q1 2007 Def: Terminal Transition System A Terminal Transition System is a structure: is the set of configurations is the transition relation T is a set of final configurations –…satisfying: –i.e. “all configurations in ‘T’ really are terminal”. , , T t T : : t

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Semantics Q1 2007 E.g.: Finite Automaton A Non-det Finite Automaton (NFA) is a 5tuple: Q finite set of states finite set of input symbols q 0 Qinitial state F Qset of acceptance states : Q P(Q)state transition relation M = Q, , q 0, F, Q = { 0,1,2 } = { = { toss,heads,tails } (0,toss) {1,2}, q 0 = 0 (1,heads) {0}, F = { 0 } (2,tails) {0} }

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Semantics Q1 2007 E.g.: NFA Transition System (1) Define Term. Trans. Syst. by: Configurations: Transition relation: –i.e., we have whenever Final Configurations: M := Q * M, M, T M T M := { | q F } M := { (, ) | q,q’ Q, a , w *, q’ (q,a) } “State component” “Data component” q’ (q,a) M Recall: M = Q, , q 0, F, Given

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Semantics Q1 2007 E.g.: NFA Trans. Sys. (Cont’d) Define behavior of M: M T M L (M) := { w * | T : * }

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Semantics Q1 2007 Def: Labelled Transition System A Labelled Transition System is a structure: is the set of configurations A is the set of actions (= labels) A is the transition relation –Note: we will write instead of , A, ’ ’ a ( , a, ’) ‘ ’

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Semantics Q1 2007 The labelled transition may be illustrated as: –The labels(/actions) add the opportunity of describing transitions: –Internally (e.g., information about what went on internally) –Externally (e.g., information about communication /w env) –…or both. A Labelled Transition ’ ’ ’’ a a

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Semantics Q1 2007 E.g.: NFA Transition System (2) Given Finite Automaton: Define Labelled Terminal Transition System: Configurations: Labels: Transition relation: Final configurations: M := Q M, A M, M, T M T M := F M := {(q,a,q’)|q’ (q,a)} A M := M = Q, , q 0, F, X ??? “Relaxed” v

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Semantics Q1 2007 E.g.: NFA Trans. Sys. (Cont’d) Behavior: Define ” *” as the reflexive transitive closure of ” ” on sequences of labels: 0 M 1 M 0 M 2 M 0 L (M) := { w A* | q T : q 0 * q w toss heads toss tails L (M) := { a a’ … a’’ A* | q T : q 0 q’ … q a a’ a’’

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Semantics Q1 2007 NFA: Machine 1 vs. Machine 2 The two transition systems are very different: Machine 1: –”I transitioned from to ” Implicit: ”…by consuming part of (internal) data component” Machine 2: –”I transitioned from q to q’ …by inputing an ’a’ symbol from the (external) environment!” M q M q’ a

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Semantics Q1 2007 More Examples… More Examples in [Plotkin, p. 22 – 26]: –Three Counter Machine (***) –Context-Free Grammars (**) –Petri Nets (*) They illustrate expressive power of transition systems –no new points...except formalizing input / output behavior (also later here…) Read these yourself…

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Semantics Q1 2007 T HE L ANGUAGE “L”

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Semantics Q1 2007 The Language ”L” Basic Syntactic Sets: Truthvalues: –Setranged over by: t, t’, t 0, … Numbers: –Setranged over by: m, n, … Variables: –Setranged over by: v, v’, … T = { tt, ff } N = { 0, 1, 2, …} VAR = { a, b, c, …, z } Meta-variables

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Semantics Q1 2007 The Language ”L” Derived Syntactic Sets: Arithmetic Expressions ( e Exp): Boolean Expressions ( b BExp): Commands ( c Com): e ::= n | v | e + e’ | e – e’ | e e’ b ::= t | e = e’ | b or b’ | ~ b c ::= nil | v := e | c ; c’ | if b then c else c’ | while b do c

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Semantics Q1 2007 Consider program: It can be (ambiguously) understood: Note: Abstract- vs. Concrete Syntax while b do c ; c’ while b do c ; c’while b do c ; c’ …either as: …or as: “Concrete syntax” while b c ; c’ while b c ; c’ “Abstract syntax” Parsing: “Concrete syntax” “Abstract syntax”

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Semantics Q1 2007 Next week Use everything…: Inference Systems, Transition Systems, Syntax, … …to: –describe –explain –analyse –compare …semantics of Expressions:

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Semantics Q1 2007 "Three minutes paper" Please spend three minutes writing down the most important things that you have learned today (now). After 1 day After 1 week After 3 weeks After 2 weeks Right away

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Semantics Q1 2007 Exercises... [ homepage ]

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Semantics Q1 2007 See you next week… Questions?

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Semantics Q1 2007

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