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Approximating partial by total: fixpoint characterizations of back-and-forth equivalences Samson Abramsky Department of Computer Science, University of Oxford Samson Abramsky (Department of Computer Science, University of Oxford) Approximating

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Approximating partial by total: fixpoint characterizations of back-and-forth equivalences

Samson Abramsky

Department of Computer Science, University of Oxford

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 1 / 21

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Relating two facets of Dana’s work

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 2 / 21

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Relating two facets of Dana’s work

As well as founding Domain theory, Dana is, of course, a pre-eminent figure in logic, with seminal contributions in model theory, set theory, modal logic, . . .

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 2 / 21

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SLIDE 4

Relating two facets of Dana’s work

As well as founding Domain theory, Dana is, of course, a pre-eminent figure in logic, with seminal contributions in model theory, set theory, modal logic, . . . This talk will, in a modest way, relate these different facets.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 2 / 21

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SLIDE 5

Relating two facets of Dana’s work

As well as founding Domain theory, Dana is, of course, a pre-eminent figure in logic, with seminal contributions in model theory, set theory, modal logic, . . . This talk will, in a modest way, relate these different facets. (Actually, I believe (thanks to Luca Reggio) that this can be taken much further; however, this will have to be left to future work!)

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 2 / 21

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SLIDE 6

Relating two facets of Dana’s work

As well as founding Domain theory, Dana is, of course, a pre-eminent figure in logic, with seminal contributions in model theory, set theory, modal logic, . . . This talk will, in a modest way, relate these different facets. (Actually, I believe (thanks to Luca Reggio) that this can be taken much further; however, this will have to be left to future work!) Based on: The pebbling comonad in finite model theory, SA, Anuj Dawar and Pengming Wang, LiCS 2017 Relating Structure to Power: comonadic semantics for computational resources, SA and Nihil Shah, to appear in CSL 2018.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 2 / 21

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Model theory and deception

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 3 / 21

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Model theory and deception

A famous notion in logic is that of Scott sentences: Lω1,ω sentences which characterize countable structures up to isomorphism.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 3 / 21

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Model theory and deception

A famous notion in logic is that of Scott sentences: Lω1,ω sentences which characterize countable structures up to isomorphism. In general though, model theory involves deception:

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 3 / 21

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Model theory and deception

A famous notion in logic is that of Scott sentences: Lω1,ω sentences which characterize countable structures up to isomorphism. In general though, model theory involves deception: In model theory, we see a structure, not “as it really is” (up to isomorphism) but only up to definable properties.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 3 / 21

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SLIDE 11

Model theory and deception

A famous notion in logic is that of Scott sentences: Lω1,ω sentences which characterize countable structures up to isomorphism. In general though, model theory involves deception: In model theory, we see a structure, not “as it really is” (up to isomorphism) but only up to definable properties. The crucial notion is equivalence of structures up to the equivalence ≡L induced by the logic L: A ≡L B

∆

⇐ ⇒ ∀ϕ ∈ L. A | = ϕ ⇐ ⇒ B | = ϕ.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 3 / 21

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Model theory and deception

A famous notion in logic is that of Scott sentences: Lω1,ω sentences which characterize countable structures up to isomorphism. In general though, model theory involves deception: In model theory, we see a structure, not “as it really is” (up to isomorphism) but only up to definable properties. The crucial notion is equivalence of structures up to the equivalence ≡L induced by the logic L: A ≡L B

∆

⇐ ⇒ ∀ϕ ∈ L. A | = ϕ ⇐ ⇒ B | = ϕ. It is always true that if a class of structures K is definable in L, then K must be saturated under ≡L.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 3 / 21

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Model theory and deception

A famous notion in logic is that of Scott sentences: Lω1,ω sentences which characterize countable structures up to isomorphism. In general though, model theory involves deception: In model theory, we see a structure, not “as it really is” (up to isomorphism) but only up to definable properties. The crucial notion is equivalence of structures up to the equivalence ≡L induced by the logic L: A ≡L B

∆

⇐ ⇒ ∀ϕ ∈ L. A | = ϕ ⇐ ⇒ B | = ϕ. It is always true that if a class of structures K is definable in L, then K must be saturated under ≡L. In most cases of interest in FMT, the converse is true too.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 3 / 21

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Model theory and deception

A famous notion in logic is that of Scott sentences: Lω1,ω sentences which characterize countable structures up to isomorphism. In general though, model theory involves deception: In model theory, we see a structure, not “as it really is” (up to isomorphism) but only up to definable properties. The crucial notion is equivalence of structures up to the equivalence ≡L induced by the logic L: A ≡L B

∆

⇐ ⇒ ∀ϕ ∈ L. A | = ϕ ⇐ ⇒ B | = ϕ. It is always true that if a class of structures K is definable in L, then K must be saturated under ≡L. In most cases of interest in FMT, the converse is true too. In descriptive complexity, we seek to characterize a complexity class C (for decision problems) as those classes of structures K (e.g. graphs) definable in L.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 3 / 21

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Syntax-independent characterizations of logical equivalence

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 4 / 21

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Syntax-independent characterizations of logical equivalence

A classic theme in Model theory: e.g. the Keisler-Shelah theorem.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 4 / 21

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Syntax-independent characterizations of logical equivalence

A classic theme in Model theory: e.g. the Keisler-Shelah theorem. Especially important in finite model theory, where model comparison games such as Ehrenfeucht-Fra¨ ıss´ e games, pebble games and bisimulation games play a central role.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 4 / 21

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Syntax-independent characterizations of logical equivalence

A classic theme in Model theory: e.g. the Keisler-Shelah theorem. Especially important in finite model theory, where model comparison games such as Ehrenfeucht-Fra¨ ıss´ e games, pebble games and bisimulation games play a central role. The EF-game between A and B. In the i’th round, Spoiler moves by choosing an element in A or B; Duplicator responds by choosing an element in the other

  • structure. Duplicator wins after k rounds if the relation {(ai, bi) | 1 ≤ i ≤ k} is a

partial isomorphism.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 4 / 21

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Syntax-independent characterizations of logical equivalence

A classic theme in Model theory: e.g. the Keisler-Shelah theorem. Especially important in finite model theory, where model comparison games such as Ehrenfeucht-Fra¨ ıss´ e games, pebble games and bisimulation games play a central role. The EF-game between A and B. In the i’th round, Spoiler moves by choosing an element in A or B; Duplicator responds by choosing an element in the other

  • structure. Duplicator wins after k rounds if the relation {(ai, bi) | 1 ≤ i ≤ k} is a

partial isomorphism. In the existential EF-game, Spoiler only plays in A, and Duplicator responds in B.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 4 / 21

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SLIDE 20

Syntax-independent characterizations of logical equivalence

A classic theme in Model theory: e.g. the Keisler-Shelah theorem. Especially important in finite model theory, where model comparison games such as Ehrenfeucht-Fra¨ ıss´ e games, pebble games and bisimulation games play a central role. The EF-game between A and B. In the i’th round, Spoiler moves by choosing an element in A or B; Duplicator responds by choosing an element in the other

  • structure. Duplicator wins after k rounds if the relation {(ai, bi) | 1 ≤ i ≤ k} is a

partial isomorphism. In the existential EF-game, Spoiler only plays in A, and Duplicator responds in B. The Ehrenfeucht-Fra¨ ıss´ e theorem says that a winning strategy for Duplicator in the k-round EF game characterizes the equivalence ≡Lk, where Lk is the fragment of first-order logic of formulas with quantifier rank ≤ k.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 4 / 21

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SLIDE 21

Syntax-independent characterizations of logical equivalence

A classic theme in Model theory: e.g. the Keisler-Shelah theorem. Especially important in finite model theory, where model comparison games such as Ehrenfeucht-Fra¨ ıss´ e games, pebble games and bisimulation games play a central role. The EF-game between A and B. In the i’th round, Spoiler moves by choosing an element in A or B; Duplicator responds by choosing an element in the other

  • structure. Duplicator wins after k rounds if the relation {(ai, bi) | 1 ≤ i ≤ k} is a

partial isomorphism. In the existential EF-game, Spoiler only plays in A, and Duplicator responds in B. The Ehrenfeucht-Fra¨ ıss´ e theorem says that a winning strategy for Duplicator in the k-round EF game characterizes the equivalence ≡Lk, where Lk is the fragment of first-order logic of formulas with quantifier rank ≤ k. Similarly, there are k-pebble games, and bismulation games played to depth k.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 4 / 21

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Pebble Games

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 5 / 21

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Pebble Games

Similar but subtly different to EF-games

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 5 / 21

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Pebble Games

Similar but subtly different to EF-games Spoiler moves by placing one from a fixed set of pebbles on an element of A or B; Duplicator responds by placing their matching pebble on an element of the other structure.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 5 / 21

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Pebble Games

Similar but subtly different to EF-games Spoiler moves by placing one from a fixed set of pebbles on an element of A or B; Duplicator responds by placing their matching pebble on an element of the other structure. Duplicator wins if after each round, the relation defined by the current positions of the pebbles is a partial isomorphism

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 5 / 21

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Pebble Games

Similar but subtly different to EF-games Spoiler moves by placing one from a fixed set of pebbles on an element of A or B; Duplicator responds by placing their matching pebble on an element of the other structure. Duplicator wins if after each round, the relation defined by the current positions of the pebbles is a partial isomorphism Thus there is a “sliding window” on the structures, of fixed size. It is this size which bounds the resource, not the length of the play.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 5 / 21

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SLIDE 27

Pebble Games

Similar but subtly different to EF-games Spoiler moves by placing one from a fixed set of pebbles on an element of A or B; Duplicator responds by placing their matching pebble on an element of the other structure. Duplicator wins if after each round, the relation defined by the current positions of the pebbles is a partial isomorphism Thus there is a “sliding window” on the structures, of fixed size. It is this size which bounds the resource, not the length of the play. Whereas the k-round EF game corresponds to bounding the quantifier rank, k-pebble games correspond to bounding the number of variables which can be used in a formula.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 5 / 21

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Pebble Games

Similar but subtly different to EF-games Spoiler moves by placing one from a fixed set of pebbles on an element of A or B; Duplicator responds by placing their matching pebble on an element of the other structure. Duplicator wins if after each round, the relation defined by the current positions of the pebbles is a partial isomorphism Thus there is a “sliding window” on the structures, of fixed size. It is this size which bounds the resource, not the length of the play. Whereas the k-round EF game corresponds to bounding the quantifier rank, k-pebble games correspond to bounding the number of variables which can be used in a formula. Just as for EF-games, there is an existential-positive version, in which Spoiler only plays in A, and Duplicator responds in B.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 5 / 21

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A new perspective

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 6 / 21

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A new perspective

We shall study these games, not as external artefacts, but as semantic constructions in their own right.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 6 / 21

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A new perspective

We shall study these games, not as external artefacts, but as semantic constructions in their own right. For each type of game G, and value of the resource parameter k, we shall define a corresponding comonad Ck on R(σ).

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 6 / 21

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SLIDE 32

A new perspective

We shall study these games, not as external artefacts, but as semantic constructions in their own right. For each type of game G, and value of the resource parameter k, we shall define a corresponding comonad Ck on R(σ). The idea is that Duplicator strategies for the existential version of G-games from A to B will be recovered as coKleisli morphisms CkA → B.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 6 / 21

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SLIDE 33

A new perspective

We shall study these games, not as external artefacts, but as semantic constructions in their own right. For each type of game G, and value of the resource parameter k, we shall define a corresponding comonad Ck on R(σ). The idea is that Duplicator strategies for the existential version of G-games from A to B will be recovered as coKleisli morphisms CkA → B. Thus the notion of local approximation built into the game is internalised into the category of σ-structures and homomorphisms.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 6 / 21

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SLIDE 34

A new perspective

We shall study these games, not as external artefacts, but as semantic constructions in their own right. For each type of game G, and value of the resource parameter k, we shall define a corresponding comonad Ck on R(σ). The idea is that Duplicator strategies for the existential version of G-games from A to B will be recovered as coKleisli morphisms CkA → B. Thus the notion of local approximation built into the game is internalised into the category of σ-structures and homomorphisms. This leads to comonadic and coalgebraic characterisations of a number of central concepts in Finite Model Theory and combinatorics.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 6 / 21

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The setting: homomorphisms of relational structures

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 7 / 21

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The setting: homomorphisms of relational structures

A relational vocabulary σ is a family of relation symbols R, each of some arity n > 0.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 7 / 21

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The setting: homomorphisms of relational structures

A relational vocabulary σ is a family of relation symbols R, each of some arity n > 0. A relational structure for σ is A = (A, {RA | R ∈ σ})), where RA ⊆ An.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 7 / 21

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The setting: homomorphisms of relational structures

A relational vocabulary σ is a family of relation symbols R, each of some arity n > 0. A relational structure for σ is A = (A, {RA | R ∈ σ})), where RA ⊆ An. A homomorphism of σ-structures f : A → B is a function f : A → B such that, for each relation R ∈ σ of arity n and (a1, . . . , an) ∈ An: (a1, . . . , an) ∈ RA ⇒ (f (a1), . . . , f (an))) ∈ RB.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 7 / 21

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SLIDE 39

The setting: homomorphisms of relational structures

A relational vocabulary σ is a family of relation symbols R, each of some arity n > 0. A relational structure for σ is A = (A, {RA | R ∈ σ})), where RA ⊆ An. A homomorphism of σ-structures f : A → B is a function f : A → B such that, for each relation R ∈ σ of arity n and (a1, . . . , an) ∈ An: (a1, . . . , an) ∈ RA ⇒ (f (a1), . . . , f (an))) ∈ RB. There notions are pervasive in logic (model theory), computer science (databases, constraint satisfaction, finite model theory) combinatorics (graphs and graph homomorphisms).

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 7 / 21

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SLIDE 40

The setting: homomorphisms of relational structures

A relational vocabulary σ is a family of relation symbols R, each of some arity n > 0. A relational structure for σ is A = (A, {RA | R ∈ σ})), where RA ⊆ An. A homomorphism of σ-structures f : A → B is a function f : A → B such that, for each relation R ∈ σ of arity n and (a1, . . . , an) ∈ An: (a1, . . . , an) ∈ RA ⇒ (f (a1), . . . , f (an))) ∈ RB. There notions are pervasive in logic (model theory), computer science (databases, constraint satisfaction, finite model theory) combinatorics (graphs and graph homomorphisms). Our setting will be R(σ), the category of relational structures and homomorphisms.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 7 / 21

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SLIDE 41

The EF comonad

Given a structure A, the universe of EkA is A≤k, the non-empty sequences of length ≤ k.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 8 / 21

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SLIDE 42

The EF comonad

Given a structure A, the universe of EkA is A≤k, the non-empty sequences of length ≤ k. The counit map εA : EkA → A sends a sequence [a1, . . . , an] to an.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 8 / 21

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SLIDE 43

The EF comonad

Given a structure A, the universe of EkA is A≤k, the non-empty sequences of length ≤ k. The counit map εA : EkA → A sends a sequence [a1, . . . , an] to an. How do we lift the relations on A to EkA?

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 8 / 21

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SLIDE 44

The EF comonad

Given a structure A, the universe of EkA is A≤k, the non-empty sequences of length ≤ k. The counit map εA : EkA → A sends a sequence [a1, . . . , an] to an. How do we lift the relations on A to EkA? Given e.g. a binary relation R, we define REkA to the set of pairs (s, t) such that

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 8 / 21

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SLIDE 45

The EF comonad

Given a structure A, the universe of EkA is A≤k, the non-empty sequences of length ≤ k. The counit map εA : EkA → A sends a sequence [a1, . . . , an] to an. How do we lift the relations on A to EkA? Given e.g. a binary relation R, we define REkA to the set of pairs (s, t) such that s ⊑ t or t ⊑ s (in prefix order)

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 8 / 21

slide-46
SLIDE 46

The EF comonad

Given a structure A, the universe of EkA is A≤k, the non-empty sequences of length ≤ k. The counit map εA : EkA → A sends a sequence [a1, . . . , an] to an. How do we lift the relations on A to EkA? Given e.g. a binary relation R, we define REkA to the set of pairs (s, t) such that s ⊑ t or t ⊑ s (in prefix order) RA(εA(s), εA(t)).

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 8 / 21

slide-47
SLIDE 47

The EF comonad

Given a structure A, the universe of EkA is A≤k, the non-empty sequences of length ≤ k. The counit map εA : EkA → A sends a sequence [a1, . . . , an] to an. How do we lift the relations on A to EkA? Given e.g. a binary relation R, we define REkA to the set of pairs (s, t) such that s ⊑ t or t ⊑ s (in prefix order) RA(εA(s), εA(t)). Given a homomorphism f : EkA → B, we define the coextension f ∗ : A≤k → B≤k by f ∗[a1, . . . , aj] = [b1, . . . , bj], where bi = f [a1, . . . , ai], 1 ≤ i ≤ j.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 8 / 21

slide-48
SLIDE 48

The EF comonad

Given a structure A, the universe of EkA is A≤k, the non-empty sequences of length ≤ k. The counit map εA : EkA → A sends a sequence [a1, . . . , an] to an. How do we lift the relations on A to EkA? Given e.g. a binary relation R, we define REkA to the set of pairs (s, t) such that s ⊑ t or t ⊑ s (in prefix order) RA(εA(s), εA(t)). Given a homomorphism f : EkA → B, we define the coextension f ∗ : A≤k → B≤k by f ∗[a1, . . . , aj] = [b1, . . . , bj], where bi = f [a1, . . . , ai], 1 ≤ i ≤ j. This is easily verified to yield a comonad on R(σ).

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 8 / 21

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SLIDE 49

CoKleisli maps are strategies

Intuitively, an element of A≤k represents a play in A of length ≤ k.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 9 / 21

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SLIDE 50

CoKleisli maps are strategies

Intuitively, an element of A≤k represents a play in A of length ≤ k. A coKleisli morphism EkA → B represents a Duplicator strategy for the existential Ehrenfeucht-Fra¨ ıss´ e game with k rounds:

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 9 / 21

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SLIDE 51

CoKleisli maps are strategies

Intuitively, an element of A≤k represents a play in A of length ≤ k. A coKleisli morphism EkA → B represents a Duplicator strategy for the existential Ehrenfeucht-Fra¨ ıss´ e game with k rounds: Spoiler plays only in A, and bi = f [a1, . . . , ai] represents Duplicator’s response in B to the i’th move by Spoiler.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 9 / 21

slide-52
SLIDE 52

CoKleisli maps are strategies

Intuitively, an element of A≤k represents a play in A of length ≤ k. A coKleisli morphism EkA → B represents a Duplicator strategy for the existential Ehrenfeucht-Fra¨ ıss´ e game with k rounds: Spoiler plays only in A, and bi = f [a1, . . . , ai] represents Duplicator’s response in B to the i’th move by Spoiler. The winning condition for Duplicator in this game is that, after k rounds have been played, the induced relation {(ai, bi) | 1 ≤ i ≤ k} is a partial homomorphism from A to B.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 9 / 21

slide-53
SLIDE 53

CoKleisli maps are strategies

Intuitively, an element of A≤k represents a play in A of length ≤ k. A coKleisli morphism EkA → B represents a Duplicator strategy for the existential Ehrenfeucht-Fra¨ ıss´ e game with k rounds: Spoiler plays only in A, and bi = f [a1, . . . , ai] represents Duplicator’s response in B to the i’th move by Spoiler. The winning condition for Duplicator in this game is that, after k rounds have been played, the induced relation {(ai, bi) | 1 ≤ i ≤ k} is a partial homomorphism from A to B.

Theorem

The following are equivalent:

1

There is a homomorphism EkA → B.

2

Duplicator has a winning strategy for the existential Ehrenfeucht-Fra¨ ıss´ e game with k rounds, played from A to B.

3

For every existential positive sentence ϕ with quantifier rank ≤ k, A | = ϕ ⇒ B | = ϕ.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 9 / 21

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SLIDE 54

The pebbling comonad

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 10 / 21

slide-55
SLIDE 55

The pebbling comonad

Given a structure A, the universe of PkA is (k × A)+, the set of finite non-empty sequences of moves (p, a). Note this will be infinite even if A is finite. We showed that this is essential!

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 10 / 21

slide-56
SLIDE 56

The pebbling comonad

Given a structure A, the universe of PkA is (k × A)+, the set of finite non-empty sequences of moves (p, a). Note this will be infinite even if A is finite. We showed that this is essential! The counit map εA : EkA → A sends a sequence [(p1, a1), . . . , (pn, an)] to an.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 10 / 21

slide-57
SLIDE 57

The pebbling comonad

Given a structure A, the universe of PkA is (k × A)+, the set of finite non-empty sequences of moves (p, a). Note this will be infinite even if A is finite. We showed that this is essential! The counit map εA : EkA → A sends a sequence [(p1, a1), . . . , (pn, an)] to an. How do we lift the relations on A to EkA?

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 10 / 21

slide-58
SLIDE 58

The pebbling comonad

Given a structure A, the universe of PkA is (k × A)+, the set of finite non-empty sequences of moves (p, a). Note this will be infinite even if A is finite. We showed that this is essential! The counit map εA : EkA → A sends a sequence [(p1, a1), . . . , (pn, an)] to an. How do we lift the relations on A to EkA? Given e.g. a binary relation R, we define RPkA to the set of pairs (s, t) such that

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 10 / 21

slide-59
SLIDE 59

The pebbling comonad

Given a structure A, the universe of PkA is (k × A)+, the set of finite non-empty sequences of moves (p, a). Note this will be infinite even if A is finite. We showed that this is essential! The counit map εA : EkA → A sends a sequence [(p1, a1), . . . , (pn, an)] to an. How do we lift the relations on A to EkA? Given e.g. a binary relation R, we define RPkA to the set of pairs (s, t) such that s ⊑ t or t ⊑ s

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 10 / 21

slide-60
SLIDE 60

The pebbling comonad

Given a structure A, the universe of PkA is (k × A)+, the set of finite non-empty sequences of moves (p, a). Note this will be infinite even if A is finite. We showed that this is essential! The counit map εA : EkA → A sends a sequence [(p1, a1), . . . , (pn, an)] to an. How do we lift the relations on A to EkA? Given e.g. a binary relation R, we define RPkA to the set of pairs (s, t) such that s ⊑ t or t ⊑ s If s ⊑ t, then the pebble index of the last move in s does not appear in the suffix of s in t; and symmetrically if t ⊑ s.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 10 / 21

slide-61
SLIDE 61

The pebbling comonad

Given a structure A, the universe of PkA is (k × A)+, the set of finite non-empty sequences of moves (p, a). Note this will be infinite even if A is finite. We showed that this is essential! The counit map εA : EkA → A sends a sequence [(p1, a1), . . . , (pn, an)] to an. How do we lift the relations on A to EkA? Given e.g. a binary relation R, we define RPkA to the set of pairs (s, t) such that s ⊑ t or t ⊑ s If s ⊑ t, then the pebble index of the last move in s does not appear in the suffix of s in t; and symmetrically if t ⊑ s. RA(εA(s), εA(t)).

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 10 / 21

slide-62
SLIDE 62

The pebbling comonad

Given a structure A, the universe of PkA is (k × A)+, the set of finite non-empty sequences of moves (p, a). Note this will be infinite even if A is finite. We showed that this is essential! The counit map εA : EkA → A sends a sequence [(p1, a1), . . . , (pn, an)] to an. How do we lift the relations on A to EkA? Given e.g. a binary relation R, we define RPkA to the set of pairs (s, t) such that s ⊑ t or t ⊑ s If s ⊑ t, then the pebble index of the last move in s does not appear in the suffix of s in t; and symmetrically if t ⊑ s. RA(εA(s), εA(t)). Given a homomorphism f : PkA → B, we define the coextension f ∗ : PkA → PkB by f ∗[(p1, a1), . . . , (pj, aj)] = [(p1, b1), . . . , (pj, bj)], where bi = f [(p1, a1), . . . , (pi, ai)], 1 ≤ i ≤ j.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 10 / 21

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SLIDE 63

The modal comonad

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 11 / 21

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SLIDE 64

The modal comonad

The flexibility of the comonadic approach is illustrated by showing that it also covers the well-known construction of unfolding a Kripke structure into a tree (“unravelling”).

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 11 / 21

slide-65
SLIDE 65

The modal comonad

The flexibility of the comonadic approach is illustrated by showing that it also covers the well-known construction of unfolding a Kripke structure into a tree (“unravelling”). For the modal case, we assume that the relational vocabulary σ contains only symbols of arity at most 2.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 11 / 21

slide-66
SLIDE 66

The modal comonad

The flexibility of the comonadic approach is illustrated by showing that it also covers the well-known construction of unfolding a Kripke structure into a tree (“unravelling”). For the modal case, we assume that the relational vocabulary σ contains only symbols of arity at most 2. We can thus regard a σ-structure as a Kripke structure for a multi-modal logic. If there are no unary symbols, such structures are exactly the labelled transition systems.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 11 / 21

slide-67
SLIDE 67

The modal comonad

The flexibility of the comonadic approach is illustrated by showing that it also covers the well-known construction of unfolding a Kripke structure into a tree (“unravelling”). For the modal case, we assume that the relational vocabulary σ contains only symbols of arity at most 2. We can thus regard a σ-structure as a Kripke structure for a multi-modal logic. If there are no unary symbols, such structures are exactly the labelled transition systems. Modal logic localizes its notion of satisfaction in a structure to a world. We reflect this by using the category of pointed relational structures (A, a).

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 11 / 21

slide-68
SLIDE 68

The modal comonad

The flexibility of the comonadic approach is illustrated by showing that it also covers the well-known construction of unfolding a Kripke structure into a tree (“unravelling”). For the modal case, we assume that the relational vocabulary σ contains only symbols of arity at most 2. We can thus regard a σ-structure as a Kripke structure for a multi-modal logic. If there are no unary symbols, such structures are exactly the labelled transition systems. Modal logic localizes its notion of satisfaction in a structure to a world. We reflect this by using the category of pointed relational structures (A, a). For k > 0 we define a comonad Mk, where Mk(A, a) corresponds to unravelling the structure A, starting from a, to depth k.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 11 / 21

slide-69
SLIDE 69

The modal comonad

The flexibility of the comonadic approach is illustrated by showing that it also covers the well-known construction of unfolding a Kripke structure into a tree (“unravelling”). For the modal case, we assume that the relational vocabulary σ contains only symbols of arity at most 2. We can thus regard a σ-structure as a Kripke structure for a multi-modal logic. If there are no unary symbols, such structures are exactly the labelled transition systems. Modal logic localizes its notion of satisfaction in a structure to a world. We reflect this by using the category of pointed relational structures (A, a). For k > 0 we define a comonad Mk, where Mk(A, a) corresponds to unravelling the structure A, starting from a, to depth k. The universe of Mk(A, a) comprises [a], which is the distinguished element, together with all sequences of the form [a0, α1, a1, . . . , αj, aj], where a = a0, 1 ≤ j ≤ k, and RA

αi(ai, ai+1), 0 ≤ i < j.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 11 / 21

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SLIDE 70

Simulation

The resource index of Mk corresponds to the level of approximation in simulation k and bisimulation ∼k.

Theorem

Let A, B be Kripke structures, with a ∈ A and b ∈ B, and k > 0. The following are equivalent:

1

There is a homomorphism f : Mk(A, a) → (B, b).

2

a k b.

3

There is a winning strategy for Duplicator in the k-round simulation game from (A, a) to (B, b).

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 12 / 21

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SLIDE 71

Logical equivalences

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 13 / 21

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SLIDE 72

Logical equivalences

For each of our three types of game, there are corresponding fragments Lk of first-order logic:

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 13 / 21

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SLIDE 73

Logical equivalences

For each of our three types of game, there are corresponding fragments Lk of first-order logic: For Ehrenfeucht-Fra¨ ıss´ e games, Lk is the fragment of quantifier-rank ≤ k.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 13 / 21

slide-74
SLIDE 74

Logical equivalences

For each of our three types of game, there are corresponding fragments Lk of first-order logic: For Ehrenfeucht-Fra¨ ıss´ e games, Lk is the fragment of quantifier-rank ≤ k. For pebble games, Lk is the k-variable fragment.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 13 / 21

slide-75
SLIDE 75

Logical equivalences

For each of our three types of game, there are corresponding fragments Lk of first-order logic: For Ehrenfeucht-Fra¨ ıss´ e games, Lk is the fragment of quantifier-rank ≤ k. For pebble games, Lk is the k-variable fragment. For bismulation games over relational vocabularies with symbols of arity at most 2, Lk is the modal fragment with modal depth ≤ k.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 13 / 21

slide-76
SLIDE 76

Logical equivalences

For each of our three types of game, there are corresponding fragments Lk of first-order logic: For Ehrenfeucht-Fra¨ ıss´ e games, Lk is the fragment of quantifier-rank ≤ k. For pebble games, Lk is the k-variable fragment. For bismulation games over relational vocabularies with symbols of arity at most 2, Lk is the modal fragment with modal depth ≤ k. In each case, we write ∃Lk for the existential positive fragment of Lk Lk(#) for the extension of Lk with counting quantifiers ∃≤n, ∃≥n

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 13 / 21

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SLIDE 77

Characterization

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 14 / 21

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SLIDE 78

Characterization

We can generically define two equivalences based on our indexed comonads Ek: A ⇄E

k B iff there are coKleisli morphisms EkA → B and EkB → A. Note

that there need be no relationship between these morphisms. A ∼ =E

k B iff A and B are isomorphic in the coKleisli category Kl(Ek).

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 14 / 21

slide-79
SLIDE 79

Characterization

We can generically define two equivalences based on our indexed comonads Ek: A ⇄E

k B iff there are coKleisli morphisms EkA → B and EkB → A. Note

that there need be no relationship between these morphisms. A ∼ =E

k B iff A and B are isomorphic in the coKleisli category Kl(Ek).

Theorem

For structures A and B: A ≡∃Lk B ⇐ ⇒ A ⇄k B. A ≡Lk(#) B ⇐ ⇒ A ∼ =Kl(Ck) B.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 14 / 21

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SLIDE 80

From Forth to Back and Forth

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 15 / 21

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SLIDE 81

From Forth to Back and Forth

To complete the picture, we need to show how to define a back-and-forth equivalence ↔k which characterizes ≡Lk purely in terms of coKleisli morphisms.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 15 / 21

slide-82
SLIDE 82

From Forth to Back and Forth

To complete the picture, we need to show how to define a back-and-forth equivalence ↔k which characterizes ≡Lk purely in terms of coKleisli morphisms. Our solution to this will have the following features:

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 15 / 21

slide-83
SLIDE 83

From Forth to Back and Forth

To complete the picture, we need to show how to define a back-and-forth equivalence ↔k which characterizes ≡Lk purely in terms of coKleisli morphisms. Our solution to this will have the following features: While not completely generic, it will be general enough to apply to all our game comonads – so we subsume EF equivalence, bisimulation equivalence and pebble game equivalence as instances of a single construction.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 15 / 21

slide-84
SLIDE 84

From Forth to Back and Forth

To complete the picture, we need to show how to define a back-and-forth equivalence ↔k which characterizes ≡Lk purely in terms of coKleisli morphisms. Our solution to this will have the following features: While not completely generic, it will be general enough to apply to all our game comonads – so we subsume EF equivalence, bisimulation equivalence and pebble game equivalence as instances of a single construction. It uses approximations and fixpoints.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 15 / 21

slide-85
SLIDE 85

From Forth to Back and Forth

To complete the picture, we need to show how to define a back-and-forth equivalence ↔k which characterizes ≡Lk purely in terms of coKleisli morphisms. Our solution to this will have the following features: While not completely generic, it will be general enough to apply to all our game comonads – so we subsume EF equivalence, bisimulation equivalence and pebble game equivalence as instances of a single construction. It uses approximations and fixpoints. The approximation is “from above”. E.g. we use total homomorphisms to approximate partial isomorphisms in the EF case.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 15 / 21

slide-86
SLIDE 86

From Forth to Back and Forth

To complete the picture, we need to show how to define a back-and-forth equivalence ↔k which characterizes ≡Lk purely in terms of coKleisli morphisms. Our solution to this will have the following features: While not completely generic, it will be general enough to apply to all our game comonads – so we subsume EF equivalence, bisimulation equivalence and pebble game equivalence as instances of a single construction. It uses approximations and fixpoints. The approximation is “from above”. E.g. we use total homomorphisms to approximate partial isomorphisms in the EF case. We assume that for each structure A, the universe CkA has a forest order ⊑ (prefix ordering on sequences in our examples). We add a root ⊥ for convenience.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 15 / 21

slide-87
SLIDE 87

From Forth to Back and Forth

To complete the picture, we need to show how to define a back-and-forth equivalence ↔k which characterizes ≡Lk purely in terms of coKleisli morphisms. Our solution to this will have the following features: While not completely generic, it will be general enough to apply to all our game comonads – so we subsume EF equivalence, bisimulation equivalence and pebble game equivalence as instances of a single construction. It uses approximations and fixpoints. The approximation is “from above”. E.g. we use total homomorphisms to approximate partial isomorphisms in the EF case. We assume that for each structure A, the universe CkA has a forest order ⊑ (prefix ordering on sequences in our examples). We add a root ⊥ for convenience. We write the covering relation for this order as ≺; thus s ≺ t iff s ⊑ t, s = t, and for all u, s ⊑ u ⊑ t implies u = s or u = t.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 15 / 21

slide-88
SLIDE 88

General back-and-forth game

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 16 / 21

slide-89
SLIDE 89

General back-and-forth game

The definition is parameterized on a set WA,B ⊆ CkA × CkB of “winning positions” for each pair of structures A, B.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 16 / 21

slide-90
SLIDE 90

General back-and-forth game

The definition is parameterized on a set WA,B ⊆ CkA × CkB of “winning positions” for each pair of structures A, B. We define the back-and-forth Ck game between A and B as follows:

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 16 / 21

slide-91
SLIDE 91

General back-and-forth game

The definition is parameterized on a set WA,B ⊆ CkA × CkB of “winning positions” for each pair of structures A, B. We define the back-and-forth Ck game between A and B as follows: At the start of each round of the game, the position is specified by (s, t) ∈ CkA × CkB. The initial position is (⊥, ⊥).

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 16 / 21

slide-92
SLIDE 92

General back-and-forth game

The definition is parameterized on a set WA,B ⊆ CkA × CkB of “winning positions” for each pair of structures A, B. We define the back-and-forth Ck game between A and B as follows: At the start of each round of the game, the position is specified by (s, t) ∈ CkA × CkB. The initial position is (⊥, ⊥). Either Spoiler chooses some s′ ≻ s, and Duplicator responds with t′ ≻ t, resulting in (s′, t′); or Spoiler chooses t′′ ≻ t and Duplicator responds with s′′ ≻ s, resulting in (s′′, t′′).

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 16 / 21

slide-93
SLIDE 93

General back-and-forth game

The definition is parameterized on a set WA,B ⊆ CkA × CkB of “winning positions” for each pair of structures A, B. We define the back-and-forth Ck game between A and B as follows: At the start of each round of the game, the position is specified by (s, t) ∈ CkA × CkB. The initial position is (⊥, ⊥). Either Spoiler chooses some s′ ≻ s, and Duplicator responds with t′ ≻ t, resulting in (s′, t′); or Spoiler chooses t′′ ≻ t and Duplicator responds with s′′ ≻ s, resulting in (s′′, t′′). Duplicator wins after k rounds if the resulting position (s, t) is in WA,B.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 16 / 21

slide-94
SLIDE 94

General back-and-forth game

The definition is parameterized on a set WA,B ⊆ CkA × CkB of “winning positions” for each pair of structures A, B. We define the back-and-forth Ck game between A and B as follows: At the start of each round of the game, the position is specified by (s, t) ∈ CkA × CkB. The initial position is (⊥, ⊥). Either Spoiler chooses some s′ ≻ s, and Duplicator responds with t′ ≻ t, resulting in (s′, t′); or Spoiler chooses t′′ ≻ t and Duplicator responds with s′′ ≻ s, resulting in (s′′, t′′). Duplicator wins after k rounds if the resulting position (s, t) is in WA,B. This is essentially bisimulation up to WA,B.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 16 / 21

slide-95
SLIDE 95

General back-and-forth game

The definition is parameterized on a set WA,B ⊆ CkA × CkB of “winning positions” for each pair of structures A, B. We define the back-and-forth Ck game between A and B as follows: At the start of each round of the game, the position is specified by (s, t) ∈ CkA × CkB. The initial position is (⊥, ⊥). Either Spoiler chooses some s′ ≻ s, and Duplicator responds with t′ ≻ t, resulting in (s′, t′); or Spoiler chooses t′′ ≻ t and Duplicator responds with s′′ ≻ s, resulting in (s′′, t′′). Duplicator wins after k rounds if the resulting position (s, t) is in WA,B. This is essentially bisimulation up to WA,B. By instantiating WA,B appropriately, we obtain the equivalences corresponding to the EF, pebbling and bisimulation games.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 16 / 21

slide-96
SLIDE 96

General back-and-forth game

The definition is parameterized on a set WA,B ⊆ CkA × CkB of “winning positions” for each pair of structures A, B. We define the back-and-forth Ck game between A and B as follows: At the start of each round of the game, the position is specified by (s, t) ∈ CkA × CkB. The initial position is (⊥, ⊥). Either Spoiler chooses some s′ ≻ s, and Duplicator responds with t′ ≻ t, resulting in (s′, t′); or Spoiler chooses t′′ ≻ t and Duplicator responds with s′′ ≻ s, resulting in (s′′, t′′). Duplicator wins after k rounds if the resulting position (s, t) is in WA,B. This is essentially bisimulation up to WA,B. By instantiating WA,B appropriately, we obtain the equivalences corresponding to the EF, pebbling and bisimulation games. For example, WEk

A,B is the set of all (s, t) which define a partial isomorphism.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 16 / 21

slide-97
SLIDE 97

Characterization by coKleisli morphisms

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 17 / 21

slide-98
SLIDE 98

Characterization by coKleisli morphisms

We define S(A, B) to be the set of all functions f : CkA → B such that, for all s ∈ CkA, (s, f ∗(s)) ∈ WA,B.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 17 / 21

slide-99
SLIDE 99

Characterization by coKleisli morphisms

We define S(A, B) to be the set of all functions f : CkA → B such that, for all s ∈ CkA, (s, f ∗(s)) ∈ WA,B. A locally invertible pair (F, G) from A to B is a pair of sets F ⊆ S(A, B), G ⊆ S(B, A), satisfying the following conditions:

1

For all f ∈ F, s ∈ CkA, for some g ∈ G, g ∗f ∗(s) = s.

2

For all g ∈ G, t ∈ CkB, for some f ∈ F, f ∗g ∗(t) = t.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 17 / 21

slide-100
SLIDE 100

Characterization by coKleisli morphisms

We define S(A, B) to be the set of all functions f : CkA → B such that, for all s ∈ CkA, (s, f ∗(s)) ∈ WA,B. A locally invertible pair (F, G) from A to B is a pair of sets F ⊆ S(A, B), G ⊆ S(B, A), satisfying the following conditions:

1

For all f ∈ F, s ∈ CkA, for some g ∈ G, g ∗f ∗(s) = s.

2

For all g ∈ G, t ∈ CkB, for some f ∈ F, f ∗g ∗(t) = t. We define A ↔C

k B iff there is a non-empty locally invertible pair from A to B.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 17 / 21

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SLIDE 101

Characterization by coKleisli morphisms

We define S(A, B) to be the set of all functions f : CkA → B such that, for all s ∈ CkA, (s, f ∗(s)) ∈ WA,B. A locally invertible pair (F, G) from A to B is a pair of sets F ⊆ S(A, B), G ⊆ S(B, A), satisfying the following conditions:

1

For all f ∈ F, s ∈ CkA, for some g ∈ G, g ∗f ∗(s) = s.

2

For all g ∈ G, t ∈ CkB, for some f ∈ F, f ∗g ∗(t) = t. We define A ↔C

k B iff there is a non-empty locally invertible pair from A to B.

Proposition

The following are equivalent:

1

A ↔C

k B.

2

There is a winning strategy for Duplicator in the Ck game between A and B.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 17 / 21

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SLIDE 102

A fixpoint characterization

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 18 / 21

slide-103
SLIDE 103

A fixpoint characterization

Write S := S(A, B), T := S(B, A).

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 18 / 21

slide-104
SLIDE 104

A fixpoint characterization

Write S := S(A, B), T := S(B, A). Define set functions Γ : P(S) → P(T), ∆ : P(T) → P(S): Γ(F) = {g ∈ T | ∀t ∈ CkB.∃f ∈ F. f ∗g ∗t = t}, ∆(G) = {f ∈ S | ∀s ∈ CkA.∃g ∈ G. g ∗f ∗s = s}.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 18 / 21

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SLIDE 105

A fixpoint characterization

Write S := S(A, B), T := S(B, A). Define set functions Γ : P(S) → P(T), ∆ : P(T) → P(S): Γ(F) = {g ∈ T | ∀t ∈ CkB.∃f ∈ F. f ∗g ∗t = t}, ∆(G) = {f ∈ S | ∀s ∈ CkA.∃g ∈ G. g ∗f ∗s = s}. These functions are monotone. Moreover, a pair of sets (F, G) is locally invertible iff F ⊆ ∆(G) and G ⊆ Γ(F).

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 18 / 21

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SLIDE 106

A fixpoint characterization

Write S := S(A, B), T := S(B, A). Define set functions Γ : P(S) → P(T), ∆ : P(T) → P(S): Γ(F) = {g ∈ T | ∀t ∈ CkB.∃f ∈ F. f ∗g ∗t = t}, ∆(G) = {f ∈ S | ∀s ∈ CkA.∃g ∈ G. g ∗f ∗s = s}. These functions are monotone. Moreover, a pair of sets (F, G) is locally invertible iff F ⊆ ∆(G) and G ⊆ Γ(F). Thus existence of a locally invertible pair is equivalent to the existence of non-empty F such that F ⊆ Θ(F), where Θ = ∆Γ.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 18 / 21

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SLIDE 107

A fixpoint characterization

Write S := S(A, B), T := S(B, A). Define set functions Γ : P(S) → P(T), ∆ : P(T) → P(S): Γ(F) = {g ∈ T | ∀t ∈ CkB.∃f ∈ F. f ∗g ∗t = t}, ∆(G) = {f ∈ S | ∀s ∈ CkA.∃g ∈ G. g ∗f ∗s = s}. These functions are monotone. Moreover, a pair of sets (F, G) is locally invertible iff F ⊆ ∆(G) and G ⊆ Γ(F). Thus existence of a locally invertible pair is equivalent to the existence of non-empty F such that F ⊆ Θ(F), where Θ = ∆Γ. Since Θ is monotone, by Knaster-Tarski this is equivalent to the greatest fixpoint

  • f Θ being non-empty. (Note that Θ(∅) = ∅).

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 18 / 21

slide-108
SLIDE 108

A fixpoint characterization

Write S := S(A, B), T := S(B, A). Define set functions Γ : P(S) → P(T), ∆ : P(T) → P(S): Γ(F) = {g ∈ T | ∀t ∈ CkB.∃f ∈ F. f ∗g ∗t = t}, ∆(G) = {f ∈ S | ∀s ∈ CkA.∃g ∈ G. g ∗f ∗s = s}. These functions are monotone. Moreover, a pair of sets (F, G) is locally invertible iff F ⊆ ∆(G) and G ⊆ Γ(F). Thus existence of a locally invertible pair is equivalent to the existence of non-empty F such that F ⊆ Θ(F), where Θ = ∆Γ. Since Θ is monotone, by Knaster-Tarski this is equivalent to the greatest fixpoint

  • f Θ being non-empty. (Note that Θ(∅) = ∅).

If A and B are finite, so is S, and we can construct the greatest fixpoint by a finite descending sequence S ⊇ Θ(S) ⊇ Θ2(S) ⊇ · · · .

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 18 / 21

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SLIDE 109

A fixpoint characterization

Write S := S(A, B), T := S(B, A). Define set functions Γ : P(S) → P(T), ∆ : P(T) → P(S): Γ(F) = {g ∈ T | ∀t ∈ CkB.∃f ∈ F. f ∗g ∗t = t}, ∆(G) = {f ∈ S | ∀s ∈ CkA.∃g ∈ G. g ∗f ∗s = s}. These functions are monotone. Moreover, a pair of sets (F, G) is locally invertible iff F ⊆ ∆(G) and G ⊆ Γ(F). Thus existence of a locally invertible pair is equivalent to the existence of non-empty F such that F ⊆ Θ(F), where Θ = ∆Γ. Since Θ is monotone, by Knaster-Tarski this is equivalent to the greatest fixpoint

  • f Θ being non-empty. (Note that Θ(∅) = ∅).

If A and B are finite, so is S, and we can construct the greatest fixpoint by a finite descending sequence S ⊇ Θ(S) ⊇ Θ2(S) ⊇ · · · . This fixpoint is non-empty iff A ↔E

k B.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 18 / 21

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SLIDE 110

Logical Equivalences

We can now complete our results on logical equivalences.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 19 / 21

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SLIDE 111

Logical Equivalences

We can now complete our results on logical equivalences.

Theorem

For structures A and B: (1) A ≡∃Lk B ⇐ ⇒ A ⇄k B. (2) A ≡Lk B ⇐ ⇒ A ↔k B. (3) A ≡Lk(#) B ⇐ ⇒ A ∼ =Kl(Ck) B.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 19 / 21

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SLIDE 112

Logical Equivalences

We can now complete our results on logical equivalences.

Theorem

For structures A and B: (1) A ≡∃Lk B ⇐ ⇒ A ⇄k B. (2) A ≡Lk B ⇐ ⇒ A ↔k B. (3) A ≡Lk(#) B ⇐ ⇒ A ∼ =Kl(Ck) B. Note that this is really a family of three theorems, one for each type of game G.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 19 / 21

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SLIDE 113

Logical Equivalences

We can now complete our results on logical equivalences.

Theorem

For structures A and B: (1) A ≡∃Lk B ⇐ ⇒ A ⇄k B. (2) A ≡Lk B ⇐ ⇒ A ↔k B. (3) A ≡Lk(#) B ⇐ ⇒ A ∼ =Kl(Ck) B. Note that this is really a family of three theorems, one for each type of game G. Thus in each case, we capture the salient logical equivalences in syntax-free, categorical form.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 19 / 21

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SLIDE 114

Coalgebraic characterization of combinatorial parameters

There is a beautiful connection between these indexed comonads and combinatorial invariants of structures.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 20 / 21

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SLIDE 115

Coalgebraic characterization of combinatorial parameters

There is a beautiful connection between these indexed comonads and combinatorial invariants of structures. We define the coalgebra number of a structure A, with respect to the indexed family of comonads Ck, to be the least k such that there is a Ck-coalgebra on A.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 20 / 21

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SLIDE 116

Coalgebraic characterization of combinatorial parameters

There is a beautiful connection between these indexed comonads and combinatorial invariants of structures. We define the coalgebra number of a structure A, with respect to the indexed family of comonads Ck, to be the least k such that there is a Ck-coalgebra on A.

Theorem

For the Ehrenfeucht-Fra¨ ıss´ e comonad, the coalgebra number of A corresponds precisely to the tree-depth of A. For the pebbling comonad, the coalgebra number of A corresponds precisely to the tree-width of A. For the modal comonad, the coalgebra number of A corresponds precisely to the synchronization tree depth of A.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 20 / 21

slide-117
SLIDE 117

Coalgebraic characterization of combinatorial parameters

There is a beautiful connection between these indexed comonads and combinatorial invariants of structures. We define the coalgebra number of a structure A, with respect to the indexed family of comonads Ck, to be the least k such that there is a Ck-coalgebra on A.

Theorem

For the Ehrenfeucht-Fra¨ ıss´ e comonad, the coalgebra number of A corresponds precisely to the tree-depth of A. For the pebbling comonad, the coalgebra number of A corresponds precisely to the tree-width of A. For the modal comonad, the coalgebra number of A corresponds precisely to the synchronization tree depth of A. The main idea behind these results is that coalgebras on A are in bijective correspondence with decompositions of A of the appropriate form.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 20 / 21

slide-118
SLIDE 118

Coalgebraic characterization of combinatorial parameters

There is a beautiful connection between these indexed comonads and combinatorial invariants of structures. We define the coalgebra number of a structure A, with respect to the indexed family of comonads Ck, to be the least k such that there is a Ck-coalgebra on A.

Theorem

For the Ehrenfeucht-Fra¨ ıss´ e comonad, the coalgebra number of A corresponds precisely to the tree-depth of A. For the pebbling comonad, the coalgebra number of A corresponds precisely to the tree-width of A. For the modal comonad, the coalgebra number of A corresponds precisely to the synchronization tree depth of A. The main idea behind these results is that coalgebras on A are in bijective correspondence with decompositions of A of the appropriate form. We thus obtain categorical characterizations of these key combinatorial parameters.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 20 / 21

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SLIDE 119

Final Remarks

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 21 / 21

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SLIDE 120

Final Remarks

Our three comonadic constructions show a striking unity, but also some very interesting differences.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 21 / 21

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SLIDE 121

Final Remarks

Our three comonadic constructions show a striking unity, but also some very interesting differences. Need to understand better what makes these constructions work, and what the scope of these ideas are.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 21 / 21

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SLIDE 122

Final Remarks

Our three comonadic constructions show a striking unity, but also some very interesting differences. Need to understand better what makes these constructions work, and what the scope of these ideas are. Currently investigating the guarded fragment. Other natural candidates include existential second-order logic, and branching quantifiers and dependence logic.

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 21 / 21

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SLIDE 123

Final Remarks

Our three comonadic constructions show a striking unity, but also some very interesting differences. Need to understand better what makes these constructions work, and what the scope of these ideas are. Currently investigating the guarded fragment. Other natural candidates include existential second-order logic, and branching quantifiers and dependence logic. Wider horizons: can we connect with significant meta-algorithms, such as decision procedures for guarded logics based on the tree model property, or algorithmic metatheorems such as Courcelle’s theorem?

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 21 / 21

slide-124
SLIDE 124

Final Remarks

Our three comonadic constructions show a striking unity, but also some very interesting differences. Need to understand better what makes these constructions work, and what the scope of these ideas are. Currently investigating the guarded fragment. Other natural candidates include existential second-order logic, and branching quantifiers and dependence logic. Wider horizons: can we connect with significant meta-algorithms, such as decision procedures for guarded logics based on the tree model property, or algorithmic metatheorems such as Courcelle’s theorem? The wider issue: can we get Structure and Power to work with each other to address genuinely deep questions?

Samson Abramsky (Department of Computer Science, University of Oxford) Approximating partial by total: fixpoint characterizations of back-and-forth equivalences 21 / 21