A Semantic Criterion for Proving Infeasibility in Conditional Rewriting

Transcription

1 A Semantic Criterion for Proving Infeasibility in Conditional Rewriting Salvador Lucas Raúl Gutiérrez DSIC, Universitat Politècnica de València, Spain 6 th International Workshop on Confluence, IWC Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 1 / 1

2 Motivation Conditional Term Rewriting Systems (CTRSs) consist of rules of the following general form: l r c where the conditional part c is used as a test for the application of (an instance of) l and r in any rewriting step σ(l) σ(r), i.e., σ(c) must be satisfied. Infeasible rules Some conditional parts cannot be used with any substitution! a b a c In this case, the test specified by the conditional part is a reachability test a c (as for oriented CTRSs) which never succeeds. 2 Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 2 / 1

3 Motivation In the analysis of computational properties of CTRSs R, the conditional part of the rules is often used to investigate them. 1 Confluence is analyzed by means of conditional critical pairs s, t c, where c is obtained from two rules l r d and l r d in R. 2 Operational termination is analyzed by means of conditional dependency pairs u v c where c is obtained from the conditional part d of a rule l r d in R. Feasibility tests In all these cases, the following question is relevant: Is there any substitution σ such that σ(c) holds using R? Or, assuming a logic-based operational semantics for CTRSs: Is there any substitution σ such that R σ(c)? 3 Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 3 / 1

4 Motivation A negative answer to this question, i.e., ensuring that R σ(c) does not hold for any substitution σ (i.e., proving c infeasible), can be useful: Rules l r c R will never be used when rewriting with R. Joinability of s and t in conditional critical pairs s, t c does not matter in the analysis of confluence of R. Conditional dependency pairs u v c can be safely discarded in the analysis of operational termination of R. Analyzing infeasibility How to prove the conditional part of a conditional rule/critical pair/dependency pair infeasible? This problem has been addressed before; several approaches have been proposed. Our contribution exploits the logical approach sketched above. 4 Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 4 / 1

5 Infeasibility problems In the following, we focus on oriented CTRSs R, with rules l r s 1 t 1,..., s n t n whose operational semantics is given by the following inference system: (Rf) x x (C) x i y i f (x 1,..., x i,..., x k ) f (x 1,..., y i,..., x k ) for all f F and 1 i k = arity(f ) (T) x z z y x y (Rp) s 1 t 1... s n t n l r for all l r s 1 t 1 s n t n R Definition Let R be a CTRS. A sequence s 1 t 1,..., s n t n, where s i and t i are terms for all 1 i n is called a feasibility sequence. It is called R-feasible if there is a substitution σ such that for all 1 i n, σ(s i ) R σ(t i). Otherwise, it is called R-infeasible. 5 Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 5 / 1

6 CTRSs as First-Order Theories The first-order theory R for a CTRS R is obtained by specializing (C) and (Rp) as above. Inference rules B 1 B n A become universally quantified implications B 1 B n A. Example For the CTRS R (from [Giesl & Arts, AAECC 01]) a b g(x) g(a) f(x) x f(a) b its associated theory R is ( x) x x ( x, y, z) x y y z x z ( x, y) x y f(x) f(y) a b f(a) b ( x) f(x) x g(x) g(a) ( x, y) x y g(x) g(y) 6 Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 6 / 1

7 A semantic criterion for infeasibility A structure (or interpretation) A for a first-order language gives meaning to function and predicate symbols as mappings and relations on a given set. The usual interpretation of first-order formulas with respect to the structure is then considered. A model for a set S of first-order sentences (i.e., formulas without free variables) is a structure that makes them all true, written A = S. Proposition Let R be a CTRS, s 1 t 1,..., s n t n be a feasibility sequence, and A be a structure with nonempty domain. The sequence is R-infeasible if holds. A = R { ( x) (s 1 t 1 s n t n )} 7 Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 7 / 1

8 A semantic criterion for infeasibility Proof by contradiction 1 R-feasiblity implies that σ(s i ) R σ(t i) (i.e., R σ(s i ) σ(t i )) for some substitution σ and all 1 i n. 8 Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 8 / 1

9 A semantic criterion for infeasibility Proof by contradiction 1 R-feasiblity implies that σ(s i ) R σ(t i) (i.e., R σ(s i ) σ(t i )) for some substitution σ and all 1 i n. 2 Since A = R, by correctness, A = ( y i ) σ(s i ) σ(t i ) for all 1 i n, where y i are the variables in Var(σ(s i )) Var(σ(t i )). 8 Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 8 / 1

10 A semantic criterion for infeasibility Proof by contradiction 1 R-feasiblity implies that σ(s i ) R σ(t i) (i.e., R σ(s i ) σ(t i )) for some substitution σ and all 1 i n. 2 Since A = R, by correctness, A = ( y i ) σ(s i ) σ(t i ) for all 1 i n, where y i are the variables in Var(σ(s i )) Var(σ(t i )). 3 Thus, A = ( y) (σ(s 1 ) σ(t 1 ) σ(s n ) σ(t n )). 8 Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 8 / 1

11 A semantic criterion for infeasibility Proof by contradiction 1 R-feasiblity implies that σ(s i ) R σ(t i) (i.e., R σ(s i ) σ(t i )) for some substitution σ and all 1 i n. 2 Since A = R, by correctness, A = ( y i ) σ(s i ) σ(t i ) for all 1 i n, where y i are the variables in Var(σ(s i )) Var(σ(t i )). 3 Thus, A = ( y) (σ(s 1 ) σ(t 1 ) σ(s n ) σ(t n )). 4 For all ν : y A, [σ(s 1 ) σ(t 1 ) σ(s n ) σ(t n )] A ν, is true. 8 Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 8 / 1

12 A semantic criterion for infeasibility Proof by contradiction 1 R-feasiblity implies that σ(s i ) R σ(t i) (i.e., R σ(s i ) σ(t i )) for some substitution σ and all 1 i n. 2 Since A = R, by correctness, A = ( y i ) σ(s i ) σ(t i ) for all 1 i n, where y i are the variables in Var(σ(s i )) Var(σ(t i )). 3 Thus, A = ( y) (σ(s 1 ) σ(t 1 ) σ(s n ) σ(t n )). 4 For all ν : y A, [σ(s 1 ) σ(t 1 ) σ(s n ) σ(t n )] A ν, is true. 5 Since A has a nonempty domain, there is a valuation ν : x A given by ν (x) = [σ(x)] A ν for all variable x in x = n i=1 Var(s i) Var(t i ), such that [s 1 t 1 s n t n ] A ν is true. 8 Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 8 / 1

13 A semantic criterion for infeasibility Proof by contradiction 1 R-feasiblity implies that σ(s i ) R σ(t i) (i.e., R σ(s i ) σ(t i )) for some substitution σ and all 1 i n. 2 Since A = R, by correctness, A = ( y i ) σ(s i ) σ(t i ) for all 1 i n, where y i are the variables in Var(σ(s i )) Var(σ(t i )). 3 Thus, A = ( y) (σ(s 1 ) σ(t 1 ) σ(s n ) σ(t n )). 4 For all ν : y A, [σ(s 1 ) σ(t 1 ) σ(s n ) σ(t n )] A ν, is true. 5 Since A has a nonempty domain, there is a valuation ν : x A given by ν (x) = [σ(x)] A ν for all variable x in x = n i=1 Var(s i) Var(t i ), such that [s 1 t 1 s n t n ] A ν is true. 6 This contradicts A = ( x) (s 1 t 1 s n t n ). 8 Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 8 / 1

14 Examples of use Infeasible rule The following structure A over N {0}: a A = 1 b A = 2 f A (x) = x + 1 g A (x) = 1 x A y y x x ( ) A y y x is a model of R { ( x) f(x) x} for our running CTRS R. Automation This model has been automatically generated by using the tool AGES: Thus, rule is proved R-infeasible. g(x) g(a) f(x) x 9 Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 9 / 1

15 Examples of use Infeasible critical pair The following CTRS R (Example 23 in [Sternagel & Sternagel, FSCD 16]) g(x) f(x, x) (1) g(x) g(x) g(x) f(a, b) (2) has a conditional critical pair f(x, x) g(x) g(x) f(a, b). The following structure A over the finite domain {0, 1}: { a A = 1 b A = 0 f A x y + 1 if x y (x, y) = y x + 1 otherwise g A (x) = 1 x A y x = y x ( ) A y x y is a model R { ( x) g(x) f(a, b)}. The critical pair is infeasible. In the FSCD 16 paper, this is proved by using unification tests together with a transformation. It is discussed that the alternative tree automata techniques investigated in the paper do not work for this example. 10 Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 10 / 1

16 Examples of use Non-joinable terms Proposition (Joinability and feasibility) Let R be a CTRS, s, t be terms, and x be a fresh variable not occurring in s or t. If s and t are joinable, then s x, t x is R-feasible. If s and t are ground and s x, t x is R-feasible, then s and t are joinable. Consider the following CTRS R (Example in Ohlebusch s book): a b (3) f(x) c x a (4) Although there is no critical pair, the system is not (locally) confluent because f(a) R f(b) and f(a) R c but c and f(b) are not joinable. The following structure A over N { 1}: a A = 0 b A = 1 c A = 1 f A (x) = x + 1 x A y x = y x ( ) A y x = y is a model of U(R, f(b), c) { ( x) (f(b) x c x)}, where U(R, f(b), c) = {(4)} is the set of usable rules (from R) for f(b) and c. Therefore, f(b) and c are proved non-joinable. 11 Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 11 / 1

17 Conclusions and future work We have presented a semantic approach to prove infeasibility in conditional rewriting. We could handle many examples coming from papers developing different specific techniques to deal with these problems We do not have a dedicated, fully automated infeasibility checker yet. Instead we just encode the problem we want to deal with (e.g., infeasibility of a critical pair, or rule; or non-joinability) as an specific infeasibility sequence and then use AGES to find a model. Future work Improving automation, and the connection of AGES as a backend for other (confluence) tools are interesting subjects for future work. 12 Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 12 / 1

18 Conclusions and future work This paper defines feasibility problems as (instantiated) reachability problems, which corresponds to the use oriented CTRSs. As shown in [Lucas, LOPSTR 2017], the current treatment can be generalized to deal with more general notions of CTRSs, with many-sorted signatures, alternative satisfiability notions for the conditions (e.g., joinability), or more general components there (e.g., memberships). Future work The use of our semantic techniques in proofs of confluence of Maude programs is also an interesting subject for future work. 13 Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 13 / 1

19 Conclusions and future work Thanks! 14 Salvador Lucas and Raúl Gutiérrez A Semantic Criterion for Proving Infeasibility IWC 17, September 8 14 / 1