Induction, Transitive Closure and Cycles Liron Cohen, Cornell - - PowerPoint PPT Presentation

SLIDE 1

Induction, Transitive Closure and Cycles

Liron Cohen, Cornell University, Reuben Rowe, University of Kent ASL North American Annual Meeting, 2018

SLIDE 2

Applications

• f Logic in CS

MKM Knowledge Reasoning Model Checking Type Theory Complexity Verification Database

Halpern, Harper, Immerman, Kolaitis, Vardi, Vianu. On the unusual effectiveness of logic in computer science, 2001

SLIDE 3

Applications

• f Logic in CS

MKM Knowledge Reasoning Model Checking Type Theory Complexity Verification Database

Inductive arguments on programs

Halpern, Harper, Immerman, Kolaitis, Vardi, Vianu. On the unusual effectiveness of logic in computer science, 2001

SLIDE 4

Applications

• f Logic in CS

MKM Knowledge Reasoning Model Checking Type Theory Complexity Verification Database

Inductive arguments on programs Expressive Query languages (WITH RECURSIVE) SQL3, IBM DB2, Datalog

Halpern, Harper, Immerman, Kolaitis, Vardi, Vianu. On the unusual effectiveness of logic in computer science, 2001

SLIDE 5

Applications

• f Logic in CS

MKM Knowledge Reasoning Model Checking Type Theory Complexity Verification Database

Inductive arguments on programs Expressive Query languages (WITH RECURSIVE) SQL3, IBM DB2, Datalog Characterization of complexity classes

Halpern, Harper, Immerman, Kolaitis, Vardi, Vianu. On the unusual effectiveness of logic in computer science, 2001

SLIDE 6

Applications

• f Logic in CS

MKM Knowledge Reasoning Model Checking Type Theory Complexity Verification Database

Inductive arguments on programs Expressive Query languages (WITH RECURSIVE) SQL3, IBM DB2, Datalog Characterization of complexity classes Inductive definition of type judgments

Halpern, Harper, Immerman, Kolaitis, Vardi, Vianu. On the unusual effectiveness of logic in computer science, 2001

SLIDE 7

Applications

• f Logic in CS

MKM Knowledge Reasoning Model Checking Type Theory Complexity Verification Database

Inductive arguments on programs Expressive Query languages (WITH RECURSIVE) SQL3, IBM DB2, Datalog Characterization of complexity classes Inductive definition of type judgments Reachability properties

Halpern, Harper, Immerman, Kolaitis, Vardi, Vianu. On the unusual effectiveness of logic in computer science, 2001

SLIDE 8

Applications

• f Logic in CS

MKM Knowledge Reasoning Model Checking Type Theory Complexity Verification Database

Inductive arguments on programs Expressive Query languages (WITH RECURSIVE) SQL3, IBM DB2, Datalog Characterization of complexity classes Inductive definition of type judgments Reachability properties Common knowledge, defined inductively

Halpern, Harper, Immerman, Kolaitis, Vardi, Vianu. On the unusual effectiveness of logic in computer science, 2001

SLIDE 9

Applications

• f Logic in CS

MKM Knowledge Reasoning Model Checking Type Theory Complexity Verification Database

Inductive arguments on programs Expressive Query languages (WITH RECURSIVE) SQL3, IBM DB2, Datalog Characterization of complexity classes Inductive definition of type judgments Reachability properties Common knowledge, defined inductively Natural numbers

Halpern, Harper, Immerman, Kolaitis, Vardi, Vianu. On the unusual effectiveness of logic in computer science, 2001

SLIDE 10

Applications

• f Logic in CS

MKM Knowledge Reasoning Model Checking Type Theory Complexity Verification Database

Inductive arguments on programs Expressive Query languages (WITH RECURSIVE) SQL3, IBM DB2, Datalog Characterization of complexity classes Inductive definition of type judgments Reachability properties Common knowledge, defined inductively Natural numbers

What Logic?

Halpern, Harper, Immerman, Kolaitis, Vardi, Vianu. On the unusual effectiveness of logic in computer science, 2001

SLIDE 11

What Logic?

FOL SOL

SLIDE 12

What Logic?

FOL SOL

No inductive machinery

SLIDE 13

What Logic?

FOL SOL

No inductive machinery Overkill

SLIDE 14

What Logic?

FOL SOL

No inductive machinery Overkill natural, effective extensions of FOL that allow inductive definitions

SLIDE 15

What Logic?

FOL SOL

No inductive machinery Overkill natural, effective extensions of FOL that allow inductive definitions Transitive Closure Logic

SLIDE 16

Transitive Closure Logic

Transitive Closure Logic = FOL + a transitive closure operator.

SLIDE 17

Transitive Closure Logic

Transitive Closure Logic = FOL + a transitive closure operator. The transitive closure R∗ of binary relation R is defined by: R∗ =

• R(n)

where R(0) = Id, R(n+1) = R(n) ◦ R.

SLIDE 18

Transitive Closure Logic

Transitive Closure Logic = FOL + a transitive closure operator. The transitive closure R∗ of binary relation R is defined by: R∗ =

• R(n)

where R(0) = Id, R(n+1) = R(n) ◦ R. Alternatively, R∗ = Id ∪

• {S | R ∪ S ◦ R ⊆ S}

(Least fixed point of the composition operator)

SLIDE 19

Why Transitive Closure Logic?

• The concept of the transitive closure is truly basic.

SLIDE 20

Why Transitive Closure Logic?

• The concept of the transitive closure is truly basic.
• Being a ‘descendent of’
• The natural numbers
• Well-formed formulas

SLIDE 21

Why Transitive Closure Logic?

• The concept of the transitive closure is truly basic.
• Being a ‘descendent of’
• The natural numbers
• Well-formed formulas
• A minimal extension.

SLIDE 22

Why Transitive Closure Logic?

• The concept of the transitive closure is truly basic.
• Being a ‘descendent of’
• The natural numbers
• Well-formed formulas
• A minimal extension.
• A special case of a least fixed point.

SLIDE 23

Why Transitive Closure Logic?

• The concept of the transitive closure is truly basic.
• Being a ‘descendent of’
• The natural numbers
• Well-formed formulas
• A minimal extension.
• A special case of a least fixed point.
• Equivalent to other extensions of FOL, but the most

convenient from a proof theoretical perspective.

SLIDE 24

Why Transitive Closure Logic?

• The concept of the transitive closure is truly basic.
• Being a ‘descendent of’
• The natural numbers
• Well-formed formulas
• A minimal extension.
• A special case of a least fixed point.
• Equivalent to other extensions of FOL, but the most

convenient from a proof theoretical perspective.

• Captures inductive principles in a uniform way.

SLIDE 25

Why Transitive Closure Logic?

• The concept of the transitive closure is truly basic.
• Being a ‘descendent of’
• The natural numbers
• Well-formed formulas
• A minimal extension.
• A special case of a least fixed point.
• Equivalent to other extensions of FOL, but the most

convenient from a proof theoretical perspective.

• Captures inductive principles in a uniform way.
• Not parametrized by a set of inductive principles.

SLIDE 26

The Language

The Language The language LTC is defined as LFOL, with the additional clause:

• (RTCx,yϕ)(s, t) is a formula,

for ϕ a formula, x, y distinct variables, and s, t terms. (x, y become bound in this formula.)

SLIDE 27

The Language

The Language The language LTC is defined as LFOL, with the additional clause:

• (RTCx,yϕ)(s, t) is a formula,

for ϕ a formula, x, y distinct variables, and s, t terms. (x, y become bound in this formula.) Allows for:

• Rich testing
• Nested RTC

SLIDE 28

The Semantics

The Intended Meaning of (RTCx,yϕ)(s, t) s = t ∨ ϕ(s, t) ∨ ∃w1.ϕ(s, w1) ∧ ϕ(w1, t) ∨ ∃w1∃w2.ϕ(s, w1) ∧ ϕ(w1, w2) ∧ ϕ(w2, t) ∨ ...

SLIDE 29

The Semantics

The Intended Meaning of (RTCx,yϕ)(s, t) s = t ∨ ϕ(s, t) ∨ ∃w1.ϕ(s, w1) ∧ ϕ(w1, t) ∨ ∃w1∃w2.ϕ(s, w1) ∧ ϕ(w1, w2) ∧ ϕ(w2, t) ∨ ... Formal Definition Let M be a structure for LTC and v an assignment in M. M, v | = (RTCx,yϕ) (s, t) iff there exist a0, ...an ∈ D s.t. v [s] = a0; v [t] = an; M, v [x := ai, y := ai+1] | = ϕ for 0 ≤ i < n.

a0 a1 a2 an−1 an

s s t t

ϕ ϕ ϕ ϕ

SLIDE 30

The Semantics

The Intended Meaning of (RTCx,yϕ)(s, t) s = t ∨ ϕ(s, t) ∨ ∃w1.ϕ(s, w1) ∧ ϕ(w1, t) ∨ ∃w1∃w2.ϕ(s, w1) ∧ ϕ(w1, w2) ∧ ϕ(w2, t) ∨ ... Formal Definition Let M be a structure for LTC and v an assignment in M. M, v | = (RTCx,yϕ) (s, t) iff there exist a0, ...an ∈ D s.t. v [s] = a0; v [t] = an; M, v [x := ai, y := ai+1] | = ϕ for 0 ≤ i < n.

a0 a1 a2 an−1 an

s s t t

ϕ ϕ ϕ ϕ

M, v | = (RTCx,yϕ) (s, t) provided for every A ⊆ D, if v (s) ∈ A and ∀a, b ∈ D : (a ∈ A ∧ M, v [x := a, y := b] | = ϕ) → b ∈ A, then v (t) ∈ A.

SLIDE 31

Expressive Power

• The reflexive and the non-reflexive TC operators are

equivalent (assuming equality).

SLIDE 32

Expressive Power

• The reflexive and the non-reflexive TC operators are

equivalent (assuming equality). Theorem [Avron, ’03] All recursive functions and relations are definable in L{0,s}

TC

SLIDE 33

Expressive Power

• The reflexive and the non-reflexive TC operators are

equivalent (assuming equality). Theorem [Avron, ’03] All recursive functions and relations are definable in L{0,s}

TC

(with pairs)

SLIDE 34

Expressive Power

• The reflexive and the non-reflexive TC operators are

equivalent (assuming equality). Theorem [Avron, ’03] All recursive functions and relations are definable in L{0,s}

TC

(with pairs)

• + is definable in L{0,s}

TC

(with pairs) by:

x = y + z ⇐ ⇒ (RTCu,vv.1 = s (u.1) ∧ v.2 = s (u.2)) ((0, y) , (z, x))

0, y

SLIDE 35

Expressive Power

• The reflexive and the non-reflexive TC operators are

equivalent (assuming equality). Theorem [Avron, ’03] All recursive functions and relations are definable in L{0,s}

TC

(with pairs)

• + is definable in L{0,s}

TC

(with pairs) by:

x = y + z ⇐ ⇒ (RTCu,vv.1 = s (u.1) ∧ v.2 = s (u.2)) ((0, y) , (z, x))

0, y 1, y + 1

SLIDE 36

Expressive Power

• The reflexive and the non-reflexive TC operators are

equivalent (assuming equality). Theorem [Avron, ’03] All recursive functions and relations are definable in L{0,s}

TC

(with pairs)

• + is definable in L{0,s}

TC

(with pairs) by:

x = y + z ⇐ ⇒ (RTCu,vv.1 = s (u.1) ∧ v.2 = s (u.2)) ((0, y) , (z, x))

0, y 1, y + 1 2, y + 2

SLIDE 37

Expressive Power

• The reflexive and the non-reflexive TC operators are

equivalent (assuming equality). Theorem [Avron, ’03] All recursive functions and relations are definable in L{0,s}

TC

(with pairs)

• + is definable in L{0,s}

TC

(with pairs) by:

x = y + z ⇐ ⇒ (RTCu,vv.1 = s (u.1) ∧ v.2 = s (u.2)) ((0, y) , (z, x))

0, y 1, y + 1 2, y + 2 z, y + x

SLIDE 38

Expressive Power

Categorical Characterization of the Natural Numbers ∀x (s (x) = 0) ∀x∀y (s (x) = s (y) → x = y) ∀x (RTCw,u (s(w) = u)) (0, x)

SLIDE 39

Expressive Power

Categorical Characterization of the Natural Numbers ∀x (s (x) = 0) ∀x∀y (s (x) = s (y) → x = y) ∀x (RTCw,u (s(w) = u)) (0, x) Corollaries:

• The upward Löwenheim-Skolem theorem fails for TC-logic.

SLIDE 40

Expressive Power

Categorical Characterization of the Natural Numbers ∀x (s (x) = 0) ∀x∀y (s (x) = s (y) → x = y) ∀x (RTCw,u (s(w) = u)) (0, x) Corollaries:

• The upward Löwenheim-Skolem theorem fails for TC-logic.
• TC-logic is not compact.

SLIDE 41

Expressive Power

Categorical Characterization of the Natural Numbers ∀x (s (x) = 0) ∀x∀y (s (x) = s (y) → x = y) ∀x (RTCw,u (s(w) = u)) (0, x) Corollaries:

• The upward Löwenheim-Skolem theorem fails for TC-logic.
• TC-logic is not compact.
• TC-logic is inherently incomplete.

SLIDE 42

Expressive Power

Categorical Characterization of the Natural Numbers ∀x (s (x) = 0) ∀x∀y (s (x) = s (y) → x = y) ∀x (RTCw,u (s(w) = u)) (0, x) Corollaries:

• The upward Löwenheim-Skolem theorem fails for TC-logic.
• TC-logic is not compact.
• TC-logic is inherently incomplete.

SLIDE 43

Proof Theory

Infinitary Systems Finitary Systems

E f f e c t i v e n e s s C

• m

p l e t e n e s s

SLIDE 44

Proof Theory

Infinitary Systems Finitary Systems

E f f e c t i v e n e s s C

• m

p l e t e n e s s

SLIDE 45

The System LK= [Gentzen, ’34]

ψ, Γ ⇒ ∆ ϕ ∧ ψ, Γ ⇒ ∆ (∧L1) ϕ, Γ ⇒ ∆ ψ, Γ ⇒ ∆ ϕ ∨ ψ, Γ ⇒ ∆ (∨L) ϕ, Γ ⇒ ∆ ϕ ∧ ψ, Γ ⇒ ∆ (∧L2) Γ ⇒ ∆, ϕ Γ ⇒ ∆, ϕ ∨ ψ (∨R1) Γ ⇒ ∆, ϕ Γ ⇒ ∆, ψ Γ ⇒ ∆, ϕ ∧ ψ (∧R) Γ ⇒ ∆, ψ Γ ⇒ ∆, ϕ ∨ ψ (∨R2) Γ ⇒ ∆, ϕ ψ, Γ ⇒ ∆ ϕ → ψ, Γ ⇒ ∆ (→ L) Γ ⇒ ∆, ϕ ¬ϕ, Γ ⇒ ∆ (¬L) ϕ t

x

• , Γ ⇒ ∆

∀xϕ, Γ ⇒ ∆ (∀L) ϕ y

x

• , Γ ⇒ ∆

∃xϕ, Γ ⇒ ∆ (∃L)∗ ϕ, Γ ⇒ ∆, ψ Γ ⇒ ∆, ϕ → ψ (→ R) ϕ, Γ ⇒ ∆ Γ ⇒ ∆, ¬ϕ (¬R) Γ ⇒ ∆, ϕ y

x

• Γ ⇒ ∆, ∀xϕ (∀R)∗

Γ ⇒ ∆, ϕ t

x

• Γ ⇒ ∆, ∃xϕ (∃R)

SLIDE 46

The System LK= [Gentzen, ’34]

Γ ⇒ ∆ ϕ, Γ ⇒ ∆ (wkL) ϕ, ϕ, Γ ⇒ ∆ ϕ, Γ ⇒ ∆ (cntL) Γ ⇒ ∆, ϕ ϕ, Γ ⇒ ∆ Γ ⇒ ∆ (cut) Γ ⇒ ∆ Γ ⇒ ∆, ϕ (wkR) Γ ⇒ ∆, ϕ, ϕ Γ ⇒ ∆, ϕ (cntR) Γ ⇒ ∆ Γ

s

• x
• ⇒ ∆

s

• x

(sub)

ϕ ⇒ ϕ (id) Γ ⇒ ∆, s = t Γ ⇒ ∆, ϕ s

x

• Γ ⇒ ∆, ϕ t

x

• (eq)

⇒ t = t (eq)

SLIDE 47

Finitary Proof System – RTCG

Reflexivity

Γ ⇒ ∆, (RTCx,yϕ) (s, s)

Step

Γ ⇒ ∆, (RTCx,yϕ) (s, r) Γ ⇒ ∆, ϕ

• r

x , t y

• Γ ⇒ ∆, (TCx,yϕ) (s, t)

Induction Γ, ψ (x) , ϕ(x, y) ⇒ ∆, ψ

y

x

• Γ, ψ

s

x

, (RTCx,yϕ)(s, t) ⇒ ∆, ψ t

x

• provided x /

∈ FV (Γ ∪ ∆) and y / ∈ FV (Γ ∪ ∆ ∪ {ψ}).

SLIDE 48

RTCG ‘Captures’ TC-logic

Γ ⇒ ∆, (RTCx,yϕ) (s, t) Γ ⇒ ∆, (RTCy,xϕ) (t, s) Γ ⇒ ∆, (RTCx,yϕ) (s, t) Γ ⇒ ∆, RTCu,vϕ u

x , v y

• (s, t)

ϕ s

x

• , Γ ⇒ ∆

(RTCx,yϕ) (s, t) , Γ ⇒ s = t, ∆ Γ ⇒ ∆, ϕ s

x , r y

• Γ ⇒ ∆, (RTCx,yϕ) (r, t)

Γ ⇒ ∆, (RTCx,yϕ) (s, t) Γ, ϕ ⇒ ∆, ψ Γ, (RTCx,yϕ) (s, t) ⇒ ∆, (RTCx,yψ) (s, t) (RTCx,yϕ) (s, t) , Γ ⇒ ∆ (RTCu,v (RTCx,yϕ) (u, v)) (s, t) , Γ ⇒ ∆ Γ ⇒ ∆, (RTCx,yϕ) (s, t) Γ ⇒ ∆, s = t, ∃z (RTCx,yϕ) (s, z) ∧ ϕ z

x , t y

SLIDE 49

Arithmetics in RTCG

TC for Arithmetics RTCG+A is obtained from RTCG by the addition of the standard axioms for successor and addition, and the axiom characterizing the natural numbers in TC-logic.

SLIDE 50

Arithmetics in RTCG

TC for Arithmetics RTCG+A is obtained from RTCG by the addition of the standard axioms for successor and addition, and the axiom characterizing the natural numbers in TC-logic. Theorem RTCG+A is equivalent to the sequent calculi of PA, i.e. there is a provability preserving translation algorithm between them.

SLIDE 51

Arithmetics in RTCG

TC for Arithmetics RTCG+A is obtained from RTCG by the addition of the standard axioms for successor and addition, and the axiom characterizing the natural numbers in TC-logic. Theorem RTCG+A is equivalent to the sequent calculi of PA, i.e. there is a provability preserving translation algorithm between them. Corollary The ordinal number of the RTCG+A is ε0.

SLIDE 52

Henkin Semantics

A σ-Henkin structure is a triple M = D, I, D′ (frame), s.t.:

• 1. D, I is a FO structure for σ
• 2. D′ ⊆ P (D) is closed under parametric definability.

SLIDE 53

Henkin Semantics

A σ-Henkin structure is a triple M = D, I, D′ (frame), s.t.:

• 1. D, I is a FO structure for σ
• 2. D′ ⊆ P (D) is closed under parametric definability.

M, v | = (RTCx,yϕ) (s, t) provided for every A ∈ D′, if v (s) ∈ A and ∀a, b ∈ D : (a ∈ A ∧ M, v [x := a, y := b] | = ϕ) → b ∈ A, then v (t) ∈ A.

SLIDE 54

Henkin Semantics

A σ-Henkin structure is a triple M = D, I, D′ (frame), s.t.:

• 1. D, I is a FO structure for σ
• 2. D′ ⊆ P (D) is closed under parametric definability.

M, v | = (RTCx,yϕ) (s, t) provided for every A ∈ D′, if v (s) ∈ A and ∀a, b ∈ D : (a ∈ A ∧ M, v [x := a, y := b] | = ϕ) → b ∈ A, then v (t) ∈ A. Completeness Theorem T ⊢RTCG ϕ ⇐ ⇒ T | =H ϕ.

SLIDE 55

So Far

standard validity Henkin validity

SLIDE 56

So Far

standard validity Henkin validity

RTCG

SLIDE 57

Proof Theory

Infinitary Systems Finitary Systems

E f f e c t i v e n e s s C

• m

p l e t e n e s s

SLIDE 58

Infinitary Systems

Infinitary ?

SLIDE 59

Infinitary Systems

Infinitary ? width

SLIDE 60

Infinitary Systems

Infinitary ? width infinite rules finite proofs

SLIDE 61

Infinitary Systems

Infinitary ? width infinite rules finite proofs height

SLIDE 62

Infinitary Systems

Infinitary ? width infinite rules finite proofs height finite rules infinite proofs

SLIDE 63

Infinitary Systems

Infinitary ? width infinite rules finite proofs height finite rules infinite proofs non-effective

SLIDE 64

Infinitary Systems

Infinitary ? width infinite rules finite proofs height finite rules infinite proofs non-effective can be effective?

SLIDE 65

Infinite Descent-Style Proof System

. . . . . . . .

• . . . . . . . .

. . . . .

(Inference)

• ·

· ·

• (Axiom)
• ·

· · · ·

• Infinite height,

not width

SLIDE 66

Infinite Descent-Style Proof System

. . . . . . . .

• . . . . . . . .

. . . . .

(Inference)

• ·

· ·

• (Axiom)
• ·

· · · ·

• Infinite height,

not width

• Proofs can be infinite, non-well-founded trees, provided that

every infinite path admits some infinite descent.

• The descent is witnessed by tracing terms/formulas

corresponding to elements of a well-founded set.

• This global trace condition is decidable using Büchi automata.
• Systems of implicit induction.

SLIDE 67

Infinitary Proof System – RTCω

G

Reflexivity

Γ ⇒ ∆, (RTCx,yϕ) (s, s)

Step

Γ ⇒ ∆, (RTCx,yϕ) (s, r) Γ ⇒ ∆, ϕ

• r

x , t y

• Γ ⇒ ∆, (TCx,yϕ) (s, t)

Case-split Γ, s = t ⇒ ∆ Γ, (RTCx,yϕ)(s, z), ϕ

• z

x , t y

• ⇒ ∆

Γ, (RTCx,yϕ)(s, t) ⇒ ∆

provided z is fresh.

SLIDE 68

Infinitary Proof System – RTCω

G

Reflexivity

Γ ⇒ ∆, (RTCx,yϕ) (s, s)

Step

Γ ⇒ ∆, (RTCx,yϕ) (s, r) Γ ⇒ ∆, ϕ

• r

x , t y

• Γ ⇒ ∆, (TCx,yϕ) (s, t)

Case-split Γ, s = t ⇒ ∆ Γ, (RTCx,yϕ)(s, z), ϕ

• z

x , t y

• ⇒ ∆

Γ, (RTCx,yϕ)(s, t) ⇒ ∆

provided z is fresh.

SLIDE 69

Soundness and Completeness

Completeness Theorem T ⊢cf

RTCω

G ϕ ⇐

⇒ T | = ϕ.

SLIDE 70

Soundness and Completeness

Completeness Theorem T ⊢cf

RTCω

G ϕ ⇐

⇒ T | = ϕ. Global soundness via an infinite descent proof-by-contradiction:

SLIDE 71

Soundness and Completeness

Completeness Theorem T ⊢cf

RTCω

G ϕ ⇐

⇒ T | = ϕ. Global soundness via an infinite descent proof-by-contradiction:

• Assume the conclusion of the proof is invalid

SLIDE 72

Soundness and Completeness

Completeness Theorem T ⊢cf

RTCω

G ϕ ⇐

⇒ T | = ϕ. Global soundness via an infinite descent proof-by-contradiction:

• Assume the conclusion of the proof is invalid
• Local soundness entails an infinite sequence of counter models

SLIDE 73

Soundness and Completeness

Completeness Theorem T ⊢cf

RTCω

G ϕ ⇐

⇒ T | = ϕ. Global soundness via an infinite descent proof-by-contradiction:

• Assume the conclusion of the proof is invalid
• Local soundness entails an infinite sequence of counter models
• Mapped to the minimal length for witnessing the transitive

closure trace.

SLIDE 74

Soundness and Completeness

Completeness Theorem T ⊢cf

RTCω

G ϕ ⇐

⇒ T | = ϕ. Global soundness via an infinite descent proof-by-contradiction:

• Assume the conclusion of the proof is invalid
• Local soundness entails an infinite sequence of counter models
• Mapped to the minimal length for witnessing the transitive

closure trace.

• Global trace condition entails the chain is infinitely descending

SLIDE 75

Soundness and Completeness

Completeness Theorem T ⊢cf

RTCω

G ϕ ⇐

⇒ T | = ϕ. Global soundness via an infinite descent proof-by-contradiction:

• Assume the conclusion of the proof is invalid
• Local soundness entails an infinite sequence of counter models
• Mapped to the minimal length for witnessing the transitive

closure trace.

• Global trace condition entails the chain is infinitely descending
• But the numbers are well-founded . . . contradiction!

SLIDE 76

So Far

standard validity Henkin validity

RTCG

SLIDE 77

So Far

standard validity Henkin validity

RTCG (cut-free) RTCω

G

SLIDE 78

Proof Theory

Infinitary Systems Finitary Systems

E f f e c t i v e n e s s C

• m

p l e t e n e s s

SLIDE 79

The Cyclic Subsystem – CRTCω

G

. . . . . . . .

• . . . . . . . .

(Inference)

• ·

· ·

• (Axiom)
• (Axiom)
• ·

· · · · · ·

• (Axiom)

SLIDE 80

The Cyclic Subsystem – CRTCω

G

. . . . . . . .

• . . . . .

(Inference)

• ·

· ·

• (Axiom)
• ·

· · ·

• An effective subsystem can be obtained by considering only

the regular infinite proofs.

• Regular proofs = represented as finite, possibly cyclic, graphs.

SLIDE 81

Implicit Induction Subsumes Explicit Induction

(Eq)

ψ

v

x

• , v = w ⇒ ψ

w

x

• .

. . . . . . Γ, ψ

v

x

• , (RTCx,y ϕ)(v, w) ⇒ ∆, ψ

w

x

• (Subst)

Γ, ψ

v

x

• , (RTCx,y ϕ)(v, z) ⇒ ∆, ψ

z

x

• Γ, ψ(x), ϕ(x, y) ⇒ ∆, ψ

y

x

• (Subst)

Γ, ψ

z

x

• , ϕ
• z

x , w y

• ⇒ ∆, ψ

w

x

• (Cut)

Γ, ψ

v

x

• , (RTCx,y ϕ)(v, z), ϕ
• z

x , w y

• ⇒ ∆, ψ

w

x

• (Case-split)

Γ, ψ

v

x

• , (RTCx,y ϕ)(v, w) ⇒ ∆, ψ

w

x

• (Subst)

Γ, ψ

s

x

• , (RTCx,y ϕ)(s, t) ⇒ ∆, ψ

t

x

SLIDE 82

Implicit Induction Subsumes Explicit Induction

(Eq)

ψ

v

x

• , v = w ⇒ ψ

w

x

• .

. . . . . . Γ, ψ

v

x

• , (RTCx,y ϕ)(v, w) ⇒ ∆, ψ

w

x

• (Subst)

Γ, ψ

v

x

• , (RTCx,y ϕ)(v, z) ⇒ ∆, ψ

z

x

• Γ, ψ(x), ϕ(x, y) ⇒ ∆, ψ

y

x

• (Subst)

Γ, ψ

z

x

• , ϕ
• z

x , w y

• ⇒ ∆, ψ

w

x

• (Cut)

Γ, ψ

v

x

• , (RTCx,y ϕ)(v, z), ϕ
• z

x , w y

• ⇒ ∆, ψ

w

x

• (Case-split)

Γ, ψ

v

x

• , (RTCx,y ϕ)(v, w) ⇒ ∆, ψ

w

x

• (Subst)

Γ, ψ

s

x

• , (RTCx,y ϕ)(s, t) ⇒ ∆, ψ

t

x

• Every infinite path (from conclusion

to premise) is eventually followed by a trace of RTC-formulas (on the left-hand side) which progresses (via case-split) infinitely often.

SLIDE 83

Implicit Induction Subsumes Explicit Induction

(Eq)

ψ

v

x

• , v = w ⇒ ψ

w

x

• .

. . . . . . Γ, ψ

v

x

• , (RTCx,y ϕ)(v, w) ⇒ ∆, ψ

w

x

• (Subst)

Γ, ψ

v

x

• , (RTCx,y ϕ)(v, z) ⇒ ∆, ψ

z

x

• Γ, ψ(x), ϕ(x, y) ⇒ ∆, ψ

y

x

• (Subst)

Γ, ψ

z

x

• , ϕ
• z

x , w y

• ⇒ ∆, ψ

w

x

• (Cut)

Γ, ψ

v

x

• , (RTCx,y ϕ)(v, z), ϕ
• z

x , w y

• ⇒ ∆, ψ

w

x

• (Case-split)

Γ, ψ

v

x

• , (RTCx,y ϕ)(v, w) ⇒ ∆, ψ

w

x

• (Subst)

Γ, ψ

s

x

• , (RTCx,y ϕ)(s, t) ⇒ ∆, ψ

t

x

• Every infinite path (from conclusion

to premise) is eventually followed by a trace of RTC-formulas (on the left-hand side) which progresses (via case-split) infinitely often.

• Normal Cyclic Proofs = non-overlapping cyclic proofs.

SLIDE 84

Cyclic Proof vs. Explicit Induction

Induction invariant

SLIDE 85

Cyclic Proof vs. Explicit Induction

Induction invariant Explicit induction requires it a priori

Major challenge for automatic proof search

SLIDE 86

Cyclic Proof vs. Explicit Induction

Induction invariant Explicit induction requires it a priori

Major challenge for automatic proof search

Cyclic proof enables its ‘discovery’

More exploratory approach to proof search

SLIDE 87

Cyclic Proof vs. Explicit Induction

Induction invariant Explicit induction requires it a priori

Major challenge for automatic proof search

Cyclic proof enables its ‘discovery’

More exploratory approach to proof search

• Complex induction schemes naturally represented by nested

and overlapping cycles.

SLIDE 88

Cyclic Proof vs. Explicit Induction

Induction invariant Explicit induction requires it a priori

Major challenge for automatic proof search

Cyclic proof enables its ‘discovery’

More exploratory approach to proof search

• Complex induction schemes naturally represented by nested

and overlapping cycles.

• Every sequent provable using the explicit induction rule is also

derivable using cyclic proof.

SLIDE 89

So Far

standard validity Henkin validity

RTCG (cut-free) RTCω

G

SLIDE 90

So Far

standard validity Henkin validity

RTCG (cut-free) RTCω

G

CRTCω

G

SLIDE 91

So Far

standard validity Henkin validity

RTCG (cut-free) RTCω

G

CRTCω

G

NCRTCω

G

SLIDE 92

Is the Cyclic System Stronger?

• For arithmetics, the explicit and cyclic systems are equivalent.

SLIDE 93

Is the Cyclic System Stronger?

• For arithmetics, the explicit and cyclic systems are equivalent.
• In general, the question of the (in)equivalence between the

systems remains open.

SLIDE 94

Is the Cyclic System Stronger?

• For arithmetics, the explicit and cyclic systems are equivalent.
• In general, the question of the (in)equivalence between the

systems remains open.

• In systems for FOL with inductive

definition, the equivalence was refuted when both systems have the same set

• f inductive definitions. [Berardi,

Tatsuta, 2017]

SLIDE 95

Is the Cyclic System Stronger?

• For arithmetics, the explicit and cyclic systems are equivalent.
• In general, the question of the (in)equivalence between the

systems remains open.

• In systems for FOL with inductive

definition, the equivalence was refuted when both systems have the same set

• f inductive definitions. [Berardi,

Tatsuta, 2017]

• In the TC framework all inductive definitions at once.

SLIDE 96

So Far

standard validity Henkin validity

(cut-free) RTCω

G

RTCG CRTCω

G

NCRTCω

G

SLIDE 97

So Far

standard validity Henkin validity

(cut-free) RTCω

G

RTCG CRTCω

G

NCRTCω

G

CRTCω

G+A

RTCG+A

SLIDE 98

Future (and Current) Work

• Resolving the open question of the (in)equivalence of RTCG

and CRTCω

G.

• Implementing CRTCω

G and investigating the practicalities of

TC-logic to support automated inductive reasoning.

• Using the uniformity of TC-logic to better study the

relationship between implicit and explicit induction.

• Cuts required in each system
• Relative complexity of proofs
• Incorporating coinductive reasoning into the formal system.

SLIDE 99

Summary

standard validity Henkin validity

(cut-free) RTCω

G

RTCG CRTCω

G

NCRTCω

G

CRTCω

G+A

RTCG+A ? ? ?

SLIDE 100

Summary

standard validity Henkin validity

(cut-free) RTCω

G

RTCG CRTCω

G

NCRTCω

G

CRTCω

G+A

RTCG+A ? ? ?

Thank you