1/32 Fixed-point elimination in Heyting algebras 1 Silvio Ghilardi, Università di Milano Maria João Gouveia, Universidade de Lisboa Luigi Santocanale, Aix-Marseille Université TACL@Praha, June 2017 1 See [Ghilardi et al., 2016]
2/32 Plan A primer on mu-calculi The intuitionistic µ-calculus The elimination procedure Bounding closure ordinals
3/32 µ-calculi Add to a given algebraic framework syntactic least and greatest fixed-point constructors.
3/32 µ-calculi Add to a given algebraic framework syntactic least and greatest fixed-point constructors. E.g., the propositional modal µ-calculus: φ := x x φ φ φ φ φ φ µ x.φ ν x.φ, when x is positive in φ.
3/32 µ-calculi Add to a given algebraic framework syntactic least and greatest fixed-point constructors. E.g., the propositional modal µ-calculus: φ := x x φ φ φ φ φ φ µ x.φ ν x.φ, when x is positive in φ. Interpret the syntactic least (resp. greatest) fixed-point as expected. µ x.φ v := least fixed-point of the monotone mapping X φ v,x /x
4/32 Alternation hierarchies in µ-calculi Let count the number of alternating blocks of fixed-points.
4/32 Alternation hierarchies in µ-calculi Let count the number of alternating blocks of fixed-points. Problem. For a given µ-calculus, does there exist n such that, for each φ with φ > n, there exists ψ with γ ψ and ψ n?
5/32 Alternation hierarchies, facts The alternation hierarchy for the modal µ-calculus is infinite (there exists no such n) [Lenzi, 1996, Bradfield, 1998]. Idem for the lattice µ-calculus [Santocanale, 2002]. The alternation hierarchy for the linear µ-calculus ( x = x) is reduced to the Büchi fragment (here n = 2). The alternation hierarchy for the modal µ-calculus on transitive frames collapses to the alternation free fragment (here n = 1.5) [Alberucci and Facchini, 2009]. The alternation hierarchy for the distributive µ-calculus is trivial (here n = 0) [Kozen, 1983].
5/32 Alternation hierarchies, facts The alternation hierarchy for the modal µ-calculus is infinite (there exists no such n) [Lenzi, 1996, Bradfield, 1998]. Idem for the lattice µ-calculus [Santocanale, 2002]. The alternation hierarchy for the linear µ-calculus ( x = x) is reduced to the Büchi fragment (here n = 2). The alternation hierarchy for the modal µ-calculus on transitive frames collapses to the alternation free fragment (here n = 1.5) [Alberucci and Facchini, 2009]. The alternation hierarchy for the distributive µ-calculus is trivial (here n = 0) [Kozen, 1983]. A research plan: Develop a theory explaining why alternation hierarchies collapses.
6/32 µ-calculi on generalized distributive lattices Theorem. [Frittella and Santocanale, 2014] There are lattice varieties (Nation s varieties) D 0 D 1... D n... with D 0 the variety of distributive lattices, such that, on D n and for any lattice term φ, φ n+2 ( ) = φ n+1 ( ) (= µ x.φ), φ n ( ) φ n+1 ( ).
6/32 µ-calculi on generalized distributive lattices Theorem. [Frittella and Santocanale, 2014] There are lattice varieties (Nation s varieties) D 0 D 1... D n... with D 0 the variety of distributive lattices, such that, on D n and for any lattice term φ, φ n+2 ( ) = φ n+1 ( ) (= µ x.φ), φ n ( ) φ n+1 ( ). Corollary. The alternation hierarchy of the lattice µ-calculus is trivial on D n, for each n 0.
7/32 Plan A primer on mu-calculi The intuitionistic µ-calculus The elimination procedure Bounding closure ordinals
8/32 The intuitionistic µ-calculus After the distributive µ-calculus, the next on the list by Pitt s quantifiers, we knew that least fixed-points and greatest fixed-points are definable. We extend the signature of Heyting algebras (i.e. Intuitionistic Logic) with least and greatest fixed-point constructors. Intuitionistic µ-terms are generated by the grammar: φ := x φ φ φ φ φ φ µ x.φ ν x.φ, when x is positive in φ.
9/32 Heyting algebra semantics We take any provability semantics of IL with fixed points: (Complete) Heyting algebras. Kripke frames. Any sequent calculus for Intuitionisitc Logic (e.g. LJ) plus Park/Kozen s rules for least and greatest fixed-points: φ[ψ/x] ψ Γ φ(µ x.φ) φ(ν x.φ) δ ψ φ[ψ/x] µ x.φ ψ Γ µ x.φ ν x.φ δ ψ ν x.φ Definition. A Heyting algebra is a bounded lattice H = H,,,, with an additional binary operation satisfying x y z iff x y z.
10/32 Ruitenburg s theorem [Ruitenburg, 1984] Theorem. For each intuitionistic formula φ, there exists n 0 such that φ n (x) φ n+2 (x).
10/32 Ruitenburg s theorem [Ruitenburg, 1984] Theorem. For each intuitionistic formula φ, there exists n 0 such that φ n (x) φ n+2 (x). Then φ n ( ) φ n+1 ( ) φ n+2 ( ) = φ n ( ), so φ n ( ) is the least fixed-point of φ.
10/32 Ruitenburg s theorem [Ruitenburg, 1984] Theorem. For each intuitionistic formula φ, there exists n 0 such that φ n (x) φ n+2 (x). Then φ n ( ) φ n+1 ( ) φ n+2 ( ) = φ n ( ), so φ n ( ) is the least fixed-point of φ. Corollary. The alternation hierarchy for the intuitionistic µ-calculus is trivial.
10/32 Ruitenburg s theorem [Ruitenburg, 1984] Theorem. For each intuitionistic formula φ, there exists n 0 such that φ n (x) φ n+2 (x). Then φ n ( ) φ n+1 ( ) φ n+2 ( ) = φ n ( ), so φ n ( ) is the least fixed-point of φ. Corollary. The alternation hierarchy for the intuitionistic µ-calculus is trivial. NB : Ruitenburg s n might not be the closure ordinal of µ x.φ.
11/32 Peirce, compatibility, strenghs and strongness Proposition. Peirce s theorem for Heyting algebras. Every term φ is compatible. In particular, for ψ, χ arbitrary terms, the equation φ[ψ/x] χ = φ[ψ χ/x] χ. holds on Heyting algebras. Corollary. Every term φ monotone in x is strong in x. That is, any the following equivalent conditions φ[ψ/x] χ φ[ψ χ/x], ψ χ φ[ψ/x] φ[χ/x], hold, for any terms ψ and χ.
12/32 Plan A primer on mu-calculi The intuitionistic µ-calculus The elimination procedure Bounding closure ordinals
13/32 Greatest fixed-points Proposition. On Heyting algebras, we have ν x.φ = φ( ).
13/32 Greatest fixed-points Proposition. On Heyting algebras, we have ν x.φ = φ( ). Using the deduction theorem and Pitts quantifiers: ν x.φ(x) = x.(x x φ(x)) = x.(x φ(x)) = φ( ).
13/32 Greatest fixed-points Proposition. On Heyting algebras, we have ν x.φ = φ( ). Using the deduction theorem and Pitts quantifiers: ν x.φ(x) = x.(x x φ(x)) = x.(x φ(x)) = φ( ). Using strongness: φ( ) = φ( ) φ( ) φ( φ( )) = φ 2 ( ).
14/32 Greatest solutions of systems of equations Proposition. On Heyting algebras, a system of equations x 1 = φ 1 (x 1,..., x n ). x n = φ n (x 1,..., x n ) has a greatest solution obtained by iterating n times from. φ := φ 1,..., φ n Proof. Using the Bekic property.
15/32 Least fixed-points: splitting the roles of variables Due to µ x.φ(x, x) = µ x.µ y.φ(x, y)
15/32 Least fixed-points: splitting the roles of variables Due to µ x.φ(x, x) = µ x.µ y.φ(x, y) we can separate computing the least fixed-points w.r.t: weakly negative variables: variables that appear within the left-hand-side of an implication, fully positive variables: those appearing only within the right-hand-side of an implication.
16/32 Weakly negative least fixed-points: an example Use µ x.(f g)(x) = f ( µ y.(g f )(y) )
16/32 Weakly negative least fixed-points: an example Use µ x.(f g)(x) = f ( µ y.(g f )(y) ) to argue that: µ x.[ (x a) b ] = [ ν y.(y b) a ] b = [ ( b) a ] b = [ b a ] b.
17/32 Weakly negative least fixed-points: reducing to greatest fixed-points If each occurrence of x in φ is weakly negative, then φ(x) = φ 0 [φ 1 (x)/y 1,..., φ n (x)/y n ] with φ 0 (y 1,..., y n ) negative in each y j. Due to µ x.(f g)(x) = f ( µ y.(g f )(y) )
17/32 Weakly negative least fixed-points: reducing to greatest fixed-points If each occurrence of x in φ is weakly negative, then φ(x) = φ 0 [φ 1 (x)/y 1,..., φ n (x)/y n ] with φ 0 (y 1,..., y n ) negative in each y j. Due to µ x.(f g)(x) = f ( µ y.(g f )(y) ) we have µ x.φ(x) = µ x.( φ 0 φ 1,..., φ n )(x) = φ 0 ( ν y1...y n.( φ 1,... φ n φ 0 )(y 1,... y n )).
18/32 Interlude: least fixed-points of strong functions If f and f i, i I, are strong, then µ x.a f (x) = a µ x.f (x), µ x. f i (x) = µ x.f i (x), i I µ x.a f (x) = a µ x.f (x).
19/32 Strongly positive fixed-points: disjunctive formulas The equation µ x. i I f i (x) = i I µ x.f i (x) allows to push least fixed-points down through conjunctions.
19/32 Strongly positive fixed-points: disjunctive formulas The equation µ x. i I f i (x) = i I µ x.f i (x) allows to push least fixed-points down through conjunctions. Once all conjunctions have been pushed up in formulas, we are left to compute least fixed-points of disjunctive formulas, generated by the grammar: φ = x β φ α φ i=1,...,n φ i, where α and β do not contain the variable x. We call α an head subformula and β a side subformula.
20/32 Least fixed-points of inflating functions All functions f denoted by such formula φ are (monotone and) inflating: x f (x). Let f i, i = 1,..., n, be a collection of monotone inflating functions. Then µ x. f i (x) = µ x.(f 1... f n )(x). i=1,...,n
21/32 Least fixed-points of disjunctive formulas Proposition. Let φ be a disjunctive formula, with Head(φ) (resp., Side(φ)) the collection of its head (resp., side) subformulas. Then µ x.φ = α β. α Head(φ) β Side(φ)
21/32 Least fixed-points of disjunctive formulas Proposition. Let φ be a disjunctive formula, with Head(φ) (resp., Side(φ)) the collection of its head (resp., side) subformulas. Then µ x.φ = α β. α Head(φ) β Side(φ) If Head(φ) = { α 1,..., α n } and Side(φ) = { β 1,..., β m }: µ x.φ = µ x.α 1 α 2... α n β 1... β m x = µ x. α β x = = α Head(φ) α Head(φ) α µ x. α β Side(φ) β Side(φ) α Head(φ) β Side(φ) β. β x
22/32 Plan A primer on mu-calculi The intuitionistic µ-calculus The elimination procedure Bounding closure ordinals
23/32 Closure ordinals Definition. (Closure ordinal). For K a class of models and φ(x) a monotone formula/term, let cl K (φ) = least ordinal α such that M = µ x.φ = φ α ( ). In general, cl K (φ) might not exist. If H is the class of Heyting algebras and φ(x) is an intuitionistic formula, then cl H (φ) < ω.
24/32 Upper bounds from fixed-point equations cl K ( f g ) cl K ( g f ) + 1, cl H ( φ 0 (φ 1 (x),..., φ n (x)) ) n + 1, when φ 0 (y 1,..., y n ) contravariant, cl H ( φ ) card(head(φ)) + 1, when φ is a disjunctive formula, cl K ( f ) n cl K (g), when n = cl K (f (x, )) and g(x) = µ y.f (x, y), cl K (f g) cl K (f ) + cl H (g) 1, when f and g are strong.
25/32 Examples 1. Weakly negative x: i=1,...,n (x a i ) b i converges after n + 1 steps. This upper bound is strict.
25/32 Examples 1. Weakly negative x: i=1,...,n (x a i ) b i converges after n + 1 steps. This upper bound is strict.
25/32 Examples 1. Weakly negative x: i=1,...,n (x a i ) b i converges after n + 1 steps. This upper bound is strict. 2. Fully positive x: b a i x i=1,...,n converges after n + 1 steps. This upper bound is strict.
26/32 Examples (II) Similarly, φ(x) := (x a i ) b i i=1,...,n converges within n + 1 steps, according to the general theory.
26/32 Examples (II) Similarly, φ(x) := (x a i ) b i i=1,...,n converges within n + 1 steps, according to the general theory. Theorem. For any n 2, φ(x) converges to its least fixed-point within 3 steps.
27/32 Back to Ruitenburg s theorem Inspection of Ruitenburg s paper shows that cl H (φ) = O(n), where n is the number of implication symbols in φ.
27/32 Back to Ruitenburg s theorem Inspection of Ruitenburg s paper shows that cl H (φ) = O(n), where n is the number of implication symbols in φ. Given a fully positive formula φ, pushing up conjuctions yields a formula δ i, δ i disjunctive, i=1,...,k where k might be exponential w.r.t. the size of φ.
27/32 Back to Ruitenburg s theorem Inspection of Ruitenburg s paper shows that cl H (φ) = O(n), where n is the number of implication symbols in φ. Given a fully positive formula φ, pushing up conjuctions yields a formula δ i, δ i disjunctive, i=1,...,k where k might be exponential w.r.t. the size of φ. Our method yields the upper bound cl H (φ) = cl H ( δ i ) 1 k + i=1,...,k exponential w.r.t. the size of φ. i=1,...,k cl H (δ i ),
28/32 Closing the gap? Problem. Let δ 1, δ 2,..., δ n be disjunctive formulas and put φ(x) := δ i (x). i=1,...,n Does µ x.φ(x) = i=1,...,n Head(δi ) Side(δ i ) φ H+1 ( ), with H = card( i=1,...,n Head(δ i))? This holds (non trivially) for n = 2.
28/32 Closing the gap? Problem. Let δ 1, δ 2,..., δ n be disjunctive formulas and put φ(x) := δ i (x). i=1,...,n Does µ x.φ(x) = i=1,...,n Head(δi ) Side(δ i ) φ H+1 ( ), with H = card( i=1,...,n Head(δ i))? This holds (non trivially) for n = 2. Not the only plausible conjecture.
29/32 After thoughts A decision procedure for the Intuitionistic µ-calculus. Axiomatization of fixed-points and of some Pitt s quantifiers. General theory of fixed-point elimination: no uniform upper bounds for closure ordinals. Relevance of strongness it looks like Pitt s quantifiers less relevant. A working path to understand Ruitenburg s theorem.
Thanks! Questions? 30/32
31/32 References I Alberucci, L. and Facchini, A. (2009). The modal µ-calculus hierarchy on restricted classes of transition systems. The Journal of Symbolic Logic, 74(4):1367 1400. Bradfield, J. C. (1998). The modal µ-calculus alternation hierarchy is strict. Theor. Comput. Sci., 195(2):133 153. Frittella, S. and Santocanale, L. (2014). Fixed-point theory in the varieties D n. In Höfner, P., Jipsen, P., Kahl, W., and Müller, M. E., editors, Relational and Algebraic Methods in Computer Science - 14th International Conference, RAMiCS 2014, Marienstatt, Germany, April 28-May 1, 2014. Proceedings, volume 8428 of Lecture Notes in Computer Science, pages 446 462. Springer. Ghilardi, S., Gouveia, M. J., and Santocanale, L. (2016). Fixed-point elimination in the intuitionistic propositional calculus. In Jacobs, B. and Löding, C., editors, Foundations of Software Science and Computation Structures - 19th International Conference, FOSSACS 2016, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2016, Eindhoven, The Netherlands, April 2-8, 2016, Proceedings, volume 9634 of Lecture Notes in Computer Science, pages 126 141. Springer. Ghilardi, S. and Zawadowski, M. W. (1997). Model completions, r-heyting categories. Ann. Pure Appl. Logic, 88(1):27 46. Kozen, D. (1983). Results on the propositional mu-calculus. Theor. Comput. Sci., 27:333 354.
32/32 References II Lenzi, G. (1996). A hierarchy theorem for the µ-calculus. In auf der Heide, F. M. and Monien, B., editors, Automata, Languages and Programming, 23rd International Colloquium, ICALP96, Paderborn, Germany, 8-12 July 1996, Proceedings, volume 1099 of Lecture Notes in Computer Science, pages 87 97. Springer. Pitts, A. M. (1992). On an interpretation of second order quantification in first order intuitionistic propositional logic. J. Symb. Log., 57(1):33 52. Ruitenburg, W. (1984). On the period of sequences (a n (p)) in intuitionistic propositional calculus. The Journal of Symbolic Logic, 49(3):892 899. Santocanale, L. (2002). The alternation hierarchy for the theory of µ-lattices. Theory and Applications of Categories, 9:166 197.