Some proof-theoretical approaches to Monadic Second-Order logic PhD - PowerPoint PPT Presentation

� � � MSO/automata correspondance ϕ �→A ϕ MSO formulas over Σ automata over Σ ϕ �→L ( ϕ ) A�→L ( A ) P (Σ ω ) Decidability [Büchi (1962)] MSO over infinite words is decidable. ◮ Proof idea: automata theoretic-construction for each logical connective. ◮ Hard case for infinite words: negation ¬ . corresponds to complementation 11 / 36

Complementation, determinization and constructivity For finite word automata: easy complementation for deterministic automata. 0 , 1 0 0 . . . but Büchi automata are hard to determinize. 12 / 36

Complementation, determinization and constructivity For finite word automata: easy complementation for deterministic automata. 0 , 1 0 0 . . . but Büchi automata are hard to determinize. Theorem [McNaughton (1968)] Non-deterministic Büchi automata can be determinized into Rabin automata . more complex acceptance condition ◮ Büchi’s original complementation procedure: w/o determinization. ◮ Effective algorithms for automata . . . 12 / 36

Complementation, determinization and constructivity For finite word automata: easy complementation for deterministic automata. 0 , 1 0 0 . . . but Büchi automata are hard to determinize. Theorem [McNaughton (1968)] Non-deterministic Büchi automata can be determinized into Rabin automata . more complex acceptance condition ◮ Büchi’s original complementation procedure: w/o determinization. ◮ Effective algorithms for automata . . . ◮ . . . but non-constructive proofs of soundness! usual proofs: infinite Ramsey theorem, weak König’s lemma 12 / 36

Complementation, determinization and constructivity For finite word automata: easy complementation for deterministic automata. 0 , 1 0 0 . . . but Büchi automata are hard to determinize. Theorem [McNaughton (1968)] Non-deterministic Büchi automata can be determinized into Rabin automata . more complex acceptance condition ◮ Büchi’s original complementation procedure: w/o determinization. ◮ Effective algorithms for automata . . . ◮ . . . but non-constructive proofs of soundness! usual proofs: infinite Ramsey theorem, weak König’s lemma Quantify how non-constructive they are? 12 / 36

Outline Monadic Second-Order logic Part I: Reverse Mathematics Reverse Mathematics Büchi’s theorem Beyond infinite words Part II: proof systems for Church’s synthesis Conclusion 13 / 36

Reverse Mathematics ◮ A framework to analyze axiomatic strength. ◮ Vast program. [Friedman, Simpson, Steele 70s] Methodology ◮ Consider a theorem T formulated in second-order arithmetic. ◮ Work in the weak theory RCA 0 . ◮ Target some natural axiom A such that RCA 0 � A . ◮ Show that RCA 0 ⊢ A ⇔ T . Essentially independence proofs. . . ◮ Similar in spirit to statements like “Tychonoff’s theorem is equivalent to the axiom of choice.” 14 / 36

The big five Outliers: infinite Ramsey for pairs, determinacy statements. 15 / 36

The big five Outliers: infinite Ramsey for pairs, determinacy statements. � Where does Büchi’s theorem sit in this hierarchy? 15 / 36

Büchi’s decidability theorem (over RCA 0 ) Weak K¨ onig’s lemma Infinite Ramsey theorem ⇓ Σ 0 ⇒ 2 -induction Additive Ramsey ⇓ ⇑ ⇑ ⇓ ⇐ MSO ( ω ) Compl. of NBA � ⇓ Bounded weak K¨ onig’s lemma Determinization of NBA The Logical Strength of Büchi’s Decidability Theorem [Kołodziejczyk, Michalewski, P., Skrzypczak, 2016] 16 / 36

Beyond infinite words Theorem [Kołodziejczyk, Michalewski (2015)] Decidability of MSO over the infinite binary tree is not provable in Π 1 2 - CA 0 . ◮ Rabin’s theorem requires much higher axiomatic strength. ◮ Roughly on par with determinacy of infinite parity games. BC (Σ 0 2 ) games 17 / 36

Beyond infinite words Theorem [Kołodziejczyk, Michalewski (2015)] Decidability of MSO over the infinite binary tree is not provable in Π 1 2 - CA 0 . ◮ Rabin’s theorem requires much higher axiomatic strength. ◮ Roughly on par with determinacy of infinite parity games. BC (Σ 0 2 ) games ◮ Intermediate cases? 17 / 36

Beyond infinite words Theorem [Kołodziejczyk, Michalewski (2015)] Decidability of MSO over the infinite binary tree is not provable in Π 1 2 - CA 0 . ◮ Rabin’s theorem requires much higher axiomatic strength. ◮ Roughly on par with determinacy of infinite parity games. BC (Σ 0 2 ) games ◮ Intermediate cases? MSO over the rationals (MSO ( Q ) ) ◮ Decidable via a reduction to the infinite tree. ◮ Cover all countable linear orders. ◮ Direct algebraic decidability proofs. [Shelah (1975)], [Carton, Colcombet, Puppis (2013)] 17 / 36

Strength of additive Ramsey over Q and MSO ( Q ) Theorem [Kołodziejczyk, Michalewski, P., Skrzypczak] Over RCA 0 , the following are equivalent: ◮ the shuffle principle [Carton, Colcombet, Puppis (2013)] ◮ Shelah’s additive Ramseyan theorem over Q [Shelah (1975)] ◮ induction for Σ 0 2 formulas 18 / 36

Strength of additive Ramsey over Q and MSO ( Q ) Theorem [Kołodziejczyk, Michalewski, P., Skrzypczak] Over RCA 0 , the following are equivalent: ◮ the shuffle principle [Carton, Colcombet, Puppis (2013)] ◮ Shelah’s additive Ramseyan theorem over Q [Shelah (1975)] ◮ induction for Σ 0 2 formulas However, does not gauge the strength of MSO ( Q ) 18 / 36

Strength of additive Ramsey over Q and MSO ( Q ) Theorem [Kołodziejczyk, Michalewski, P., Skrzypczak] Over RCA 0 , the following are equivalent: ◮ the shuffle principle [Carton, Colcombet, Puppis (2013)] ◮ Shelah’s additive Ramseyan theorem over Q [Shelah (1975)] ◮ induction for Σ 0 2 formulas However, does not gauge the strength of MSO ( Q ) Expressivity The classical theory MSO ( Q ) has a sentence equivalent to Π 1 1 - CA 0 . 18 / 36

Strength of additive Ramsey over Q and MSO ( Q ) Theorem [Kołodziejczyk, Michalewski, P., Skrzypczak] Over RCA 0 , the following are equivalent: ◮ the shuffle principle [Carton, Colcombet, Puppis (2013)] ◮ Shelah’s additive Ramseyan theorem over Q [Shelah (1975)] ◮ induction for Σ 0 2 formulas However, does not gauge the strength of MSO ( Q ) Expressivity The classical theory MSO ( Q ) has a sentence equivalent to Π 1 1 - CA 0 . Conjecture Over RCA 0 , the following are equivalent: ◮ The axiom of finite Π 1 1 -recursion. ◮ Determinacy of infinite weak parity games. BC (Σ 0 1 ) games ◮ Soundness of the decision algorithm for MSO ( Q ) . 18 / 36

Outline Monadic Second-Order logic Part I: Reverse Mathematics Reverse Mathematics Büchi’s theorem Beyond infinite words Part II: proof systems for Church’s synthesis Church’s synthesis and witness extraction Constructive proof systems Categorical/syntactic approach Conclusion 19 / 36

Church’s synthesis (1/2): causal functions Causal/synchronous stream functions f : Σ ω → Γ ω 1 ◮ Interpret n ∈ N as time steps . b | b b | a , a | a f : Σ + → Γ as ◮ Lifted from functions ˆ ˆ f : Σ ω → Γ ω a | a 0 s �→ n �→ f ( s ( 0 ) . . . s ( n )) i.e., the output does not depend on the future. ◮ Focus on finite-state causal functions. (Correspond to Mealy machines ) ◮ All f.s. causal functions are recursive. ◮ All causal functions are continuous. ◮ Some recursive functions are not causal. w �− → n �→ w n + 1 20 / 36

Church’s synthesis (2/2): the Büchi-Landweber theorem Church’s synthesis problem Given a formula ϕ ( X , Y ) , find a f. s. causal f : Σ ω → Γ ω such that ∀ w ϕ ( w , f ( w )) 21 / 36

Church’s synthesis (2/2): the Büchi-Landweber theorem Church’s synthesis problem Given a formula ϕ ( X , Y ) , find a f. s. causal f : Σ ω → Γ ω such that ∀ w ϕ ( w , f ( w )) Example (inspired from [Thomas (2008)]) : ◮ ϕ ( X , Y ) ≡ ( X infinite ⇒ Y infinite) and ∀ i ( i ∈ Y ⇒ i + 1 / ∈ Y ) 1 | 0 , 0 | 0 1 0 0 | 0 1 | 1 Theorem [Büchi-Landweber (1969)] Algorithmic solution for ϕ ( X , Y ) in MSO. ◮ Algorithmically costly. . . 21 / 36

MSO and proofs MSO can also be seen as a classical axiomatic theory Theorem [Siefkes (1970)] MSO is completely axiomatized by the axioms of second-order arithmetic. 22 / 36

MSO and proofs MSO can also be seen as a classical axiomatic theory Theorem [Siefkes (1970)] MSO is completely axiomatized by the axioms of second-order arithmetic. Church’s synthesis reminiscent of extraction from proofs: ? MSO ⊢ ∀ x ∃ y ϕ ( x , y ) = ⇒ ∃ f f.s. causal ∀ x ϕ ( x , f ( x )) 22 / 36

MSO and proofs MSO can also be seen as a classical axiomatic theory Theorem [Siefkes (1970)] MSO is completely axiomatized by the axioms of second-order arithmetic. Church’s synthesis reminiscent of extraction from proofs: MSO ⊢ ∀ x ∃ y ϕ ( x , y ) �⇒ ∃ f f.s. causal ∀ x ϕ ( x , f ( x )) Classical theorems in MSO (subtle point { 0 , 1 } ω vs P ( N ) ) ◮ Excluded middle ◮ The infinite pigeonhole principle ◮ Instances of additive Ramsey � No algorithmic witnesses for ∀∃ theorems. 22 / 36

Extraction from proofs Goal : a refinement of MSO ( N ) with extraction for causal functions. ◮ Toward semi-automatic approach to synthesis. ◮ Approach inspired by realizability. [Kleene (1945), . . . ] 23 / 36

Extraction from proofs Goal : a refinement of MSO ( N ) with extraction for causal functions. ◮ Toward semi-automatic approach to synthesis. ◮ Approach inspired by realizability. [Kleene (1945), . . . ] Analogous example: extraction for intuitionistic arithmetic (HA) If HA ⊢ ∀ x ∃ y ϕ ( x , y ) , there is an algorithm computing f : N → N recursive such that ∀ x ϕ ( x , f ( x )) ◮ A subset of classical arithmetic (PA). ◮ As expressive as classical arithmetic. ( ϕ �→ ϕ ¬¬ ) ◮ Can be refined to System T functions. [Gödel (1930s)] Analogy Classical system MSO ( N ) PA Realizers Causal functions System T Intuitionistic system ??? HA 23 / 36

Synchronous MSO (SMSO) [P., Riba (2017)] Intuitionistic version of MSO ϕ, ψ ::= α | ϕ ∧ ψ | ∃ X ϕ | ¬ ϕ Quantification over individuals encoded as usual 24 / 36

Synchronous MSO (SMSO) [P., Riba (2017)] Intuitionistic version of MSO ϕ, ψ ::= α | ϕ ∧ ψ | ∃ X ϕ | ¬ ϕ Quantification over individuals encoded as usual Glivenko’s theorem for SMSO MSO ⊢ ϕ if and only if SMSO ⊢ ¬¬ ϕ ◮ Negation erases computational contents. 24 / 36

Synchronous MSO (SMSO) [P., Riba (2017)] Intuitionistic version of MSO ϕ, ψ ::= α | ϕ ∧ ψ | ∃ X ϕ | ¬ ϕ Quantification over individuals encoded as usual Glivenko’s theorem for SMSO MSO ⊢ ϕ if and only if SMSO ⊢ ¬¬ ϕ ◮ Negation erases computational contents. Extraction of f.s. causal functions SMSO ⊢ ∃ y ¬¬ ϕ ( x , y ) iff there is a f.s. causal f s.t. MSO ⊢ ∀ x ϕ ( x , f ( x )) ◮ Proofs ϕ ⊢ ψ interpreted as simulations between ND automata. 24 / 36

Synchronous MSO (SMSO) [P., Riba (2017)] Intuitionistic version of MSO ϕ, ψ ::= α | ϕ ∧ ψ | ∃ X ϕ | ¬ ϕ Quantification over individuals encoded as usual Glivenko’s theorem for SMSO MSO ⊢ ϕ if and only if SMSO ⊢ ¬¬ ϕ ◮ Negation erases computational contents. Extraction of f.s. causal functions SMSO ⊢ ∃ y ¬¬ ϕ ( x , y ) iff there is a f.s. causal f s.t. MSO ⊢ ∀ x ϕ ( x , f ( x )) ◮ Proofs ϕ ⊢ ψ interpreted as simulations between ND automata. No interpretation for ⇒ and ∀ 24 / 36

Synchronous MSO (SMSO) [P., Riba (2017)] Intuitionistic version of MSO ϕ, ψ ::= α | ϕ ∧ ψ | ∃ X ϕ | ¬ ϕ Quantification over individuals encoded as usual Glivenko’s theorem for SMSO MSO ⊢ ϕ if and only if SMSO ⊢ ¬¬ ϕ ◮ Negation erases computational contents. Extraction of f.s. causal functions SMSO ⊢ ∃ y ¬¬ ϕ ( x , y ) iff there is a f.s. causal f s.t. MSO ⊢ ∀ x ϕ ( x , f ( x )) ◮ Proofs ϕ ⊢ ψ interpreted as simulations between ND automata. No interpretation for ⇒ and ∀ Polarity restriction 24 / 36

A linear refinement LMSO [P., Riba (2018)] ◮ Polarized system with dualities. ◮ Requires the introduction of linear connectives. Linear MSO (LMSO) α | ϕ ⊗ ψ | ϕ ` ψ | ϕ ⊸ ψ | ∀ X ϕ | ∃ X ϕ | ! ϕ − | ? ϕ + | ϕ, ψ ::= . . . 25 / 36

A linear refinement LMSO [P., Riba (2018)] ◮ Polarized system with dualities. ◮ Requires the introduction of linear connectives. Linear MSO (LMSO) α | ϕ ⊗ ψ | ϕ ` ψ | ϕ ⊸ ψ | ∀ X ϕ | ∃ X ϕ | ! ϕ − | ? ϕ + | ϕ, ψ ::= . . . Alternating ( ∀ , ∃ , ⊗ , ` , ⊸ ) ⊗ , ` , ⊸ ⊗ , ` , ∃ ⊗ , ` , ∀ ?( − ) !( − ) Deterministic ( ± ) Universal Non-deterministic ( − ) (+) ( − ) ⊥ 25 / 36

A linear refinement LMSO [P., Riba (2018)] ◮ Polarized system with dualities. ◮ Requires the introduction of linear connectives. Linear MSO (LMSO) α | ϕ ⊗ ψ | ϕ ` ψ | ϕ ⊸ ψ | ∀ X ϕ | ∃ X ϕ | ! ϕ − | ? ϕ + | ϕ, ψ ::= . . . Alternating ( ∀ , ∃ , ⊗ , ` , ⊸ ) ⊗ , ` , ⊸ ⊗ , ` , ∃ ⊗ , ` , ∀ ?( − ) !( − ) Deterministic ( ± ) Universal Non-deterministic ( − ) (+) ( − ) ⊥ SMSO ≈ restriction to positives 25 / 36

A linear refinement LMSO [P., Riba (2018)] ◮ Polarized system with dualities. ◮ Requires the introduction of linear connectives. Linear MSO (LMSO) α | ϕ ⊗ ψ | ϕ ` ψ | ϕ ⊸ ψ | ∀ X ϕ | ∃ X ϕ | ! ϕ − | ? ϕ + | ϕ, ψ ::= . . . Alternating ( ∀ , ∃ , ⊗ , ` , ⊸ ) ⊗ , ` , ⊸ ⊗ , ` , ∃ ⊗ , ` , ∀ ?( − ) !( − ) Deterministic ( ± ) Universal Non-deterministic ( − ) (+) ( − ) ⊥ SMSO ≈ restriction to positives 25 / 36

� � � Expressivity and proof extraction for LMSO Conservativity Expressivity LMSO → MSO MSO → LMSO �→ ⌈ ϕ ⌉ ϕ L ϕ ϕ �→ If LMSO ⊢ ϕ , then MSO ⊢ ⌈ ϕ ⌉ . If MSO ⊢ ϕ , then LMSO ⊢ ϕ L . ϕ �→ A ϕ LMSO Alternating automata ϕ �→ � ϕ � Acceptance game Simulation games 26 / 36

� � � Expressivity and proof extraction for LMSO Conservativity Expressivity LMSO → MSO MSO → LMSO �→ ⌈ ϕ ⌉ ϕ L ϕ ϕ �→ If LMSO ⊢ ϕ , then MSO ⊢ ⌈ ϕ ⌉ . If MSO ⊢ ϕ , then LMSO ⊢ ϕ L . ϕ �→ A ϕ LMSO Alternating automata ϕ �→ � ϕ � Acceptance game Simulation games Extraction of f.s. causal functions LMSO ⊢ ∀ x ∃ y ϕ L ( x , y ) iff there is a f.s causal f s.t. MSO ⊢ ∀ x ϕ ( x , f ( x )) 26 / 36

Simulation model: logical aspects ◮ LMSO includes Full Intuitionistic Multiplicative Linear Logic . [Hyland, de Paiva (1993)] ◮ Similarities with Dialectica categories DC: [de Paiva (1989,1991)] 27 / 36

Simulation model: logical aspects ◮ LMSO includes Full Intuitionistic Multiplicative Linear Logic . [Hyland, de Paiva (1993)] ◮ Similarities with Dialectica categories DC: [de Paiva (1989,1991)] Realized principles ◮ Linear Markov principle and independence of premise . 27 / 36

Simulation model: logical aspects ◮ LMSO includes Full Intuitionistic Multiplicative Linear Logic . [Hyland, de Paiva (1993)] ◮ Similarities with Dialectica categories DC: [de Paiva (1989,1991)] Realized principles ◮ Linear Markov principle and independence of premise . ◮ A classically false choice-like scheme ∀ x ∈ Σ ω ∃ y ∈ Γ ω ϕ ( x , y ) ∃ f ∈ (Σ → Γ) ω ∀ x ∈ Σ ω ϕ ( x , f ( x )) − ⊸ f ( x ) for pointwise application 27 / 36

Simulation model: logical aspects ◮ LMSO includes Full Intuitionistic Multiplicative Linear Logic . [Hyland, de Paiva (1993)] ◮ Similarities with Dialectica categories DC: [de Paiva (1989,1991)] Realized principles ◮ Linear Markov principle and independence of premise . ◮ A classically false choice-like scheme ∀ x ∈ Σ ω ∃ y ∈ Γ ω ϕ ( x , y ) ∃ f ∈ (Σ → Γ) ω ∀ x ∈ Σ ω ϕ ( x , f ( x )) − ⊸ f ( x ) for pointwise application Double linear-negation elimination For every ϕ , there is a realizer ( ϕ ⊸ ⊥ ) ⊸ ⊥ − ϕ ⊸ 27 / 36

Simulation model: logical aspects ◮ LMSO includes Full Intuitionistic Multiplicative Linear Logic . [Hyland, de Paiva (1993)] ◮ Similarities with Dialectica categories DC: [de Paiva (1989,1991)] Realized principles ◮ Linear Markov principle and independence of premise . ◮ A classically false choice-like scheme ∀ x ∈ Σ ω ∃ y ∈ Γ ω ϕ ( x , y ) ∃ f ∈ (Σ → Γ) ω ∀ x ∈ Σ ω ϕ ( x , f ( x )) − ⊸ f ( x ) for pointwise application Double linear-negation elimination For every ϕ , there is a realizer ( ϕ ⊸ ⊥ ) ⊸ ⊥ − ϕ ⊸ but no canonical iso in general! ◮ Also holds in DC if the base satisfies choice. 27 / 36

Why automata? The above logic can be defined without reference to automata. ◮ ω -word automata guarantee decidability properties. . . ◮ But they are not needed to extract realizers. 28 / 36

Why automata? The above logic can be defined without reference to automata. ◮ ω -word automata guarantee decidability properties. . . ◮ But they are not needed to extract realizers. � A purely logical reformulation of LMSO using categorical semantics. Goals ◮ Purely syntactic transformations. ◮ Understand links with typed realizability and Dialectica. 28 / 36

Finite-state causal functions as terms Define the category M of causal functions ◮ Objects: sets of streams Σ ω for Σ finite ◮ Morphisms: finite-state causal functions ◮ Cartesian products Σ ω × Γ ω ≃ (Σ × Γ) ω , but not cartesian-closed 29 / 36

Finite-state causal functions as terms Define the category M of causal functions ◮ Objects: sets of streams Σ ω for Σ finite ◮ Morphisms: finite-state causal functions ◮ Cartesian products Σ ω × Γ ω ≃ (Σ × Γ) ω , but not cartesian-closed Inductive presentation f : Σ ω × Γ ω → Γ ω f : Σ → Γ b 0 ∈ Γ f ω : Σ ω → Γ ω fix b 0 ( f ) : Σ ω → Γ ω + closure under composition Σ ω Γ ω f Γ ω ≈ guarded recursion fix : A ◮ A → A b 0 topos of trees fix b 0 ( f ) 29 / 36

MSO ( N ) as an equational logic over M FOM (First-Order Mealy) t = Σ ω u | ϕ ∧ ψ | ¬ ϕ | ∃ x ∈ Σ ω ϕ ϕ, ψ ::= ◮ Typed variables stand for streams, terms for every f.s. causal functions. Proposition FOM and MSO ( N ) are interpretable in one another. ◮ Justifies focusing on FOM. 30 / 36

MSO ( N ) as an equational logic over M FOM (First-Order Mealy) t = Σ ω u | ϕ ∧ ψ | ¬ ϕ | ∃ x ∈ Σ ω ϕ ϕ, ψ ::= ◮ Typed variables stand for streams, terms for every f.s. causal functions. Proposition FOM and MSO ( N ) are interpretable in one another. ◮ Justifies focusing on FOM. Tarskian semantics (categorical logic) ◮ Regard M as a multi-sorted Lawvere theory. � Tarskian semantics ≈ indexed category, from global section functor Γ Γ : Σ ω �− → Hom M ( 1 ω , Σ ω ) Σ ω �− → ( P ( Γ (Σ ω )) , ⊆ ) 30 / 36

� � � � � � SMSO and the simple fibration Simple slice C // X = full subcategory of C / X with objects π X × Y − → X � the simple fibration s ( C ) → C The construction Sum ◮ Sum ( p ) -predicate: ( U , ϕ ( a , u )) Sum ( E ) E � U object of C , ϕ over A × U (in p ) p ≈ ∃ u : U ϕ ( a , u ) × Sum ( p ) s ( C ) C ◮ Freely adds existential quantifications (simple sums) C ◮ Reminiscent of typed realizability realizers in C 31 / 36

� � � � � � SMSO and the simple fibration Simple slice C // X = full subcategory of C / X with objects π X × Y − → X � the simple fibration s ( C ) → C The construction Sum ◮ Sum ( p ) -predicate: ( U , ϕ ( a , u )) Sum ( E ) E � U object of C , ϕ over A × U (in p ) p ≈ ∃ u : U ϕ ( a , u ) × Sum ( p ) s ( C ) C ◮ Freely adds existential quantifications (simple sums) C ◮ Reminiscent of typed realizability realizers in C Reconstructing SMSO Simulations of non-determinstic automata ≈ Sum applied to FOM 31 / 36

� � � � Linking LMSO with Dialectica Fibered Dialectica [Hyland (2001)] Dial ∼ = Sum ◦ Prod Prod ( p ) ∼ = Sum ( p op ) op [Hofstra (2011)] ◮ Dial ( p ) -predicate over A ≈ ( U , X , ϕ ( a , u , x )) think ∃ u ∀ x ϕ ( a , u , x ) ◮ interprets full intuitionistic MLL+FO LNL-adjunction Sum ( p ) Dial ( p ) Prod ( p ) ⊥ ⊤ 32 / 36

� � � � Linking LMSO with Dialectica Fibered Dialectica [Hyland (2001)] Dial ∼ = Sum ◦ Prod Prod ( p ) ∼ = Sum ( p op ) op [Hofstra (2011)] ◮ Dial ( p ) -predicate over A ≈ ( U , X , ϕ ( a , u , x )) think ∃ u ∀ x ϕ ( a , u , x ) ◮ interprets full intuitionistic MLL+FO and exponentials !( U , X , ϕ ( u , x )) = ( U , 1 , ∀ x ϕ ( u , x ) LNL-adjunction Sum ( p ) Dial ( p ) Prod ( p ) ⊥ ⊤ 32 / 36

� � � � Linking LMSO with Dialectica Fibered Dialectica [Hyland (2001)] Dial ∼ = Sum ◦ Prod Prod ( p ) ∼ = Sum ( p op ) op [Hofstra (2011)] ◮ Dial ( p ) -predicate over A ≈ ( U , X , ϕ ( a , u , x )) think ∃ u ∀ x ϕ ( a , u , x ) ◮ interprets full intuitionistic MLL+FO and exponentials !( U , X , ϕ ( u , x )) = ( U , 1 , ∀ x ϕ ( u , x ) LNL-adjunction Dial ◮ ( p ) Sum ( p ) Prod ( p ) ⊥ ⊤ Realized Dialectica-like construction Dial ◮ 32 / 36

� � � � Linking LMSO with Dialectica Fibered Dialectica [Hyland (2001)] Dial ∼ = Sum ◦ Prod Prod ( p ) ∼ = Sum ( p op ) op [Hofstra (2011)] ◮ Dial ( p ) -predicate over A ≈ ( U , X , ϕ ( a , u , x )) think ∃ u ∀ x ϕ ( a , u , x ) ◮ interprets full intuitionistic MLL+FO and exponentials !( U , X , ϕ ( u , x )) = ( U , 1 , ∀ x ϕ ( u , x ) LNL-adjunction Dial ◮ ( p ) Sum ( p ) Prod ( p ) ⊥ ⊤ Realized Dialectica-like construction Dial ◮ ◮ Only over a CCC extension of M !( U , X , ϕ ( u , x )) = ( U ◮ X , 1 , ∀ x ϕ ( f ( ◮ x ) , x ) 32 / 36

� � � � Linking LMSO with Dialectica Fibered Dialectica [Hyland (2001)] Dial ∼ = Sum ◦ Prod Prod ( p ) ∼ = Sum ( p op ) op [Hofstra (2011)] ◮ Dial ( p ) -predicate over A ≈ ( U , X , ϕ ( a , u , x )) think ∃ u ∀ x ϕ ( a , u , x ) ◮ interprets full intuitionistic MLL+FO and exponentials !( U , X , ϕ ( u , x )) = ( U , 1 , ∀ x ϕ ( u , x ) LNL-adjunction Dial ◮ ( p ) Sum ( p ) Prod ( p ) ⊥ ⊤ Realized Dialectica-like construction Dial ◮ ◮ Only over a CCC extension of M !( U , X , ϕ ( u , x )) = ( U ◮ X , 1 , ∀ x ϕ ( f ( ◮ x ) , x ) ◮ Relationship with Dial via a “feedback” monad exploits fix : A ◮ A → A 32 / 36

� � � � Linking LMSO with Dialectica Fibered Dialectica [Hyland (2001)] Dial ∼ = Sum ◦ Prod Prod ( p ) ∼ = Sum ( p op ) op [Hofstra (2011)] ◮ Dial ( p ) -predicate over A ≈ ( U , X , ϕ ( a , u , x )) think ∃ u ∀ x ϕ ( a , u , x ) ◮ interprets full intuitionistic MLL+FO and exponentials !( U , X , ϕ ( u , x )) = ( U , 1 , ∀ x ϕ ( u , x ) LNL-adjunction Dial ◮ ( p ) Sum ( p ) Prod ( p ) ⊥ ⊤ Realized Dialectica-like construction Dial ◮ ◮ Only over a CCC extension of M !( U , X , ϕ ( u , x )) = ( U ◮ X , 1 , ∀ x ϕ ( f ( ◮ x ) , x ) ◮ Relationship with Dial via a “feedback” monad exploits fix : A ◮ A → A ◮ Polarity restrictions ≈ model of LMSO (restricted exponentials) 32 / 36

Outline Monadic Second-Order logic Part I: Reverse Mathematics Part II: proof systems for Church’s synthesis Conclusion 33 / 36

Part I : the logical strength of MSO Summary Axiomatic strength of two classical MSO theories. ◮ In the context of Reverse Mathematics. ◮ Strong link between Σ 0 2 -induction and MSO ( N ) . ◮ Preliminary results on MSO ( Q ) . 34 / 36

Part I : the logical strength of MSO Summary Axiomatic strength of two classical MSO theories. ◮ In the context of Reverse Mathematics. ◮ Strong link between Σ 0 2 -induction and MSO ( N ) . ◮ Preliminary results on MSO ( Q ) . Related work ◮ Characterizations of the topological complexity of MSO-definable sets. ◮ Extension to the Reverse-mathematical analysis to intuitionistic logic. [Lichter and Smolka (2018)] ◮ Conservativity results for cyclic arithmetic. [Simpson (2017), Das (2019)] 34 / 36

Part II: Curry-Howard for MSO ( N ) Summary ◮ Realizability models based on simulations between automata ◮ Abstract reformulation link with Dialectica and typed realizability ◮ Complete extension of LMSO omitted from the talk [P., Riba (2019)] 35 / 36

Part II: Curry-Howard for MSO ( N ) Summary ◮ Realizability models based on simulations between automata ◮ Abstract reformulation link with Dialectica and typed realizability ◮ Complete extension of LMSO omitted from the talk [P., Riba (2019)] Related work ◮ Fibrations of tree automata [Riba (2015)] ◮ Good-for-games automata [Henziger, Piterman (2006), Kuperberg Skrzypczak (2015)] 35 / 36

Final word Some further questions ◮ Realizability for continuous functions Σ ω → Γ ω ? ◮ Extensions of Dial ◮ for fibrations over the topos of trees? Fam ( Fam ( p op ) op ) instead of Dial ( p ) ◮ Undecidability of the equational logic of higher-order extensions of FOM? 36 / 36

Final word Some further questions ◮ Realizability for continuous functions Σ ω → Γ ω ? ◮ Extensions of Dial ◮ for fibrations over the topos of trees? Fam ( Fam ( p op ) op ) instead of Dial ( p ) ◮ Undecidability of the equational logic of higher-order extensions of FOM? Thanks for your attention! Questions? 36 / 36

Induction and comprehension RCA 0 is defined by restricting induction and comprehension Comprehension axiom For every formula φ ( n ) (with X / ∈ FV ( φ ) ∃ X ∀ n ∈ N ( φ ( n ) ⇔ n ∈ X ) ◮ RCA 0 : restricted to ∆ 0 1 formulas recursive comprehension Induction axiom To prove that ∀ n ∈ N φ ( n ) it suffices to show ◮ φ ( 0 ) holds ◮ for every n ∈ N , φ ( n ) implies φ ( n + 1 ) ◮ RCA 0 : restricted to Σ 0 1 formulas. ∃ n δ ( n ) with δ ∈ ∆ 0 1 ◮ Equivalent to minimization principles and comprehension for finite sets. 1 / 6

Additive Ramsey over ω Additive Ramsey Let M be a monoid. For every map f : [ N ] 2 → M such that f ( i , j ) f ( j , k ) = f ( i , k ) , there exists an infinite set X ⊆ N and c ∈ M such that f ( i , j ) = c for i , j ∈ X . Theorem Over RCA 0 , additive Ramsey is equivalent to Σ 0 2 - IND. 2 / 6

Combinatorics for coloring over Q Let D be a dense linear order ( ≃ Q ). A function f : D → X is called homogeneous if f − 1 ( x ) is either dense or empty for every x ∈ X . The shuffle principle � For any coloring c : Q → � 0 , n � , there is ] x , y [ such that c ] x , y [ is homogeneous . � ◮ the key additional principle behind the usual inductive argument in [Carton, Colcombet, Puppis (2015)] Shelah’s additive Ramseyan theorem Let M be a monoid. For every map f : [ Q ] 2 → M such that f ( q , r ) f ( r , s ) = f ( q , s ) , there exists an interval I ⊆ Q and a finite partition into finitely many dense sets D i of I such that f is constant over each [ D i ] 2 . ◮ the key additional principle behind the usual inductive argument in [Shelah (1975)] 3 / 6

The Büchi-Landweber theorem Consider a formula ϕ ( u , x ) . ( u ∈ U ω , x ∈ X ω ) � Infinite 2-player game G ϕ between P and O . P wins O x 0 x 1 x n . . . . . . ⇐ ⇒ P u 0 u 1 u n ϕ ( u , x ) holds 4 / 6

The Büchi-Landweber theorem Consider a formula ϕ ( u , x ) . ( u ∈ U ω , x ∈ X ω ) � Infinite 2-player game G ϕ between P and O . P wins O x 0 x 1 x n . . . . . . ⇐ ⇒ P u 0 u 1 u n ϕ ( u , x ) holds X + → U U ∗ → X P -strategies ≃ O -strategies ≃ causal functions eager causal functions 4 / 6

The Büchi-Landweber theorem Consider a formula ϕ ( u , x ) . ( u ∈ U ω , x ∈ X ω ) � Infinite 2-player game G ϕ between P and O . P wins O x 0 x 1 x n . . . . . . ⇐ ⇒ P u 0 u 1 u n ϕ ( u , x ) holds X + → U U ∗ → X P -strategies ≃ O -strategies ≃ causal functions eager causal functions Theorem [Büchi-Landweber (1969)] Suppose ϕ is MSO-definable. The game G ϕ is determined: ◮ Either there exists a finite-state P -strategy s P ( x ) s.t. ∀ x ∈ X ω ϕ ( s P ( x ) , x ) ∀ u ∈ U ω ¬ ϕ ( u , s O ( u )) ◮ Or there exists a finite-state O -strategy s O ( u ) s.t. 4 / 6

The realizability notion for SMSO Uniform non-deterministic automata Tuples A = ( Q , q 0 , U , δ A , Ω A ) : Σ where ◮ U a set of moves ≃ amount of non-determinism ◮ transition function δ A : Σ × Q × U → Q A : Σ ω × U ω → Q ω induces δ ∗ ◮ Ω A ⊆ Q ω reasonable acceptance condition (parity, Muller, . . . ) ◮ Same definable languages L ( A ) = { w | ∃ u δ ∗ A ( w , u ) } U ≃ Q 5 / 6

The realizability notion for SMSO Uniform non-deterministic automata Tuples A = ( Q , q 0 , U , δ A , Ω A ) : Σ where ◮ U a set of moves ≃ amount of non-determinism ◮ transition function δ A : Σ × Q × U → Q A : Σ ω × U ω → Q ω induces δ ∗ ◮ Ω A ⊆ Q ω reasonable acceptance condition (parity, Muller, . . . ) ◮ Same definable languages L ( A ) = { w | ∃ u δ ∗ A ( w , u ) } U ≃ Q Simulations A � f : B Finite-state causal function f : Σ ω × U ω → V ω such that δ ∗ δ ∗ ∀ w ∈ Σ ω ∀ u ∈ U ω A ( w , u ) ∈ Ω A ⇒ A ( w , f ( w , u )) ∈ Ω B 5 / 6