Post on 25-May-2020
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Abstract Valuation Semantics
Carlos Caleiro1,2
1Dept. Matematica, IST, TU Lisbon2SQIG - Instituto de Telecomunicacoes
Portugal
XVI EBL, May 2011Petropolis, Brasil
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Outline
1 Motivation
2 AlgebraizationLogicAlgebraizationAALLimitations
3 Behavioral algebraizationMany-sorted signaturesBehavioral equivalenceBehavioral algebraizationBehavioral AAL
4 Semantical considerationsAbstract valuation semanticsBridge results
5 ExamplesK/2C1
6 Conclusions and future work
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Motivation
Theory of algebraization of logics (AAL)[Lindenbaum & Tarski, Blok & Pigozzi, Czelakowski, Nemeti et al]
Representation of logics in algebraic setting (restricted topropositional based logics)
Many-sorted and non-truth-functional logics
Propositional based logics are often not enough for reasoningabout complex systemsTruth-functionality is not ubiquitous
Generalization of the notion of algebraizable logic
Many-sorted behavioral logic
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Motivation
AAL _? // bAAL
Matrices ???
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Motivation
AAL _? // bAAL
Matrices ???
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Motivation
AAL _? // bAAL
Matrices _? // ???
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Logic
Logic
Definition
A structural propositional logic is a pair L = 〈Σ,`〉, where Σ is apropositional signature and `⊆ P(LΣ(X))× LΣ(X) is aconsequence relation satisfying the following conditions, for everyT1 ∪ T2 ∪ ϕ ⊆ LΣ(X):
Reflexivity: if ϕ ∈ T1 then T1 ` ϕCut: if T1 ` ϕ for all ϕ ∈ T2, and T2 ` ψ then T1 ` ψ
Weakening: if T1 ` ϕ and T1 ⊆ T2 then T2 ` ϕStructurality: if T1 ` ϕ then σ[T1] ` σ(ϕ)
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Algebraization
Algebraization
L
unisorted
ksStrong representation +3Equational logic
unisorted
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Algebraization
Algebraizable logic
L is algebraizable if there are translations θ and ∆, and a class ofalgebras K s.t.
LΘ(x)
.. EqnΣK
∆(x1,x2)
mm
T ` ϕ iff Θ[T ] EqnΣK
Θ(ϕ)
∆(δi, εi) : i ∈ I ` ∆(ϕ1, ϕ2) iff δi ≈ εi : i ∈ I EqnΣKϕ1 ≈ ϕ2
ϕ a` ∆[Θ(ϕ)] ϕ1 ≈ ϕ2 =||=EqnΣK
Θ[∆(ϕ1, ϕ2)]
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Algebraization
Examples
Example (I)
CPC Boolean algebras
Example (II)
IPC Heyting algebras
In both casesΘ(x) = x ≈ > and ∆(x, y) = x↔ y
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
AAL
Leibniz operator - The main tool
Definition (Leibniz operator)
Let L = 〈Σ,`〉 be a structural propositional logic. Then theLeibniz operator on the formula algebra can be given by:
Ω : ThL → Congr(LΣ(X))
T 7→ largest congruence compatible with T.
θ is compatible with T if 〈ϕ,ψ〉 ∈ θ implies (ϕ ∈ T ) iff (ψ ∈ T )
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
AAL
The Leibniz hierarchy
finitely algebraizablemonotone
injective
continuous
wwooooooooooooo
''OOOOOOOOOOO
finitely equivalentialmonotone
continuous
algebraizablemonotone
injective
commutes with inverse substitutions
ssgggggggggggggggggggggggggg
equivalentialmonotone
commutes with inverse substitutions
))RRRRRRRRRRRRRR
weakly algebraizablemonotone
injective
uullllllllllllll
protoalgebraicmonotone
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
AAL
Logical matrices
Consider fixed a propositional signature Σ.
Definition
A Σ-matrix is a tuple m = 〈A, D〉 such that:
A is a Σ-algebra,
D ⊆ A (designated values).
m with an assignment h over A satisfies ϕ, denoted by m,h ϕ, if h(ϕ) ∈ D.
The class of Σ-matrix models of L will be denoted by Matr(L).
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
AAL
Canonical semantics
A congruence of a matrix m = 〈A, D〉 is a congruence of A compatible with D.The Leibniz congruence Ω(m) of m is simply the largest congruence of m.
Reduced matrix is m∗ = 〈A/Ω(m), [D]Ω(m)〉.
Definition
The canonical matrix semantics of L is Matr∗(L).
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
AAL
Bridge results
L is complete for Matr(L), Matr∗(L)(Bloom) L is finitary iff Matr(L) is closed under ultraproducts
(Blok& Pigozzi) L is protoalgebraic iff Matr∗(L) is closedunder subdirect products
(Czelakowski) L is equivalential iff Matr∗(L) is closed undersubmatrices and products
(Herrmann) L is finitely equivalential iff Matr∗(L) is aquasivariety
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Limitations
Limitations
First-order logic
“Its” algebraization gives rise to cylindric algebras (Henkin,Monk & Tarski) or polyadic algebras (Halmos)
Limitation applicable to other many-sorted logics
Even at the propositional level there are some “relatively”well-behaved logics that are not algebraizable
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Limitations
Paraconsistent logic C1 of da Costa
C1 (da Costa 1963) given by a structural Hilbert-style axiomatization
C1 is paraconsistent: the explosion principle ϕ,¬ϕ ` ψ is not valid in C1
Mortensen 1980: there are no non-trivial congruences on the formulaalgebra compatible with the set of theoremsExample:`C1 (p ∧ q)⇔ (q ∧ p) but 0C1 ¬(p ∧ q)⇔¬(q ∧ p)
Lewin, Mikenberg & Schwarze 1991: there exists a model in which theLeibniz operator is not injective
da Costa 1966, Carnielli & Alcantara 1984 have proposed an ”algebraiccounterpart” called da Costa algebras (paraconsistent algebras)
Caleiro et al 2003: alternative approach using two-sorted algebras
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Limitations
A simpler counterexample
The logic K/2 of Beziau 1999 is not algebraizable!
Σ contains the connectives ¬, ⇒semantics defined by the set of all v : LΣ(X)→ 0, 1 st.
v(¬A) = 0 if v(A) = 1v(A⇒B) = 0 iff v(A) = 1 and v(B) = 0
Axiomatization (Humberstone)
` A⇒ (B⇒A)` (A⇒ (B⇒ C))⇒ ((A⇒B)⇒ (A⇒ C))
` ((A⇒B)⇒A)⇒A
` A⇒ ((¬A)⇒B)A,A⇒B ` B
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Many-sorted signatures
Many-sorted signatures
Definition (Many-sorted signature)
A many-sorted signature is a pair Σ = 〈S, F 〉 where S is a set (of sorts)and F = Fwsw∈S∗,s∈S is an indexed family of sets (of operations).
Example (FOL)
ΣFOL = 〈S, F 〉 such that
S = φ, tFφφ = ∀x : x ∈ X ∪ ¬Fφφφ = ⇒,∧,∨Ftnφ = P : P n-ary predicate symbolFtnt = f : f n-ary function symbol
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Many-sorted signatures
Many-sorted signatures
Definition (Many-sorted signature)
A many-sorted signature is a pair Σ = 〈S, F 〉 where S is a set (of sorts)and F = Fwsw∈S∗,s∈S is an indexed family of sets (of operations).
Example (FOL)
ΣFOL = 〈S, F 〉 such that
S = φ, tFφφ = ∀x : x ∈ X ∪ ¬Fφφφ = ⇒,∧,∨Ftnφ = P : P n-ary predicate symbolFtnt = f : f n-ary function symbol
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Behavioral equivalence
Behavioral Equivalence: motivation
Dichotomy Visible vs Hidden - The OO paradigm
For security reasons and/or to simplify the process of updating andimproving program implementations, computational systems dealwith heterogeneous data and split it in two different kinds:
hidden data (internal data or states);
visible data (the real data).
Hidden data can only be indirectly accessed by programs withvisible output taking them as input; they are only compared bybehavioral equivalence. Intuitively, two data elements are said tobe behaviorally equivalent if they cannot be distinguished by anyprogram with visible output. [Reichel 1984]
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Behavioral algebraization
Many-sorted logics
We will only consider many-sorted signatures with a distinguished sort φ
(of formulas). We will call formulas to the elements of TΣ,φ(X).
Definition (Many-sorted logics)
A many-sorted logic is a pair L = 〈Σ,`〉 where Σ is a many-sortedsignature and `⊆ P(TΣ,φ(X))× (TΣ,φ(X)) is a structuralconsequence relation over the set of formulas.
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Behavioral algebraization
How to generalize?
L
unsorted
ksStrong representation +3Equational logic
unsorted
Lmany-sorted
ksStrong representation +3
Behavioral equational logicmany-sorted
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Behavioral algebraization
How to generalize?
L
unsorted
ksStrong representation +3Equational logic
unsorted
Lmany-sorted
ksStrong representation +3
Behavioral equational logicmany-sorted
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Behavioral algebraization
How to generalize?
L
unsorted
ksStrong representation +3Equational logic
unsorted
Lmany-sorted
ksStrong representation +3 Behavioral equational logicmany-sorted
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Behavioral algebraization
Γ-Behavioral equivalence
Consider given a subsignature Γ of Σ.
Given a Σo-algebra A, two elements a1, a2 ∈ As are Γ-behavioralequivalent a1 ≡Γ a2, if for every formula ϕ(x : s, u : r) ∈ TΓ(X),called a context or an experiment, and every tuple〈b1, . . . , bn〉 ∈ Abr:
ϕA(a1, b1, . . . , bn) = ϕA(a2, b1, . . . , bn).
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Behavioral algebraization
Behavioral logic
Definition (Many-sorted behavioral equational logic BEqnΣK,Γ)
ti ≈ t′i : i ∈ I BEqnΣK,Γ
t ≈ t′
ifffor every A ∈ K and homomorphism h : TΣ(X)→ A,h(t)≡Γh(t′) whenever h(ti)≡Γh(t′i) for every i ∈ I
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Behavioral algebraization
Behavioral algebraization
LΘ(x)
.. BEqnΣK,Γ
∆(x1,x2)
mm
T ` ϕ iff Θ[T ] BEqnΣK,Γ
Θ(ϕ)
∆(δi, εi) : i ∈ I ` ∆(ϕ1, ϕ2) iff δi ≈ εi : i ∈ I BEqnΣK,Γ
ϕ1 ≈ ϕ2
ϕ a` ∆[Θ(ϕ)] ϕ1 ≈ ϕ2 =||=BEqnΣK,Γ
Θ[∆(ϕ1 ≈ ϕ2)]
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Behavioral algebraization
Examples
Example (Standard algebraization)
Algebraizable logics are Γ-behaviorally algebraizable by taking Γ = Σ.
Example (da Costa’s paraconsistent logic C1)
C1 is Γ-behaviorally algebraizable with Γ = ΣC1 \ ¬.
Example (Lewis’s modal logic S5)
S5 is Γ-behaviorally algebraizable with Γ = ΣS5 \ .
Example (Nelson’s constructive logic with strong negation)
N is Γ-behaviorally algebraizable with Γ = ΣN \ ∼. Makes explicit the keyrole that Heyting algebras play in the algebraic counterpart of N .
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Behavioral AAL
Behavioral Leibniz operator
Definition (Behavioral Leibniz operator)
Let L = 〈Σ,`〉 be a structural many-sorted logic and Γ a subsignature of Σ.The Γ-behavioral Leibniz operator on the term algebra,
ΩbhvΓ : ThL → CongΓ(TΣ(X))
T 7→ largest Γ-congruence compatible with T.
We have:
ΩbhvΣ = Ω.
The operator ΩbhvΓ is monotone iff Ω is.
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Behavioral AAL
The behavioral Leibniz hierarchy
beh. finitely algebraizablemonotone
injective
continuous
wwnnnnnnnnnnnnnn
((PPPPPPPPPPPP
beh. finitely equivalentialmonotone
continuous
beh. algebraizablemonotone
injective
commutes with inverse substitutions
ssffffffffffffffffffffffffffffff
beh. equivalentialmonotone
commutes with inverse substitutions
))SSSSSSSSSSSSSSS
beh. weakly algebraizablemonotone
injective
uukkkkkkkkkkkkkkk
protoalgebraicmonotone
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Behavioral AAL
The two hierarchies
algebraizable
6=
uujjjjjjjjjjjjjjj
**UUUUUUUUUUUUUUUU
equivalential
6=
===
====
====
====
====
====
=== weakly algebraizable
?
~~
behaviorally algebraizable
uujjjjjjjjjjjjjjj
**UUUUUUUUUUUUUUUU
behaviorally equivalential
))TTTTTTTTTTTTTTT behaviorally weakly algebraizable
ttiiiiiiiiiiiiiiii
protoalgebraic
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Abstract valuation semantics
The problem
AAL _? // bAAL
Matrices _? // ???
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Abstract valuation semantics
The problem
Γ-congruence is in general not a congruence
How to reduce a model?
=⇒ Use an algebraically well-behaved notion of valuationsemantics (da Costa & Beziau 1994)
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Abstract valuation semantics
The problem
Γ-congruence is in general not a congruence
How to reduce a model?
=⇒ Use an algebraically well-behaved notion of valuationsemantics (da Costa & Beziau 1994)
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Abstract valuation semantics
Abstract valuation semantics
Consider fixed a signature Σ = 〈S, F 〉 and a subsignature Γ of Σ.
Definition
An abstract Γ-valuation is a tuple α = 〈A, ϑ, 〈B, D〉〉 such that:
A is a (concrete) Σ-algebra,
〈B, D〉 is a (abstract) Γ-matrix, and
ϑ : A|Γ → B is a surjective Γ-homomorphism.
An abstract Γ-valuation semantics over Σ is a collection of abstractΓ-valuations.
α with an assignment h over A satisfies ϕ, denoted by α, h ϕ, if ϑ(h(ϕ)) ∈ D.
The class of abstract Γ-valuation models of L will be denoted by AValΓ(L).
=⇒ isomorphism, direct product, subvaluation, subdirect product, ultraproduct, . . .
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Abstract valuation semantics
Completeness
For each set Φ ⊆ LΣ(X) of formulas, we can define the abstract Γ-valuationλΦ
Γ = 〈LΣ(X)|Γ , id, 〈LΣ(X),Φ〉 〉.
Lindenbaum abstract Γ-bundle of L : LindΓ(L) = λΦΓ : Φ is a theory of L.
Proposition
For every logic L,
LindΓ(L) ⊆ AValΓ(L);
L is complete with respect to its Lindenbaum abstract Γ-bundleLindΓ(L);
L is complete with respect to AValΓ(L).
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Abstract valuation semantics
Canonical semantics
A congruence of an abstract Γ-valuation α = 〈A, ϑ, 〈B, D〉〉 is simply acongruence of the Γ-matrix 〈B, D〉.The Leibniz Γ-congruence ΩΓ(α) of α is simply the Leibniz congruence of〈B, D〉.
Reduced abstract Γ-valuation is α∗ = 〈A, [ ]ΩΓ(α) ϑ, 〈B/ΩΓ(α), [D]ΩΓ(α)〉〉.
Definition
The canonical abstract Γ-valuation semantics of L is AVal∗Γ(L).
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Abstract valuation semantics
Canonical completeness
If 〈A, D〉 is a Σ-matrix then αΓ(m) = 〈A, id, 〈A|Γ, D〉〉 is an abstract
Γ-valuation.
Proposition
AVal∗Γ(L) is the closure for isomorphisms of α∗Γ(m) | m ∈ Matr(L).
Proposition
For every logic L,
L is complete with respect to its reduced Lindenbaum Γ-bundleLind∗Γ(L);
L is complete with respect to AVal∗Γ(L).
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Bridge results
Bridge results
Let L be a many-sorted logic, and Γ a subsignature of Σ.
Theorem (Bloom generalized)
L is finitary iff AValΓ(L) is closed under ultraproducts.
Theorem (Blok-Pigozzi generalized)
L is Γ-behaviorally protoalgebraic iff AVal∗Γ(L) is closed undersubdirect products.
Proposition
Let L be a Γ-behaviorally algebraizable logic with behaviorallyequivalent algebraic semantics K. Then,
AVal∗Γ(L) = AVal∗Γ,K(L).
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Bridge results
Bridge results
Let L be a many-sorted logic, and Γ a subsignature of Σ.
Theorem (Czelakowski generalized)
L is Γ-behaviorally equivalential iff AVal∗Γ(L) is closed undersubvaluations and direct products.
Theorem (half-way)
If L is Γ-behaviorally finitely algebraizable with Θ(x) and ∆(x, y),and has a complete Hilbert-calculus with axioms Ax and rules Inf ,then α = 〈A, ϑ, 〈B, D〉〉 ∈ AVal∗Γ(L) iff α satisfies
Θ(ϕ) for each ϕ ∈ Ax ,
Θ(ψ1)& . . .&Θ(ψn)→ Θ(ϕ) for 〈ψ1, . . . , ψn, ϕ〉 ∈ Inf ,
Θ(∆(x, y))→ x ≈ y,
D = ϑ(a) | α Θ(a).
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
K/2
K/2
Theorem
K/2 is (Σ \ ¬)-behaviorally algebraizable withΘ(x) = x ≈ (x⇒ x), ∆(x1, x2) = x1⇒ x2, x2⇒ x1, and itscanonical semantics is the class of abstract valuations
〈A, ϑ, 〈B, >〉〉
st.
B is a Boolean algebra
ϑ(x) u ϑ(¬x) = ⊥
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
C1
Signature of C1
Signature ΣC1 = 〈φ, F 〉 such that:
Fεφ = t, f, Fφφ = ¬ and Fφφφ = ∧,∨,⇒.
ϕ is an abbreviation of ¬(ϕ ∧ (¬ϕ));
∼ ϕ is an abbreviation of ¬ϕ ∧ ϕ
Subsignature Γ = 〈φ, FΓ〉 of ΣC1 such that:
FΓφφ = and FΓ
ws = Fws for every ws 6= φφ.
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
C1
Hilbert-style axiomatization
Axioms:
ξ1⇒ (ξ2⇒ ξ1)
(ξ1⇒ (ξ2⇒ ξ3))⇒ ((ξ1⇒ ξ2)⇒ (ξ1⇒ ξ3))
(ξ1 ∧ ξ2)⇒ ξ1
(ξ1 ∧ ξ2)⇒ ξ2
ξ1⇒ (ξ2⇒ (ξ1 ∧ ξ2))
ξ1⇒ (ξ1 ∨ ξ2)
ξ2⇒ (ξ1 ∨ ξ2)
(ξ1⇒ ξ3)⇒ ((ξ2⇒ ξ3)⇒ ((ξ1 ∨ ξ2)⇒ ξ3))
¬¬ξ1⇒ ξ1
ξ1 ∨ ¬ξ1ξ1 ⇒ (ξ1⇒ (¬ξ1⇒ ξ2))
(ξ1 ∧ ξ2)⇒ (ξ1 ∧ ξ2)
(ξ1 ∧ ξ2)⇒ (ξ1 ∨ ξ2)
(ξ1 ∧ ξ2)⇒ (ξ1⇒ ξ2)
t⇔ (ξ1⇒ ξ1)
f⇔ (ξ1 ∧ (ξ1 ∧ ¬ξ1))
Rule:
ξ1 ξ1⇒ξ2ξ2
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
C1
Extended signature
Signature ΣC1 :
φ¬,∨,∧,⇒ 99
Extended signature ΣoC1 :
v
φ¬,∨,∧,⇒ 99
o
OO
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
C1
Behavioral algebraization
Theorem
C1 is Γ-behaviorally algebraizable with Θ(x) = x ≈ t,∆(x1, x2) = x1⇒ x2, x2⇒ x1 and the equivalent Γ-hiddenquasivariety semantics, KC1 , is the class of two-sorted algebrasdescribed below.
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
C1
The class KC1
The class KC1 can be presented as the class of all ΣoC1 -algebras
A = 〈Aφ, Av, oA,∧A,∨A,⇒A, tA, fA,∼A,uA,tA,AA,−A,>A,⊥A〉
such that Av = 〈Av,uA,tA,AA,−A,>A,⊥A〉 is a Boolean algebra
A also Γ-behaviorally satisfies the following set of hidden equations:
ξ1⇒ (ξ2⇒ ξ1) ≈ t
(ξ1⇒ (ξ2⇒ ξ3))⇒ ((ξ1⇒ ξ2)⇒ (ξ1⇒ ξ3)) ≈ t
(ξ1 ∧ ξ2)⇒ ξ1 ≈ t
(ξ1 ∧ ξ2)⇒ ξ2 ≈ t
ξ1⇒ (ξ2⇒ (ξ1 ∧ ξ2)) ≈ t
ξ1⇒ (ξ1 ∨ ξ2) ≈ t
ξ2⇒ (ξ1 ∨ ξ2) ≈ t
(ξ1⇒ ξ3)⇒ ((ξ2⇒ ξ3)⇒ ((ξ1 ∨ ξ2)⇒ ξ3)) ≈ t
¬¬ξ1⇒ ξ1 ≈ t
ξ1 ∨ ¬ξ1 ≈ t
ξ1 ⇒ (ξ1⇒ (¬ξ1⇒ ξ2)) ≈ t
(ξ1 ∧ ξ2)⇒ (ξ1 ∧ ξ2) ≈ t
(ξ1 ∧ ξ2)⇒ (ξ1 ∨ ξ2) ≈ t
(ξ1 ∧ ξ2)⇒ (ξ1⇒ ξ2) ≈ t
t ≈ (ξ1⇒ ξ1)
f ≈ (ξ1 ∧ (ξ1 ∧ ¬ξ1))
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
C1
da Costa algebras
By a da Costa algebra we mean a structure
U = 〈S, 0, 1,≤,∧,∨,⇒,∼〉,
such that 0, 1 ∈ S and, for every a, b, c ∈ S, the following conditions hold:
≤ ⊆ S × S is a quasi-order;
a ∧ b ≤ a and a ∧ b ≤ b ;
a ∧ a ' a and a ∨ a ' a, where a ' b iff a ≤ b andb ≤ a;
a ∧ (b ∨ c) ' (a ∧ b) ∨ (a ∧ c);
a ≤ a ∨ b, b ≤ a ∨ b;if a ≤ c and b ≤ c then a ∨ b ≤ c;a ≤ (∼ a), where a =∼ (a∧ ∼ a);
a ≤ (b⇒ a)⇒ ((b⇒ ∼ a)⇒ ∼ b);
0 ≤ a, a ≤ 1;
a ∧ (a⇒ b) ≤ b;a ∧ c ≤ b then c ≤ (a⇒ b);
a∨ ∼ a ' 1;
∼ (∼ x) ≤ x;
a∧ ∼ (a) ' 0;
a ∧ b ≤ (a ∧ b);a ∧ b ≤ (a ∨ b);a ∧ b ≤ (a⇒ b).
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
C1
KC1 and da Costa algebras
Let A = 〈Aφ, Av, oA,∧A,∨A,⇒A,∼A, tA, fA〉 ∈ KC1 and consider thefollowing structure,
UA = 〈SA, 0, 1,≤A,∧A,∨A,⇒A,∼A〉,
obtained from A in the following way:
SA = Aφ
1 = tA and 0 = fA
≤A⊆ SA × SA is such that a ≤A b iff (a⇒A b) ≡Γ tA
Consider 'UA on SA defined by: a 'UA b if a ≤UA b and b ≤UA a.
'UA coincides with behavioral equivalence ≡Γ.
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
C1
KC1 and da Costa algebras
Let U = 〈S, 0, 1,≤U ,∧U ,∨U ,⇒U ,∼U 〉 be a da Costa algebra.
Consider the equivalence relation 'U on S defined by: a 'U b if a ≤U b and b ≤U a.
Consider now the two-sorted algebra,
AU = 〈Aφ, Av , oA,∧A,∨A,⇒A,∼A, tA, fA〉
obtained from U in the following way:
Aφ = S
Av = S/'U
oA(a) = [a]'U
∧A = ∧U ∨A = ∨U ⇒A =⇒U ∼A=∼U
tA = 1 and fA = 0.
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
C1
KC1 and da Costa algebras
Theorem
Given A ∈ KC1 then UA is a da Costa algebra.
Given a da Costa algebra U then AU ∈ KC1 .
Given A ∈ KC1 then AUA∼= A.
Given a da Costa algebra U then UAU∼= U .
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
C1
Valuation semantics
The complete reduced abstract valuation semantics for C1 resulting from its behavioralalgebraization is the class of valuations 〈A, ϑ, 〈B, D〉〉 such that B is a Booleanalgebra, D = >B, and:
ϑ(t) ≈ >B and ϑ(f) ≈ ⊥B;
ϑ(x ∧ y) ≈ ϑ(x) uB ϑ(y);
ϑ(x ∨ y) ≈ ϑ(x) tB ϑ(y);
ϑ(x ⊃ y) ≈ ϑ(x)⇒B ϑ(y);
−Bϑ(x) ≤B ϑ(¬x);
ϑ(¬¬x) ≤B ϑ(x);
ϑ(x) uB ϑ(x) uB ϑ(¬x) ≈ ⊥B;
ϑ(x) uB ϑ(y) ≤B ϑ((x ∧ y));
ϑ(x) uB ϑ(y) ≤B ϑ((x ∨ y));
ϑ(x) uB ϑ(y) ≤B ϑ((x ⊃ y)).
The well-known bivaluation semantics of C1 can be easily recovered from these
valuations by looking at the underlying irreducible Boolean algebras.
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Conclusions
Generalization of the notion of algebraizable logic
Characterization results using the Leibniz operator
Covering many-sorted logics as well as some non-algebraizablelogics (according to the old notion)
Abstract valuations as behavioral logical matrices andapplications
More on the Suszko operator ...
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
Future work
Connections between different behavioral algebraizations, including thestandard notion ...
... with applications to the theory of combined logics
Establish semantic characterizations for each of the classes on the Leibnizhierarchy
Development of the strong behavioral approach, towards acharacterization of quasivarieties
Examples separating classes in the behavioral Leibniz hierarchy
Other interesting examples, such as first-order logic with many-sortedterms and exogenous probabilistic and quantum logics
Connections to Avron’s non-deterministic matrices and dyadic semantics
Motivation Algebraization Beh. algebraization Semantics Examples Conclusions
References
C. Caleiro and R. Goncalves, On the algebraization of many-sorted logics.Recent Trends in Algebraic Development Techniques, LNCS 4409:21–36, 2007.
R. Goncalves, Behavioral algebraization of logics. PhD, IST - TU Lisbon, 2009.
C. Caleiro and M. Martins and R. Goncalves, Behavioral algebraization oflogics. Studia Logica, 91(1):63–111, 2009.
C. Caleiro and R. Goncalves, Behavioral algebraization of da Costa’sC-systems, Journal of Applied Non-Classical Logics, 19(2):127–148, 2009.
C. Caleiro and R. Goncalves, Algebraic Valuations as Behavioral LogicalMatrices. WoLLIC 2009, LNAI 5514:13–25, 2009.
C. Caleiro and R. Goncalves, Towards a behavioral algebraic theory of logicalvaluations, Fundamenta Informaticae, 106(2-4):191-209, 2011.
C. Caleiro and R. Goncalves, Abstract valuation semantics, to appear in StudiaLogica.