The unification type of Workshop on Admissible Rules and Unification - PowerPoint PPT Presentation

The uni�cation type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . of Łukasiewicz logic Łukasiewicz Luca Spada Introduction Uni�cation Łukasiewicz logic The approach through projectivity MV-algebras Uni�cation type of logic . Appendix Nullary uni�cation type u which is not less general than any most general unifier. . . . . . . . . . . . . . Then the unification type of E is called nullary. None of the above, i.e. there exists a pair ( s , t ) and a unifier

The uni�cation type . . . . . . . . . . . . . . . . . . . . . . . . of Łukasiewicz logic . . . . . . . . . . . . . . . . . . . . . . . . Then the unification type of E is called nullary. . Luca Spada Introduction Uni�cation Łukasiewicz logic The approach through projectivity MV-algebras Uni�cation type of Łukasiewicz logic Appendix Nullary uni�cation type u which is not less general than any most general unifier. . . . . . . . . . . . . . . None of the above, i.e. there exists a pair ( s , t ) and a unifier µ 1 µ 2 µ 3

The uni�cation type . . . . . . . . . . . . . . . . . . . . . . . . of Łukasiewicz logic . . . . . . . . . . . . . . . . . . . . . . . . Then the unification type of E is called nullary. . Luca Spada Introduction Uni�cation Łukasiewicz logic The approach through projectivity MV-algebras Uni�cation type of Łukasiewicz logic Appendix Nullary uni�cation type u which is not less general than any most general unifier. . . . . . . . . . . . . . . None of the above, i.e. there exists a pair ( s , t ) and a unifier µ 1 µ 2 µ 3

The uni�cation type of Łukasiewicz logic , (A4) , (A3) , (A2) , (A1) by the four axiom schemata: non-classical system going back to the 1920’s which may be Łukasiewicz (infinite-valued propositional) logic is a Łukasiewicz logic Appendix logic Łukasiewicz type of Uni�cation MV-algebras projectivity through The approach logic Łukasiewicz Uni�cation Introduction Luca Spada with modus ponens as the only deduction rule. axiomatised using the primitive connectives → (implication) and ¬ (negation)

The uni�cation type Uni�cation non-classical system going back to the 1920’s which may be Łukasiewicz (infinite-valued propositional) logic is a Łukasiewicz logic Appendix logic of Łukasiewicz logic type of Łukasiewicz MV-algebras projectivity through The approach logic Łukasiewicz Uni�cation Introduction Luca Spada with modus ponens as the only deduction rule. axiomatised using the primitive connectives → (implication) and ¬ (negation) by the four axiom schemata: (A1) α → ( β → α ) , (A2) ( α → β ) → (( β → γ ) → ( α → γ )) , (A3) ( α → β ) → β ) → (( β → α ) → α ) , (A4) ( ¬ α → ¬ β ) → ( β → α ) ,

Łukasiewicz logic, and called them MV-algebras. This The uni�cation type Łukasiewicz allowed him to obtain an algebraic proof of the completeness Chang first considered the Tarski-Lindenbaum algebras of under every such assignment. Tautologies are defined as those formulæ that evaluate to formulæ via The semantics of Łukasiewicz logic is many-valued: Semantics of Łukasiewicz logic Appendix of Łukasiewicz logic logic type of Uni�cation MV-algebras projectivity through The approach logic Łukasiewicz Uni�cation Introduction Luca Spada theorem. assignments µ to atomic formulæ range in the unit interval [0 , 1] ⊆ R ; they are extended compositionally to compound µ ( α → β ) = min { 1 , 1 − µ ( α ) + µ ( β ) } , µ ( ¬ α ) = 1 − µ ( α ) .

The uni�cation type type of allowed him to obtain an algebraic proof of the completeness Chang first considered the Tarski-Lindenbaum algebras of under every such assignment. formulæ via The semantics of Łukasiewicz logic is many-valued: Semantics of Łukasiewicz logic Appendix of Łukasiewicz logic Łukasiewicz logic Uni�cation MV-algebras Luca Spada Introduction Uni�cation Łukasiewicz The approach logic through projectivity theorem. assignments µ to atomic formulæ range in the unit interval [0 , 1] ⊆ R ; they are extended compositionally to compound µ ( α → β ) = min { 1 , 1 − µ ( α ) + µ ( β ) } , µ ( ¬ α ) = 1 − µ ( α ) . Tautologies are defined as those formulæ that evaluate to 1 Łukasiewicz logic, and called them MV-algebras. This

The uni�cation type is an involution . . . . . A is a commutative monoid, the interaction between those two operations is of Łukasiewicz logic described by the following two axioms: x x y y y x . . . projectivity Luca Spada Introduction Uni�cation Łukasiewicz logic The approach through MV-algebras De�nition Uni�cation type of Łukasiewicz logic Appendix MV-algebras . x An MV-algebra is a structure A = ⟨ A , ⊕ , ∗ , 0 ⟩ such that:

The uni�cation type is an involution . . . . . . the interaction between those two operations is of Łukasiewicz logic described by the following two axioms: x x y y y x . . De�nition projectivity Luca Spada Introduction Uni�cation Łukasiewicz logic The approach through MV-algebras . Uni�cation type of Łukasiewicz logic Appendix MV-algebras x An MV-algebra is a structure A = ⟨ A , ⊕ , ∗ , 0 ⟩ such that: ◮ A = ⟨ A , ⊕ , 0 ⟩ is a commutative monoid,

The uni�cation type the interaction between those two operations is . . . . . . described by the following two axioms: of Łukasiewicz logic x x y y y x . . De�nition projectivity Luca Spada Introduction Uni�cation Łukasiewicz logic The approach through x . MV-algebras Uni�cation type of Łukasiewicz logic Appendix MV-algebras An MV-algebra is a structure A = ⟨ A , ⊕ , ∗ , 0 ⟩ such that: ◮ A = ⟨ A , ⊕ , 0 ⟩ is a commutative monoid, ◮ ∗ is an involution

The uni�cation type . . . . . . . described by the following two axioms: De�nition x x y y y x of Łukasiewicz logic . . MV-algebras Luca Spada Introduction Uni�cation Łukasiewicz logic The approach through x projectivity MV-algebras Uni�cation type of Łukasiewicz logic Appendix An MV-algebra is a structure A = ⟨ A , ⊕ , ∗ , 0 ⟩ such that: ◮ A = ⟨ A , ⊕ , 0 ⟩ is a commutative monoid, ◮ ∗ is an involution ◮ the interaction between those two operations is

The uni�cation type Appendix described by the following two axioms: . . . . . . . . De�nition of Łukasiewicz logic MV-algebras . logic logic Luca Spada Introduction Uni�cation Łukasiewicz Łukasiewicz The approach through projectivity MV-algebras Uni�cation type of An MV-algebra is a structure A = ⟨ A , ⊕ , ∗ , 0 ⟩ such that: ◮ A = ⟨ A , ⊕ , 0 ⟩ is a commutative monoid, ◮ ∗ is an involution ◮ the interaction between those two operations is ◮ x ⊕ 0 ∗ = 0 ∗ ◮ ( x ∗ ⊕ y ) ∗ ⊕ y = ( y ∗ ⊕ x ) ∗ ⊕ x

t n x s n x The uni�cation type s Furthermore, in MV-algebras, solving a system of equation of unification problems is equivalent to solve a single unification problem: t x t x unifying two terms reduces to finding a substitution that x s x u x identifies a term with a constant (in our case either 0 or 1). In MV-alegrbas (as in several other cases) the problem of of Łukasiewicz logic through Luca Spada Introduction Uni�cation Łukasiewicz logic The approach projectivity Some small technical considerations MV-algebras Uni�cation type of Łukasiewicz logic Appendix .

t n x s n x The uni�cation type s Furthermore, in MV-algebras, solving a system of equation of unification problems is equivalent to solve a single unification problem: t x t x unifying two terms reduces to finding a substitution that x s x u x identifies a term with a constant (in our case either 0 or 1). In MV-alegrbas (as in several other cases) the problem of of Łukasiewicz logic through Luca Spada Introduction Uni�cation Łukasiewicz logic The approach projectivity Some small technical considerations MV-algebras Uni�cation type of Łukasiewicz logic Appendix .

The uni�cation type logic of Łukasiewicz logic unification problem: of unification problems is equivalent to solve a single Furthermore, in MV-algebras, solving a system of equation identifies a term with a constant (in our case either 0 or 1). unifying two terms reduces to finding a substitution that In MV-alegrbas (as in several other cases) the problem of Some small technical considerations Appendix Łukasiewicz type of Uni�cation MV-algebras projectivity through The approach logic Łukasiewicz Uni�cation Introduction Luca Spada  t 1 (¯ x ) = s 1 (¯ x )     t 1 (¯ x ) = s 2 (¯ x )  u (¯ ⇐ ⇒ x ) = 1 . ...    t n (¯ x ) = s n (¯  x ) 

The uni�cation type . of some free algebra of that variety, i.e., . . P . Free P equivalently, projective algebras. . # . s . r . An algebra is called projective (in a variety) if it is a retract The key concept is given by projective formulas or, of Łukasiewicz logic projectivity Luca Spada Introduction Uni�cation Łukasiewicz logic The approach through MV-algebras unification which has several advantages. Uni�cation type of Łukasiewicz logic Appendix Projective objects In 1997 Ghilardi proposed an alternative approach to id

The uni�cation type . of some free algebra of that variety, i.e., . . P . Free P of Łukasiewicz logic . # . s . r . equivalently, projective algebras. id unification which has several advantages. projectivity Luca Spada Introduction Uni�cation Łukasiewicz logic The approach through MV-algebras In 1997 Ghilardi proposed an alternative approach to Uni�cation type of Łukasiewicz logic Appendix Projective objects The key concept is given by projective formulas or, An algebra is called projective (in a variety) if it is a retract

The uni�cation type . of some free algebra of that variety, i.e., . . P . Free P of Łukasiewicz logic . # . s . r . equivalently, projective algebras. id unification which has several advantages. projectivity Luca Spada Introduction Uni�cation Łukasiewicz logic The approach through MV-algebras In 1997 Ghilardi proposed an alternative approach to Uni�cation type of Łukasiewicz logic Appendix Projective objects The key concept is given by projective formulas or, An algebra is called projective (in a variety) if it is a retract

The uni�cation type of Łukasiewicz logic A 2. u is an arrow from A to P , u 1. P is a finitely presented projective E-algebra and pair P u where is a s t n An algebraic E -unifier for the problem A Algebraic uni�ers Appendix logic Łukasiewicz type of Uni�cation MV-algebras projectivity through The approach logic Łukasiewicz Uni�cation Introduction Luca Spada P . Let us think of a generic E -unification problem ( s , t ) as a finitely presented E -algebra A = F n / ⟨ ( s , t ) ⟩ .

The uni�cation type MV-algebras 1. P is a finitely presented projective E-algebra and Algebraic uni�ers Appendix logic Łukasiewicz of Łukasiewicz logic Uni�cation type of projectivity through The approach logic Łukasiewicz Uni�cation Introduction Luca Spada Let us think of a generic E -unification problem ( s , t ) as a finitely presented E -algebra A = F n / ⟨ ( s , t ) ⟩ . An algebraic E -unifier for the problem A = F n / ⟨ ( s , t ) ⟩ is a pair ( P , u ) where 2. u is an arrow from A to P , u : A − → P .

The uni�cation type u . . A . P . . if P . u . t . there exists an arrow t s.t. is more general than P u of Łukasiewicz logic through Luca Spada Introduction Uni�cation Łukasiewicz logic The approach projectivity MV-algebras Uni�cation type of Łukasiewicz logic Appendix Most general algebraic uni�ers # An algebraic E -unifier ( P , u )

The uni�cation type of Łukasiewicz logic . t . . . u . P . A . . there exists an arrow t s.t. if # Most general algebraic uni�ers Appendix Luca Spada Introduction Uni�cation Łukasiewicz logic The approach through projectivity MV-algebras Uni�cation type of Łukasiewicz logic An algebraic E -unifier ( P , u ) is more general than ( P ′ , u ′ ) u ′ P ′

The uni�cation type of Łukasiewicz logic . t . . . u . P . A . . there exists an arrow t s.t. # Most general algebraic uni�ers Appendix Luca Spada Introduction Uni�cation Łukasiewicz logic The approach through projectivity MV-algebras Uni�cation type of Łukasiewicz logic An algebraic E -unifier ( P , u ) is more general than ( P ′ , u ′ ) if u ′ P ′

The uni�cation type The two approaches define two pre-orders which can be . categories). The syntactic approach and the algebraic are equivalent (as . . . . . . . . Theorem (1997 Ghilardi) . thought of as categories. Equivalence of the two approaches of Łukasiewicz logic Appendix logic Łukasiewicz type of Uni�cation MV-algebras projectivity through The approach logic Łukasiewicz Uni�cation Introduction Luca Spada . Jump to the proof

The uni�cation type The two approaches define two pre-orders which can be . categories). The syntactic approach and the algebraic are equivalent (as . . . . . . . . Theorem (1997 Ghilardi) . thought of as categories. Equivalence of the two approaches of Łukasiewicz logic Appendix logic Łukasiewicz type of Uni�cation MV-algebras projectivity through The approach logic Łukasiewicz Uni�cation Introduction Luca Spada . Jump to the proof

The uni�cation type of Łukasiewicz logic This was re-proved explicitly and generalised to any Ghilardi himself noticed that finite-valued Łukasiewicz logic Unitarity of �nite-valued Łukasiewicz logic Appendix logic Łukasiewicz type of Uni�cation MV-algebras Duality for f.p. 1 variable Commutative Intermezzo: Non-unitarity MV-algebras Introduction Luca Spada finite-valued extension of Basic Logic by Dzik. ℓ -groups has unitary type.

The uni�cation type of Łukasiewicz logic Ghilardi himself noticed that finite-valued Łukasiewicz logic Unitarity of �nite-valued Łukasiewicz logic Appendix logic Łukasiewicz type of Uni�cation MV-algebras Duality for f.p. 1 variable Commutative Intermezzo: Non-unitarity MV-algebras Introduction Luca Spada finite-valued extension of Basic Logic by Dzik. ℓ -groups has unitary type. This was re-proved explicitly and generalised to any

at least not unitary. The uni�cation type of Łukasiewicz logic the substitution x x (hence it must be ) or (hence it is the substitution x x , then it must unify either x is a unifier for x Indeed if This entails the unification type of Łukasiewicz logic to be admissible.) Łukasiewicz logic has a weak disjunction property. Namely: logic Non unitarity of the uni�cation in Łukasiewicz Appendix logic Commutative Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: 1 variable Łukasiewicz Duality for f.p. MV-algebras Uni�cation type of ). if φ ∨ ¬ φ is derivable then either φ or ¬ φ must be derivable. ℓ -groups (in other words the multiple-conclusion rule φ ∨ ¬ φ / φ, ¬ φ is

The uni�cation type of Łukasiewicz logic the substitution x x (hence it must be ) or (hence it is the substitution x x , then it must unify either x is a unifier for x Indeed if This entails the unification type of Łukasiewicz logic to be admissible.) Łukasiewicz logic has a weak disjunction property. Namely: logic Non unitarity of the uni�cation in Łukasiewicz Appendix logic Łukasiewicz Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative ). 1 variable Duality for f.p. MV-algebras Uni�cation type of if φ ∨ ¬ φ is derivable then either φ or ¬ φ must be derivable. ℓ -groups (in other words the multiple-conclusion rule φ ∨ ¬ φ / φ, ¬ φ is at least not unitary.

The uni�cation type Uni�cation This entails the unification type of Łukasiewicz logic to be admissible.) Łukasiewicz logic has a weak disjunction property. Namely: logic Non unitarity of the uni�cation in Łukasiewicz Appendix logic of Łukasiewicz logic type of Łukasiewicz MV-algebras Duality for f.p. 1 variable Commutative Intermezzo: Non-unitarity MV-algebras Introduction Luca Spada if φ ∨ ¬ φ is derivable then either φ or ¬ φ must be derivable. ℓ -groups (in other words the multiple-conclusion rule φ ∨ ¬ φ / φ, ¬ φ is at least not unitary. Indeed if σ is a unifier for x ∨ ¬ x , then it must unify either x (hence it is the substitution x �→ 1 ) or ¬ x (hence it must be the substitution x �→ 0 ).

The uni�cation type . . . The category of MV-alegrbas is equivalent to the category of preserving the strong unit). In 1975, Beynon, expanding previous results by Baker, established a categorical duality which enabled a geometrical study of finitely presented -groups. This duality led to the following purely algebraic result. Theorem (1977 Beynon) . . . . . . . . . Finitely generated projective -groups are exactly the finitely . . of Łukasiewicz logic . Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. MV-algebras Uni�cation type of Łukasiewicz logic Appendix . Theorem (Mundici 1986) . . presented -groups. Intermezzo: Commutative ℓ -groups... Abelian ℓ -groups with strong unit (with ℓ -morphisms ℓ -groups

The uni�cation type . . . . The category of MV-alegrbas is equivalent to the category of preserving the strong unit). In 1975, Beynon, expanding previous results by Baker, established a categorical duality which enabled a geometrical This duality led to the following purely algebraic result. Theorem (1977 Beynon) of Łukasiewicz logic . . . . . . . . Finitely generated projective -groups are exactly the finitely . . . . Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. MV-algebras Uni�cation type of Łukasiewicz logic Appendix . Theorem (Mundici 1986) . presented -groups. Intermezzo: Commutative ℓ -groups... Abelian ℓ -groups with strong unit (with ℓ -morphisms ℓ -groups study of finitely presented ℓ -groups.

The uni�cation type . . . . The category of MV-alegrbas is equivalent to the category of preserving the strong unit). In 1975, Beynon, expanding previous results by Baker, established a categorical duality which enabled a geometrical This duality led to the following purely algebraic result. Theorem (1977 Beynon) of Łukasiewicz logic . . . . . . . . . . . . Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. MV-algebras Uni�cation type of Łukasiewicz logic Appendix . Theorem (Mundici 1986) . Intermezzo: Commutative ℓ -groups... Abelian ℓ -groups with strong unit (with ℓ -morphisms ℓ -groups study of finitely presented ℓ -groups. Finitely generated projective ℓ -groups are exactly the finitely presented ℓ -groups.

The uni�cation type . . Theorem . . . . . In the light of the Beynon’s and Ghilardi’s results, one easily . . The unification type of the theory of -groups is unitary. In a forthcoming paper with V. Marra, we exploit Beynon’s geometrical duality to give an algorithm that, taken any (system of) term in the language of -groups, outputs its gets: most general unifier. of Łukasiewicz logic ...and their uni�cation type. Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. MV-algebras Uni�cation type of Łukasiewicz logic Appendix In the theory of ℓ -groups all system of equations are solvable. ℓ -groups

(system of) term in the language of -groups, outputs its The uni�cation type . gets: . Theorem . . . . of Łukasiewicz logic . . . In a forthcoming paper with V. Marra, we exploit Beynon’s geometrical duality to give an algorithm that, taken any In the light of the Beynon’s and Ghilardi’s results, one easily most general unifier. ...and their uni�cation type. Introduction logic Łukasiewicz type of Uni�cation MV-algebras Duality for f.p. 1 variable Appendix Commutative Intermezzo: Non-unitarity Luca Spada MV-algebras In the theory of ℓ -groups all system of equations are solvable. ℓ -groups The unification type of the theory of ℓ -groups is unitary.

The uni�cation type of Łukasiewicz logic geometrical duality to give an algorithm that, taken any In a forthcoming paper with V. Marra, we exploit Beynon’s . . . . . . . . Theorem . gets: In the light of the Beynon’s and Ghilardi’s results, one easily most general unifier. ...and their uni�cation type. Introduction logic Łukasiewicz type of Uni�cation MV-algebras Duality for f.p. 1 variable Appendix Commutative Intermezzo: Non-unitarity MV-algebras Luca Spada In the theory of ℓ -groups all system of equations are solvable. ℓ -groups The unification type of the theory of ℓ -groups is unitary. (system of) term in the language of ℓ -groups, outputs its

In particular the proof shows that there are at most two The uni�cation type . . which is continuous, piece-wise linear and with integer coefficients. These functions are named after McNaughton, who first proved that they exactly correspond to formulæ of Łukasiewicz logic. . Proposition . . of Łukasiewicz logic . . . . . The unification type of the 1-variable fragment of Łukasiewicz logic is finitary. most general unifiers, for any given formula. . . . . Uni�cation Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. MV-algebras type of . Łukasiewicz logic Appendix Finitarity result . De�nition . . . . Jump to the proof A McNaughton function is a function φ : [0 , 1] n → [0 , 1] ℓ -groups

In particular the proof shows that there are at most two The uni�cation type . . which is continuous, piece-wise linear and with integer coefficients. These functions are named after McNaughton, who first proved that they exactly correspond to formulæ of Łukasiewicz logic. . Proposition . . of Łukasiewicz logic . . . . . The unification type of the 1-variable fragment of Łukasiewicz logic is finitary. most general unifiers, for any given formula. . . . . Uni�cation Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. MV-algebras type of . Łukasiewicz logic Appendix Finitarity result . De�nition . . . . Jump to the proof A McNaughton function is a function φ : [0 , 1] n → [0 , 1] ℓ -groups

The uni�cation type . . which is continuous, piece-wise linear and with integer coefficients. These functions are named after McNaughton, who first proved that they exactly correspond to formulæ of Łukasiewicz logic. . Proposition . . of Łukasiewicz logic . . . . . The unification type of the 1-variable fragment of Łukasiewicz logic is finitary. In particular the proof shows that there are at most two most general unifiers, for any given formula. . . . . Uni�cation Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. MV-algebras type of . Łukasiewicz logic Appendix Finitarity result . De�nition . . . . Jump to the proof A McNaughton function is a function φ : [0 , 1] n → [0 , 1] ℓ -groups

The uni�cation type . . which is continuous, piece-wise linear and with integer coefficients. These functions are named after McNaughton, who first proved that they exactly correspond to formulæ of Łukasiewicz logic. . Proposition . . of Łukasiewicz logic . . . . . The unification type of the 1-variable fragment of Łukasiewicz logic is finitary. In particular the proof shows that there are at most two most general unifiers, for any given formula. . . . . Uni�cation Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. MV-algebras type of . Łukasiewicz logic Appendix Finitarity result . De�nition . . . . Jump to the proof A McNaughton function is a function φ : [0 , 1] n → [0 , 1] ℓ -groups

The uni�cation type . . De�nition . . . . . . However, McNaughton functions are only an instance of a . A rational polytope is the convex hull of a finite set of rational points. A rational polyhedron is the union of a finite number of rational polytopes. A -map is a continuous piecewise linear function with stronger link between Łukasiewicz logic and Geometry. logic. of Łukasiewicz logic understanding the dynamic of substitutions in Łukasiewicz Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. MV-algebras Uni�cation type of Łukasiewicz logic Appendix A geometrical view on uni�cation McNaughton functions prove to be a useful tool for integer coefficients. ℓ -groups

The uni�cation type . . De�nition . . . . . . However, McNaughton functions are only an instance of a . A rational polytope is the convex hull of a finite set of rational points. A rational polyhedron is the union of a finite number of rational polytopes. A -map is a continuous piecewise linear function with stronger link between Łukasiewicz logic and Geometry. logic. of Łukasiewicz logic understanding the dynamic of substitutions in Łukasiewicz Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. MV-algebras Uni�cation type of Łukasiewicz logic Appendix A geometrical view on uni�cation McNaughton functions prove to be a useful tool for integer coefficients. ℓ -groups

The uni�cation type . . De�nition . . . . . . However, McNaughton functions are only an instance of a . A rational polytope is the convex hull of a finite set of rational points. A rational polyhedron is the union of a finite number of rational polytopes. A -map is a continuous piecewise linear function with stronger link between Łukasiewicz logic and Geometry. logic. of Łukasiewicz logic understanding the dynamic of substitutions in Łukasiewicz Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. MV-algebras Uni�cation type of Łukasiewicz logic Appendix A geometrical view on uni�cation McNaughton functions prove to be a useful tool for integer coefficients. ℓ -groups

The uni�cation type . . De�nition . . . . . . However, McNaughton functions are only an instance of a . A rational polytope is the convex hull of a finite set of rational points. A rational polyhedron is the union of a finite number of rational polytopes. A -map is a continuous piecewise linear function with stronger link between Łukasiewicz logic and Geometry. logic. of Łukasiewicz logic understanding the dynamic of substitutions in Łukasiewicz Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. MV-algebras Uni�cation type of Łukasiewicz logic Appendix A geometrical view on uni�cation McNaughton functions prove to be a useful tool for integer coefficients. ℓ -groups

The uni�cation type . stronger link between Łukasiewicz logic and Geometry. . De�nition . . . . logic. . . . A rational polytope is the convex hull of a finite set of rational points. A rational polyhedron is the union of a finite number of rational polytopes. However, McNaughton functions are only an instance of a understanding the dynamic of substitutions in Łukasiewicz of Łukasiewicz logic McNaughton functions prove to be a useful tool for Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. MV-algebras Uni�cation type of Łukasiewicz logic Appendix A geometrical view on uni�cation integer coefficients. ℓ -groups A Z -map is a continuous piecewise linear function with

The uni�cation type of Łukasiewicz logic These functors operate very similarly to the classical ones in fp and fp I will define a pair of (contravariant) functors: between them. -maps be the category of rational polyhedra and Let with their homomorphisms. Rational polyhedra and MV-algebras Appendix logic Łukasiewicz type of Uni�cation MV-algebras Duality for f.p. 1 variable Commutative Intermezzo: Non-unitarity MV-algebras Introduction Luca Spada algebraic geometry that associate ideals with varieties. Let MV fp be the category of finitely presented MV-algebras ℓ -groups

The uni�cation type of Łukasiewicz logic These functors operate very similarly to the classical ones in fp and fp I will define a pair of (contravariant) functors: between them. with their homomorphisms. Rational polyhedra and MV-algebras Appendix logic Łukasiewicz type of Uni�cation MV-algebras Duality for f.p. 1 variable Commutative Intermezzo: Non-unitarity MV-algebras Introduction Luca Spada algebraic geometry that associate ideals with varieties. Let MV fp be the category of finitely presented MV-algebras Let P Z be the category of rational polyhedra and Z -maps ℓ -groups

The uni�cation type type of These functors operate very similarly to the classical ones in and I will define a pair of (contravariant) functors: between them. with their homomorphisms. Rational polyhedra and MV-algebras Appendix logic of Łukasiewicz logic Łukasiewicz Uni�cation MV-algebras Duality for f.p. 1 variable Commutative Intermezzo: Non-unitarity MV-algebras Introduction Luca Spada Let MV fp be the category of finitely presented MV-algebras Let P Z be the category of rational polyhedra and Z -maps ℓ -groups I : MV fp → P Z V : P Z → MV fp . algebraic geometry that associate ideals with varieties.

The uni�cation type type of Free n P generators, so it makes sense to set The set I P is a congruence of the free MV-algebras on n such that objects Interpreting MV-algebras inside polyhedra: Appendix logic of Łukasiewicz logic Łukasiewicz Uni�cation MV-algebras Duality for f.p. 1 variable Commutative Intermezzo: Non-unitarity MV-algebras Introduction Luca Spada I P Let P ∈ P Z . ℓ -groups Let us write I ( P ) for the collection of all pair MV-terms ( s , t ) s ( x ) = t ( x ) for all x ∈ P

The uni�cation type Uni�cation generators, so it makes sense to set such that objects Interpreting MV-algebras inside polyhedra: Appendix logic Łukasiewicz of Łukasiewicz logic type of MV-algebras Duality for f.p. 1 variable Commutative Intermezzo: Non-unitarity MV-algebras Introduction Luca Spada Let P ∈ P Z . ℓ -groups Let us write I ( P ) for the collection of all pair MV-terms ( s , t ) s ( x ) = t ( x ) for all x ∈ P The set I ( P ) is a congruence of the free MV-algebras on n I ( P ) = Free n I ( P ) .

The uni�cation type of Łukasiewicz logic P is a homomorphism Q Then, the function P f M Q f as Define arrows Interpreting polyhedra inside MV-algebras: Appendix logic Commutative Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: 1 variable Łukasiewicz Duality for f.p. MV-algebras Uni�cation type of of MV-algebras. Let ζ : P → Q be a diagram in P Z . ℓ -groups I ( ζ ): I ( Q ) → I ( P )

The uni�cation type type of P is a homomorphism Q Then, the function as Define arrows Interpreting polyhedra inside MV-algebras: Appendix of Łukasiewicz logic Łukasiewicz logic Uni�cation Intermezzo: Luca Spada Introduction MV-algebras MV-algebras Non-unitarity 1 variable Commutative Duality for f.p. of MV-algebras. Let ζ : P → Q be a diagram in P Z . ℓ -groups I ( ζ ): I ( Q ) → I ( P ) M ( ζ ) f ∈ I ( Q ) �− → f ◦ ζ ∈ I ( P ) .

The uni�cation type Uni�cation as Define arrows Interpreting polyhedra inside MV-algebras: Appendix logic of Łukasiewicz logic type of Łukasiewicz MV-algebras Non-unitarity Luca Spada Duality for f.p. MV-algebras Introduction Intermezzo: Commutative 1 variable of MV-algebras. Let ζ : P → Q be a diagram in P Z . ℓ -groups I ( ζ ): I ( Q ) → I ( P ) M ( ζ ) f ∈ I ( Q ) �− → f ◦ ζ ∈ I ( P ) . Then, the function I ( ζ ): I ( Q ) → I ( P ) is a homomorphism

The uni�cation type Uni�cation A is a rational polyhedron, so we set The set V objects Interpreting polyhedra inside MV-algebras: Appendix logic of Łukasiewicz logic type of Łukasiewicz MV-algebras Duality for f.p. Luca Spada Introduction MV-algebras V Non-unitarity Intermezzo: Commutative 1 variable Let A = Free n ∈ MV fp . θ ℓ -groups Let us write V ( θ ) for the collection of all real points p in [0 , 1] n such that s ( p ) = t ( p ) for all ( s , t ) ∈ θ

The uni�cation type MV-algebras objects Interpreting polyhedra inside MV-algebras: Appendix logic Łukasiewicz of Łukasiewicz logic Uni�cation type of Duality for f.p. MV-algebras 1 variable Introduction Luca Spada Non-unitarity Intermezzo: Commutative Let A = Free n ∈ MV fp . θ ℓ -groups Let us write V ( θ ) for the collection of all real points p in [0 , 1] n such that s ( p ) = t ( p ) for all ( s , t ) ∈ θ The set V ( θ ) is a rational polyhedron, so we set V ( A ) = V ( θ ) .

t i p The uni�cation type of Łukasiewicz logic A is a B h Then, the function A i I h A p as Suppose that h sends the generators of A into the elements arrows Interpreting MV-algebras inside polyhedra: Appendix 1 variable Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative Duality for f.p. logic MV-algebras Uni�cation type of Łukasiewicz -map. Let h : A → B be a diagram in MV fp . ℓ -groups { t i } i ∈ I of B , then define V ( h ): V ( B ) → V ( A )

The uni�cation type Łukasiewicz A is a B h Then, the function as Suppose that h sends the generators of A into the elements arrows Interpreting MV-algebras inside polyhedra: of Łukasiewicz logic logic Appendix type of Uni�cation Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. MV-algebras -map. Let h : A → B be a diagram in MV fp . ℓ -groups { t i } i ∈ I of B , then define V ( h ): V ( B ) → V ( A ) V ( h ) p ∈ V ( A ) �− → ⟨ t i ( p ) ⟩ i ∈ I ∈ V ( A ) .

The uni�cation type Uni�cation as Suppose that h sends the generators of A into the elements arrows Interpreting MV-algebras inside polyhedra: Appendix logic of Łukasiewicz logic type of Łukasiewicz MV-algebras Non-unitarity Luca Spada Duality for f.p. MV-algebras Introduction Intermezzo: Commutative 1 variable Let h : A → B be a diagram in MV fp . ℓ -groups { t i } i ∈ I of B , then define V ( h ): V ( B ) → V ( A ) V ( h ) p ∈ V ( A ) �− → ⟨ t i ( p ) ⟩ i ∈ I ∈ V ( A ) . Then, the function V ( h ): V ( B ) → V ( A ) is a Z -map.

The uni�cation type . constitutes a contravariant equivalence between the two fp and fp The pair of functors . . . . . . . . Theorem (Folklore) Duality for �nitely presented MV-algebras of Łukasiewicz logic Commutative Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: 1 variable Appendix Duality for f.p. MV-algebras Uni�cation type of Łukasiewicz logic categories. ℓ -groups

The uni�cation type of Łukasiewicz logic constitutes a contravariant equivalence between the two and The pair of functors . . . . . . . . Theorem (Folklore) . Duality for �nitely presented MV-algebras Appendix logic Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. MV-algebras Uni�cation type of Łukasiewicz categories. ℓ -groups I : MV fp → P Z V : P Z → MV fp .

n (i.e. a retract by The uni�cation type logic 2. u is a some n , and -maps) for -retract of 1. P is a A co-unifier for A is a pair P u , where rational polyhedra A . A co-unification problem, in this setting, is now given by a Co-uni�cation problems Appendix Łukasiewicz of Łukasiewicz logic type of Uni�cation MV-algebras Duality for f.p. 1 variable Commutative Intermezzo: Non-unitarity MV-algebras Introduction Luca Spada -map from P to A . ℓ -groups

The uni�cation type of Łukasiewicz logic some n , and rational polyhedra A . A co-unification problem, in this setting, is now given by a Co-uni�cation problems Appendix logic Łukasiewicz type of Uni�cation MV-algebras Duality for f.p. 1 variable Commutative Intermezzo: Non-unitarity MV-algebras Introduction Luca Spada ℓ -groups A co-unifier for A is a pair ( P , u ) , where 1. P is a Z -retract of [0 , 1] n (i.e. a retract by Z -maps) for 2. u is a Z -map from P to A .

is more general than Q w if there exists The uni�cation type # u . P . w . t . . P Remark . . . . . . . . The unification type of Łukasiewicz logic and the . . of Łukasiewicz logic MV-algebras Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. Uni�cation A type of Łukasiewicz logic Appendix Co-uni�cation type t such that: . . co-unification type of rational prolyhedra coincide. A co-unifier ( P , u ) ℓ -groups

The uni�cation type . . u . . w . t . # Remark of Łukasiewicz logic . . . . . . . . The unification type of Łukasiewicz logic and the P . A . Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. MV-algebras Uni�cation type of Łukasiewicz logic Appendix Co-uni�cation type if there exists t such that: . co-unification type of rational prolyhedra coincide. A co-unifier ( P , u ) is more general than ( Q , w ) ℓ -groups P ′

The uni�cation type . u . . w . t . # Remark P . . . . . . . . The unification type of Łukasiewicz logic and the . . of Łukasiewicz logic A Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. MV-algebras Uni�cation type of Łukasiewicz logic Appendix Co-uni�cation type t such that: . . co-unification type of rational prolyhedra coincide. A co-unifier ( P , u ) is more general than ( Q , w ) if there exists ℓ -groups P ′

The uni�cation type . u . . w . t . # Remark P . . . . . . . . The unification type of Łukasiewicz logic and the . . of Łukasiewicz logic A Luca Spada Introduction MV-algebras Non-unitarity Intermezzo: Commutative 1 variable Duality for f.p. MV-algebras Uni�cation type of Łukasiewicz logic Appendix Co-uni�cation type t such that: . . co-unification type of rational prolyhedra coincide. A co-unifier ( P , u ) is more general than ( Q , w ) if there exists ℓ -groups P ′

The uni�cation type x . . . . . The full Łukasiewicz logic has nullary unification type. Proof. Consider the unification problem given by x y . y The rational polyhedron V associated to the finitely presented MV-algebra Free is the following union of four rational polytopes: . . . . of Łukasiewicz logic A co�nal Luca Spada Introduction MV-algebras Uni�cation type of Łukasiewicz logic Statement sequence Theorem Universal covers Strictness and unboundness Co�nal Appendix Nullarity of Łukasiewicz logic . A

The uni�cation type The full Łukasiewicz logic has nullary unification type. . . . . . . . Proof. Consider the unification problem given by Theorem The rational polyhedron V associated to the finitely presented MV-algebra Free is the following union of four rational polytopes: . . . . of Łukasiewicz logic Statement Luca Spada Introduction MV-algebras Uni�cation type of Łukasiewicz logic A co�nal Nullarity of Łukasiewicz logic sequence Universal covers Strictness and unboundness Co�nal Appendix A θ = ( x ∨ x ∗ ∨ y ∨ y ∗ , 1)

The uni�cation type . Theorem . . . . . . of Łukasiewicz logic . The full Łukasiewicz logic has nullary unification type. Proof. Consider the unification problem given by four rational polytopes: . . . Nullarity of Łukasiewicz logic Appendix Co�nal Luca Spada Introduction MV-algebras Uni�cation type of Łukasiewicz logic Statement A co�nal sequence Universal covers Strictness and unboundness A θ = ( x ∨ x ∗ ∨ y ∨ y ∗ , 1) The rational polyhedron V ( θ ) associated to the finitely presented MV-algebra Free 2 / ⟨ θ ⟩ is the following union of

i is a retract of m for some m , so the pairs i are co-unifiers for A . . . . . . . Let us consider the following sequence of rational polyhedra, . . . of Łukasiewicz logic Together with the projections i i A into A . It can be proved (cfr. Cabrer and Mundici) that each i . The uni�cation type . Step 1. Luca Spada Introduction MV-algebras Uni�cation type of Łukasiewicz logic Statement A co�nal sequence Universal covers Strictness and unboundness Co�nal Appendix Proof Cont.'d . . t 3 t 2 t 1

i are co-unifiers for A . . of Łukasiewicz logic . . . . . . . Let us consider the following sequence of rational polyhedra, . . . . so the pairs i Step 1. The uni�cation type Proof Cont.'d Statement Luca Spada Introduction MV-algebras Uni�cation type of Appendix logic Łukasiewicz A co�nal sequence Universal covers Strictness and unboundness Co�nal . t 3 t 2 t 1 Together with the projections ζ i : t i → A into A . It can be proved (cfr. Cabrer and Mundici) that each t i is a retract of [0 , 1] m for some m ,

The uni�cation type . Step 1. . . . . . . Proof Cont.'d . Let us consider the following sequence of rational polyhedra, . . . . of Łukasiewicz logic . Appendix logic Luca Spada Introduction MV-algebras Uni�cation type of Co�nal Łukasiewicz Statement A co�nal sequence Universal covers Strictness and unboundness t 3 t 2 t 1 Together with the projections ζ i : t i → A into A . It can be proved (cfr. Cabrer and Mundici) that each t i is a retract of [0 , 1] m for some m , so the pairs ( t i , ζ i ) are co-unifiers for A .

ij such that the following diagram commutes. . j . i ij is the embedding of i in The uni�cation type . . . . The sequence is increasing, i.e. for any i j , . there exists . . A . j . i . ij Indeed . . of Łukasiewicz logic sequence Luca Spada Introduction MV-algebras Uni�cation type of Łukasiewicz logic Statement A co�nal Universal . covers Strictness and unboundness Co�nal Appendix Proof Cont.'d . Step 2. j

ij is the embedding of i in The uni�cation type . of Łukasiewicz logic . . . . . A . . . . . . . Indeed . . Step 2. sequence Luca Spada Introduction MV-algebras Uni�cation type of Łukasiewicz logic . A co�nal Statement Universal covers Strictness and unboundness Co�nal Appendix Proof Cont.'d j The sequence is increasing, i.e.for any i < j , there exists ι ij such that the following diagram commutes. t j ζ j ι ij ζ i t i

The uni�cation type . . of Łukasiewicz logic . . . . . . . . A . . . . . Step 2. . Proof Cont.'d Statement Luca Spada Introduction MV-algebras Uni�cation type of Łukasiewicz Appendix logic A co�nal sequence Universal covers Strictness and unboundness Co�nal The sequence is increasing, i.e.for any i < j , there exists ι ij such that the following diagram commutes. t j ζ j ι ij ζ i t i Indeed ι ij is the embedding of t i in t j

i and an arrow . i The uni�cation type (called the . . . . and for any arrow P A , there exists some lift of . ) making the following diagram commute. . . A . i . P . . . of Łukasiewicz logic sequence Luca Spada Introduction MV-algebras Uni�cation type of Łukasiewicz logic Statement A co�nal Universal . covers Strictness and unboundness Co�nal Appendix Proof Cont.'d . Step 3: The lifting of functions Lemma. . For any Z -retract P of some cube [0 , 1] n

i and an arrow . i The uni�cation type lift of . . . . . there exists some (called the ) making the following diagram commute. of Łukasiewicz logic . . A . i . P . . . . . sequence Luca Spada Introduction MV-algebras Uni�cation type of Łukasiewicz logic Statement A co�nal Universal Step 3: The lifting of functions Lemma. covers Strictness and unboundness Co�nal Appendix Proof Cont.'d . For any Z -retract P of some cube [0 , 1] n and for any arrow φ : P → A , φ

The uni�cation type ) making the following diagram commute. . . . . . and an arrow (called the lift of . . . A . . P . . . of Łukasiewicz logic . . Step 3: The lifting of functions Lemma. Luca Spada Introduction MV-algebras Uni�cation type of Łukasiewicz logic Statement A co�nal sequence Universal covers Strictness and unboundness Co�nal Appendix Proof Cont.'d . For any Z -retract P of some cube [0 , 1] n and for any arrow φ : P → A , there exists some t i t i ζ i φ

The uni�cation type . . of Łukasiewicz logic . . . . . . . . A . . P . . Step 3: The lifting of functions Lemma. . Proof Cont.'d Introduction Uni�cation Luca Spada type of Łukasiewicz logic Statement Appendix MV-algebras A co�nal sequence Universal covers Strictness and unboundness Co�nal For any Z -retract P of some cube [0 , 1] n and for any arrow φ : P → A , there exists some t i and an arrow ˜ φ (called the lift of φ ) making the following diagram commute. t i φ . ˜ ζ i φ

i is the piecewise linear correspondent of the universal cover of the circle, a space that, indeed, enjoys The uni�cation type topology. . . . . . The above lemma is the piecewise linear version of the “ Lifting of functions ” Lemma , widely used in algebraic spiral The reason why the above lemma works is that the infinite . i this factorisation property for any continuous maps from a simply connected spaces into the circle. Such a lift is known to be unique even in the rather general case of continuous maps, so the fact that in our setting such a map is actually a . . of Łukasiewicz logic A co�nal Luca Spada Introduction MV-algebras Uni�cation type of Łukasiewicz logic Statement sequence Intermezzo 2 Universal covers Strictness and unboundness Co�nal Appendix The universal cover of the circle . -map is a quite pleasant discovery.

The uni�cation type The above lemma is the piecewise linear version of the . . . . . . . “ Lifting of functions ” Lemma , widely used in algebraic of Łukasiewicz logic topology. The reason why the above lemma works is that the infinite this factorisation property for any continuous maps from a simply connected spaces into the circle. Such a lift is known to be unique even in the rather general case of continuous maps, so the fact that in our setting such a map is actually a . Intermezzo 2 . The universal cover of the circle Luca Spada Introduction MV-algebras Uni�cation type of Łukasiewicz logic Statement A co�nal sequence Universal covers Strictness and unboundness Co�nal Appendix -map is a quite pleasant discovery. spiral t ∞ = ∪ i ∈ ω t i is the piecewise linear correspondent of the universal cover of the circle, a space that, indeed, enjoys