Representation theorems and the semantics of (semi)lattice based - PowerPoint PPT Presentation

Examples • positive logics [Dunn 1995] • (modal) intuitionistic logic DLO • G¨ odel logics [G¨ odel 1930] • SH n , SHK n logics [Iturrioz 1982] • Post logics and generalizations HAO • modal logic, dynamic logic, ... BAO

Examples • positive logics [Dunn 1995] • (modal) intuitionistic logic DLO • G¨ odel logics [G¨ odel 1930] • SH n , SHK n logics [Iturrioz 1982] RDO • Post logics and generalizations HAO • modal logic, dynamic logic, ... • relevant logic RL [Urquhart’96] BAO • fuzzy logics G¨ odel, � Lukasiewicz, product

Examples • positive logics [Dunn 1995] SLO LO • (modal) intuitionistic logic DLO • G¨ odel logics [G¨ odel 1930] • SH n , SHK n logics [Iturrioz 1982] RDO • Post logics and generalizations HAO • modal logic, dynamic logic, ... • relevant logic RL [Urquhart’96] BAO • fuzzy logics G¨ odel, � Lukasiewicz, product • BCC and related logics • Lambek calculus; linear logic ...

� � Motivation. Semantics f Algebraic models � A ( A, D ) Var � � � � � � � f � � Fma ( Var )

� � Motivation. Semantics f Algebraic models � A ( A, D ) Var � � � � � � � f � � Fma ( Var ) Kripke-style models m : Var → P ( X ) ( W, { R W } R ∈ Rel ) meaning function

� � Motivation. Semantics f Algebraic models � A ( A, D ) Var � � � � � � � f � � Fma ( Var ) Kripke-style models m : Var → P ( X ) ( W, { R W } R ∈ Rel ) meaning function Relational models algebras of relations

Motivation. Decidability results Logical calculi ◦ Gentzen-style calculi ◦ natural deduction ◦ hypersequent calculi [Avron 1991]

Motivation. Decidability results Logical calculi ◦ Gentzen-style calculi ◦ natural deduction ◦ hypersequent calculi [Avron 1991] Semantics ◦ Algebraic semantics ◦ Kripke-style semantics ◦ Relational semantics

Motivation. Decidability results Logical calculi ◦ Gentzen-style calculi ◦ natural deduction ◦ hypersequent calculi [Avron 1991] Semantics ◦ Algebraic semantics ◦ Kripke-style semantics ◦ Relational semantics

Motivation. Decidability results Logical calculi ◦ Gentzen-style calculi ◦ natural deduction ◦ hypersequent calculi [Avron 1991] Semantics ◦ Algebraic semantics ◦ Kripke-style semantics ◦ Relational semantics Automated theorem proving ◦ embedding into FOL + resolution ◦ tableau methods ◦ natural deduction; labelled deductive systems

Motivation. Decidability results Logical calculi ◦ Gentzen-style calculi ◦ natural deduction ◦ hypersequent calculi [Avron 1991] Semantics ◦ Algebraic semantics ◦ Kripke-style semantics ◦ Relational semantics Automated theorem proving ◦ embedding into FOL + resolution ◦ tableau methods ◦ natural deduction; labelled deductive systems

Connections between classes of models Algebraic models � � ������������������� � � � � � � � � � � � � � � � � � � Kripke models Relational models

Connections between classes of models Algebraic models � � ������������������� � � � � representation theorems � � � � � � (algebras of sets) � � � � � � � � Kripke models Relational models

Connections between classes of models Algebraic models � � ������������������� � � � � representation theorems � representation theorems � � � � � (algebras of sets) � (algebras of relations) � � � � � � � Kripke models Relational models

Connections between classes of models Algebraic models � � ������������������� � � � � representation theorems � representation theorems � � � � � (algebras of sets) � (algebras of relations) � � � � � � � Kripke models Relational models

Algebraic and Kripke-style semantics Algebraic models Kripke-style models

� � Algebraic and Kripke-style semantics Algebraic models Kripke-style models D (C) A R E (i) E ( K ) ⊆ P ( K ) algebra of subsets of K (ii) i : A → E ( D ( A )) injective homomorphism

� � Algebraic and Kripke-style semantics Algebraic models Kripke-style models D (C) A R E (i) E ( K ) ⊆ P ( K ) algebra of subsets of K (ii) i : A → E ( D ( A )) injective homomorphism Kripke-style models

� � Algebraic and Kripke-style semantics Algebraic models Kripke-style models D (C) A R E (i) E ( K ) ⊆ P ( K ) algebra of subsets of K (ii) i : A → E ( D ( A )) injective homomorphism ( K, m ) K ∈ R ; m : Var → E ( K ) ⊆ P ( K ) Kripke-style models

� � Algebraic and Kripke-style semantics Algebraic models Kripke-style models D (C) A R E (i) E ( K ) ⊆ P ( K ) algebra of subsets of K (ii) i : A → E ( D ( A )) injective homomorphism ( K, m ) K ∈ R ; m : Var → E ( K ) ⊆ P ( K ) Kripke-style models r ( K, m ) | = x φ iff x ∈ m ( φ )

� � Algebraic and Kripke-style semantics Algebraic models Kripke-style models D (C) A R E (i) E ( K ) ⊆ P ( K ) algebra of subsets of K (ii) i : A → E ( D ( A )) injective homomorphism ( K, m ) K ∈ R ; m : Var → E ( K ) ⊆ P ( K ) Kripke-style models r ( K, m ) | = x φ iff x ∈ m ( φ ) r a | = φ iff E ( K ) | = φ = 1. K

� � Algebraic and Kripke-style semantics Algebraic models Kripke-style models D (C) A R E (i) E ( K ) ⊆ P ( K ) algebra of subsets of K (ii) i : A → E ( D ( A )) injective homomorphism ( K, m ) K ∈ R ; m : Var → E ( K ) ⊆ P ( K ) Kripke-style models r | = a r Theorem If A , R satisfy (C)(i,ii) then A | = φ iff R | = φ .

� � Algebraic and relational semantics Algebraic models Relational models D (C) A R E (i) E ( K ) algebra of relations (ii) i : A → E ( D ( A )) injective homomorphism K ∈ R ; f : Var → E ( K ) ( K, f ) Relational models a = | a a Theorem If A , R satisfy (C)(i,ii) then = φ iff R = φ . A | |

Representation theorems Stone 1940: Bool Priestley 1972: D 01 B ∼ L ∼ → Clopen ( F m ( B ) , τ ) → ClopenOF ( F p ( L ) , ⊆ , τ ) η B ( x ) = { F ∈ F m ( L ) | x ∈ F } η L ( x ) = { F ∈ F p ( L ) | x ∈ F }

Representation theorems Natural Dualities: V = ISP ( P ) A ∼ → Hom Rel ( D ( A ) , P ) P ’alter-ego’ of P D ( A ) = Hom V ( A, P ) Stone 1940: Bool Priestley 1972: D 01 B ∼ L ∼ → Clopen ( F m ( B ) , τ ) → ClopenOF ( F p ( L ) , ⊆ , τ ) η B ( x ) = { F ∈ F m ( L ) | x ∈ F } η L ( x ) = { F ∈ F p ( L ) | x ∈ F }

Representation theorems Natural Dualities: V = ISP ( P ) A ∼ → Hom Rel ( D ( A ) , P ) P ’alter-ego’ of P D ( A ) = Hom V ( A, P ) Stone 1940: Bool = ISP ( B 2 ) Priestley 1972: D 01 B ∼ L ∼ → Hom St ( D ( B ) , B 2 ) → ClopenOF ( F p ( L ) , ⊆ , τ ) η B ( x )( h ) = h ( x ) η L ( x ) = { F ∈ F p ( L ) | x ∈ F }

Representation theorems Natural Dualities: V = ISP ( P ) A ∼ → Hom Rel ( D ( A ) , P ) P ’alter-ego’ of P D ( A ) = Hom V ( A, P ) Stone 1940: Bool = ISP ( B 2 ) Priestley 1972: D 01 = ISP ( L 2 ) B ∼ L ∼ → Hom St ( D ( B ) , B 2 ) → Hom Pr ( D ( L ) , L 2 ) η B ( x )( h ) = h ( x ) η L ( x )( h ) = h ( x )

� � Representation theorems Natural Dualities: V = ISP ( P ) A ∼ → Hom Rel ( D ( A ) , P ) P ’alter-ego’ of P D ( A ) = Hom V ( A, P ) � � �������������������� � � � � � � � � � � � � � � � � � � � Stone 1940: Bool = ISP ( B 2 ) Priestley 1972: D 01 = ISP ( L 2 ) B ∼ L ∼ → Hom St ( D ( B ) , B 2 ) → Hom Pr ( D ( L ) , L 2 ) η B ( x )( h ) = h ( x ) η L ( x )( h ) = h ( x ) Semilattices: SL = ISP ( S 2 ) S ∼ → Hom ts ( D ( S ) , S 2 ) η S ( x )( h ) = h ( x )

� � Representation theorems Natural Dualities: V = ISP ( P ) A ∼ → Hom Rel ( D ( A ) , P ) P ’alter-ego’ of P D ( A ) = Hom V ( A, P ) � � �������������������� � � � � � � � � � � � � � � � � � � � Stone 1940: Bool = ISP ( B 2 ) Priestley 1972: D 01 = ISP ( L 2 ) → OF ( D ( L )) → P ( D ( B )) L ֒ B ֒ η L ( x ) = { F ∈ D ( L ) | x ∈ F } η B ( x ) = { F ∈ D ( B ) | x ∈ F } Semilattices: SL = ISP ( S 2 ) ( S, ∧ ) ֒ → ( SF ( D ( S )) , ∩ ) η S ( x ) = { F ∈ D ( S ) | x ∈ F } Lattices: η L : ( L, ∧ , ∨ ) ֒ → ( SF ( D ( L )) , ∩ , ∨ ) η L ( x ) := { F ∈ D ( L ) | x ∈ F }

Example 1. Boolean algebras

Example 2. Distributive lattices

Example 3. Semilattices

Example 4. Lattices

Other representation theorems Boolean algebras with operators • J´ onsson and Tarski (1951)

Other representation theorems Boolean algebras with operators • J´ onsson and Tarski (1951) Distributive lattices with operators • Goldblatt (1986), VS (2000)

Other representation theorems Boolean algebras with operators • J´ onsson and Tarski (1951) Distributive lattices with operators • Goldblatt (1986), VS (2000) Lattices (with operators) • Urquhart (1978) • Allwein and Dunn (1993) • Dunn and Hartonas (1997) • Hartonas (1997)

Other representation theorems Boolean algebras with operators General Idea: • A �→ D ( A ) topological space • J´ onsson and Tarski (1951) with additional structure Distributive lattices with operators • Goldblatt (1986), VS (2000) • A ∼ = ClosedSubsets of D ( A ) Lattices (with operators) • Urquhart (1978) closed wrt: topological structure order structure • Allwein and Dunn (1993) ... • Dunn and Hartonas (1997) • operators �→ relations on D ( A ) • Hartonas (1997)

Other representation theorems Boolean algebras with operators General Idea: • A �→ D ( A ) topological space • J´ onsson and Tarski (1951) with additional structure Distributive lattices with operators • Goldblatt (1986), VS (2000) • A ∼ = ClosedSubsets of D ( A ) Lattices (with operators) • Urquhart (1978) closed wrt: topological structure order structure • Allwein and Dunn (1993) ... • Dunn and Hartonas (1997) • operators �→ relations on D ( A ) • Hartonas (1997) “Gaggles”, “tonoids” Dunn (1990, 1993)

Representation theorems f A : A ε 1 × · · · × A εn → A ε join-hemimorphism f ∈ Σ ε 1 ...εn → ε :

� � Representation theorems f A : A ε 1 × · · · × A εn → A ε join-hemimorphism f ∈ Σ ε 1 ...εn → ε : D D � Rp Σ � SLSp Σ DLO Σ SLO Σ E E R f ( F 1 , . . . , F n , F ) iff f ( F ε 1 1 , . . . , F εn n ) ⊆ F ε D ( A ) ( R − 1 ( U ε 1 1 , . . . , U εn n )) ε E ( X ) f R ( U 1 , . . . , U n ) =

� � Representation theorems f A : A ε 1 × · · · × A εn → A ε join-hemimorphism f ∈ Σ ε 1 ...εn → ε : D D � Rp Σ � SLSp Σ DLO Σ SLO Σ E E R f ( F 1 , . . . , F n , F ) iff f ( F ε 1 1 , . . . , F εn n ) ⊆ F ε D ( A ) ( R − 1 ( U ε 1 1 , . . . , U εn n )) ε E ( X ) f R ( U 1 , . . . , U n ) = Example x ◦ y ≤ z iff x ≤ y → z has type + 1 , +1 → + 1 R ◦ ( F 1 , F 2 , F 3 ) iff F 1 ◦ F 2 ⊆ F 3 ◦ R → ( F 1 , F 2 , F 3 ) iff F 1 → F c 2 ⊆ F c → has type + 1 , − 1 → − 1 3

� � Representation theorems f A : A ε 1 × · · · × A εn → A ε join-hemimorphism f ∈ Σ ε 1 ...εn → ε : D D � Rp Σ � SLSp Σ DLO Σ SLO Σ E E R f ( F 1 , . . . , F n , F ) iff f ( F ε 1 1 , . . . , F εn n ) ⊆ F ε D ( A ) ( R − 1 ( U ε 1 1 , . . . , U εn n )) ε E ( X ) f R ( U 1 , . . . , U n ) = Example x ◦ y ≤ z iff x ≤ y → z has type + 1 , +1 → + 1 R ◦ ( F 1 , F 2 , F 3 ) iff F 1 ◦ F 2 ⊆ F 3 ◦ R → ( F 1 , F 2 , F 3 ) iff F 1 → F c 2 ⊆ F c → has type + 1 , − 1 → − 1 3 R → ( F 1 , F 2 , F 3 ) iff R ◦ ( F 3 , F 1 , F 2 )

� � Algebraic and Kripke-style semantics D SLO (C) A R LO E DLO (i) E ( K ) ⊆ P ( K ) algebra of subsets of K RDO (ii) i : A ֒ → E ( D ( A )) HAO BAO

� � Algebraic and Kripke-style semantics D SLO (C) A R LO E DLO (i) E ( K ) ⊆ P ( K ) algebra of subsets of K RDO (ii) i : A ֒ → E ( D ( A )) HAO ( K, m ), m : Var → E ( K ) r ( K, m ) | = x φ iff x ∈ m ( φ ) BAO

� � Algebraic and Kripke-style semantics D SLO (C) A R LO E (i) E ( K ) ⊆ P ( K ) DLO algebra of subsets of K RDO (ii) i : A ֒ → E ( D ( A )) ( K, m ), m : Var → E ( K ) HAO r ( K, m ) | = x φ iff x ∈ m ( φ ) BAO DLO Priestley representation η A : A → OF ( D ( A )) SLO , LO Representation for (semi)lattices η A : A → SF ( D ( A ))

Logic Algebraic Kripke-style meaning functions models models DLO Σ Rp Σ m : Var → OF ( X ) Positive ( L, ∨ , ∧ , 0 , 1 , { f } f ∈ Σ ) ( X, ≤ , { R } R ∈ Σ ) HAO Σ Rp Σ m : Var → OF ( X ) Post-style ( L, ∨ , ∧ , ⇒ , 0 , 1 , { f } f ∈ Σ )( X, ≤ , { R } R ∈ Σ ) BAO Σ BAO Σ m : Var → P ( X ) ( B, ∨ , ∧ , 0 , 1 , ¬ , { f } f ∈ Σ ) ( X, { R } R ∈ Σ ) RDO RSp m : Var → OF ( X ) Lukasiewicz � -style ( L, ∨ , ∧ , 0 , 1 , ◦ , → ) ( X, ≤ , R ◦ ) RSO , RLO RSO , RLO m : Var → SF ( X ) ( S, ∧ , 0 , 1 , ◦ , → ) ( X, ∧ , R ◦ ) ( S, ∨ , ∧ , 0 , 1 , ◦ , → ) ( X, ∧ , R ◦ )

Overview • Motivation • Connection between different classes of models • Representation theorems • Examples • Decidability results • Automated theorem proving • Conclusions

Class u.w.p. References Lattices PTIME Skolem (1920), Burris (1995) decidable Blok, Van Alten (1999) ResLatMon decidable Blok, Van Alten (1999) ResLatIntMon decidable Blok, Van Alten (1999) BCK → Modular Lattices undecidable Freese (1980), Herrmann (1983) co-NP complete Bloniarz et al.(1987) D 01 DLO Σ , RDO Σ EXPTIME VS (1999, 2001) decidable Andreka DLSgr ∨ , d subclasses undecidable Urquhart (1995) Heyting Algebras DEXP VS (1999) undecidable Kurucz, Nemeti et al. (1993) HASgr ∨ , d Boolean Algebras co-NP complete Cook (1971) undecidable Kurucz, Nemeti et al. (1993) ResBoolMon undecidable Kurucz, Nemeti et al. (1993) BoolSgr ∨ , d decidable Gyuris (1992) BoolSgr ∨

Decidability results Semantics • Algebraic semantics finite model property (uniform) word problem decidable

Decidability results Semantics • Algebraic semantics finite model property (uniform) word problem decidable • Kripke-style semantics finite model property embedding into decidable fragments of FOL devise sound and complete decision procedure

Decidability results Semantics • Algebraic semantics finite model property (uniform) word problem decidable • Kripke-style semantics finite model property embedding into decidable fragments of FOL devise sound and complete decision procedure • Relational semantics relational proof systems

Decidability results Semantics • Algebraic semantics finite model property (uniform) word problem decidable • Kripke-style semantics finite model property embedding into decidable fragments of FOL devise sound and complete decision procedure • Relational semantics relational proof systems Automated theorem proving ◦ embedding into FOL + ATP in first-order logic ◦ tableau methods ◦ natural deduction; labelled deductive systems

Decidability results Semantics • Algebraic semantics finite model property (uniform) word problem decidable • Kripke-style semantics finite model property embedding into decidable fragments of FOL devise sound and complete decision procedure • Relational semantics relational proof systems Automated theorem proving ◦ embedding into FOL + ATP in first-order logic ◦ tableau methods ◦ natural deduction; labelled deductive systems

Decidability results Semantics • Algebraic semantics finite model property (uniform) word problem decidable • Kripke-style semantics finite model property embedding into decidable fragments of FOL devise sound and complete decision procedure • Relational semantics relational proof systems Automated theorem proving ◦ embedding into FOL + ATP in first-order logic ◦ tableau methods ◦ natural deduction; labelled deductive systems

Class u.w.p. References Lattices PTIME Skolem (1920), Burris (1995) decidable Blok, Van Alten (1999) ResLatMon decidable Blok, Van Alten (1999) ResLatIntMon decidable Blok, Van Alten (1999) BCK → Modular Lattices undecidable Freese (1980), Herrmann (1983) co-NP complete Bloniarz et al.(1987) D 01 DLO Σ , RDO Σ EXPTIME VS (1999, 2001) decidable Andreka DLSgr ∨ , d subclasses undecidable Urquhart (1995) Heyting Algebras DEXP VS (1999) undecidable Kurucz, Nemeti et al. (1993) HASgr ∨ , d Boolean Algebras co-NP complete Cook (1971) undecidable Kurucz, Nemeti et al. (1993) ResBoolMon undecidable Kurucz, Nemeti et al. (1993) BoolSgr ∨ , d decidable Gyuris (1992) BoolSgr ∨

Class u.w.p. References Lattices PTIME Skolem (1920), Burris (1995) decidable Blok, Van Alten (1999) ResLatMon decidable Blok, Van Alten (1999) ResLatIntMon decidable Blok, Van Alten (1999) BCK → Modular Lattices undecidable Freese (1980), Herrmann (1983) co-NP complete Bloniarz et al.(1987) D 01 DLO Σ , RDO Σ EXPTIME VS (1999, 2001) decidable Andreka DLSgr ∨ , d subclasses undecidable Urquhart (1995) Heyting Algebras DEXP VS (1999) undecidable Kurucz, Nemeti et al. (1993) HASgr ∨ , d Boolean Algebras co-NP complete Cook (1971) undecidable Kurucz, Nemeti et al. (1993) ResBoolMon undecidable Kurucz, Nemeti et al. (1993) BoolSgr ∨ , d decidable Gyuris (1992) BoolSgr ∨

Resolution-based methods Advantages

Resolution-based methods Advantages • direct encoding • restricted (hence efficient) calculi – ordering, selection – simplification/elimination of redundancies • allow use of efficient implementations (SPASS, Saturate) • in many cases better than equational reasoning AC operators �→ logical operations

Automated Theorem Proving: DLO Σ Theorem DLO Σ | = φ 1 ≤ φ 2 iff the following conjunction is unsatisfiable: (Dom) x ≤ x ; x ≤ y, y ≤ z → x ≤ z R f ( x 1 , . . . , x n , x ) , x ⊲ ⊳ ǫ y ⇒ R f ( x 1 , . . . , x n , y ) if f ∈ Σ ε → ε (Her) x ≤ y, P e ( x ) ⇒ P e ( y ) (Ren)(0 , 1) ¬ P 0 ( x ) P 1 ( x ) ( ∧ ) P e 1 ∧ e 2 ( x ) ⇔ P e 1 ( x ) ∧ P e 2 ( x ) ( ∨ ) P e 1 ∨ e 2 ( x ) ⇔ P e 1 ( x ) ∨ P e 2 ( x ) (Σ) P f ( e 1 ,...,en ) ( x ) ⇔ ( ∃ x 1 , . . . ∃ x n f ∈ Σ ε 1 ...εn → ε ( P e 1 ( x 1 ) ε 1 ∧ · · · ∧ P en ( x n ) εn ∧ R f ( x 1 , . . . , x n , x ))) ε (N) ∃ c ∈ X : P φ 1 ( c ) ∧ ¬ P φ 2 ( c )

Automated Theorem Proving: DLO Σ Theorem DLO Σ | = φ 1 ≤ φ 2 iff the following conjunction is unsatisfiable: (Dom) (Her) (Ren)(0 , 1) ¬ P 0 ( x ) P 1 ( x ) ( ∧ ) P e 1 ∧ e 2 ( x ) ⇔ P e 1 ( x ) ∧ P e 2 ( x ) ( ∨ ) P e 1 ∨ e 2 ( x ) ⇔ P e 1 ( x ) ∨ P e 2 ( x ) (Σ) P f ( e 1 ,...,en ) ( x ) ⇔ ( ∃ x 1 , . . . ∃ x n f ∈ Σ ε 1 ...εn → ε ( P e 1 ( x 1 ) ε 1 ∧ · · · ∧ P en ( x n ) εn ∧ R f ( x 1 , . . . , x n , x ))) ε (N) ∃ c ∈ X : P φ 1 ( c ) ∧ ¬ P φ 2 ( c )

Automated Theorem Proving: HAO Σ Theorem HAO Σ | = φ = 1 iff the following conjunction is unsatisfiable: (Dom) x ≤ x ; x ≤ y, y ≤ z → x ≤ z R f ( x 1 , . . . , x n , x ) , x ⊲ ⊳ ǫ y ⇒ R f ( x 1 , . . . , x n , y ) if f ∈ Σ ε → ε (Her) x ≤ y, P e ( x ) ⇒ P e ( y ) (Ren)(0 , 1) ¬ P 0 ( x ) P 1 ( x ) ( ∧ ) P e 1 ∧ e 2 ( x ) ⇔ P e 1 ( x ) ∧ P e 2 ( x ) ( ∨ ) P e 1 ∨ e 2 ( x ) ⇔ P e 1 ( x ) ∨ P e 2 ( x ) (Σ) P f ( e 1 ,...,en ) ( x ) ⇔ ( ∃ x 1 , . . . ∃ x n f ∈ Σ ε 1 ...εn → ε ( P e 1 ( x 1 ) ε 1 ∧ · · · ∧ P en ( x n ) εn ∧ R f ( x 1 , . . . , x n , x ))) ε ( → ) P e 1 → e 2 ( x ) ⇔ ∀ y ( y ≥ x ∧ P e 1 ( y ) ⇒ P e 2 ( y )) (N) ∃ c ∈ X : ¬ P φ ( c )

Automated Theorem Proving Class of algebras Complexity (refinements of resolution) DLO Σ EXPTIME RDO Σ EXPTIME BAO Σ EXPTIME HA DEXPTIME HAO Σ ? RSO Σ , RLO Σ ?

Overview • Representation theorems • Connection between different classes of models • Examples • Decidability results • Automated theorem proving

Overview • Representation theorems • Connection between different classes of models • Examples • Decidability results • Automated theorem proving

Overview • Representation theorems • Connection between different classes of models • Examples • Decidability results • Automated theorem proving

Overview • Representation theorems • Connection between different classes of models • Examples • Decidability results • Automated theorem proving

Overview • Representation theorems • Connection between different classes of models • Examples • Decidability results • Automated theorem proving