Hierarchic reasoning in local theory extensions

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1 Hierarchic reasoning in local theory extensions Viorica Sofronie-Stokkermans Max-Planck-Institut für Informatik Saarbrücken CDE 20, Tallinn, July 24 28,

2 Motivation Hierarchic reasoning in extensions of theories Example (Mathematics, Verification) T 0 = real numbers with ordering T 1 = free or monotone functions on real numbers Task: prove some property of free (monotone) functions use a prover for the real numbers as a black-box 2

3 Motivation Hierarchic reasoning in extensions of theories Example (Knowledge representation) T 0 = (semi)lattices (distributive lattices, Boolean algebras,...) T 1 = free or monotone functions on T 0 Task: prove some property of free (monotone) functions use a prover for T 0 as a black-box 3

4 Motivation T 1 T 1 : Σ 1 -theory; T 0 T 1 Σ 1 extension of Σ 0. T 0 T 0 : Σ 0 -theory. Can use a prover for T 0 as a black-box to prove theorems in T 1? 4

5 Motivation T 1 T 1 : Σ 1 -theory; T 0 T 1 Σ 1 extension of Σ 0. T 0 T 0 : Σ 0 -theory. Can use a prover for T 0 as a black-box to prove theorems in T 1? extensions with relations [Bachmair,Ganzinger,Waldmann 94] extensions with partial functions [Ganzinger,Waldmann,VS 04] - constraint superposition calculus hierarchic reasoning strong conditions on base theory: compact, universal. 5

6 This talk Extensions with function symbols: Identify situations when hierarchic reasoning is possible... and simple: - no sophisticated implementations are necessary 6

7 This talk Extensions with function symbols: Identify situations when hierarchic reasoning is possible... and simple: - no sophisticated implementations are necessary - complexity of the of complexity of the fragment of extension expressible in terms (or E ) fragment of base theory 7

8 Example: The sum of two Lipschitz functions R (L f c,λ 1 ) (L g c,λ 2 ) = x f (x)+g(x) (f (c)+g(c)) (λ 1 +λ 2 ) x c (L f c,λ 1 ) (L g c,λ 2 ) x f (x) f (c) λ 1 x c x g(x) g(c) λ 2 x c Problems: prover for R does not know about f, g prover for first-order logic may have problems with the reals Nelson-Oppen reasoning in theory combinations not possible 8

9 Example: The sum of two Lipschitz functions R (L f c,λ 1 ) (L g c,λ 2 ) = x f (x)+g(x) (f (c)+g(c)) (λ 1 +λ 2 ) x c (L f c,λ 1 ) (L g c,λ 2 ) x f (x) f (c) λ 1 x c x g(x) g(c) λ 2 x c Hierarchic reasoning reduce to the problem of checking a family of constraints over R Modular reasoning if we separate the information about f and g at the beginning: no need to combine the information again at a later point 9

10 Idea Σ 1 extension of Σ 0 with function symbols K set of Σ 1 -clauses; T 0 T 1 = T 0 K Task: Check whether T 1 G = pproach: Consider a relational approximation of the problem: (approximate extension functions with functional relations) T 0 K G = 10

11 Idea Σ 1 extension of Σ 0 with function symbols K set of Σ 1 -clauses; T 0 T 1 = T 0 K Task: Check whether T 1 G = pproach: Consider a relational approximation of the problem: (approximate extension functions with functional relations) T 0 K G = Soundness: T 0 K G = = T 1 G = 11

12 Idea Σ 1 extension of Σ 0 with function symbols K set of Σ 1 -clauses; T 0 T 1 = T 0 K Task: Check whether T 1 G = pproach: Consider a relational approximation of the problem: (approximate extension functions with functional relations) T 0 K G = Soundness: T 0 K G = = T 1 G = Completeness: T 0 K G = = E partial model If every partial model can be embedded into a total model of T 0 K then T 1 G = 12

13 Example: The sum of two Lipschitz functions R (L f c,λ 1 ) (L g c,λ 2 ) = x f (x)+g(x) (f (c)+g(c)) (λ 1 +λ 2 ) x c (L f c,λ 1 ) (L g c,λ 2 ) x f (x) f (c) λ 1 x c x g(x) g(c) λ 2 x c Partial models can be made total - approximate extension functions with functional relations T 0 K G = sound and complete Local theory extension - can even do better: T 0 K[G] G = 13

14 Overview Local theory extensions - various notions of locality of an extension - link with embeddability of partial algebras into total algebras - hierarchic reasoning (direct argument) - parameterized complexity 14

15 Locality, tractability, embeddability T 1 := K set of equational Horn clauses; K is local, if for ground Horn clauses C, K = C iff K[C] = C Local theories [Givan, Mcllester 92] capture PTIME 15

16 Locality, tractability, embeddability T 1 := K set of equational Horn clauses; K is local, if for ground Horn clauses C, K = C iff K[C] = C T 1 local theory Horn theory of T 1 in PTIME [Mcllester et al. 92, 93] [Ganzinger 01] Emb(T 1 ) [Skolem 20] [Evans 53,Burris 95] 16

17 Locality, tractability, embeddability T 1 := K set of equational Horn clauses; K is local, if for ground Horn clauses C, K = C iff K[C] = C T 1 local theory Horn theory of T 1 in PTIME [Mcllester et al. 92, 93] [Ganzinger 01] Emb(T 1 ) [Skolem 20] [Evans 53,Burris 95] K is stably local, if for ground Horn clauses C, K = C iff K [C] = C 17

18 Local extensions K set of equational clauses; T 0 theory; T 1 = T 0 K (Loc) T 0 T 1 is local, if for ground clauses G, T 0 K G = iff T 0 K[G] G has no (partial) model T 1 local extension of T 0 parameterized complexity for T 1 Emb(T 0, T 1 ) 18

19 Local extensions K set of equational clauses; T 0 theory; T 1 = T 0 K (Loc) T 0 T 1 is local, if for ground clauses G, T 0 K G = iff T 0 K[G] G has no (partial) model T 1 local extension of T 0 parameterized complexity for T 1 Emb(T 0, T 1 ) (SLoc) T 0 T 1 is stably local, if for ground clauses G, T 0 K G = iff T 0 K [G] G has no (partial) model 19

20 Local extensions K set of equational clauses; T 0 theory; T 1 = T 0 K (Loc) T 0 T 1 is local, if for ground clauses G, T 0 K G = iff T 0 K[G] G has no (partial) model T 1 local extension of T 0 parameterized complexity for T 1 Emb(T 0, T 1 ) (SLoc) T 0 T 1 is stably local, if for ground clauses G, T 0 K G = iff T 0 K [G] G has no (partial) model 20

21 Validity of clauses in partial algebras - Evans validity (equational case: [Evans 53]) - weak equality Horn clauses: [Burris 95], [Ganzinger 01] 21

22 Evans validity C := _ i u i v i _ j s j t j _ k L k C true iff one of its literals is true s t true if - s, t defined and equal, or - s and t undefined, or - s or t irrelevant (proper subterm undefined) u v true if - u, v defined and different, or - u or v undefined [ ]P(t 1,...t n ) similarly Example: Hold in standard partial model? Yes: car(nil) cdr(nil) car(nil) cdr(nil) No: car(nil) nil car(cdr(nil)) nil car(cdr(nil)) nil 22

23 Weak validity C := _ i u i v i _ j s j t j _ k L k C true iff one of its literals is true s t true if - s, t defined and equal, or - s or t undefined u v true if - u, v defined and different, or - u or v undefined P(t 1,...t n ) similarly P(t 1,...t n ) similarly Example: Hold in standard partial model? Yes: car(nil) cdr(nil) car(nil) cdr(nil) Yes: car(nil) nil car(cdr(nil)) nil car(cdr(nil)) nil 23

24 Embeddability of partial into total models T 1 = T 0 K, K set of clauses; Π 0 = (Σ 0, Pred), Π = (Σ 0 Σ 1, Pred) Notations: PMod(Σ 1, T 1 ) Evans partial models of K which are total models of T 0 PMod w (Σ 1, T 1 ) Weak partial models of K which are total models of T 0 Embeddability conditions (Emb) Every PMod(Σ 1, T 1 ) weakly embeds into a total model of T 1. (Emb w ) Every PMod w (Σ 1, T 1 ) weakly embeds into a total model of T 1. (Emb) f, (Emb) f w refer only to finite partial algebras Lemma: PMod(Σ 1, T 1 ) PMod w (Σ 1, T 1 ). Therefore (Emb w ) (Emb) 24

25 Embeddability implies locality Theorem K set of Σ 1 -flat, Σ 1 -linear clauses. Then for T 0 T 1 = T 0 K: (1) (Emb w ) (Loc). (2) T 0 locally finite, universal & K: finitely many ground terms: (Emb f w) (Loc f ). Similar relationships between embeddability of Evans partial models and stable locality (T 0 required to be universal; K can be arbitrary). 25

26 Embeddability implies locality Theorem K set of Σ 1 -flat, Σ 1 -linear clauses. Then for T 0 T 1 = T 0 K: (1) (Emb w ) (Loc). (2) T 0 locally finite, universal & K finitely many ground terms: (Emb f w) (Loc f ). Similar relationships between embeddability of Evans partial models and stable locality (T 0 required to be universal; K can be arbitrary). Consequence: Can identify local and stably local extensions by proving that embedding properties hold 26

27 Examples of local extensions Extensions of a theory T 0 : - with free function symbols - with selectors {s 1,..., s n } for an n-ary function c, injective in T 0. s i (c(x 1,..., x n )) = x i x = c(x 1,..., x n ) c(s 1 (x),..., s n (x)) = x - with monotone functions for: T 0 {Posets, TotOrd,DenseTotOrd,Lat, SLat, DLat, Boollg} T 0 = R theory of real numbers - with λ-lipschitz functions at c for: T 0 = R theory of real numbers 27

28 Example R (L f c,λ 1 ) (L g c,λ 2 ) = x f (x)+g(x) (f (c)+g(c)) (λ 1 +λ 2 ) x c (L f c,λ 1 ) (L g c,λ 2 ) x f (x) f (c) λ 1 x c x g(x) g(c) λ 2 x c 9 = ; set of clauses K Solution: R K f (d) + g(d) (f (c) + g(c)) (λ 1 + λ 2 ) x c {z } G = 28

29 Example R (L f c,λ 1 ) (L g c,λ 2 ) = x f (x)+g(x) (f (c)+g(c)) (λ 1 +λ 2 ) x c (L f c,λ 1 ) (L g c,λ 2 ) x f (x) f (c) λ 1 x c x g(x) g(c) λ 2 x c 9 = ; set of clauses K Solution: R K f (d) + g(d) (f (c) + g(c)) (λ 1 + λ 2 ) x c {z } G = Locality condition: R K[G] G has no partial model where f (c), f (d), g(c), g(d) defined 29

30 Example R (L f c,λ 1 ) (L g c,λ 2 ) = x f (x)+g(x) (f (c)+g(c)) (λ 1 +λ 2 ) x c (L f c,λ 1 ) (L g c,λ 2 ) x f (x) f (c) λ 1 x c x g(x) g(c) λ 2 x c 9 = ; set of clauses K Solution: R K f (d) + g(d) (f (c) + g(c)) (λ 1 + λ 2 ) x c {z } G = Locality condition: R K[G] G has no partial model where f (c), f (d), g(c), g(d) defined K[G] : f (d) f (c) λ 1 d c f (c) f (c) λ 1 c c g(d) g(c) λ 2 d c g(c) g(c) λ 2 c c size: polynomial in st(g) for a fixed K 30

31 Relational translations T 1 = T 0 K, K set of clauses; Π 0 = (Σ 0, Pred), Π = (Σ 0 Σ 1, Pred) Step 1: purify and flatten (abstract out Σ 1 -terms) 8 c + d = c 1 >< f (c 1 ) = c 2 Example: f (c + d) f (d) l c f (d) = c 3 >: c 2 + c 3 l c 31

32 Relational translations T 1 = T 0 K, K set of clauses; Π 0 = (Σ 0, Pred), Π = (Σ 0 Σ 1, Pred) Step 1: purify and flatten (abstract out Σ 1 -terms) 8 c + d = c 1 >< f (c 1 ) = c 2 Example: f (c + d) f (d) l c f (d) = c 3 >: c 2 + c 3 l c Step 2: encode functions in Σ 1 as relations Example: f (d) = c 3 r f (d, c 3 ) Step 3: express functionality of relations r f (ground instances [G]) 32

33 Example Show: R K[G] G no partial model with f (c), f (d), g(c), g(d) defined K[G] f (d) f (c) λ 1 d c f (c) f (c) λ 1 c c g(d) g(c) λ 2 d c g(c) g(c) λ 2 c c G f (d) + g(d) (f (c) + g(c)) (λ 1 + λ 2 ) d c 1. Flatten K[G] G f (d) = d 1 d 1 c 1 λ 1 d c f (c) = c 1 c 1 c 1 λ 1 c c g(d) = d 2 d 2 c 2 λ 2 d c g(c) = c 2 c 2 c 2 λ 2 c c d 1 + d 2 c 1 c 2 (λ 1 + λ 2 ) d c 33

34 Example Show: R K[G] G no partial model with f (c), f (d), g(c), g(d) defined K[G] f (d) f (c) λ 1 d c f (c) f (c) λ 1 c c g(d) g(c) λ 2 d c g(c) g(c) λ 2 c c G f (d) + g(d) (f (c) + g(c)) (λ 1 + λ 2 ) d c 2. Replace functions with functional relations in K[G] G r f (d, d 1 ) d 1 c 1 λ 1 d c c = d r f (c, c 1 ) r f (d, d 1 ) c 1 = d 1 r f (c, c 1 ) c 1 c 1 λ 1 c c r g (d, d 2 ) d 2 c 2 λ 2 d c c = d r g (c, c 2 ) r g (d, d 2 ) c 2 = d 2 r g (c, c 2 ) c 2 c 2 λ 2 c c d 1 + d 2 c 1 c 2 (λ 1 + λ 2 ) d c 34

35 Example Show: R K[G] G no partial model with f (c), f (d), g(c), g(d) defined K[G] f (d) f (c) λ 1 d c f (c) f (c) λ 1 c c g(d) g(c) λ 2 d c g(c) g(c) λ 2 c c G f (d) + g(d) (f (c) + g(c)) (λ 1 + λ 2 ) d c (3) resolve away the functional relations r f (d, d 1 ) d 1 c 1 λ 1 d c c = d c 1 = d 1 r f (c, c 1 ) c 1 c 1 λ 1 c c r g (d, d 2 ) d 2 c 2 λ 2 d c c = d c 2 = d 2 r g (c, c 2 ) c 2 c 2 λ 2 c c d 1 + d 2 c 1 c 2 (λ 1 + λ 2 ) d c 35

36 Example Show: R K[G] G no partial model with f (c), f (d), g(c), g(d) defined K[G] f (d) f (c) λ 1 d c f (c) f (c) λ 1 c c g(d) g(c) λ 2 d c g(c) g(c) λ 2 c c G f (d) + g(d) (f (c) + g(c)) (λ 1 + λ 2 ) d c (4) check the satisfiability of a set of constraints over R r f (d, d 1 ) d 1 c 1 λ 1 d c c = d c 1 = d 1 r f (c, c 1 ) c 1 c 1 λ 1 c c r g (d, d 2 ) d 2 c 2 λ 2 d c c = d c 2 = d 2 r g (c, c 2 ) c 2 c 2 λ 2 c c d 1 + d 2 c 1 c 2 (λ 1 + λ 2 ) d c 36

37 Parameterized complexity T 0 K G T 0 K[G] G polynomial in st(g) T 0 K 0 G 0 D Fun(D ) quadratic T 0 K 0 G 0 N 0 constant constraint solver for base theory to check satisfiability N 0 = { V n i=1 c i d i c = d r f (c, c), r f (d, d) D }. Theorem. The following are equivalent (1) T 0 K[G] G has a partial model (all ground terms defined) (2) T 0 K 0 G 0 N 0 has a (total) model. 37

38 Parameterized complexity T 0 K G T 0 K[G] G polynomial in st(g) T 0 K 0 G 0 D Fun(D ) quadratic T 0 K 0 G 0 N 0 constant constraint solver for base theory to check satisfiability N 0 = { V n i=1 c i d i c = d r f (c, c), r f (d, d) D }. Theorem. The following are equivalent (1) T 0 K[G] G has a partial model (all ground terms defined) (2) T 0 K 0 G 0 N 0 has a (total) model. ground ground if all variables in K shielded by an extension function with free variables otherwise 38

39 Parameterized complexity T 0 K G T 0 K[G] G polynomial in st(g) T 0 K 0 G 0 D Fun(D ) quadratic T 0 K 0 G 0 N 0 constant constraint solver for base theory to check satisfiability N 0 = { V n i=1 c i d i c = d r f (c, c), r f (d, d) D }. Theorem. ssumptions: (1) T 0 T 1 has (Loc f ) or (2) T 0 T 1 has (SLoc f ), T 0 locally finite. (a) If variables in K shielded: Th (b) Otherwise: Th E (T 0 ) decidable Th (T 0 ) decidable Th (T 1 ) decidable. (T 1 ) decidable. 39

40 Conclusions Local theory extensions - hierarchical reasoning - parameterized complexity results Future work - go beyond the universal fragment - modular reasoning: combinations of local theory extensions - use results to generate interpolants (applications in verification) 40

Locality and subsumption testing in EL and some of its extensions

Locality and subsumption testing in EL and some of its extensions Viorica Sofronie-Stokkermans abstract. In this paper we show that subsumption problems in many lightweight description logics (including