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Reasoning in Complex Theories and Applications Viorica Sofronie-Stokkermans Max-Planck-Institut f ur Informatik Saarbr ucken, Germany Tutorial KI 2008, September 23, 2008 1 Motivation Long-term goal of research in computer science -

Reminder: Variables Predicate logic admits the formulation of abstract, schematic assertions. (Object) variables are the technical tool for schematization. We assume that X is a given countably infinite set of symbols which we use for (the denotation of) variables. 21

Terms Terms over Σ (Σ-terms) are formed according to these syntactic rules: , x ∈ X s , t , u , v ::= x (variable) | f ( s 1 , ..., s n ) , f / n ∈ Ω (functional term) By T Σ ( X ) we denote the set of Σ-terms (over X ). A term not containing any variable is called a ground term. Example 1: Σ 1 = ( { f /2, g /1, c /0 } , { P /2 } ) Terms: f ( g ( x ), y ), g ( f ( x , c )), c Example 2: Σ 2 = ( { +/2, s /1, 0/0 } , {≤ /2 } ) Terms: x + s ( y ), s (0 + s ( z )), 0 22

Reminder: Atoms Atoms (atomic formulas) over Σ are formed according to this syntax: , p / m ∈ Π A , B ::= p ( s 1 , ..., s m ) h i | ( s ≈ t ) (equation) Whenever we admit equations as atomic formulas we are in the realm of first-order logic with equality. Example 1: Σ 1 = ( { f /2, g /1, c /0 } , { P /2 } ) Atoms: P ( f ( g ( x ), y ), g ( f ( x , c ))), P ( c , c ) Example 2: Σ 2 = ( { +/2, s /1, 0/0 } , {≤ /2 } ) Atoms: x + s ( y ) ≤ s (0 + s ( z )), s (0) ≤ 0 23

Reminder: Literals and clauses Literals: L ::= A (positive literal) | ¬ A (negative literal) Clauses ⊥ C , D ::= (empty clause) | L 1 ∨ . . . ∨ L k , k ≥ 1 (non-empty clause) 24

Conventions In what follows we will use the following conventions: constants (0-ary function symbols) are denoted with a , b , c , d , ... function symbols with arity ≥ 1 are denoted • f , g , h , ... if the formulae are interpreted into arbitrary algebras • +, − , s , ... if the intended interpretation is into numerical domains predicate symbols with arity 0 are denoted p , q , r , s , ... predicate symbols with arity ≥ 1 are denoted • P , Q , R , ... if the formulae are interpreted into arbitrary algebras • ≤ , ≥ , < , > if the intended interpretation is into numerical domains variables are denoted x , y , z , ... 26

Reminder: Σ -algebras, valuations Σ -algebra A = ( A , { f A : A a ( f ) → A } f ∈ Ω , { P A ⊆ A a ( P ) } P ∈ Pred ) Valuation in a Σ -algebra A = ( A , { f A } f ∈ Ω , { P A } P ∈ Pred ) : β : X → A extends in a canonical way to terms: Example 1: Σ 1 = ( { f /2, g /1, c /0 } , { P /2 } ) A 1 = ( { a , b } , { f A , g A , c A } , { P A } ) f A : A 2 → A f A ( a , a ) = f A ( a , b ) = a , f A ( b , a ) = f A ( b , b ) = b , g A : A → A g A ( a ) = b , g A ( b ) = a , c A = a ∈ A P A ⊆ A 2 P A = { ( b , a ) } . Valuation in A 1 . β : X → { a , b } with β ( x ) = a , β ( y ) = b ( A 1 , β )( f ( g ( x ), y )) = f A ( g A ( β ( x )), β ( y )) = f A ( g A ( a ), b ) = b Example 2: Σ 2 = ( { +/2, s /1, 0/0 } , {≤ /2 } ) A 2 = ( Z , { +, s , 0 } , {≤} ) Valuation in A 2 . β : X → Z with β ( x ) = 4, β ( y ) = 5 ( A 2 , β )(+( s ( x ), y )) = s ( β ( x )) + β ( y ) = s (4) + 5 = 5 + 5 = 10 29

Reminder: Σ -algebras, valuations Σ -algebra A = ( A , { f A : A a ( f ) → A } f ∈ Ω , { P A ⊆ A a ( P ) } P ∈ Pred ) Valuation in a Σ -algebra A = ( A , { f A } f ∈ Ω , { P A } P ∈ Pred ) : β : X → A extends in a canonical way to formulae: P A ⊆ A 2 P A = { ( b , a ) } ; Example 1: A 1 = ( { a , b } , { f A , g A , c A } , { P A } ) f A : A 2 → A f A ( a , a ) = f A ( a , b ) = a , f A ( b , a ) = f A ( b , b ) = b , g A : A → A g A ( a ) = b , g A ( b ) = a , c A = a ∈ A β : X → { a , b } with β ( x ) = a , β ( y ) = b A ( A 1 , β )( xP ( f ( g ( x ), x ), c )) = min { ( A 1 , β [ x �→ d ]) P ( f ( g ( x ), x ), c ) | d ∈{ a , b }} = min { P A ( f A ( g A ( a ), a ), c A ), P A ( f A ( g A ( b ), b ), c A ) } = min { P A ( f A ( b , a ), a ), P A ( f A ( a , b ), a ) } = min(1, 0) = 0 E ( A 1 , β )( xP ( f ( g ( x ), x ), c )) = max { ( A 1 , β [ x �→ d ]) P ( f ( g ( x ), x ), c ) | d ∈{ a , b }} = max { P A ( f A ( g A ( a ), a ), c A ), P A ( f A ( g A ( b ), b ), c A ) } = max { P A ( f A ( b , a ), a ), P A ( f A ( a , b ), a ) } = max(1, 0) = 1 30

Reminder: Satisfiability, Validity, Entailment ( A , β ) | = F iff ( A , β )( F ) = 1 A | = F iff ( A , β )( F ) = 1 for all β : X → A F is valid iff | = F iff ( A , β )( F ) = 1 for all A and all β : X → A Validity Satisfiability F is satisfiable iff ( A , β )( F ) = 1 for some A and β : X → A Note: F valid iff ¬ F is unsatisfiable Entailment: A A F | = G iff ( A β : X → A if ( A , β ) | = F then ( A , β ) | = G ). A A N | A β : X → A if ( A , β ) | = F for all F ∈ N then ( A , β ) | = G iff ( = G ). Note: F | F ∧ ¬ G unsatisfiable F ∧ ¬ G | = ⊥ = G iff iff Validity/Satisfiability/Entailment are undecidable in first-order logic 31

Reminder: Satisfiability, Validity, Entailment ( A , β ) | = F iff ( A , β )( F ) = 1 A | = F iff ( A , β )( F ) = 1 for all β : X → A F is valid iff | = F iff ( A , β )( F ) = 1 for all A and all β : X → A Validity Satisfiability F is satisfiable iff ( A , β )( F ) = 1 for some A and β : X → A Note: F valid iff ¬ F is unsatisfiable Entailment: A A F | = G iff ( A β : X → A if ( A , β ) | = F then ( A , β ) | = G ). A A N | A β : X → A if ( A , β ) | = F for all F ∈ N then ( A , β ) | = G iff ( = G ). In applications, when testing validity/satisfiability/entailment we want to test it w.r.t. a specific class of models (e.g. only those satisfying a set of axioms) Note: F | F ∧ ¬ G unsatisfiable F ∧ ¬ G | = ⊥ = G iff iff ... associated with the theory corresponding to the application. Validity/Satisfiability/Entailment are undecidable in first-order logic 32

Logical theories Syntactic view Axiomatized by a set F of (closed) first-order Σ-formulae. the models of F : Mod( F ) = {A ∈ Σ-alg | A | = G , for all G in F} F ⊆ Th(Mod( F )) (typically strict) Semantic view M ⊆ Mod(Th( M )) (typically strict) given a class M of Σ-algebras the first-order theory of M : Th( M ) = { G ∈ F Σ ( X ) closed | M | = G } Th(Mod( F )) the set of formulae true in all models of F represents exactly the set of consequences of F 33

Examples 1. Groups. Σ = ( { e /0, ∗ /2, i /1 } , ∅ ) 8 A x ∗ ( y ∗ z ) ≈ ( x ∗ y ) ∗ z x , y , z > > F = < A x ∗ i ( x ) ≈ e ∧ i ( x ) ∗ x ≈ e x (group axioms) > > A : x ∗ e ≈ x ∧ e ∗ x ≈ x x Mod( F ): the class of all groups; F ⊂ Th(Mod( F )) 2. Linear integer arithmetic. Σ = ( { 0/0, s /1, +/2 } , {≤ /2 } ) Z + = ( Z , 0, s , +, ≤ ) the standard interpretation of integers. { Z + } ⊂ Mod(Th( Z + )) 34

Examples 3. Uninterpreted function symbols. Σ = (Ω, Pred) M = Σ-alg: the class of all Σ-structures The theory of uninterpreted function symbols is Th(Σ-alg) the family of all first-order formulae which are true in all Σ-algebras. 4. Lists. Σ = ( { car/1, cdr/1, cons/2 } , ∅ ) 8 car(cons( x , y )) ≈ x > > < F = ≈ cdr(cons( x , y )) y > > : cons(car( x ), cdr( x )) ≈ x Mod( F ): the class of all models of F Th Lists = Th(Mod( F )) theory of lists (axiomatized by F ) 35

Decidable theories Σ = (Ω, Pred) be a signature. M : class of Σ-algebras. T = Th( M ) is decidable iff there is an algorithm which, for every closed first-order formula φ , can decide (after a finite number of steps) whether φ is in T or not. F : class of (closed) first-order formulae. The theory T = Th(Mod( F )) is decidable iff there is an algorithm which, for every closed first-order formula φ , can decide (in finite time) whether F | = φ or not. 36

Examples Undecidable theories • Peano arithmetic A Axiomatized by: x ¬ ( x + 1 ≈ 0) (zero) A A x y ( x + 1 ≈ y + 1 → x ≈ y (successor) A A F [0] ∧ ( x ( F [ x ] → F [ x + 1]) → xF [ x ]) (induction) A x ( x + 0 ≈ x ) (plus zero) A x , y ( x + ( y + 1) ≈ ( x + y ) + 1) (plus successor) A x , y ( x ∗ 0 ≈ 0) (times zero) A x , y ( x ∗ ( y + 1) ≈ x ∗ y + x ) (times successor) E 3 ∗ y + 5 > 2 ∗ y expressed as z ( z � = 0 ∧ 3 ∗ y + 5 ≈ 2 ∗ y + z ) Intended interpretation: ( N , { 0, 1, +, ∗} , {≈ , ≤} ) (does not capture true arithmetic by G¨ odel’s incompleteness theorem) • Th(( Z , { 0, 1, +, ∗} , {≤} )) • Th(Σ-alg) 37

Examples In order to obtain decidability results: • Restrict the signature • Enrich axioms • Look at certain fragments 38

Examples In order to obtain decidability results: • Restrict the signature • Enrich axioms • Look at certain fragments Decidable theories • Presburger arithmetic decidable in 3EXPTIME [Presburger’29] Signature: ( { 0, 1, + } , {≈ , ≤} ) (no ∗ ) Axioms { (zero), (successor), (induction), (plus zero), (plus successor) } • Th( Z + ) Z + = ( Z , 0, s , +, ≤ ) the standard interpretation of integers. 39

Examples In order to obtain decidability results: • Restrict the signature • Enrich axioms • Look at certain fragments Decidable theories • The theory of real numbers (with addition and multiplication) is decidable in 2EXPTIME [Tarski’30] 40

Examples In order to obtain decidability results: • Restrict the signature • Enrich axioms • Look at certain fragments 41

Problems T : first-order theory in signature Σ; L class of (closed) Σ-formulae Given φ in L , is it the case that T | = φ ? Common restrictions on L Pred = ∅ { φ ∈ L | T | = φ } A L = { xA ( x ) | A atomic } word problem A L = { x ( A 1 ∧ . . . ∧ A n → B ) | A i , B atomic } uniform word problem Th A Horn A L = { xC ( x ) | C ( x ) clause } clausal validity problem Th A ,cl A L = { x φ ( x ) | φ ( x ) unquantified } universal validity problem Th A E L = { xA 1 ∧ . . . ∧ A n | A i atomic } unification problem Th E A E L = { xA 1 ∧ . . . ∧ A n | A i atomic } unification with constants Th x A E 42

T -validity T first-order theory in signature Σ; L class of (closed) Σ-formulae Common restrictions on L Pred = ∅ { φ ∈ L | T | = φ } A L = { x ( A 1 ∧ . . . ∧ A n → B ) | A i , B atomic } uniform word problem Th A Horn A L = { xC ( x ) | C ( x ) clause } clausal validity problem Th A ,cl A L = { x φ ( x ) | φ ( x ) unquantified } universal validity problemTh A Remarks: • T -validity for Th ,cl decidable iff T -validity for Th decidable A A • For convex theories: T -validity for Th ,Horn decidable iff T -validity for Th ,cl decidable. A A 43

Convex theories Definition A first-order Σ-theory T is Σ 0 -convex (Σ 0 ⊆ Σ) iff n m ^ _ whenever T | = A i → B j , where A 1 , . . . , A n are Σ-atoms, i =1 j =1 and B 1 , . . . , B m are Σ 0 -atoms n ^ there exists k ∈ { 1, . . . , m } such that T | A i → B k , = i =1 Examples: LI ( Q ) – theory of rational numbers with linear arithmetic: 1. LI ( Q ) is convex with respect to equality atoms, i.e. n ^ _ ^ t i ≈ t ′ A i → t k ≈ t ′ if LI ( Q ) | = A i → i then LI ( Q ) | = k for some k i j i =1 2. LI ( Q ) is not convex with respect to inequality atoms: LI ( Q ) | = x ≤ y ∨ y ≤ x but Q �| = x ≤ y and Q �| = y ≤ x . 3. Theories axiomatized by sets of Horn clauses are convex 4. Any theory T such that Mod( T ) closed under products is convex. 44

T -validity vs. T -satisfiability T -validity: Let T be a first-order theory in signature Σ Let L be a class of (closed) Σ-formulae Given φ in L , is it the case that T | = φ ? Remark: T | = φ iff T ∪ ¬ φ unsatisfiable Every T -validity problem has a dual T -satisfiability problem: T -satisfiability: Let T be a first-order theory in signature Σ Let L be a class of (closed) Σ-formulae ¬L = {¬ φ | φ ∈ L} Given ψ in ¬L , is it the case that T ∪ ψ is satisfiable? 45

T -validity vs. T -satisfiability Common restrictions on L / ¬L L ¬L A E { xA ( x ) | A atomic } { x ¬ A ( x ) | A atomic } A E { x ( A 1 ∧ . . . ∧ A n → B ) | A i , B atomic } { x ( A 1 ∧ . . . ∧ A n ∧¬ B ) | A i , B atomic } x W L i | L i literals } x V L ′ A E { { i | L ′ i literals } A E { x φ ( x ) | φ ( x ) unquantified } { x φ ′ ( x ) | φ ′ ( x ) unquantified } validity problem for universal formulae ground satisfiability problem 46

SMT tools SAT problems Given: conjunction φ of prop. clauses Task: check if φ satisfiable Method: DPLL • deterministic choices first unit resolution pure literal assignment • case distinction (splitting) • heuristics selection criteria for splitting backtracking conflict-driven learning 48

SMT tools SAT problems SMT problems Given: conjunction φ of prop. clauses Given: conjunction φ of clauses Task: check if φ | = T ⊥ Task: check if φ satisfiable Method: DPLL Method: DPLL( T ) • deterministic choices first • Boolean assignment found unit resolution using DPLL • ... and checked for T -satisfiability pure literal assignment • case distinction (splitting) • the assignment can be partial • heuristics and checked before splitting selection criteria for splitting • usual heuristics are used: backtracking non-chronological backtracking conflict-driven learning learning 49

SMT tools SAT problems SMT problems Given: conjunction φ of prop. clauses Given: conjunction φ of clauses Task: check if φ satisfiable Task: check if φ | = T ⊥ Method: DPLL( T ) Method: DPLL • deterministic choices first • Boolean assignment found unit resolution using DPLL pure literal assignment • ... and checked for T -satisfiability • case distinction (splitting) • the assignment can be partial • heuristics and checked before splitting • usual heuristics are used: selection criteria for splitting backtracking non-chronological backtracking conflict-driven learning learning Systems implementing such specialized satisfiability problems: Yices, Barcelogic Tools, CVC lite, haRVey, Math-SAT,... are called (S)atisfiability (M)odulo (T)heory solvers. 50

T -satisfiability vs. Constraint Solving The field of constraint solving also deals with satisfiability problems But be careful: • in constraint solving one is interested if a formula is satisfiable in a given, fixed model of T . • in T -satisfiability one is interested if a formula is satisfiable in any model of T at all. 51

Useful theories Many example of theories in which ground satisfiability is decidable: • The empty theory (no axioms) UIF (Σ) • linear (rational or integer) arithmetic • theories axiomatizing common datatypes (lists, arrays) • algebraic counterparts of modal logics (Boolean algebras with operators) 52

Useful theories Example 1: McCarthy’s theory of arrays. Sorts: a (arrays), i (index) and e (elements) Functions: write : a × i × e → a, read : a × i → e Axioms: read(write( a , i , e ), i ) = e i � = j → read(write( a , i , e ), j ) = read( a , j ) A a = b ↔ i (read( a , i ) = read( b , i )) - the full first-order theory of arrays is undecidable - the ground satisfiability problem is decidable (in NP) 53

Useful theories Example 2: The theory of acyclic lists Axioms: car(cons( x , y ) = x cdr(cons( x , y ) = y cons(car( x ), cdr( x )) = x t ( x ) � = x t contains only cons - the full first-order theory is decidable (but non-elementary) - the ground satisfiability problem is decidable (in PTIME) 54

Tractability Main issue: Reduce the search space Here: Datalog Locality 56

Example 1: Deductive Databases Deductive database Inference rules: Facts: Query: 57

Example 1: Deductive Databases Deductive database Example: reachability in graphs S ( x ) R ( x ) E ( x , y ) Inference rules: R ( x ) R ( y ) Facts: S ( a ), E ( a , c ), E ( c , d ), E ( d , c ), E ( b , c ) Query: R ( d ) b S ( a ), E ( a , c ), E ( c , d ), E ( d , c ), E ( b , c ) c a d 58

Example 1: Deductive Databases Deductive database �→ Datalog (Horn clauses, no function symbols) S ( x ) → R ( x ) R ( x ) ∧ E ( x , y ) → R ( y ) Inference rules: | {z } set K of Horn clauses Facts: S ( a ), E ( a , c ), E ( c , d ), E ( d , c ), E ( b , c ) | {z } set F of ground atoms Query: R ( d ) | {z } ground atom G F | K ∪ F | K ∪ F ∪ ¬ G | = ⊥ = K G iff = G iff 62

Example 1: Deductive Databases Deductive database �→ Datalog (Horn clauses, no function symbols) Inference rules: S ( x ) → R ( x ) R ( x ) ∧ E ( x , y ) → R ( y ) | {z } set K of Horn clauses Facts: S ( a ), E ( a , c ), E ( c , d ), E ( d , c ), E ( b , c ) | {z } set F of ground atoms Query: R ( d ) | {z } ground atom G S ( a ) S ( x ) → R ( x ) R ( a ) E ( a , c ) R ( x ) ∧ E ( x , y ) → R ( y ) R ( c ) E ( c , d ) R ( x ) ∧ E ( x , y ) → R ( y ) Ex: R ( d ) 63

Example 1: Deductive Databases Deductive database �→ Datalog (Horn clauses, no function symbols) S ( x ) → R ( x ) R ( x ) ∧ E ( x , y ) → R ( y ) Inference rules: | {z } set K of Horn clauses Facts: S ( a ), E ( a , c ), E ( c , d ), E ( d , c ), E ( b , c ) | {z } set F of ground atoms Query: R ( d ) | {z } Necessary: instances of Rules containing ground atom G ground. terms in Facts, Query S ( a ) S ( x ) → R ( x ) R ( a ) E ( a , c ) R ( x ) ∧ E ( x , y ) → R ( y ) R ( c ) E ( c , d ) R ( x ) ∧ E ( x , y ) → R ( y ) Ex: R ( d ) 64

Example 2: A theory of doubly-linked lists Reasoning about doubly-linked lists cf. also [Necula, McPeak 2005] A p ( p � = null ∧ p .next � = null → p .next.prev = p ) A p ( p � = null ∧ p .prev � = null → p .prev.next = p ) ∧ c � =null ∧ c .next � =null ∧ d � =null ∧ d .next � =null ∧ c .next= d .next ∧ c � = d | = ⊥ 65

Example 2: A theory of doubly-linked lists Reasoning about doubly-linked lists cf. also [Necula, McPeak 2005] ( c � =null ∧ c .next � =null → c .next.prev= c ) ( c .next � =null ∧ c .next.next � =null → c .next.next.prev= c .next) ( d � =null ∧ d .next � =null → d .next.prev= d ) ( d .next � =null ∧ d .next.next � =null → d .next.next.prev= d .next) ∧ c � =null ∧ c .next � =null ∧ d � =null ∧ d .next � =null ∧ c .next= d .next ∧ c � = d | = ⊥ 66

Local Reasoning K set of Horn clauses; K is local, if for ground Horn clauses C , K | iff K [ C ] | = C = C ⇐ = always ⇒ = locality 67

Local Reasoning K set of Horn clauses; K is local, if for ground Horn clauses C , K | iff K [ C ] | = C = C Local theories [Givan, McAllester’92] capture PTIME 68

Local Reasoning K set of Horn clauses; K is local, if for ground Horn clauses C , K | iff K [ C ] | = C = C Local theories [Givan, McAllester’92] capture PTIME Many examples (also for the equational case) 8 - theory of free function symbols s ( x ) = y → p ( y ) = x > > < - theory of successor / predecessor p ( y ) = x → s ( x ) = y > > : s ( x ) = s ( y ) → x = y - theory of lists with cons, car, cdr - a theory of lattices [Skolem 1920] 69

� � Locality, tractability K set of Horn clauses; K is local, if for ground Horn clauses C , K | iff K [ C ] | = C = C K local theory � � � � � [McAllester et al.’92,’93] � � � � � � � [Basin, Ganzinger’96] Horn theory of K in PTIME Saturation 70

Locality, tractability, embeddability K set of Horn clauses; K is local, if for ground Horn clauses C , K | iff K [ C ] | = C = C ⇐ = always ⇒ = locality K [ C ] ∧ ¬ C has model ⇒ K ∪ ¬ C has model. All terms → ֒ Subterms of C Subterms of C 71

� � � Locality, tractability, embeddability K set of Horn clauses; K is local, if for ground Horn clauses C , K | iff K [ C ] | = C = C [Ganzinger’01] � Emb( K ) K local theory � � � �� [Skolem’20] � [McAllester et al.’92,’93] � � � � � � � [Evans’53,Burris’95] [Basin, Ganzinger’96] Horn theory of K in PTIME Saturation 72

Structure • Reasoning in standard theories Preliminaries: Logic, theories, models Decidable logical theories and theory fragments Tractability • Reasoning in combinations of theories disjoint signature non-disjoint signature • Theory extensions • Applications 73

Reasoning in combinations of theories A Here: L = { xC ( x ) | C ( x ) clause } We focus on the following (equivalent) problems: • T -validity of clauses • T -satisfiability of (existentially quantified) conjunctions of literals where T is a combination of theories 74

Problems The combined decidability problem • let T i be a first-order theory in signature Σ i For i = 1, 2 • assume the T i -validity problem for universal formulae is decidable L T 2 be a combination of T 1 and T 2 Let T 1 Question: L T 2 -validity problem for universal formulae decidable? Is the T 1 75

Problems The combined decidability problem • let T i be a first-order theory in signature Σ i For i = 1, 2 • assume the T i ground satisfiability problem is decidable L T 2 be a combination of T 1 and T 2 Let T 1 Question: L T 2 ground satisfiability problem decidable? Is the T 1 76

Goal: Modularity Modular Reasoning Example: T 0 : Σ 0 -theory. lists ( R ) ∪ arrays ( R ) T 1 T 0 T 2 T i : Σ i -theory; T 0 ⊆ T i Σ 0 ⊆ Σ i . Can use provers for T 1 , T 2 as blackboxes to prove theorems in T 1 ∪ T 2 ? Which information needs to be exchanged between the provers? 77

The combined decidability problem L T 2 defined? Main issue: How is T 1 Here: put together the axioms Syntactic view: T 1 + T 2 = T 1 ∪ T 2 ⊆ F Σ 1 ∪ Σ 2 ( X ) Mod( T 1 ∪ T 2 ) = {A ∈ (Σ 1 ∪ Σ 2 )-alg | A | = G , for all G in T 1 ∪ T 2 } where Σ 1 ∪ Σ 2 = (Ω 1 , Pred 1 ) ∪ (Ω 2 , Pred 2 ) = (Ω 1 ∪ Ω 2 , Pred 1 ∪ Pred 2 ) Semantic view: Let M i = Mod( T i ), i = 1, 2 M 1 + M 2 = {A ∈ (Σ 1 ∪ Σ 2 )-alg | A | Σ i ∈ M i for i = 1, 2 } 79

The combined decidability problem L T 2 defined? Main issue: How is T 1 Here: put together the axioms Syntactic view: T 1 + T 2 = T 1 ∪ T 2 ⊆ F Σ 1 ∪ Σ 2 ( X ) Mod( T 1 ∪ T 2 ) = {A ∈ (Σ 1 ∪ Σ 2 )-alg | A | = G , for all G in T 1 ∪ T 2 } where Σ 1 ∪ Σ 2 = (Ω 1 , Pred 1 ) ∪ (Ω 2 , Pred 2 ) = (Ω 1 ∪ Ω 2 , Pred 1 ∪ Pred 2 ) A ∈ Mod( T 1 ∪ T 2 ) iff A ∈ M 1 + M 2 Semantic view: Let M i = Mod( T i ), i = 1, 2 M 1 + M 2 = {A ∈ (Σ 1 ∪ Σ 2 )-alg | A | Σ i ∈ M i for i = 1, 2 } 80

Combinations of theories Definition. A theory is consistent if it has at least one model. Question: Is the union of two consistent theories always consistent? Answer: No. (Not even when the two theories have disjoint signatures) Σ 1 = (Ω 1 , ∅ ), Σ 2 = ( { c /0, d /0 } , ∅ ), c , d �∈ Ω 1 Example: E T 1 = { x , y , z ( x �≈ y ∧ x �≈ z ∧ y �≈ z ) } A T 2 = { x ( x ≈ c ∨ x ≈ d ) } A ∈ Mod( T 1 ) | A | ≥ 3. iff B ∈ Mod( T 2 ) iff | A | ≤ 2. 81

Combinations of theories • let T i be a first-order theory in signature Σ i For i = 1, 2 • s.t. the ground satisfiability problem for T i is decidable Question: Is the ground decidability problem for T 1 ∪ T 2 decidable? 82

Combinations of theories For i = 1, 2 • let T i be a first-order theory in signature Σ i • s.t. the ground satisfiability problem for T i is decidable Question: Is the ground decidability problem for T 1 ∪ T 2 decidable? In general: No (restrictions needed for affirmative answer) Theorem [Bonacina, Ghilardi et.al, IJCAR 2006] There are theories T 1 , T 2 with disjoint signatures and decidable ground satisfiability problem such that ground satisfiability in T 1 ∪ T 2 is unsatisfiable. Idea: It is undecidable whether a constraint Γ 1 is satisfiable in an infinite model of T 1 . If T 2 has only infinite models then the ground satisfiability problem for T 1 ∪ T 2 requires exactly that. 83

Combination of theories over disjoint signatures The Nelson/Oppen procedure Given: T 1 , T 2 first-order theories with signatures Σ 1 , Σ 2 Assume that Σ 1 ∩ Σ 2 = ∅ (share only ≈ ) P i decision procedures for satisfiability of ground formulae w.r.t. T i φ quantifier-free formula over Σ 1 ∪ Σ 2 Task: Check whether φ is satisfiable w.r.t. T 1 ∪ T 2 Note: Restrict to conjunctive quantifier-free formulae φ �→ DNF ( φ ) DNF ( φ ) satisfiable in T iff one of the disjuncts satisfiable in T 84

Example [Nelson & Oppen, 1979] Theories R theory of rationals Σ R = {≤ , +, − , 0, 1 } ≈ L Σ L = { car, cdr, cons } ≈ theory of lists E theory of equality (UIF) Σ: free function and predicate symbols ≈ 85

Example [Nelson & Oppen, 1979] Theories R theory of rationals Σ R = {≤ , +, − , 0, 1 } ≈ L theory of lists Σ L = { car, cdr, cons } ≈ E ≈ theory of equality (UIF) Σ: free function and predicate symbols Problems: A 1. R∪L∪E | = x , y ( x ≤ y ∧ y ≤ x +car(cons(0, x )) ∧ P ( h ( x ) − h ( y )) → P (0)) 2. Is the following conjunction: c ≤ d ∧ d ≤ c + car(cons(0, c )) ∧ P ( h ( c ) − h ( d )) ∧ ¬ P (0) satisfiable in R ∪ L ∪ E ? 86

An Example R L E {≤ , +, − , 0, 1 } { car, cdr, cons } F ∪ P Σ Axioms x + 0 ≈ x car(cons( x , y )) ≈ x x − x ≈ 0 cdr(cons( x , y )) ≈ y at( x ) ∨ cons(car( x ), cdr( x )) ≈ x (univ. + is A , C quantif.) ≤ is R , T , A ¬ at(cons( x , y )) x ≤ y ∨ y ≤ x x ≤ y → x + z ≤ y + z Is the following conjunction: c ≤ d ∧ d ≤ c + car(cons(0, c )) ∧ P ( h ( c ) − h ( d )) ∧ ¬ P (0) satisfiable in R ∪ L ∪ E ? 87

Step 1: Purification Given: φ conjunctive quantifier-free formula over Σ 1 ∪ Σ 2 Task: Find φ 1 , φ 2 s.t. φ i is a pure Σ i -formula and φ 1 ∧ φ 2 equivalent with φ f ( s 1 , . . . , s n ) ≈ g ( t 1 , . . . , t m ) �→ u ≈ f ( s 1 , . . . , s n ) ∧ u ≈ g ( t 1 , . . . , t m ) f ( s 1 , . . . , s n ) �≈ g ( t 1 , . . . , t m ) �→ u ≈ f ( s 1 , . . . , s n ) ∧ v ≈ g ( t 1 , . . . , t m ) ∧ u �≈ v ( ¬ ) P ( . . . , s i , . . . ) �→ ( ¬ ) P ( . . . , u , . . . ) ∧ u ≈ s i ( ¬ ) P ( . . . , s i [ t ], . . . ) �→ ( ¬ ) P ( . . . , s i [ t �→ u ], . . . ) ∧ u ≈ t where t ≈ f ( t 1 , . . . , t n ) Termination: Obvious Correctness: φ 1 ∧ φ 2 and φ satisfiable in exactly the same models of T 1 ∪ T 2 88

Step 1: Purification c ≤ d ∧ d ≤ c + car(cons(0, c )) ∧ P ( h ( c ) − h ( d )) ∧ ¬ P (0) 89

Step 1: Purification c ≤ d ∧ d ≤ c + car(cons(0, c )) ∧ P ( h ( c ) − h ( d )) ∧ ¬ P (0) | {z } c 1 90

Step 1: Purification c ≤ d ∧ d ≤ c + car(cons(0, c )) ∧ P ( h ( c ) − h ( d ) ) ∧ ¬ P (0) | {z } | {z } c 1 c 2 91

Step 1: Purification c ≤ d ∧ d ≤ c + car(cons(0, c )) ∧ P ( h ( c ) − h ( d ) ) ∧ ¬ P ( 0 ) |{z} | {z } |{z} |{z} c 5 c 1 c 3 c 4 | {z } c 2 92

Step 1: Purification c ≤ d ∧ d ≤ c + car(cons(0, c )) ∧ P ( h ( c ) − h ( d ) ) ∧ ¬ P ( 0 ) |{z} | {z } |{z} |{z} c 5 c 1 c 3 c 4 | {z } c 2 R L E c ≤ d c 1 ≈ car(cons( c 5 , c )) P ( c 2 ) d ≤ c + c 1 ¬ P ( c 5 ) c 2 ≈ c 3 − c 4 c 3 ≈ h ( c ) c 5 ≈ 0 c 4 ≈ h ( d ) 93

Step 2: Propagation c ≤ d ∧ d ≤ c + car(cons(0, c )) ∧ P ( h ( c ) − h ( d ) ) ∧ ¬ P ( 0 ) |{z} | {z } |{z} |{z} c 5 c 1 c 3 c 4 | {z } c 2 R L E c ≤ d c 1 ≈ car(cons( c 5 , c )) P ( c 2 ) d ≤ c + c 1 ¬ P ( c 5 ) c 2 ≈ c 3 − c 4 c 3 ≈ h ( c ) c 5 ≈ 0 c 4 ≈ h ( d ) deduce and propagate equalities between constants entailed by components 95

Step 2: Propagation c ≤ d ∧ d ≤ c + car(cons(0, c )) ∧ P ( h ( c ) − h ( d ) ) ∧ ¬ P ( 0 ) |{z} | {z } |{z} |{z} c 5 c 1 c 3 c 4 | {z } c 2 R L E c ≤ d c 1 ≈ car(cons( c 5 , c )) P ( c 2 ) d ≤ c + c 1 ¬ P ( c 5 ) c 2 ≈ c 3 − c 4 c 3 ≈ h ( c ) c 5 ≈ 0 c 4 ≈ h ( d ) c 1 ≈ c 5 c 1 ≈ c 5 c ≈ d c ≈ d c 3 ≈ c 4 98

Step 2: Propagation c ≤ d ∧ d ≤ c + car(cons(0, c )) ∧ P ( h ( c ) − h ( d ) ) ∧ ¬ P ( 0 ) |{z} | {z } |{z} |{z} c 5 c 1 c 3 c 4 | {z } c 2 R L E c ≤ d c 1 ≈ car(cons( c 5 , c )) P ( c 2 ) d ≤ c + c 1 ¬ P ( c 5 ) c 2 ≈ c 3 − c 4 c 3 ≈ h ( c ) c 5 ≈ 0 c 4 ≈ h ( d ) c 1 ≈ c 5 c 1 ≈ c 5 c ≈ d c ≈ d c 3 ≈ c 4 c 2 ≈ c 5 ⊥ 99

The Nelson-Oppen algorithm φ conjunction of literals Step 1. Purification T 1 ∪ T 2 ∪ φ �→ ( T 1 ∪ φ 1 ) ∪ ( T 2 ∪ φ 2 ): where φ i is a pure Σ i -formula and φ 1 ∧ φ 2 is equisatisfiable with φ . Step 2. Propagation. The decision procedure for ground satisfiability for T 1 and T 2 fairly exchange information concerning entailed unsatisfiability of constraints in the shared signature i.e. clauses over the shared variables. until an inconsistency is detected or a saturation state is reached. 100