Winter 2004/5Pls – expresssion-lang – Catriel Beeri22 Informal Analysis Goal: prove the semantics is deterministic Stages: Values --- we know they have no transitions Minimal non-values --- consider the axioms General composite expressions --- consider the propagation rules

Winter 2004/5Pls – expresssion-lang – Catriel Beeri23 “Minimal” non-value expressions Reducible expression (redex): provided it has a transition (condition?) (always has a transition) by (op-apply): 3+4  7 by (if_) : if true 3 4  3 Both are reductions Claim: there is a a unique transition from a redex. (both ) Q: Why “minimal”?

Winter 2004/5Pls – expresssion-lang – Catriel Beeri24 Intuition: The rules of  define paths in a composite expression E; each descends to a “minimal” non- value sub-expression F A path defined by the rules: a selection path The sub-expression at its end: a selected sub-expression If F is a redex, has a transition F  F’,  E has a transition E  {F’/F}E (replacement) Arbitrary composite expressions

Winter 2004/5Pls – expresssion-lang – Catriel Beeri25 Examples applic tuple applic +tuple 3545 (eval-rand) (eval-tuple) (2,1) (op-apply)

Winter 2004/5Pls – expresssion-lang – Catriel Beeri26 applic tuple applic +tuple 3545 (eval-rand) (eval-tuple) (2,2) (op-apply)

Winter 2004/5Pls – expresssion-lang – Catriel Beeri27 For , an expression may have several selected sub- expressions, several transitions But, for  d, there is a unique selected sub- expression, hence at most one transition How do argue this is the case?

Winter 2004/5Pls – expresssion-lang – Catriel Beeri28 Consider application E1 E2 E1 is not a value: – (op-apply), (eval-rand) cannot be used. – (eval-rator) can be used, if E1 has a transition E1 is a value, E2 is not: –(op-apply), (eval-rator) cannot be used (why?) –(eval-rand) can be used, if E2 has a transition Both E1 E2 are values: –(eval-rator),(eval-rand) cannot be used, –Possibly (op-apply) can be used

Winter 2004/5Pls – expresssion-lang – Catriel Beeri29 This “shows” that for  d : For each composite expression E, there is at most one transition The binary relation  d is functional (a partial function) This transition semantics is deterministic

Winter 2004/5Pls – expresssion-lang – Catriel Beeri30 Natural (evaluation) Semantics Transition semantics deals with one step Natural semantics deals with a complete computation EV transition Natural:

Winter 2004/5Pls – expresssion-lang – Catriel Beeri31 Natural semantics rules (nat-int)(nat-bool) (nat-op) (nat-applic-op)(nat-tuple)(nat-if-true)(nat-if-false) if defined

Winter 2004/5Pls – expresssion-lang – Catriel Beeri32 A proof tree in natural semantics Q: What is the difference in shape of trees between transition and natural?

Winter 2004/5Pls – expresssion-lang – Catriel Beeri33 A (faulty) proof tree in natural semantics: (what is missing?)

Winter 2004/5Pls – expresssion-lang – Catriel Beeri34 Properties of natural semantics For each value V, For each E, if then E’ is a value (The value variables in the rules are not needed!) For each E, there is at most one proof tree, hence at most one value (main idea of proof: For each expression category, except if, a unique rule.) Natural semantics is a functional relation --- it defines a partial function

Winter 2004/5Pls – expresssion-lang – Catriel Beeri35  On run-time errors High level pl’s do not allow run-time errors to occur w/o taking notice: prevent, detect and treat A semantics should specify explicitly both what are errors, error handling

Winter 2004/5Pls – expresssion-lang – Catriel Beeri36 A computation from E may get stuck in a non-value final expression 3/0, true + 2, if 3 4 5, 2 3 We shall: Determine the forms of error expressions Introduce terminology Add rules that specify error treatment to all specs

Winter 2004/5Pls – expresssion-lang – Catriel Beeri37 Candidate for a reduction: a expression of the forms For both, proper rules certainly do not apply If there is no transition: a run-time error 4/0, 4 2, false + 3, + (2, 3, 4) (All, except the first, are run-time type errors) Candidate for a reduction: either a redex or a run-time error Claim: there is a at most one transition from a candidate for a reduction (for both ,  d).

Winter 2004/5Pls – expresssion-lang – Catriel Beeri38 The run-time errors: 3/0, 2+false, + (2,3,4) 2 4 if 5 4 6

Winter 2004/5Pls – expresssion-lang – Catriel Beeri39 For each composite expression: each selected sub-expression is a candidate for a reduction Def: E contains an error: a selected sub-expression is a run-time error E is an error: ( a stuck expression) all its selected sub-expressions are run-time errors Discuss differences between ,  d on these concepts

Winter 2004/5Pls – expresssion-lang – Catriel Beeri40 Error treatment --- error rules We add a special “constant” ER (not a regular expression/value) We add error rules to specify error handling An expression may transform in one step to ER iff It is a run-time error (error axioms) It contains an error (error propagation rules) Def: Exval - extended values: Val {ER} extended expressions: Exp {ER}

Winter 2004/5Pls – expresssion-lang – Catriel Beeri41 Need to change assumption about built-in: For each built-in operation o, is a total function from Val to Exval (returns a value or ER) Takes care of

Winter 2004/5Pls – expresssion-lang – Catriel Beeri42 Some error rules: For other constructs –left to you For  d -- left for you The new set of rules Is there a change in selected sub-expressions? Can rules be improved upon by using v instead of e?

Winter 2004/5Pls – expresssion-lang – Catriel Beeri43 Example:  ER applic tuple applic seq (error-rand) (error-tuple) (2,1) (apply- error)

Winter 2004/5Pls – expresssion-lang – Catriel Beeri44 Rules for natural semantics: The other constructs – left to you The new set:

Winter 2004/5Pls – expresssion-lang – Catriel Beeri45 Additional properties for semantics with error rules: Soundness: an expression cannot evaluate to both a value and ER Completeness/Progress: If E is not in Exval, then E evaluates to some XV in Exval (progress --- evaluation does not get stuck) Determinism implies soundness (we argue that is deterministic, as before) Completeness: Argue that a candidate for a reduction is never stuck, extend to all expressions