Image schemas via finite-state methods structured category-theoretically

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1 Objective predication Subjective predication Reconciliation Image schemas via finite-state methods structured category-theoretically Tim Fernando Bolzano, 22 August 2016, PROSECCO a recurring structure within our cognitive processes which establishes patterns of understanding and reasoning (Wikipedia) Ontological logs (ologs) Institutions D. Spivak J. Goguen 1 / 21 Predication Donald is in trouble. Chelsea has, as mother, Hillary. Aristotle subject predicate containment Frege/RDF triple subject predicate object Graph (directed multi) vertex edge vertex Automata state label state? source path goal Category domain morphism ;= codomain category functor category Spivak olog interpretation I Set.... Linguistic motivation: individual-, kind-, stage-level predication (G. Carlson)

2 Agenda 1 Objective predication - declarative (logical) - extensional: concept c as a set I (c), where I is a functor from an olog C to Set 2 Subjective predication - procedural (cognitive) - attributional: concept c traced by strings over Σ 3 Reconciliation - institutions (schemas =) trace(c) Σ Goal: justify & develop finite-state methods for containment, path,... (image schemas) via some category theory Objective predication Subjective predication Reconciliation 1 Objective predication 2 Subjective predication 3 Reconciliation 4 / 21

3 Types vs instances : ologs & functors (1) Sylvester is a cat. cat(sylvester) sylvester I (cat) (2) Lions are cats. lion cat I (lion) I(cat) (3) Pat lives in }{{} Cork. f : A B in C (olog/database schema) I : C Set for I (f ) : I (A) I (B) in Set Grothendieck construction {}}{ (pat, f, cork) : (A, pat) (B, cork) in I pat I (A) & I (f )(pat) = cork Triplestores & database schemas person lives-in pat cork ann rome place state cork ireland rome italy (pat,lives-in,cork), (cork, state, ireland),... in I Olog/category C as top row + I for remaining rows person lives-in place place state state plus C-identities A = A A pat I (person) (pat, = person, pat) in I cork I (place) (cork, = place, cork) in I... composition - (pat, lives-in;state, ireland) in I

4 Predicates as transition labels, not morphisms Sylvester is a cat sylv I (cat) Lions are cats (sylv, = cat, sylv) in I I (lion) I (cat) Given I : C Set, - the RDF triple (a, f, b) is determined by (a, f ) with label f for transitions b = I (f )(a) a f I (f )(a) from a to I (f )(a) - I (A) I (B) can be encoded as (A, in(b)) with label in(b) for transitions A in(b) B from A to B. Objective predication Subjective predication Reconciliation 1 Objective predication 2 Subjective predication 3 Reconciliation 8 / 21

5 Traces q α q as (q, α, q ), where Q Σ Q For q 0 Q, let trace(q 0 ) := n 0{α 1 α n Σ n ( q 1 q n Q n ) α q i i 1 qi for 1 i n}. is deterministic if for all α Σ and q Q, there is at most one q s.t. q α q. Fact. If is deterministic, then trace equivalence q q trace(q) = trace(q ) is well-suited to (bisimulation equivalence). Brzozowski derivatives & finite-state matters For deterministic and q α q, trace(q ) = trace(q) α where For s = α 1 α n, L α = {s αs L} L s := {s ss L} L ɛ = L = {s ɛ L s } α s L L 1 α ɛ 2 Lα1 Lα1 α 2 αn L α1 α n = L s and ɛ L s. Myhill-Nerode L is regular {L s s Σ } is finite L s = L s ( w Σ ) (sw L s w L)

6 Q(Σ): composition as typed concatenation Always ɛ trace(q) and whenever sα trace(q), s trace(q). A Σ-state is a non-empty language L Σ that is prefix-closed sα L s L A Q(Σ)-morphism is a pair (L, s) of a Σ-state L and an s L with identities (L, ɛ) labeled by ɛ. dom(l, s) := L cod(l, s) := L s (L, s) ; (L s, s ) := (L, ss ) A B as A in(b) B allows trace(b) trace(a) (contra (L, ɛ)) but should we not expect L in(b) L? Labeled transitions from C I Set Yes, for built from I and on C A f B in C a f I (f )(a) a I (A) A, B C I (A) I (B) A in[b] B Are there enough labels for trace(pat) trace(ann)? Brute force differentiation via labels = a A C a = a a a I (A) Cognitive processes may pick up only some of these transitions. Connect instances and types by A C a in[a] A a I (A) A f B in C A f B raising the question L in(a) L (also for L = trace(a))?

7 Non-extensionality & non-monotonicity (a) Cats are widespread. (Carlson kind-level predication) Sylvester is widespread?? (b) Birds fly. Penguins are birds that don t. (Defaults...) Tweety is a penguin. (Penguin principle... DATR) Not only may types be more than sets of their instances, these instances may vary with time. (c) Sylvester was hungry this morning. (C stage-level) (d) Pat moved from Belfast to Cork in (events) Vary not only extension I : C Set described (change/events) but also granularity Σ (Vendler classes; Fernando 2016). Partiality: presheaves Fix a large set Θ of labels, with fragments in Fin(Θ) := {Σ Θ Σ is finite} Contravariant functor Q : Fin(Θ) op Cat for Σ Σ Fin(Θ), Q(Σ, Σ) : Q(Σ ) Q(Σ) L L Σ (L, s) (L Σ, s Σ ) where s Σ is the longest prefix of s in Σ. Morphisms in an olog vs Q(Σ): free monoid of strings (L, s) ; (L s, s ) := (L, ss ) Time as change: stutter-free strings from block compression bc( rain rain rain rain,sun sun sun ) = rain rain,sun sun (Fernando 2016)

8 Objective predication Subjective predication Reconciliation 1 Objective predication 2 Subjective predication 3 Reconciliation 15 / 21 Back to image schema (broadly) olog a { recurring }} structure { within our cognitive processes which establishes patterns of understanding and reasoning }{{} M = S ϕ Institution (Goguen): = as a schema instantiatied by a signature S relating S -models M and S -sentences ϕ Example 1. S is an olog C C-model is a functor I : C Set C-sentence is a path equation (commuting diagram) A f 1 f n g 1 g m B = A B Example 2. S is a finite alphabet Σ Σ-model is a Σ-state L Σ-sentence from Hennessy-Milner logic L = Σ α ϕ iff α L and L α = Σ ϕ

9 Three functors F : A Cat for F f Given A B in A, a F (A) and b F (B) (f,x) x (A, a) (B, b) in F F (f )(a) b in F (B) (i) F = I : }{{} C Set with sets as discrete categories olog Sign (only morphisms are identities) I is a C-model (ii) (iii) F = Q : Fin(Θ) op Cat with Q(Σ)-composition as typed concatenation Sign from Q (L s and s ϕ; Fernando 2016a) F = Th : Sign Cat with Th(S ) pre-ordered Γ S Γ {M M = S ϕ} {M M = S ϕ} ϕ Γ ϕ Γ Lattice of theories (LOT, Sowa) & ologs Spivak & Kent 2012 In the Olog formalism, LOT is locally represented by the entailment preorders spec(g)... the entailment ordering defines paths to the more generalized ologs above and the more specialized ologs below. Sowa defines four ways for moving along paths from one olog to another: contraction, expansion, revision and analogy Olog specification/equation A f 1 is broken down by Q to f n g 1 g m B = A B L f1 f n = L g1 g m for all L s.t. L in(a) or in Hennessy-Milner (interpreted by traces) ( in(a) f 1 f n ϕ g 1 g m ϕ) ( in(a) g 1 g m ϕ f 1 f n ϕ)

10 A finite-state calculus L = α Θ αl α + o(l) where o(l) = { ɛ if ɛ L otherwise both Taylor s theorem & the mean value theorem in this theory (Conway 1971) Finite approximability hypothesis: finite subset of Θ will do open-ended signature: Σ Fin(Θ) (directed poset) Sign = Q (inverse limit) Identity as indiscernibility (Leibniz) wrt Σ: bounded granularity Σ-sentences from Hennessy-Milner, Monadic Second-Order Logic (Fernando 2015, 2016, 2016a) Predication on kinds & stages via strings Extension I changes along path reflecting - inheritance hierarchy (finite) tweety penguin bird ϕ ϕ ϕ inheritable/inertial ϕ - time (bounded granularity) Ed explained E S E S Ed explaining E V E E,V E Reichenbach tense aspect it rained E,R S R S E,R it has rained E R,S R,S E R projection E,R S R S...

11 Some references G. Carlson, A unified analysis of the English bare plural, Linguistics & Philosophy 1(3):413 58, J.H. Conway, Regular Algebra & Finite Machines T. Fernando, Two perspectives on change & institutions, 2015 ( FOfAI paper 2.pdf). T. Fernando, On regular languages over power sets, Journal of Language Modelling 4(1):29 56, T. Fernando, Types from frames as finite automata, Formal Grammar, Springer LNCS 9804, 2016a, pp J. Goguen & R. Burstall, Institutions: Abstract model theory for specifications & programming. J. ACM 39(1):95 146, D.I. Spivak & R.E. Kent, Ologs: A categorical framework for knowledge representation. PLoS ONE 7(1), D.I. Spivak, Category Theory for the Sciences. MIT Press, 2014.