Diagrams for Meaning Preservation
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1 Diagrams for Meaning Preservation Joe Wells Detlef Plump Fairouz Kamareddine Heriot-Watt University University of York det Diagrams for Meaning Preservation p.1/28
2 Overview Motivation. The AES framework. Methods for proving meaning preservation. Discussion. Diagrams for Meaning Preservation p.2/28
3 Some Notation The notation t 1 R t 2 means t 1 is related to t 2 by zero or more steps of the relation R. Diagrams for Meaning Preservation p.3/28
4 Some Notation The notation t 1 R t 2 means t 1 is related to t 2 by zero or more steps of the relation R. The notation t 1 R,nf t 2 means t 1 R t 2 and furthermore t 2 is not related by 1 further step of R to any other term. Diagrams for Meaning Preservation p.3/28
5 Some Notation The notation t 1 R t 2 means t 1 is related to t 2 by zero or more steps of the relation R. The notation t 1 R,nf t 2 means t 1 R t 2 and furthermore t 2 is not related by 1 further step of R to any other term. The notation has-nf(r, t 1 ) means t 1 R,nf t 2 for some t 2. Diagrams for Meaning Preservation p.3/28
6 Some Notation The notation t 1 R t 2 means t 1 is related to t 2 by zero or more steps of the relation R. The notation t 1 R,nf t 2 means t 1 R t 2 and furthermore t 2 is not related by 1 further step of R to any other term. The notation has-nf(r, t 1 ) means t 1 R,nf t 2 for some t 2. Trm, Conf, LConf, and Std are short names for termination, confluence local confluence, and standardization. Diagrams for Meaning Preservation p.3/28
7 An Example Operational Semantics Example: functions with call-by-name evaluation to weak head normal form. Diagrams for Meaning Preservation p.4/28
8 An Example Operational Semantics Example: functions with call-by-name evaluation to weak head normal form. Terms: t T ::= x (λx. t) (t 1 t 2 ) Diagrams for Meaning Preservation p.4/28
9 An Example Operational Semantics Example: functions with call-by-name evaluation to weak head normal form. Terms: t T ::= x (λx. t) (t 1 t 2 ) Evaluation contexts: E EvalContext ::= (E t) Diagrams for Meaning Preservation p.4/28
10 An Example Operational Semantics Example: functions with call-by-name evaluation to weak head normal form. Terms: t T ::= x (λx. t) (t 1 t 2 ) Evaluation contexts: E EvalContext ::= (E t) Evaluation rewriting: E[(λx. t 1 ) t 2 ] E E[t 1 [x := t 2 ]] Diagrams for Meaning Preservation p.4/28
11 An Example Operational Semantics Example: functions with call-by-name evaluation to weak head normal form. Terms: t T ::= x (λx. t) (t 1 t 2 ) Evaluation contexts: E EvalContext ::= (E t) Evaluation rewriting: E[(λx. t 1 ) t 2 ] E E[t 1 [x := t 2 ]] Operational meaning: result(t) = diverges if has-nf(e, t) halt if t E,nf λx. t stuck if t E,nf t λx. t Diagrams for Meaning Preservation p.4/28
12 Rewriting for Program Equivalences Suppose we want to use the evaluation rewrite rule in arbitrary contexts C, i.e., the usual β rule: C[(λx. t 1 ) t 2 ] C[t 1 [x := t 2 ]] Diagrams for Meaning Preservation p.5/28
13 Rewriting for Program Equivalences Suppose we want to use the evaluation rewrite rule in arbitrary contexts C, i.e., the usual β rule: C[(λx. t 1 ) t 2 ] C[t 1 [x := t 2 ]] Example rewrite steps in non-evaluation position: (λy. (λw. w) y) (λy. y) Diagrams for Meaning Preservation p.5/28
14 Rewriting for Program Equivalences Suppose we want to use the evaluation rewrite rule in arbitrary contexts C, i.e., the usual β rule: C[(λx. t 1 ) t 2 ] C[t 1 [x := t 2 ]] Example rewrite steps in non-evaluation position: (λy. (λw. w) y) (λy. y) (λx. xx) (λy. (λw. w) y) (λx. xx) (λy. y) Diagrams for Meaning Preservation p.5/28
15 Rewriting for Program Equivalences Suppose we want to use the evaluation rewrite rule in arbitrary contexts C, i.e., the usual β rule: C[(λx. t 1 ) t 2 ] C[t 1 [x := t 2 ]] Example rewrite steps in non-evaluation position: (λy. (λw. w) y) (λy. y) (λx. xx) (λy. (λw. w) y) (λx. xx) (λy. y) Is this meaning-preserving, i.e., does t 1 t 2 imply that result(t 1 ) = result(t 2 )? Diagrams for Meaning Preservation p.5/28
16 Rewriting for Program Equivalences Suppose we want to use the evaluation rewrite rule in arbitrary contexts C, i.e., the usual β rule: C[(λx. t 1 ) t 2 ] C[t 1 [x := t 2 ]] Example rewrite steps in non-evaluation position: (λy. (λw. w) y) (λy. y) (λx. xx) (λy. (λw. w) y) (λx. xx) (λy. y) Is this meaning-preserving, i.e., does t 1 t 2 imply that result(t 1 ) = result(t 2 )? Is it an observational equivalence, i.e., does t 1 t 2 imply result(c[t 1 ]) = result(c[t 2 ]) for any context C? Diagrams for Meaning Preservation p.5/28
17 Overview Motivation. The AES framework. Methods for proving meaning preservation. Discussion. Diagrams for Meaning Preservation p.6/28
18 Abstract Evaluation Systems To discuss the issues, the notion of abstract evaluation system (AES) will be used. Diagrams for Meaning Preservation p.7/28
19 Abstract Evaluation Systems To discuss the issues, the notion of abstract evaluation system (AES) will be used. An AES is a 6-tuple: Diagrams for Meaning Preservation p.7/28
20 Abstract Evaluation Systems To discuss the issues, the notion of abstract evaluation system (AES) will be used. An AES is a 6-tuple: (T, set of terms Diagrams for Meaning Preservation p.7/28
21 Abstract Evaluation Systems To discuss the issues, the notion of abstract evaluation system (AES) will be used. An AES is a 6-tuple: (T, set of terms S, set of rewrite steps Diagrams for Meaning Preservation p.7/28
22 Abstract Evaluation Systems To discuss the issues, the notion of abstract evaluation system (AES) will be used. An AES is a 6-tuple: (T, set of terms S, set of rewrite steps R, set of evaluation results Diagrams for Meaning Preservation p.7/28
23 Abstract Evaluation Systems To discuss the issues, the notion of abstract evaluation system (AES) will be used. An AES is a 6-tuple: (T, set of terms S, set of rewrite steps R, set of evaluation results endpoints, maps S to T T Diagrams for Meaning Preservation p.7/28
24 Abstract Evaluation Systems To discuss the issues, the notion of abstract evaluation system (AES) will be used. An AES is a 6-tuple: (T, set of terms S, set of rewrite steps R, set of evaluation results endpoints, maps S to T T E, a subset of S, the evaluation steps Diagrams for Meaning Preservation p.7/28
25 Abstract Evaluation Systems To discuss the issues, the notion of abstract evaluation system (AES) will be used. An AES is a 6-tuple: (T, set of terms S, set of rewrite steps R, set of evaluation results endpoints, maps S to T T E, a subset of S, the evaluation steps result) maps T to R Diagrams for Meaning Preservation p.7/28
26 Abstract Evaluation Systems To discuss the issues, the notion of abstract evaluation system (AES) will be used. An AES is a 6-tuple: (T, set of terms S, set of rewrite steps R, set of evaluation results endpoints, maps S to T T E, a subset of S, the evaluation steps result) maps T to R Let the non-evaluation steps be N = S \ E. Diagrams for Meaning Preservation p.7/28
27 Comments on AES Framework (1) There is a separate set S of steps which are not just term pairs. Diagrams for Meaning Preservation p.8/28
28 Comments on AES Framework (1) There is a separate set S of steps which are not just term pairs. This helps distinguish between different redexes that reach the same term, e.g.: ((λx. xx) (λx. xx)) ((λx. xx) (λx. xx)) Diagrams for Meaning Preservation p.8/28
29 Comments on AES Framework (1) There is a separate set S of steps which are not just term pairs. This helps distinguish between different redexes that reach the same term, e.g.: ((λx. xx) (λx. xx)) ((λx. xx) (λx. xx)) So an AES definer does not need to reason about all other redexes in the same term when deciding whether a step is in E or N. Diagrams for Meaning Preservation p.8/28
30 Comments on AES Framework (1) There is a separate set S of steps which are not just term pairs. This helps distinguish between different redexes that reach the same term, e.g.: ((λx. xx) (λx. xx)) ((λx. xx) (λx. xx)) So an AES definer does not need to reason about all other redexes in the same term when deciding whether a step is in E or N. Rewriting notation for a subset S S: t 1 S t 2 s S. endpoints(s) = (t 1, t 2 ) Diagrams for Meaning Preservation p.8/28
31 Comments on AES Framework (1) There is a separate set S of steps which are not just term pairs. This helps distinguish between different redexes that reach the same term, e.g.: ((λx. xx) (λx. xx)) ((λx. xx) (λx. xx)) So an AES definer does not need to reason about all other redexes in the same term when deciding whether a step is in E or N. Rewriting notation for a subset S S: t 1 S t 2 s S. endpoints(s) = (t 1, t 2 ) t 1 S 1,S 2 t 2 t 1 S 1 S 2 t 2 Diagrams for Meaning Preservation p.8/28
32 Comments on AES Framework (2) Evaluation rewriting (i.e., E ) must be subcommutative. Often, it will be deterministic, but the extra flexibility is there for when needed. Diagrams for Meaning Preservation p.9/28
33 Comments on AES Framework (2) Evaluation rewriting (i.e., E ) must be subcommutative. Often, it will be deterministic, but the extra flexibility is there for when needed. The special result value diverges is reserved for terms with non-halting evaluation. Diagrams for Meaning Preservation p.9/28
34 Comments on AES Framework (2) Evaluation rewriting (i.e., E ) must be subcommutative. Often, it will be deterministic, but the extra flexibility is there for when needed. The special result value diverges is reserved for terms with non-halting evaluation. Evaluation steps must preserve results, i.e., t 1 E t 2 must imply that result(t 1 ) = result(t 2 ). Diagrams for Meaning Preservation p.9/28
35 Comments on AES Framework (2) Evaluation rewriting (i.e., E ) must be subcommutative. Often, it will be deterministic, but the extra flexibility is there for when needed. The special result value diverges is reserved for terms with non-halting evaluation. Evaluation steps must preserve results, i.e., t 1 E t 2 must imply that result(t 1 ) = result(t 2 ). The intention is to model execution where the only way to observe a result is to do evaluation steps as long as possible and then inspect the halted term, which is unique even when evaluation is non-deterministic (a deliberate AES framework limitation). Diagrams for Meaning Preservation p.9/28
36 Example AES Terms: t T ::= x (λx. t) (t 1 t 2 ) Diagrams for Meaning Preservation p.10/28
37 Example AES Terms: t T ::= x (λx. t) (t 1 t 2 ) Contexts: C Context ::= x (λx. t) (t 1 t 2 ) Rewrite steps: s S ::= (C, ((λx. t 1 ) t 2 )) Diagrams for Meaning Preservation p.10/28
38 Example AES Terms: t T ::= x (λx. t) (t 1 t 2 ) Contexts: C Context ::= x (λx. t) (t 1 t 2 ) Rewrite steps: s S ::= (C, ((λx. t 1 ) t 2 )) Evaluation results: r R = {diverges, stuck, halt} Diagrams for Meaning Preservation p.10/28
39 Example AES Terms: t T ::= x (λx. t) (t 1 t 2 ) Contexts: C Context ::= x (λx. t) (t 1 t 2 ) Rewrite steps: s S ::= (C, ((λx. t 1 ) t 2 )) Evaluation results: r R = {diverges, stuck, halt} Rewrite step endpoints: endpoints(c, (λx. t 1 ) t 2 ) = (C[(λx. t 1 ) t 2 ], C[t 1 [x := t 2 ]]) Diagrams for Meaning Preservation p.10/28
40 Example AES Terms: t T ::= x (λx. t) (t 1 t 2 ) Contexts: C Context ::= x (λx. t) (t 1 t 2 ) Rewrite steps: s S ::= (C, ((λx. t 1 ) t 2 )) Evaluation results: r R = {diverges, stuck, halt} Rewrite step endpoints: endpoints(c, (λx. t 1 ) t 2 ) = (C[(λx. t 1 ) t 2 ], C[t 1 [x := t 2 ]]) Evaluation contexts: E EvalContext ::= (E t) Evaluation steps: s E ::= (E, ((λx. t 1 ) t 2 )) Diagrams for Meaning Preservation p.10/28
41 Example AES Terms: t T ::= x (λx. t) (t 1 t 2 ) Contexts: C Context ::= x (λx. t) (t 1 t 2 ) Rewrite steps: s S ::= (C, ((λx. t 1 ) t 2 )) Evaluation results: r R = {diverges, stuck, halt} Rewrite step endpoints: endpoints(c, (λx. t 1 ) t 2 ) = (C[(λx. t 1 ) t 2 ], C[t 1 [x := t 2 ]]) Evaluation contexts: E EvalContext ::= (E t) Evaluation steps: s E ::= (E, ((λx. t 1 ) t 2 )) diverges if has-nf(e, t) Operational meaning: result(t) = halt if t E,nf λx. t stuck if t E,nf t λx. Diagrams for Meaning Preservation p.10/28
42 Overview Motivation. The AES framework. Methods for proving meaning preservation. Previous high-level proof methods. New high-level proof methods. Low-level proof methods with elementary diagrams. Marks (e.g., finite developments). Discussion. Diagrams for Meaning Preservation p.11/28
43 Proving Program Equivalences Denotational methods (semantic models). Diagrams for Meaning Preservation p.12/28
44 Proving Program Equivalences Denotational methods (semantic models). Logical relations. Requires a type system, so hard to use for the full untyped calculus. (An intersection type system can sometimes cover an entire untyped system, but this is difficult.) Diagrams for Meaning Preservation p.12/28
45 Proving Program Equivalences Denotational methods (semantic models). Logical relations. Requires a type system, so hard to use for the full untyped calculus. (An intersection type system can sometimes cover an entire untyped system, but this is difficult.) Operational techniques. Applicative bisimulation and co-induction. Howe s method. Diagrams for Meaning Preservation p.12/28
46 Proving Program Equivalences Denotational methods (semantic models). Logical relations. Requires a type system, so hard to use for the full untyped calculus. (An intersection type system can sometimes cover an entire untyped system, but this is difficult.) Operational techniques. Applicative bisimulation and co-induction. Howe s method. This talk will focus on rewriting-based methods: Plotkin [1975], Machkasova and Turbak [2000], Odersky [1993], Ariola and Blom [2002]. Diagrams for Meaning Preservation p.12/28
47 Suppose t 1 t 2. Plotkin s Method Diagrams for Meaning Preservation p.13/28
48 Suppose t 1 t 2. Plotkin s Method If both diverge, they are assigned same meaning (important!). Diagrams for Meaning Preservation p.13/28
49 Suppose t 1 t 2. Plotkin s Method If both diverge, they are assigned same meaning (important!). Suppose evaluation of one halts, maybe t 1. Then: E,nf E t 1 t 3 t 5 Std Conf t 6 N Std t 2 t 4 By definition, result(t 1 ) = result(t 5 ) and result(t 2 ) = result(t 4 ). E N Diagrams for Meaning Preservation p.13/28
50 Suppose t 1 t 2. Plotkin s Method If both diverge, they are assigned same meaning (important!). Suppose evaluation of one halts, maybe t 1. Then: E,nf E t 1 t 3 t 5 Std Conf t 6 N Std t 2 t 4 By definition, result(t 1 ) = result(t 5 ) and result(t 2 ) = result(t 4 ). Because t 5 has halted and N-conversion preserves both this fact and results (important!), result(t 5 ) = result(t 4 ). E N Diagrams for Meaning Preservation p.13/28
51 Comments on Plotkin s Method Plotkin [1975] originally developed it for the call-by-name and call-by-value λ-calculus. Diagrams for Meaning Preservation p.14/28
52 Comments on Plotkin s Method Plotkin [1975] originally developed it for the call-by-name and call-by-value λ-calculus. Other researchers have used it for variations on the λ-calculus, e.g., λ-calculus plus assignments and continuations [Felleisen and Hieb, 1992] and the call-by-need λ-calculus [Ariola and Felleisen, 1997; Maraist, Odersky, and Wadler, 1998]. Diagrams for Meaning Preservation p.14/28
53 Comments on Plotkin s Method Plotkin [1975] originally developed it for the call-by-name and call-by-value λ-calculus. Other researchers have used it for variations on the λ-calculus, e.g., λ-calculus plus assignments and continuations [Felleisen and Hieb, 1992] and the call-by-need λ-calculus [Ariola and Felleisen, 1997; Maraist, Odersky, and Wadler, 1998]. Requiring confluence prevents using some approaches for reasoning about mutually recursive bindings [Ariola and Klop, 1997]. Diagrams for Meaning Preservation p.14/28
54 Comments on Plotkin s Method Plotkin [1975] originally developed it for the call-by-name and call-by-value λ-calculus. Other researchers have used it for variations on the λ-calculus, e.g., λ-calculus plus assignments and continuations [Felleisen and Hieb, 1992] and the call-by-need λ-calculus [Ariola and Felleisen, 1997; Maraist, Odersky, and Wadler, 1998]. Requiring confluence prevents using some approaches for reasoning about mutually recursive bindings [Ariola and Klop, 1997]. Requiring standardization tends to force the evaluation contexts and rewrite rules to look arbitrarily deep into the term and inspect an arbitrary number of tree nodes. Diagrams for Meaning Preservation p.14/28
55 Lift & Project Method Machkasova and Turbak [2000] introduced the Lift and Project diagrams: Lift : t 1 t 4 E E t 2 t 3 N Proj : t 1 t 2 t 4 E E E t 3 t 5 N Diagrams for Meaning Preservation p.15/28
56 Lift & Project Method Machkasova and Turbak [2000] introduced the Lift and Project diagrams: Lift : t 1 t 4 E E t 2 t 3 N Proj : t 1 t 2 t 4 E E E t 3 t 5 N Lift and Project can be substituted for confluence and standardization when proving meaning preservation. Diagrams for Meaning Preservation p.15/28
57 Lift & Project Method Machkasova and Turbak [2000] introduced the Lift and Project diagrams: Lift : t 1 t 4 E E t 2 t 3 N Proj : t 1 t 2 t 4 E E E t 3 t 5 N Lift and Project can be substituted for confluence and standardization when proving meaning preservation. The key benefit is that Lift and Project do not imply confluence, although Lift is equivalent to standardization. Diagrams for Meaning Preservation p.15/28
58 Lift & Project Method Machkasova and Turbak [2000] introduced the Lift and Project diagrams: Lift : t 1 t 4 E E t 2 t 3 N Proj : t 1 t 2 t 4 E E E t 3 t 5 N Lift and Project can be substituted for confluence and standardization when proving meaning preservation. The key benefit is that Lift and Project do not imply confluence, although Lift is equivalent to standardization. In particular, Lift and Project can be used to prove correctness of Ariola/Blom/Klop-style rewrite rules for letrec. Diagrams for Meaning Preservation p.15/28
59 Comparison of Previous Proof Methods Lift & Project can handle cases for which confluence & standardization fail, e.g.: t 1 t 2 N t 3 N Diagrams for Meaning Preservation p.16/28
60 Comparison of Previous Proof Methods Lift & Project can handle cases for which confluence & standardization fail, e.g.: t 1 t 2 N t 3 (New result:) Confluence & standardization can handle cases for which Lift & Project fail, e.g.: N t 1 t 2 N E t 3 N E E Diagrams for Meaning Preservation p.16/28
61 Overview Motivation. The AES framework. Methods for proving meaning preservation. Previous high-level proof methods. New high-level proof methods. Low-level proof methods with elementary diagrams. Marks (e.g., finite developments). Discussion. Diagrams for Meaning Preservation p.17/28
62 Weakening the Proof Burden (1) By carefully inspecting how confluence & standardization and Lift & Project prove meaning preservation, we obtained the following weaker Lift/Project (LP) diagram which implies meaning preservation: t 1 E t 3 E t 5 LP : N E t 2 t 4 Diagrams for Meaning Preservation p.18/28
63 Weakening the Proof Burden (1) By carefully inspecting how confluence & standardization and Lift & Project prove meaning preservation, we obtained the following weaker Lift/Project (LP) diagram which implies meaning preservation: t 1 E t 3 E t 5 LP : N E t 2 t 4 We further weakened LP to the following Lift/Project when Terminating (LPT) diagram: LPT : t 1 t 3 E,nf E t 2 t 4 N Diagrams for Meaning Preservation p.18/28
64 Weakening the Proof Burden (2) The LPT diagram gives the fundamental essence of this family of proof methods and is the weakest condition we know of its kind. It is weaker than similar conditions in the related work by Ariola and Blom [2002]. Diagrams for Meaning Preservation p.19/28
65 Weakening the Proof Burden (2) The LPT diagram gives the fundamental essence of this family of proof methods and is the weakest condition we know of its kind. It is weaker than similar conditions in the related work by Ariola and Blom [2002]. We further weaken the conditions by parameterizing all diagrams on rewrite steps or step sets. Diagrams for Meaning Preservation p.19/28
66 Weakening the Proof Burden (2) The LPT diagram gives the fundamental essence of this family of proof methods and is the weakest condition we know of its kind. It is weaker than similar conditions in the related work by Ariola and Blom [2002]. We further weaken the conditions by parameterizing all diagrams on rewrite steps or step sets. E.g., actual definition of LPT is: s LPT s t 1 t 3 E,nf E t 2 t 4 N Diagrams for Meaning Preservation p.19/28
67 Weakening the Proof Burden (2) The LPT diagram gives the fundamental essence of this family of proof methods and is the weakest condition we know of its kind. It is weaker than similar conditions in the related work by Ariola and Blom [2002]. We further weaken the conditions by parameterizing all diagrams on rewrite steps or step sets. E.g., actual definition of LPT is: s LPT s t 1 t 3 E,nf E t 2 t 4 N This is important because sometimes different methods are needed for different parts of a rewriting system. Diagrams for Meaning Preservation p.19/28
68 Overview Motivation. The AES framework. Methods for proving meaning preservation. Previous high-level proof methods. New high-level proof methods. Low-level proof methods with elementary diagrams. Marks (e.g., finite developments). Discussion. Diagrams for Meaning Preservation p.20/28
69 Proving the High-Level Diagrams Diagrams like confluence, standardization, Lift, Proj, LP, and LPT can be used to prove meaning preservation, but they are themselves quite hard to prove, because the diagrams are quite high-level and abstract. Diagrams for Meaning Preservation p.21/28
70 Proving the High-Level Diagrams Diagrams like confluence, standardization, Lift, Proj, LP, and LPT can be used to prove meaning preservation, but they are themselves quite hard to prove, because the diagrams are quite high-level and abstract. We present 2 new meaning preservation proof methods which are low-level because their conditions are only elementary diagrams and simple (to understand, not necessarily to prove) kinds of termination. Diagrams for Meaning Preservation p.21/28
71 Proving the High-Level Diagrams Diagrams like confluence, standardization, Lift, Proj, LP, and LPT can be used to prove meaning preservation, but they are themselves quite hard to prove, because the diagrams are quite high-level and abstract. We present 2 new meaning preservation proof methods which are low-level because their conditions are only elementary diagrams and simple (to understand, not necessarily to prove) kinds of termination. It is expected that a rewriting system will be partitioned into step sets that are closed under residuals w.r.t. evaluation and the right method will be used for each partition. Often, each partition will contain all of the steps for some subset of the rewrite rules. Diagrams for Meaning Preservation p.21/28
72 Low-level Method 1 Well Behaved with Standardization: WB+Std(S) Trm(E S) WL1(S, S) WL1(S, S) WP1( Weak Lift 1-Step: Weak Project 1-Step: WL1(S, S ) t 1 t E,S 4 N,S E,S S WP1(S) t 2 t 3 t 1 t 2 t 3 t 4 Useful for difficult rewrite step sets which do not have finite developments, e.g., Ariola/Blom/Klop-style letrec rewrite rules. N,S E E S Diagrams for Meaning Preservation p.22/28
73 Low-level Method 2 Well Behaved without Standardization: WB Std(S) Trm(S) LConf(S) WLP1(S) NE(S) Weak Lift/Project 1-Step: N-Steps Do Not Create E-Step WLP1(S) N,S t 1 t 4 E E t 2 t 3 S NE(S) N,S Useful for difficult rewrite step sets which do not have standardization but do have termination. t 1 t 4 E,S E,S t 2 t 3 Diagrams for Meaning Preservation p.23/28
74 Overview Motivation. The AES framework. Methods for proving meaning preservation. Previous high-level proof methods. New high-level proof methods. Low-level proof methods with elementary diagrams. Marks (e.g., finite developments). Discussion. Diagrams for Meaning Preservation p.24/28
75 When Termination Properties Fail Sometimes, a desired termination property fails for a rewrite step set S generated by some rewrite rule(s), but holds for S M where M is a set of marked steps. Diagrams for Meaning Preservation p.25/28
76 When Termination Properties Fail Sometimes, a desired termination property fails for a rewrite step set S generated by some rewrite rule(s), but holds for S M where M is a set of marked steps. The marks typically force termination by forbidding contracting unmarked redexes and ensuring created redexes are unmarked. Diagrams for Meaning Preservation p.25/28
77 When Termination Properties Fail Sometimes, a desired termination property fails for a rewrite step set S generated by some rewrite rule(s), but holds for S M where M is a set of marked steps. The marks typically force termination by forbidding contracting unmarked redexes and ensuring created redexes are unmarked. The rewriting system is embedded in a larger marked system with additional marked terms and rewrite steps. Proving the larger system correct is enough. Diagrams for Meaning Preservation p.25/28
78 When Termination Properties Fail Sometimes, a desired termination property fails for a rewrite step set S generated by some rewrite rule(s), but holds for S M where M is a set of marked steps. The marks typically force termination by forbidding contracting unmarked redexes and ensuring created redexes are unmarked. The rewriting system is embedded in a larger marked system with additional marked terms and rewrite steps. Proving the larger system correct is enough. We give conditions on marking such that proving LPT for S M (i.e., the marked fragment of the larger marked system) is sufficient to prove LPT for S. Diagrams for Meaning Preservation p.25/28
79 Overview Motivation. The AES framework. Methods for proving meaning preservation. Discussion. Diagrams for Meaning Preservation p.26/28
80 Related Work Our work is a direct successor to the work of Machkasova and Turbak [2000]. Diagrams for Meaning Preservation p.27/28
81 Related Work Our work is a direct successor to the work of Machkasova and Turbak [2000]. The work of Ariola and Blom [2002] has important similarities at a deep level. Their framework does not make it easy to prove operational properties, e.g., the user must prove a connection between infinite normal forms and operational behavior. Also, there are no low-level abstract proof methods. Diagrams for Meaning Preservation p.27/28
82 Related Work Our work is a direct successor to the work of Machkasova and Turbak [2000]. The work of Ariola and Blom [2002] has important similarities at a deep level. Their framework does not make it easy to prove operational properties, e.g., the user must prove a connection between infinite normal forms and operational behavior. Also, there are no low-level abstract proof methods. Odersky [1993] gives conditions that a transformation is an observational equivalence. Despite similarities, the formal presentation is quite different and tied to a particular syntactic formalism. Diagrams for Meaning Preservation p.27/28
83 Conclusions Overall, the meaning-preservation proof methods we present gather together the strengths of existing methods and improve on them in a number of ways. Diagrams for Meaning Preservation p.28/28
84 Conclusions Overall, the meaning-preservation proof methods we present gather together the strengths of existing methods and improve on them in a number of ways. Our proof methods are designed to be easy for someone who is not a rewriting specialist to read, understand, and apply to their programming language calculi. Diagrams for Meaning Preservation p.28/28
85 Conclusions Overall, the meaning-preservation proof methods we present gather together the strengths of existing methods and improve on them in a number of ways. Our proof methods are designed to be easy for someone who is not a rewriting specialist to read, understand, and apply to their programming language calculi. We expect that our methods will help in making the expertise of the rewriting community accessible and useful to the outside world. Diagrams for Meaning Preservation p.28/28
86 References Zena M. Ariola and Stefan Blom. Skew confluence and the lambda calculus with letrec. Ann. Pure Appl. Logic, 117(1 3): , Zena M. Ariola and Matthias Felleisen. The call-by-need lambda calculus. J. Funct. Programming, 3(7), May Zena M. Ariola and Jan Willem Klop. Lambda calculus with explicit recursion. Inform. & Comput., 139: , Matthias Felleisen and Robert Hieb. The revised report on the syntactic theories of sequential control and state. Theoret. Comput. Sci., 102: , Elena Machkasova and Franklyn A. Turbak. A calculus for link-time compilation. In Programming Languages & Systems, 9th European Symp. Programming, volume 1782 of LNCS, pages Springer-Verlag, ISBN URL John Maraist, Martin Odersky, and Philip Wadler. The call-byneed lambda calculus J. Funct. Programming, 8(3), May Martin Odersky. A syntactic method for proving observational 28-1
87 equivalences. Research Report YALEU/DCS/RR-964, Yale Univ., Dept. of Comp. Science, May Gordon D. Plotkin. Call-by-name, call-by-value and the lambda calculus. Theoret. Comput. Sci., 1: ,
Diagrams for Meaning Preservation
Diagrams for Meaning Preservation J. B. Wells 1, Detlef Plump 2, and Fairouz Kamareddine 1 1 Heriot-Watt University http://www.cee.hw.ac.uk/ultra/ 2 University of York http://www.cs.york.ac.uk/~det/ Abstract.
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