Games and Analytic Proof Systems Part 2 Chris Ferm uller Vienna - - PowerPoint PPT Presentation

PhDs in Logic X Prague, Czech Republic, May 1, 2018

Games and Analytic Proof Systems Part 2

Chris Ferm¨ uller Vienna University of Technology Theory and Logic Group www.logic.at/people/chrisf/

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Overview

Part 1 the most basic logic game: Hintikka’s game for classical logic from Hintikka’s game to sequent calculus via disjunctive states Hintikka’s game and many truth values:

◮ many-valued truth tables, Nmatrices ◮ Giles’s game for

Lukasiewicz logic

analyzing a hypersequent calculus using games Part 2 modelling resource conciousness calls for games (briefly:) alternative forms of game semantics Lorenzen: dialogues as foundations Information extraction interpreting ILL and other substructural logics many(!) topics for future research

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Background/Motivation

substructural logics are often motivated by resource consciousness this motivation usually remains metaphorical think of Girard’s cigarette example: “For ✩1 you get a pack of Camels, but also a pack of Marlboro” “but also”: multiplicative in contrast to additive conjunction Gentzen’s sequent calculus (LK/LI) is a (the?) natural starting point for connecting inference and resource consciousness – this leads to (fragments of) linear logic, possibly even Lambek calculus, etc to breathe life into the resource metaphor, we need dynamics = ⇒ game semantics for substructural sequent calculi

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Alternative forms of game semantics

Blass (APAL 1992): game semantics for affine linear logic – new paradigm: ‘logical connectives as game operators’ – only additive connectives, otherwise ‘counter examples’ – negation as role switch Abramsky/Jagadeesan (JSL 1994): full completeness – paradigm: formulas = games, strategies = proofs – multiplicative connectives are covered – high level of abstraction: games are not assertions/statements Girard’s Locus Solum (‘ludics’) (2001): ‘loci’: pointers to subformulas, ‘designs’: corresponding proofs attempts to provide a logic of inference rules as interactions Japaridze’s computability logic CL (since 2003) – games as a general model of interactive computation – computational constructions induce (many) connectives – certain principles of linear logic get invalidated . . . many more related to the above — none about assertions/statements/rational dialogue

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The oldest form of ‘provability games’: Lorenzen’s dialogues

“According to Lorenzen, valid arguments are those patterns from premises to conclusions in which the proponent of the conclusion has a winning strategy against any opponent granting the premises. Thus, there is a third independent pragmatic intuition of logical validity, based on viewing argumentation as a game. I have been converted to that view ever since, even though most of my professional life has been under camouflage as a model theorist, or occasionally a proof theorist.” Johan van Benthem

in: Game Theory – Five Questions V.F. Hendricks, P.G. Hansen (eds.)

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Dialogues as foundations

Imagine a dialogue, where a Proponent P tries to defend a logi- cally complex statement against attacks by an Opponent O. The dialogue stepwise reduces complex assertions to their components. Lorenzen’s central idea (‘Logik und Agon’, late 1950s): G logically follows from F1, . . . , Fn means: P can always win an antagonistic, rational dialogue starting with her assertion of G, if O has granted F1, . . . , Fn Some basic features of Lorenzen style dialogues: attack moves and corresponding defense moves refer to outermost connectives and quantifiers of assertions both, P and O, may launch attacks and defend against attacks during the course of a dialogue moves alternate strictly between P and O

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Logical dialogue rules:

X/Y stands for P/O or O/P statement by X attack by Y defense by X A ∧ B l? or r? (Y chooses) A or B, accordingly A ∨ B ? A or B (X chooses) A ⊃ B A B ∀xA(x) ?c (Y chooses) A(c) ∃xA(x) ? A(c) (Y chooses c) Note: ¬A abbreviates A ⊃ ⊥, and ⊥ cannot be defended Winning conditions for P: W: O has already granted P’s active formula W⊥: O has granted ⊥ NB: Lorenzen attempted to justify constructive logic, the dialogue game allegedly ‘showed’ why even classical logic is wrong! = ⇒ A different idea is needed for substructural reasoning

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Information Extraction

Guiding Idea: Information is closer to Lorenzen’s ‘assertions’, than ‘formulas as games’ resource conciousness: information can be stored/accessed/consumed

Different ways of accessing information:

A Server may provide information to a Client e.g. as follows: You can have all of those (once): {a, b, c, . . .} (multiset) You can have any one of those: (a, b, c, . . .) (C’s choice) I (Server) give you (just) one of those: [a, b, c, . . .] (S’s choice) You can have the first of those: a, b, c, . . . | (stack) You can have the first or last of those: a, b, . . . e, f (deque) You can have these as often as you want: a, b, c, . . . (‘protected’) In principle, arbitrary nestings are conceivable: for example: {a, (b, c|, [d, e, f ])}

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Accessing/extracting information as a game

Information extraction game(s) (formerly C/S game – [F/Lang, 2017]): Proponent claims that S provides the information that C wants, while Opponent seeks to refute P’s claim Remark: In [F/Lang 2017] we identified C with P and S with O (‘C/S game’) We now prefer to keep S passive and let P act in C’s behalf, opposed by O States of the game: Γ ⊲ F Γ . . . bundle of information provided by S F . . . information wanted by C (possibly structured, as explained below) Two possible interpretations: Strict reading: F is equivalent to Γ Affine reading: F is (modulo equivalence) contained in Γ NB: ‘equivalence’ is (implicitly) defined by the rules of the game

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Where are the logical connectives?

(Some) logical connectives directly correspond to access structures. Following tradition, we formulate rules for binary connectives. {a, b} multiset offered by S = ⇒ a ⊗ b (multiplicative conjunction) (a, b) any – P’s choice (for C) = ⇒ a ∨ b (additive disjunction) [a, b] any – O’s choice (against C) = ⇒ a ∧ b (additive conjunction) a, b| first a, then b = ⇒ a; b (a new connective) a ‘protected’ a = ⇒ !a (‘bang’, ‘of course’) To emphasize our ‘information extraction’ perspective, we will speak of Information Packages (IPs), rather than formulas. In order to make these corespondences precise and make them work in full generality we need to provide precise specifications of game rules!

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Rules of the (standard) Information Extraction Game

The rules stepwise reduce states to simpler states in round: Step 1 P, as scheduler, chooses an IP F of the state Γ ⊲ H Step 2 two cases: – F in Γ = ⇒ Unpack the IP provided by S – F = H = ⇒ Check the IP wanted by C corresponding choices by P or by O determine the next state We focus on the case, where the IPs of S form a multiset Γ = [G1, . . . , Gn] (More general ‘deep inference’ style rules could be obtained analogously) Unpack-rules (F among S’s IPs) (U∨) F = F1 ∨ F2: P chooses i, Fi replaces F in Γ (U∧) F = F1 ∧ F2: O chooses i, Fi replaces F in Γ (U⊗) F = F1 ⊗ F2: F1 and F2 replace F in Γ Check-rules (F is C’s current IP — rules are dual) (C∨) F = F1 ∨ F2: O chooses i, Fi replaces F as C’s wanted IP (C∧) F = F1 ∧ F2: P chooses i, Fi replaces F as C’s wanted IP (C⊗) F = F1 ⊗ F2: P has declares which part of Γ will be used for extracting F1 and F2, respectively; O chooses correspondingly

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Did we loose implication?

F1 → F2 is interpreted as conditional information: F2 given F1 The corresponding Check-rules (state Γ ⊲ F1 → F2) is obvious : (C→) F2 becomes C’s current IP, F1 is added to Γ For F1 → F2 provided by S, the following is obvious too: (U→) If F1 as well as F1 → F2 are in Γ, the P may choose to replace these two IP-occurrences by F2 More generally, F1 only needs to be contained in information in Γ: (U→) P has declares which part (Γ1) of Γ is to be used for extracting F1 and which part (Γ2), augmented by F2, allows to extract C’s wanted IP; O chooses correspondingly Written in sequent style: F1, Γ ⊲ F2 Γ ⊲ F1 → F2 (C→) Γ1 ⊲ F1 F2, Γ2 ⊲ H F1 → F2, Γ1, Γ2 ⊲ H (U→)

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Rules for Protected IP’s

Simple form of interpreting ‘protection’:

• nly relevant for information provided by S

F1, . . . , Fn: Fi remain in Γ for each reduction, no splitting necessary Reflective form of interpreting ‘protection’: (linear logic style) we also allow F and [F, F], etc, also for C’s wanted IP the corresponding connective is ! (‘bang’ of linear logic) (U!) if P picks !F, then P may either replace if by F, delete it

• r add another copy of F in Γ, as wished

(C!) If C’s wanted !F is picked, P may replace it by F if all formulas in Γ are protected

Final States (Winning Conditions)

recall the two possible interpretations of Γ ⊲ F: equivalent / contained in corresponding winning states for P: F, Γ ⊲ F

• r

F ⊲ F Adding the clearly contradictory IP ⊥ renders ⊥, Γ ⊲ F winning for P, too

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Instances of the game matching well known calculi

Full completeness and soundness is straightforward for some calculi. For the game with reflective form of protection (strict and affine reading):

Theorem

Each of P’s winning strategies for G1, . . . , Gn ⊲ F translates into a cut-free proof of G1, . . . , Gn ⊢ F in (affine) ILL, and vice versa. For the game with simple form of protection (S: ... instead of [...]):

Theorem

Each of P’s winning strategies for G1, . . . , Gn ⊲ F translates into a cut-free proof of G1, . . . , Gn ⊢ F in Gentzen’s LI, and vice versa. Note: explicit weakening corresponds to dismissal of information by P In a similar vain, many other calculi, eg. Lambek’s, can modelled!

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Beyond interpreting known calculi

A central motive for the whole research program: The game approach invites the exploration of many new perspectives! Some examples of new aspects – (the ‘new research paradigm’) new logics, induced by other ways of organizing S: stack, . . . new connectives and information structures suggested by alternative interpretations of information extraction the game from O’s perspective: strategies providing counter-examples refined analysis of roles, also of O: eg., who schedules? natural restrictions of strategies: eg. uniform, of low complexity, etc random moves, corresponding to ‘blind’ (uniform) strategies imperfect information: leads to equilibrium semantics interpreting cut and other structural rules: strategy construction disjunctive strategies: backtracking options − → hypersequents We will discuss just 2 examples, each touching several items . . .

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(1) An alternative interpretation of conjunctive information

Consider a, b ⊢ (a ∧ b) ⊗ (a ∨ b) Not derivable in (I)LL: – ⊗ calls for immediate distribution of a and b; – clearly, one of a, b is needed for each side – but a ⊢ a ∧ b, b ⊢ a ∧ b Inspired by the information extraction game: a, b ⊢ (a ∧ b) ◦ (a ∨ b) could mean: a, b is sufficient to extract (a ∧ b) ◦ (a ∨ b): – some x ∈ {a, b}, chosen by O, suffices for a ∧ b – whatever y ∈ {a, b} remains after O’s choice suffices for a ∨ b In other words: if O is forced to decide first, C can get what she wants!

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(1 ctd.) Different ways to model our new conjunction:

(1) allow for backtracking (which is natural for games): O’s decision is recorded and informs P’s reconsideration of splitting (2) use ◦ as structural connective |◦, bundling C’s wanted information: Using a corresponding new rule D|◦, we obtain the following winning strategy for P (written in sequent format): b ⊲ b b ⊲ a ∨ b C∨ a, b ⊲ a |◦ a ∨ b D|◦ a ⊲ a a ⊲ a ∨ b C∨ a, b ⊲ b |◦ a ∨ b D|◦ a, b ⊲ a ∧ b |◦ a ∨ b C∧ NB: depends on P acting as scheduler! (3) turn ◦ into a logical connective: Check-rule Γ ⊲ F |◦G |◦ ∆ Γ ⊲ F ◦ G |◦ ∆ C◦

• pen problem: what is the corresponding Unpack-rule?

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(2) Random Moves

Randomization may appear in various ways for both players/different roles A few examples/observations: If we only care about P’s winning strategies, nothing seems to change if we let act O randomly. However: equilibrium values (various types) induce extended semantics! Let O choose subformulas randomly, but P’s obligations strategically: – equilibrium value for a ⊲ a ∧ b is 0.5 (for P) – equilibrium value for a ⊲ a ⊗ b is 0 (for P) Can an all rationals ∈ [0, 1] be realized as equilibrium values? A positive answer arises from [F/Majer2016] (lifted from IF logic) What happens if scheduling is random (rather than P’s task)? A simple, but powerful general observation:

Theorem

P has a winning strategy for G1, . . . , Gn ⊲ F under random scheduling iff there is a cut-free proof of G1, . . . , Gn ⊢ F in a corresponding sequent calculus, where all used rules are invertible.

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Conclusion

Information extraction games provide semantics for substructural logics. More importantly: a whole landscape for new research directions emerges.

Further topics:

game based interpretations of (different types of) subexponentials interpreting full (classical) linear logic (‘the challenge for ’) algebraic semantics: eg. information as lattice elements parallel games as interpretations of hypersequent calculi new connectives arising from randomization information extraction games with incomplete information . . . (many more topics) . . .

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Join the game: make logic (even more) sexy!

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