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Motivation Decision Problems Functional Representation H ajeks Basic Logic: Decision and Representation Simone Bova bova@unisi.it Department of Mathematics and Computer Science University of Siena (Italy) December 19, 2007 Logic

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Motivation Decision Problems Functional Representation

H´ ajek’s Basic Logic: Decision and Representation

Simone Bova bova@unisi.it

Department of Mathematics and Computer Science University of Siena (Italy)

December 19, 2007 Logic Seminar University of Barcelona (Spain)

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation

Outline

1

Motivation Vague Notions Basic Logic

2

Decision Problems Derivability and Validity Complexity and Algorithms

3

Functional Representation Free Algebras and Normal Forms Open Problems

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Vague Notions Basic Logic

Outline

1

Motivation Vague Notions Basic Logic

2

Decision Problems Derivability and Validity Complexity and Algorithms

3

Functional Representation Free Algebras and Normal Forms Open Problems

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Vague Notions Basic Logic

Sorite’s Paradox

Xi ⇋ “a collection of i grains is a heap”, N ⇋ 1000000 Tentative axiomatization of the notion of heap (i = 0, . . . , N − 1): (H1) XN (H2) ¬X0 (H3.i) Xi+1 → Xi

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Vague Notions Basic Logic

Sorite’s Paradox

Xi ⇋ “a collection of i grains is a heap”, N ⇋ 1000000 Tentative axiomatization of the notion of heap (i = 0, . . . , N − 1): (H1) XN (H2) ¬X0 (H3.i) Xi+1 → Xi The theory is inconsistent: 1 XN 2 XN−1 . . . . . . N + 1 X0 N + 2 ¬X0

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Vague Notions Basic Logic

Bivalence versus Vagueness

We can either reject vagueness . . . 0 = X0 = · · · = X500000 < X500001 = ... = XN = 1 “500001 grains form a heap, whether 500000 do not”

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Vague Notions Basic Logic

Bivalence versus Vagueness

We can either reject vagueness . . . 0 = X0 = · · · = X500000 < X500001 = ... = XN = 1 “500001 grains form a heap, whether 500000 do not” . . . or abjure bivalence: 0 = X0 < X1 < ... < XN−1 < XN = 1 “i grains form a heap” is less true than “j grains form a heap”, if i < j

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Vague Notions Basic Logic

H´ ajek’s Paradigm | Fuzzy Logic

Fuzzy logics are propositional logics over ⊤, ⊥, ⊙, → st: variables X, Y, . . . are interpreted over [0, 1]; ⊤ and ⊥ are interpreted over 1 and 0; ⊙ and → are interpreted over binary functions on [0, 1]; ¬X ⇋ X → ⊥. Fuzzy conjunction and implication must maintain: the behavior of Boolean counterparts over {0, 1}2; intuitive properties of Boolean counterparts over [0, 1]2; the validity of fuzzy modus ponens.

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Vague Notions Basic Logic

H´ ajek’s Paradigm | Boolean Logic

Intuitive properties of Boolean conjunction and implication:

1 x 1 y 1 1 x 1 y

Boolean conjunction is commutative, associative, weakly increasing in both arguments, and has 1 as unit.

1 x 1 y 1 1 x 1 y

Boolean implication, x implies y, is 1 iff x ≤ y, weakly decreasing in x, weakly increasing in y.

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Vague Notions Basic Logic

H´ ajek’s Paradigm | t-Norms and Residua

Definition (Continuous t-Norm, Residuum) A continuous t-norm ⊙∗ is a continuous binary function on [0, 1] that is associative, commutative, monotone (x ≤ y implies x ⊙∗ z ≤ y ⊙∗ z) and has 1 as unit (x ⊙∗ 1 = x). Given a continuous t-norm ⊙∗, its residuum is the binary function →∗ on [0, 1] defined by x →∗ y = max{z : x ⊙∗ z ≤ y}. t-norms and their residua provide suitable interpretations for fuzzy conjunction and implication.

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Vague Notions Basic Logic

H´ ajek’s Paradigm | G¨

  • del Logic

⊙G and →G over [0, 1]2:

1 x 1 y 1 x

x ⊙G y = min(x, y)

1 x 1 y 1 x

x →G y =

  • 1

if x ≤ y y

  • therwise

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Vague Notions Basic Logic

H´ ajek’s Paradigm | Łukasiewicz Logic

⊙L and →L over [0, 1]2:

1 x 1 y 1 x

x ⊙L y = max(0, x + y − 1)

1 x 1 y 1 x

x →L y = min(1, −x + y + 1)

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Vague Notions Basic Logic

Basic Logic | Logical Calculus

⊢BL φ iff φ is derivable in the following Hilbert calculus: (A1) (φ → χ) → ((χ → ψ) → (φ → ψ)) (A2) (φ ⊙ χ) → φ (A3) (φ ⊙ χ) → (χ ⊙ φ) (A4) (φ ⊙ (φ → χ)) → (χ ⊙ (χ → φ)) (A5) ((φ ⊙ χ) → ψ) ↔ (φ → (χ → ψ)) (A6) ((φ → χ) → ψ) → (((χ → φ) → ψ) → ψ) (A7) ⊥ → φ (R1) φ, φ → χ ⊢ χ

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Vague Notions Basic Logic

Basic Logic | Semantic Completeness

BL is the logic of all continuous t-norms and their residua [Cignoli et al., 2000]: (i) ⊢BL χ iff, for every t-norm ⊙∗ and every assignment v, χ evaluates to 1 with respect to ⊙∗ and v. (ii) φ1, . . . , φn ⊢BL χ iff, for every t-norm ⊙∗ and every assignment v, if φ1, . . . , φn evaluate to 1 with respect to ⊙∗ and v, then χ evaluates to 1 with respect to ⊙∗ and v.

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Derivability and Validity Complexity and Algorithms

Outline

1

Motivation Vague Notions Basic Logic

2

Decision Problems Derivability and Validity Complexity and Algorithms

3

Functional Representation Free Algebras and Normal Forms Open Problems

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Derivability and Validity Complexity and Algorithms

Derivability and Validity

Let φ1, . . . , φn, χ be formulas over X1, . . . , Xn. BL-CONSn = {({φ1, . . . , φm}, {χ}) : φ1, . . . , φm ⊢BL χ} BL-TAUTn = {χ : (∅, {χ}) ∈ BL-CONSn} ⊆ BL-CONSn

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Derivability and Validity Complexity and Algorithms

Derivability and Validity

Let φ1, . . . , φn, χ be formulas over X1, . . . , Xn. BL-CONSn = {({φ1, . . . , φm}, {χ}) : φ1, . . . , φm ⊢BL χ} BL-TAUTn = {χ : (∅, {χ}) ∈ BL-CONSn} ⊆ BL-CONSn There are infinitely many t-norms and infinitely many assignments of n propositional variables over [0, 1].

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Derivability and Validity Complexity and Algorithms

Derivability and Validity

Let φ1, . . . , φn, χ be formulas over X1, . . . , Xn. BL-CONSn = {({φ1, . . . , φm}, {χ}) : φ1, . . . , φm ⊢BL χ} BL-TAUTn = {χ : (∅, {χ}) ∈ BL-CONSn} ⊆ BL-CONSn There are infinitely many t-norms and infinitely many assignments of n propositional variables over [0, 1]. Question: Is BL-TAUTn decidable? And BL-CONSn?

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Motivation Decision Problems Functional Representation Derivability and Validity Complexity and Algorithms

Generic t-Norms | 2-Variate Fragment

3[0, 1]MV = ([0, 3], ⊙2, →2, ⊥):

1 2 3 x_1 1 2 3 x_2 1 2 3 1 2 x_1

x1 ⊙2 x2 = 8 > > > < > > > : max(x1 + x2 − 1, 0) if 0 ≤ x1, x2 < 1 max(x1 + x2 − 2, 1) if 1 ≤ x1, x2 < 2 max(x1 + x2 − 3, 2) if 2 ≤ x1, x2 ≤ 3 min(x1, x2) if ⌊x1⌋ = ⌊x2⌋

1 2 3 x_1 1 2 3 x_2 1 2 3 1 2 x_1

x1 →2 x2 = 8 > > > > > < > > > > > : 3 if x1 ≤ x2 x2 − x1 + 1 if 0 ≤ x1, x2 < 1 x2 − x1 + 2 if 1 ≤ x1, x2 < 2 x2 − x1 + 3 if 2 ≤ x1, x2 ≤ 3 x2 if ⌊x2⌋ < ⌊x1⌋

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Motivation Decision Problems Functional Representation Derivability and Validity Complexity and Algorithms

Generic t-Norms | n-Variate Fragment

(n + 1)[0, 1]MV = ([0, n + 1], ⊙n, →n, ⊥): x ⊙n y =

  • max(x + y − (i + 1), i)

if ⌊x⌋ = ⌊y⌋ = i min(x, y) if ⌊x⌋ = ⌊y⌋ x →n y =      n + 1 if x ≤ y y + (i + 1) − x if ⌊x⌋ = ⌊y⌋ = i y if ⌊y⌋ < ⌊x⌋ Let ⊥0 ⇌ ⊥, ⊥1 ⇌ 1, . . . , ⊥n ⇌ n, ⊥n+1 ⇌ ⊤ = n + 1.

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Motivation Decision Problems Functional Representation Derivability and Validity Complexity and Algorithms

Generic t-Norms | Decidability and Complexity

Theorem (∼ Aglian`

  • and Montagna, 2003)

⊙n is generic for the n-variate fragment of BL, that is: (i) ⊢BL χ(X1, . . . , Xn) iff, for every assignment v, v(χ) = n + 1 wrt ⊙n. (ii) φ1(X1, . . . , Xn), . . . , φm(X1, . . . , Xn) ⊢BL χ(X1, . . . , Xn) iff, for every assignment v, if v(φ1) = · · · = v(φn) = n + 1 wrt ⊙n, then v(χ) = n + 1 wrt ⊙n.

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Motivation Decision Problems Functional Representation Derivability and Validity Complexity and Algorithms

Generic t-Norms | Decidability and Complexity

Theorem (∼ Aglian`

  • and Montagna, 2003)

⊙n is generic for the n-variate fragment of BL, that is: (i) ⊢BL χ(X1, . . . , Xn) iff, for every assignment v, v(χ) = n + 1 wrt ⊙n. (ii) φ1(X1, . . . , Xn), . . . , φm(X1, . . . , Xn) ⊢BL χ(X1, . . . , Xn) iff, for every assignment v, if v(φ1) = · · · = v(φn) = n + 1 wrt ⊙n, then v(χ) = n + 1 wrt ⊙n. Corollary (Baaz et al., 2002; ∼ Aguzzoli and Gerla, 2002) BL-CONSn ∈ coNP. “No” instances of BL-CONSn have small witnesses wrt ⊙n.

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Motivation Decision Problems Functional Representation Derivability and Validity Complexity and Algorithms

Generic t-Norms | Decidability and Complexity

Example: ((X1 → X2) → X2) → X1 ⇋ ψ ∈ BL-TAUT2?

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Motivation Decision Problems Functional Representation Derivability and Validity Complexity and Algorithms

Generic t-Norms | Decidability and Complexity

Example: ((X1 → X2) → X2) → X1 ⇋ ψ ∈ BL-TAUT2? No:

1 2 3 x_1 1 2 3 x_2 1 2 3 x_1

= ψBL2 =

1 2 3 x_1 1 2 3 x_2 1 2 3 x_1

Sample witnesses of ψ / ∈ BL-TAUT2: (i) v(X1) = v(X2) = 4/3; (ii) any v st 0 ≤ v(X2) < 1 < v(X1) < n + 1; (iii) any v st 2 ≤ v(X1), v(X2) ≤ 3 and v(X1) < v(X2); (iv) ⊥ ≤ X2 = X1 → X2 < ⊥1 < X1 = ψ < ⊤ = (X1 → X2) → X2.

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Motivation Decision Problems Functional Representation Derivability and Validity Complexity and Algorithms

Generic t-Norms | Decidability and Complexity

Definition (Subformulae Order) Let χ(X1, . . . , Xn) be a formula with l connectives. A subformulae

  • rder for χ is a partition of the subformulae of χ, ⊥0, . . . , ⊥n+1 and ⊤

into ≤ n + 2 blocks. For j = 0, . . . , n + 1, the block Bj is linearly

  • rdered with least element ⊥j, and holds a linear program of O(l)

constraints over x1, . . . , xn. The order is consistent if and only if there exists an assignment v of the variables in [0, n + 1] that satisfies the linear orders and the linear programs. Fact χ < ⊤ holds in a consistent order iff, for some v, v(φ) < v(⊤).

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Motivation Decision Problems Functional Representation Derivability and Validity Complexity and Algorithms

Generic t-Norms | Decidability and Complexity

Question: How many witnesses do we have to check, in the worst case, to conclude that a given instance χ of size l is not in BL-TAUTn? What about BL-CONSn?

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Derivability and Validity Complexity and Algorithms

Generic t-Norms | Decidability and Complexity

Question: How many witnesses do we have to check, in the worst case, to conclude that a given instance χ of size l is not in BL-TAUTn? What about BL-CONSn? We know that testing 23l ≤ l! witnesses suffices wrt BL-TAUTn [Bova and Montagna, 2007]. Wrt BL-CONSn, the bound l! still resists [Baaz et al., 2002].

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Free Algebras and Normal Forms Open Problems

Outline

1

Motivation Vague Notions Basic Logic

2

Decision Problems Derivability and Validity Complexity and Algorithms

3

Functional Representation Free Algebras and Normal Forms Open Problems

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Free Algebras and Normal Forms Open Problems

Algebraic Logic

Definition (BL-Algebras) A BL-algebra is an algebra (A, ∨, ∧, ⊙, →, ⊤, ⊥) of type (2, 2, 2, 2, 0, 0) such that: (i) (A, ⊙, ⊤) is a commutative monoid; (ii) (A, ∨, ∧, ⊤, ⊥) is a bounded lattice; (iii) x ⊙ y ≤ z if and only if y ≤ x → z (residuation); (iv) (x → y) ∨ (y → x) (prelinearity). BL-algebras form an algebraic variety.

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Free Algebras and Normal Forms Open Problems

Algebraic Logic

The variety of BL-algebras forms the algebraic semantics of BL. Thus, the free n-generated BL-algebra, BLn, encodes the n-variate fragment of BL, in the precise sense that BLn is isomorphic to the Lindenbaum-Tarski algebra of the n-variate fragment of BL.

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Free Algebras and Normal Forms Open Problems

Algebraic Logic

The variety of BL-algebras forms the algebraic semantics of BL. Thus, the free n-generated BL-algebra, BLn, encodes the n-variate fragment of BL, in the precise sense that BLn is isomorphic to the Lindenbaum-Tarski algebra of the n-variate fragment of BL. Question: Is there an explicit description of BLn?

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Motivation Decision Problems Functional Representation Free Algebras and Normal Forms Open Problems

Functional Definition

Fact (Aglian`

  • and Montagna, 2002)

The free n-generated BL-algebra, BLn, is the subalgebra of ((n + 1)[0, 1]MV)((n+1)[0,1]MV )n, generated by the projections, with pointwise defined operations. The explicit description of BLn amounts to the characterization

  • f the class F of functions f : [0, n + 1]n → [0, n + 1] st:

(i) f is either a projection x1, . . . , xn or the constant 0; (ii) f has the form g1 ◦n g2, where g1, g2 ∈ F, ◦n ∈ {⊙n, →n}, and (g1 ◦n g2)(·) = g1(·) ◦n g2(·).

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Motivation Decision Problems Functional Representation Free Algebras and Normal Forms Open Problems

BL1 | Functional Characterization

The explicit description of BL1 amounts to the characterization

  • f the functions f : [0, 2] → [0, 2] that are definable as arbitrary

compositions of the projection x and the constant 0 via the

  • perations ⊙1 and →1:

x ⊙1 y =      max(x + y − 1, 0) if 0 ≤ x, y < 1 max(x + y − 2, 1) if 1 ≤ x, y ≤ 2 min(x, y) if ⌊x⌋ = ⌊y⌋ x →1 y =            2 if x ≤ y y + 1 − x if 0 ≤ x, y < 1 y + 2 − x if 1 ≤ x, y ≤ 2 y if ⌊y⌋ < ⌊x⌋

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Free Algebras and Normal Forms Open Problems

BL1 | McNaughton Functions

Definition (McNaughton Function) A continuous n-variate function over [0, 1] is a McNaughton function iff there are linear polynomials p1, . . . , pk with integer coefficients such that, for every x ∈ [0, 1]n, there is j ∈ [k] such that f(x) = pj(x).

1

  • 2

1 x_1 1 1

  • 3

2

  • 3

5

  • 6

1 x_1 1

  • 2

1 2

  • 3

1 x_1 1

Figure: 1-variate McNaughton functions f, g1, g2 : [0, 1] → [0, 1].

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Free Algebras and Normal Forms Open Problems

BL1 | Lifting

1

  • 2

1 x_1 1 1

  • 3

2

  • 3

5

  • 6

1 x_1 1

  • 2

1 2

  • 3

1 x_1 1

Figure: f, g1, g2 : [0, 1] → [0, 1].

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Free Algebras and Normal Forms Open Problems

BL1 | Lifting

1

  • 2

1 x_1 1 1

  • 3

2

  • 3

5

  • 6

1 x_1 1

  • 2

1 2

  • 3

1 x_1 1

Figure: f, g1, g2 : [0, 1] → [0, 1].

1

  • 2

1 2 x_1 1 2 1

  • 3

2

  • 3

5

  • 6

1 4

  • 3

5

  • 3

11

  • 6

2 x_1 1

  • 2

1 3

  • 2

2 2

  • 3

1 5

  • 3

2 x_1 1 2

Figure: lift1(f), lift2(g1), lift2(g2) : [0, 2] → [0, 2].

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Free Algebras and Normal Forms Open Problems

BL1 | Masking

1

  • 3

2

  • 3

5

  • 6

1 4

  • 3

5

  • 3

11

  • 6

2 x_1 1

  • 2

1 3

  • 2

2 2

  • 3

1 5

  • 3

2 x_1 1 2

Figure: lift2(g1), lift2(g2) : [0, 2] → [0, 2].

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Free Algebras and Normal Forms Open Problems

BL1 | Masking

1

  • 3

2

  • 3

5

  • 6

1 4

  • 3

5

  • 3

11

  • 6

2 x_1 1

  • 2

1 3

  • 2

2 2

  • 3

1 5

  • 3

2 x_1 1 2

Figure: lift2(g1), lift2(g2) : [0, 2] → [0, 2].

1

  • 3

2

  • 3

5

  • 6

1 2 x_1 1

  • 2

1 2 1 5

  • 3

2 x_1 1 2

Figure: mask1(lift2(g1)), mask2(lift2(g2)) : [0, 2] → [0, 2].

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Free Algebras and Normal Forms Open Problems

BL1 | ∧’ing

1

  • 3

2

  • 3

5

  • 6

1 2 x_1 1

  • 2

1 2 1 5

  • 3

2 x_1 1 2

Figure: mask1(lift2(g1)), mask2(lift2(g2)) : [0, 2] → [0, 2].

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Free Algebras and Normal Forms Open Problems

BL1 | ∧’ing

1

  • 3

2

  • 3

5

  • 6

1 2 x_1 1

  • 2

1 2 1 5

  • 3

2 x_1 1 2

Figure: mask1(lift2(g1)), mask2(lift2(g2)) : [0, 2] → [0, 2].

1

  • 3

2

  • 3

5

  • 6

1 5

  • 3

2 x_1 1

  • 2

1 2

Figure: mask1(lift2(g1)) ∧ mask2(lift2(g2)) : [0, 2] → [0, 2].

Simone Bova H´ ajek’s Basic Logic: Decision and Representation

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Motivation Decision Problems Functional Representation Free Algebras and Normal Forms Open Problems

BL1 | Explicit Description

Theorem (∼ Montagna, 2000) Let f, g1, g2 be McNaughton functions, st f(1) = 0, g1(1) = g2(1) = 1. The free 1-generated BL-algebra, BL1, is the algebra of 1-variate functions over [0, 2] of the form lift1(f)

  • r mask1(lift2(g1)) ∧ mask2(lift2(g2)):

1

  • 2

1 2 x_1 1 2 1

  • 3

2

  • 3

5

  • 6

1 5

  • 3

2 x_1 1

  • 2

1 2

with pointwise defined operations ⊙1 and →1.

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Motivation Decision Problems Functional Representation Free Algebras and Normal Forms Open Problems

Open Problems

(i) Give the functional characterization of BLn for 2 ≤ n < ω. (ii) Compute deductive interpolants in BL. (iii) Provide a combinatorial characterization of finite n-generated free BL-algebras.

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References

P . Aglian`

  • and F

. Montagna. Varieties of BL-Algebras I: General Properties. Journal of Pure and Applied Algebra, 181:105–129, 2003.

  • M. Baaz, P

. H´ ajek, F . Montagna and H. Veith. Complexity of t-Tautologies. Annals of Pure and Applied Logic, 113:3–11, 2002.

  • S. Bova and F

. Montagna. Proof Search in H´ ajek’s Basic Logic. To appear in ACM Transactions on Computational Logic.

Simone Bova H´ ajek’s Basic Logic: Decision and Representation