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Lecture 10. Many-valued logics

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Logical methods for AI

Lecture 10

Many-valued logics

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Many-valued logics

Background

  • Classical assumption: bivalence with $0,1$
  • Many AI applications conflict with that.
  • We look at some many-valued logics.

Syntax

$$\langle prop\rangle::= p_1\mid \dots\mid p_n$$ $$\langle fml\rangle::=\langle prop\rangle\mid\neg\langle fml\rangle\mid (\langle fml\rangle\land \langle fml\rangle)\mid (\langle fml\rangle\lor \langle fml\rangle)$$

Kleene/Łukasiewicz logics

Motivation

  • What's the truth-value of: $$\mathsf{RAIN\_TOMORROW}?$$
  • Proposal: indeterminate $i$.
  • Open world assumption
  • SQL null

3-valued valuations

$$\nu:\langle prop\rangle\to \Set{0,i,1}$$

Recursive truth

$$\nu(\neg A)=-\nu(A)$$ $$\nu(A\land B)=\nu(A)\times \nu(B)$$ $$\nu(A\lor B)=\nu(A)+\nu(B)$$

Valid inference

$$[A]:=\Set{\nu:\nu(A)=1}$$ $$P_1,P_2,\dots\vDash C\Leftrightarrow [P_1]\cap [P_2]\cap \dots\subseteq [C]$$
$$\mathsf{RAIN}\lor \mathsf{BIKE},\neg\mathsf{RAIN}\vDash\mathsf{BIKE}$$   $$!!!$$

Changes?

  • Complementation 1: $x\times -x=0$
    • $$i\times -i=i\times i=i$$
  • Complementation 2: $x+ -x=1$
    • $$i+ -i=i+i=i$$

Changes?

  • No logical truths:
    • $$\nvDash \mathsf{RAIN\_TOMORROW}\lor\neg\mathsf{RAIN\_TOMORROW}$$
  • Open world assumption!

Contradictions

  • No contradictions:
    • $$x\times - x\neq 1$$ $$1\times - 1=1\times 0=0$$ $$i\times - i=i\times i=i$$ $$0\times - 0=0\times 1=0$$
  • $\leadsto$ paraconsistent logics

Kleene

  • $A\to B$ "means" $\neg A\lor B$
  • But: $$\nvDash \mathsf{RAIN\_TOMORROW}\to \mathsf{RAIN\_TOMORROW}$$
  • Just set: $$\nu(\mathsf{RAIN\_TOMORROW})=i$$

Łukasiewicz

  • $A\to B$ becomes a logical conditional
  • We get: $$\vDash \mathsf{RAIN\_TOMORROW}\to \mathsf{RAIN\_TOMORROW}$$

    • Fuzzy logics

    Vaguenes

    • What's the truth-value of $\mathsf{WARM}$?
    • Fuzzy:
      • At 30℃ close to 1
      • At 15℃ close to 0.5
      • At 0℃ close to 0

    Fuzzy systems

    • Rules like: $$\mathbf{IF}\ \mathsf{WARM}\lor\mathsf{HUMID}\ \mathbf{ THEN }\ \mathsf{POWER}$$
    • Turn the power on to the extend that it's warm and humid
    • Big in Japan!
      • AC
      • Autofocus
      • Self-driving trains

    Fuzzy valuations

    $$\nu:\langle prop\rangle\to [0,1]$$

    Fuzzy truth-functions

    $$f_\neg(x)=1-x$$ $$f_\land(x,y)=min(x,y)$$ $$f_\lor(x,y)=max(x,y)$$

    Recursive values

    $$\nu(\neg A)=1-\nu(A)$$ $$\nu(A\land B)=min(\nu(A),\nu(B))$$ $$\nu(A\lor B)=max(\nu(A),\nu(B))$$

    Valid inference

    Modelled after fuzzy rules:

    $$P_1,P_2,\dots\vDash C$$ $$\Leftrightarrow$$ $$\text{for all }\nu, min(\nu(P_1),\dots,\nu(P_n))\leq \nu(C)$$
    $$\mathsf{WARM}\vDash\mathsf{WARM}\lor\mathsf{HUMID}$$ $$\nu(\mathsf{WARM})\leq max(\nu(\mathsf{WARM})$$ $$\mathsf{WARM}\land\mathsf{HUMID}\vDash \mathsf{WARM}$$ $$\min(\nu(\mathsf{WARM}),\nu(\mathsf{HUMID}))\leq\nu(\mathsf{WARM})$$
    $$\neg\mathsf{WARM},\mathsf{WARM}\lor \mathsf{HUMID}\nvDash \mathsf{HUMID}$$ $$\nu(\neg\mathsf{WARM})=0.5$$ $$\nu(\mathsf{WARM}\lor \mathsf{HUMID})=min(\nu(\mathsf{WARM}),\nu(\mathsf{HUMID}))=$$ $$max(0.5,0.25)=0.5$$ $$min(\nu(\neg\mathsf{WARM}), \nu(\mathsf{HUMID}\lor\mathsf{WARM}))=0.5$$ $$\nleq$$ $$\nu(\mathsf{HUMID})=0.25$$

    Outlook

    • Many many-valued logics:
      • 3-valued Logic of Pardox
      • 4-valued FDE
      • 8-valued, 16-valued logics,...
    • Fuzzy logic is a huge field (especially in Japan).

    Thanks!

    Highlights
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