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Gen2sat:A Generic Tool for Reasoning with Non-classical Logics

Yoni Zohar – Tel Aviv University

Joint work with Ori Lahav and Anna Zamansky

AIC @ SRIAugust 24, 2017

Yoni Zohar Gen2sat: A Generic Tool for Reasoning with Non-classical Logics

Motivation

Propositional classical logic:A lot of researchUsed in applications

Propositional Non-classical logics:Many-valued and fuzzy logics: v(A) = 0.5Paraconsistent logics: A,¬A 6` BIntuitionistic logic: 6` A ∨ ¬AModal logics: �AA lot of researchFew are used in applications

A possible explanation:Lack of available tools for reasoning with non-classical logicsOne has to develop a reasoning tool from scratch for each logic


Tools for Non-classical Logics

Formula

SequentCalculus

Logic ADecision Procedure

Valid?

Formula

SequentCalculus

Logic BDecision Procedure

Valid?

Formula

SequentCalculus

Logic CDecision Procedure

Valid?

...Yoni Zohar Gen2sat: A Generic Tool for Reasoning with Non-classical Logics

Tool for Non-classical Logics

Formula

LogicDecision Procedure Valid?



Sequent

SequentCalculus

Gen2sat Derivable?



Sequent

SequentCalculus

Gen2sat Derivable?

Works for Propositional Pure Analytic Sequent Calculiwith “Next” Operators

IN:classical logic3-valued logics4-valued logicsparaconsistent logics...

OUT:intuitionistic logicrelevance logicsfuzzy logicsfirst-order logics...



Sequent

SequentCalculus

Gen2sat Derivable?



Sequent

SequentCalculus

Reduction SATSolver Derivable?



Sequent

SequentCalculus

ReductionSemanticInterpretation

SATSolver Derivable?



Sequent

SequentCalculus

ReductionSemanticInterpretation Derivable?


What Are Sequents?

Sequents have the form Γ⇒ ∆, where Γ, ∆ are finite sets of formulas.Intuition:

A1, . . . , An ⇒ B1, . . . , Bm ! A1 ∧ . . . ∧ An → B1 ∨ . . . ∨ Bm

Example

A, B ⇒ A ∧ BA⇒ A ∨ B⇒ A ∨ ¬A

A,¬A⇒⇒ A,¬AA⇒ A, B, C



Sequent

SequentCalculus



Sequent Calculi

Sequent CalculiProof systems that manipulate sequentsSequent Calculus = finite set of sequent derivation rules

Γ1 ⇒ ∆1, . . . , Γn ⇒ ∆nΓ0 ⇒ ∆0

Example (Thinking of a Sequent Rule)Γ⇒ A Γ⇒ B

Γ⇒ A ∧ BA⇒ ∆

A ∧ B ⇒ ∆B ⇒ ∆

A ∧ B ⇒ ∆


Sequent Calculi

Sequent CalculiProof systems that manipulate sequentsSequent Calculus = finite set of sequent derivation rules

Γ1 ⇒ ∆1, . . . , Γn ⇒ ∆nΓ0 ⇒ ∆0

Example (Thinking of a Sequent Rule)Γ⇒ A, ∆ Γ⇒ B, ∆

Γ⇒ A ∧ B, ∆Γ, A⇒ ∆

Γ, A ∧ B ⇒ ∆Γ, B ⇒ ∆

Γ, A ∧ B ⇒ ∆


Examples of Sequent Calculi

The Propositional Fragment of LK [Gentzen 1934]Structural Rules:

(id) Γ, A⇒ A, ∆ (cut)Γ⇒ A, ∆ Γ, A⇒ ∆

Γ⇒ ∆ (weak)Γ⇒ ∆

Γ, Γ′ ⇒ ∆, ∆′

Logical Rules:

(¬ ⇒)Γ⇒ A, ∆

Γ,¬A⇒ ∆ (⇒ ¬)Γ, A⇒ ∆

Γ⇒ ¬A, ∆

(∧ ⇒)Γ, A, B ⇒ ∆

Γ, A ∧ B ⇒ ∆ (⇒ ∧)Γ⇒ A, ∆ Γ⇒ B, ∆

Γ⇒ A ∧ B, ∆

(∨ ⇒)Γ, A⇒ ∆ Γ, B ⇒ ∆

Γ, A ∨ B ⇒ ∆ (⇒ ∨)Γ⇒ A, B, ∆

Γ⇒ A ∨ B, ∆

(⊃ ⇒)Γ⇒ A, ∆ Γ, B ⇒ ∆

Γ, A ⊃ B ⇒ ∆ (⇒⊃)Γ, A⇒ B, ∆

Γ⇒ A ⊃ B, ∆

Analyticity (“subformula property”)Only subformulas of the proved sequent are used


Examples of Sequent Calculi

Lukasiewicz 3-valued Logic [Avron ’03]A sequent calculus for L3 is obtained by augmenting the positive fragmentof LK with some pure rules for negation. For example:

(¬ ⊃⇒)Γ, A,¬B ⇒ ∆

Γ,¬(A ⊃ B)⇒ ∆

(⇒ ¬ ⊃)Γ⇒ A, ∆ Γ⇒ ¬B, ∆

Γ⇒ ¬(A ⊃ B), ∆

¬-AnalyticityOnly subformulas of the proved sequent and their negations are used


Examples of Sequent CalculiCalculus for Primal Infon Logic [Gurevich,Neeman ’09]

(∧ ⇒)Γ, A, B ⇒ ∆

Γ, A ∧ B ⇒ ∆ (⇒ ∧)Γ⇒ A, ∆ Γ⇒ B, ∆

Γ⇒ A ∧ B, ∆

(∨ ⇒)��

��

��XXXXXXXXXXXX

Γ, A⇒ ∆ Γ, B ⇒ ∆Γ, A ∨ B ⇒ ∆ (⇒ ∨)

Γ⇒ A, B, ∆Γ⇒ A ∨ B, ∆

(⊃ ⇒)Γ⇒ A, ∆ Γ, B ⇒ ∆

Γ, A ⊃ B ⇒ ∆ (⇒⊃)Γ,��HHA ⇒ B, ∆Γ⇒ A ⊃ B, ∆

Γ⇒ ∆q said Γ⇒ q said ∆ for every principal q

An extremely efficient propositional logic.One of the main logical engines behind MSR DKAL

AnalyticityOnly subformulas of the proved sequent are used


Scope of Gen2sat

Pure Analytic sequent calculus with “Next” OperatorsPropositional and structuralInclude pure logical rules that allow any Γ and ∆:

X Γ, A⇒ B, ∆Γ⇒ A ⊃ B, ∆ X Γ, A⇒ B

Γ⇒ A ⊃ B

May include impure rules of the form: Γ⇒ ∆∗Γ⇒ ∗∆

Analytic (subformula property and its variants)

TheoremThere is a polynomial reduction from the derivability problem of any suchcalculus to (the complement of) SAT.

corollaryThe derivability problem for every such calculus is in co-NP.



Sequent

SequentCalculus



Semantics for Pure CalculivaluationsA valuation is a function v : WFF → {t, f}

WarningValuations are defined over all formulas, not only the atomic ones!

G-legal valuationsA valuation is G-legal if it respects the “semantic reading” of G .

Example (Classical Conjunction)⇒ A ⇒ B⇒ A ∧ B

A⇒A ∧ B ⇒

B ⇒A ∧ B ⇒

1 If v(A) = t and v(B) = t then v(A ∧ B) = t2 If v(A) = f then v(A ∧ B) = f3 If v(B) = f then v(A ∧ B) = f


Semantics for Pure CalculivaluationsA valuation is a function v : WFF → {t, f}

WarningValuations are defined over all formulas, not only the atomic ones!

G-legal valuationsA valuation is G-legal if it respects the “semantic reading” of G .

Example (Sequent Calculus for C1)A⇒⇒ ¬A

A⇒¬¬A⇒

¬A⇒ ¬B ⇒¬(A ∧ B)⇒

1 If v(A) = f then v(¬A) = t2 If v(A) = f then v(¬¬A) = f3 If v(¬A) = f and v(¬B) = f then v(¬(A ∧ B)) = f

This semantics is non-deterministic.Yoni Zohar Gen2sat: A Generic Tool for Reasoning with Non-classical Logics

Semantics for Pure Calculi

Soundness and Completeness [Beziau ‘01]s is provable in G

⇐⇒s is satisfied by every G-legal valuation



Soundness and Completenesss is provable in G using F ⊆WFF

⇐⇒s is satisfied by every G-legal valuation with domain F



Soundness and Completenesss is provable in G using sub(s)

⇐⇒s is satisfied by every G-legal valuation with domain sub(s)



Soundness and Completenesss is provable in G

⇐⇒s is satisfied by every G-legal valuation with domain sub(s)



Sequent

SequentCalculus



Reduction to SAT

s

Greduction SAT solver

X

X

“I am a G-legalcountermodel of s”

UNSAT

SAT

TheoremCorrectness: s is provable in G iff the generated instance is UNSAT.Complexity:

Translating: O(nk); k depends on G (“usually” k = 1)Solving: Exp in the worse case. Linear for “Horn calculi”

In the presence of Next operators, we use Kripke modelsCorrectness is more challengingYoni Zohar Gen2sat: A Generic Tool for Reasoning with Non-classical Logics


Sequent

SequentCalculus



Reduction

A1, . . . , An ⇒ B1, . . . , Bm

rule1

rule2

rule3

rule4

clause 1clause 2

clause 3clause 4clause 5

clause 6

clause 7clause 8

clause 9clause 10

SAT assignment


Reduction

A1, . . . , An ⇒ B1, . . . , Bm

rule1

rule2

rule3

rule4

clause 1clause 2


clause 6

clause 7clause 8

clause 9clause 10

SATSAT assignment

A1 = false, A2 = true, . . .


Reduction

A1, . . . , An ⇒ B1, . . . , Bm

rule1

rule2

rule3

rule4

clause 1clause 2


clause 6

clause 7clause 8

clause 9clause 10

UNSATUNSAT core

2, 4, 8


Evaluation

Gen2sat vs. MetTeLinput: The calculus for Lukasiewicz 3-valued logicStructured Problems for Lukasiewicz infinite-valued logic [Rothenberg’07]Random Problems generated by MetTeL

100 200 300

5,000

10,000

N

time

inm

s

Structured Problems

10 15 20100

101

102

103

104

depth

time

inm

s

Random Problems


An Idea: Logic Education

Motivation:Gen2sat can be useful for teaching sequent calculiStudents can focus solely on the logical aspectsHeuristics and search are left for the SAT solver

Preliminary Pilot:13 logic students were given a bonus assignment:present a minimal test plan with maximal coverage for Gen2satThey all got 70%-85% coverageSome used 0-ary and 3-ary connectives.Some found (intentionally planted) bugsFeedback from students was encouraging

“it helped me see the variety of different connectives and rules”“for me thinking of the extreme cases was really illuminating”“I wish all of the course assignments were more of this type”...


Conclusion

We have seen:A generic tool for deciding derivability in analytic pure (and someimpure) sequent calculiThe actual search is done by a SAT-solverBased on a semantic interpretation

Future work:ApplicationsSupport more logics and calculiAutomatically detect analyticity (when possible)Integrate with a theorem prover

Thank you!


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