, We distinguish two cases
, If b j a 1 then h (s 1 (x)) = h(s 1 (x)), and s 1 (x) {a 1 , . . . , a i }, thus h(s 1 (x) {a 2 , . . . , a i+1 }, hence is false in both structures. , with s 1 (y) = s 1 (y) {a 2 , . . . , a i }. We have h (s 1 (x)) = a o+1 and h(s 1 (x)) = a o , thus h (s 1 (x)), h(s 1 (x)) ? {a 3, hence has the same truth value in, vol.15
, ) necessarily coincide on ?, and consequently
, ? be a prenex formula of SL 1 of free variables x 1 , . . . , x n (with n > 0) where ? is a boolean combination of universe-independent test formulas, Lemma 17. Let ? = ?y 1, vol.3
, Corollary 18. The finite and infinite satisfiability problems for formulas of BSR
, Indeed, this proof does not depend on the universe being infinite or the fact that k = 2. There remains to show PSPACE-membership for both problems. Observe that this does not directly follow from Lemmas 4 and 17, because (i) the sets µ inf (?) and µ fin (?) are of exponential size hence no efficient algorithm can compute them and, (ii) Lemma 17 only holds for universe-independent formulas. W.l.o.g., we assume that the considered formula contains at least one free variable and is of the form ?y 1
, x i+1 ) ? i+1 j=1 ¬alloc(x j ), -|h| ? |U| ? i ? ?x 1 , . . . , x i . dist(x 1 , . . . , x i ) ? i j=1 ¬alloc(x j ), -|U| ? i ? ?x 1 , . . . , x i+1 ¬dist(x 1 , . . . , x i+1 ). Let ? be the conjunction of all formulas ?( ) where ? C. Note that ? contains (up to redundancy) at most 3L + 2 existential variables and 3L + 2 universal variables. Now consider the formula ? obtained from ? by replacing every test formula such that ? C (resp. ? C) by (resp. ?). Let ? be the formula obtained by putting ?y 1 , . . . , y m . ¬? ? ? in prenex form, By Theorem 8, ?y 1 , . . . , y m . ? has an infinite model iff ?y 1 , . . . , y m . ? ? ? n+m has a finite model, where the size of ? n+m is quadratic in n + m. Moreover, since ? n+m is a BSR(SL) formula, ?y 1 , . . . , y m . ? ? ? n+m is also a BSR(SL) formula. Hence infinite satisfiability can be reduced polynomially to finite satisfiability. Let ? = M?µ fin (¬?) M (note that the size of ? is exponential w.r.t. that of ?). Let L be the maximal number l such that a test formula |h| ? l or |h| ? |U| ? l occurs in µ inf (?)
, Conclusion We have shown that the prenex fragment of Separation Logic over heaps with one selector, denoted as SL 1 , is decidable in time not elementary recursive. Moreover, the Bernays-Schönfinkel-Ramsey BSR(SL 1 ) is PSPACE-complete. These results settle an open question raised in [6] and allow one to draw a precise boundary between decidable and undecidable cases inside BSR(SL k ), Theorem 8, relating infinite and finite satisfiability, holds for
, The Classical Decision Problem. Perspectives in Mathematical Logic, 1997.
, On the almighty wand, Information and Computation, vol.211, pp.106-137, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01905158
, Infer: An Automatic Program Verifier for Memory Safety of C Programs, Proc. of NASA Formal Methods'11, vol.6617, 2011.
, Computability and complexity results for a spatial assertion language for data structures, FST TCS 2001, Proceedings, pp.108-119, 2001.
, Separation Logic with One Quantified Variable, CSR'14, vol.8476, pp.125-138
URL : https://hal.archives-ouvertes.fr/hal-01258821
, , 2014.
, The Bernays-Schönfinkel-Ramsey Class of Separation Logic on Arbitrary Domains, Foundations of Software Science and Computation Structures -22nd International Conference, FOSSACS 2019, Held as part of ETAPS 2019, pp.242-259, 2019.
, First-Order Logic and Automated Theorem Proving. Texts and Monographs in Computer Science, 1990.
, Bi as an assertion language for mutable data structures, ACM SIGPLAN Notices, vol.36, pp.14-26, 2001.
, Expressivité des logiques spatiales, 2004.
, Decidability of Second-Order Theories and Automata on Infinite Trees, Transactions of the American Mathematical Society, vol.141, pp.1-35, 1969.
, Separation Logic: A Logic for Shared Mutable Data Structures, Proc. of LICS'02, 2002.