Equational Theories for Real-Time
Coalgebraic State Machines
Sergey Goncharova Stefan Miliusa Alexandra Silvab
ICTAC 2018, October 15-19, Stellenbosch
aFriedrich-Alexander-Universitat Erlangen-NurnbergbUniversity College London
Some Related Work
• Deterministic automata as coalgebras [Rutten, 1998]
• Generalized regular expressions and Kleene theorem for
Kripke-polynomial functors [Silva, 2010]
• Generalized powerset construction [Silva, Bonchi, Bonsangue, and
Rutten, 2010]
• Regular expressions for equationally presented functors and
monads [Myers, 2013]
• Context-free languages, coalgebraically [Winter, Bonsangue, and
Rutten, 2013]
This talk is based on
• Goncharov, Milius, and Silva 2014, Towards a Coalgebraic Chomsky
Hierarchy (TCS 2014)
• Goncharov, Milius, and Silva 2018, Towards a Uniform Theory of
Effectful State Machines (ArXiv preprint)
1/15
Overview
What’s in here:
• A theory of effectful real-time state machines and expressions
• Semantics over the space of formal power series
• Equational theories for effects
• Kleene theorem
What isn’t here:
• Results on the equational theory of fixpoint expressions, e.g.
completeness
• Non-linear semantics (e.g. with trees instead of words)
• Infinite trace semantics
• Unguarded expressions/non-real-time machines (but see the paper)
2/15
Prelude: Coalgebraic Powerset
Construction
T-automata and Generalized Powerset Construction
For a monad T and a T-algebra am : TB Ñ B, we dub a triple of maps
om : X Ñ B, tm : X ˆ AÑ TX , am : TB Ñ B
a T-automaton. Equivalently, it is a coalgebra m : X Ñ B ˆ pTX qA
X TX BA‹
B ˆ pTX qA B ˆ pBA‹qA
m
ηpm7
m7out
idˆppm7qA
Factorization m “ m 7η is unique and we define JxKm “ JηpxqKm7 P BA‹ ,
meaning that the formal power series JxKm : A‹ Ñ B is the semantics
of m in state x
3/15
T-automata and Generalized Powerset Construction
For a monad T and a T-algebra am : TB Ñ B, we dub a triple of maps
om : X Ñ B, tm : X ˆ AÑ TX , am : TB Ñ B
a T-automaton. Equivalently, it is a coalgebra m : X Ñ B ˆ pTX qA
X PX PA‹
2ˆ pPX qA 2ˆ pPA‹qA
m
ηpm7
m7out
idˆppm7qA
Factorization m “ m 7η is unique and we define JxKm “ JηpxqKm7 P BA‹ ,
meaning that the formal power series JxKm : A‹ Ñ B is the semantics
of m in state x
3/15
Towards Kleene Theorem
• We seek a syntactic counterpart of the generalized powerset
construction
• Hence, we need a syntax for the corresponding fixpoint expressions,
corresponding to the classical regular expressions
• Such expressions must include
• reactive constructs for actions from A and final outputs from B
(Think of prefixing a. -- and 0, 1 of Kleene algebra)
• effectful constructs for representing side-effecting transitions of
the corresponding T-automaton
(Think of nondeterministic ` and 0)
4/15
Equational Theories for Effects
Monads for Effects
Mac Lane: A monad T is just a monoid in the category of endofunctors
Any monad T supports inclusion of a value into a computation
η : X Ñ TX and a Kleisli lifting pf : X Ñ TY q ÞÑ pf ‹ : TX Ñ TY q
Examples:
• (finitary) nondeterminism: TX “ PωX ; “nondeterministic
functions” AÑ PωB are relations
• probabilistic nondeterminism: TX “
ρ : X Ñ r0, 1s |ř
ρ “ 1(
;
“probabilistic functions” AÑ TB are “probabilistic relations”
• (finite) background store: TX “ pX ˆ SqS ; side-effecting functions
AÑ TB are functions Aˆ S Ñ B ˆ S
5/15
Monads for Effects
Mac Lane: A monad T is just a monoid in the category of endofunctors
Moggi: A monad T is a (generalized) computational effect
Any monad T supports inclusion of a value into a computation
η : X Ñ TX and a Kleisli lifting pf : X Ñ TY q ÞÑ pf ‹ : TX Ñ TY q
Examples:
• (finitary) nondeterminism: TX “ PωX ; “nondeterministic
functions” AÑ PωB are relations
• probabilistic nondeterminism: TX “
ρ : X Ñ r0, 1s |ř
ρ “ 1(
;
“probabilistic functions” AÑ TB are “probabilistic relations”
• (finite) background store: TX “ pX ˆ SqS ; side-effecting functions
AÑ TB are functions Aˆ S Ñ B ˆ S
5/15
Monads for Effects
Moggi: A monad T is a (generalized) computational effect
Any monad T supports inclusion of a value into a computation
η : X Ñ TX and a Kleisli lifting pf : X Ñ TY q ÞÑ pf ‹ : TX Ñ TY q
Examples:
• (finitary) nondeterminism: TX “ PωX ; “nondeterministic
functions” AÑ PωB are relations
• probabilistic nondeterminism: TX “
ρ : X Ñ r0, 1s |ř
ρ “ 1(
;
“probabilistic functions” AÑ TB are “probabilistic relations”
• (finite) background store: TX “ pX ˆ SqS ; side-effecting functions
AÑ TB are functions Aˆ S Ñ B ˆ S
5/15
Algebraic Theories
Moggi: A monad is a (generalized) computational effect
An algebraic theory E can be presented by a signature Σ and a set of
equations. Any E defines a monad:
• TEX “ ‘set of Σ-terms over X modulo E ’
• η coerces a variable to a term
• σ‹ptq applies substitution σ : X Ñ TEY to t : TEX
Example: Finite powerset monad Pω ðñ join semilattices with
bottom ðñ idempotent commutative monoids
Example: Finite probability distributions Dω ðñ barycentric algebras;
Σ “ t`p | p P r0, 1su, satisfying e.g. “biased associativity”:
px `p yq `q z “ x `p{pp`q´pqq py `p`q´pq zq
6/15
Algebraic Theories
Moggi: A monad is a (generalized) computational effect
Plotkin & Power: A monad is a (generalized) algebraic theory
An algebraic theory E can be presented by a signature Σ and a set of
equations. Any E defines a monad:
• TEX “ ‘set of Σ-terms over X modulo E ’
• η coerces a variable to a term
• σ‹ptq applies substitution σ : X Ñ TEY to t : TEX
Example: Finite powerset monad Pω ðñ join semilattices with
bottom ðñ idempotent commutative monoids
Example: Finite probability distributions Dω ðñ barycentric algebras;
Σ “ t`p | p P r0, 1su, satisfying e.g. “biased associativity”:
px `p yq `q z “ x `p{pp`q´pqq py `p`q´pq zq
6/15
Algebraic Theories
Plotkin & Power: A monad is a (generalized) algebraic theory
An algebraic theory E can be presented by a signature Σ and a set of
equations. Any E defines a monad:
• TEX “ ‘set of Σ-terms over X modulo E ’
• η coerces a variable to a term
• σ‹ptq applies substitution σ : X Ñ TEY to t : TEX
Example: Finite powerset monad Pω ðñ join semilattices with
bottom ðñ idempotent commutative monoids
Example: Finite probability distributions Dω ðñ barycentric algebras;
Σ “ t`p | p P r0, 1su, satisfying e.g. “biased associativity”:
px `p yq `q z “ x `p{pp`q´pqq py `p`q´pq zq
6/15
Stack Monad
The store monad TX “ pΓ‹ ˆ X qΓ‹
with Γ “ tγ1, . . . , γnu could be
regarded as a monad for stack transformations. But it contains too much!
Definition: The stack monad is the submonad Tstk of the store monad
p--ˆΓ‹qΓ‹
formed by xr : Γ‹ Ñ X , t : Γ‹ Ñ Γ‹y, which satisfy restriction:
Dk.w P Γk .@u P Γ‹. rpwuq “ rpwq ^ tpwuq “ tpwqu
Intuitively, this ensures that the underlying stack may only be finitely
read. E.g. nonexample: empty “ xλ -- . ‹, λ -- . εy R Tstk1
The stack signature consists of unary pushi (i ď n) and (n ` 1)-ary pop:
JpushiKpp P TstkX qpwq “ ppγiwq
JpopKpp1 P TstkX , . . . , pn P TstkX , q P TstkX qpγiwq “ pi pwq
JpopKpp1 P TstkX , . . . , pn P TstkX , q P TstkX qpεq “ pi pεq
7/15
Stack Theory
Theorem: The following stack theory is complete w.r.t. the stack monad
pushi ppoppx1, . . . , xn, yqq “ xi
popppush1pxq, . . . ,pushnpxq, xq “ x
poppx1, . . . , xn,poppy1, . . . , yn, zqq “ poppx1, . . . , xn, zq
Moreover, each TstkX is the free algebra of the stack theory over X
Proof: Interpreting the axioms as a rewriting system (it is strongly
normalizing, no non-trivial critical pairs) and identifying each TstkX with
the set of normal forms
8/15
Adding Nondeterminism
Store-like effects can be sensibly combined by tensoring1: a tensor
product of two theories E1 and E2 is obtaining by joining signatures and
equations and adding the tensor laws: for all f P E1, g P E2
f pgpx11 , . . . , x
1k q, . . . gpx
n1 , . . . , x
nk qq “ gpf px1
1 , . . . , xn1 q, . . . f px
nk , . . . , x
nk qq
Theorem [Freyd]: Tensor product of any theory with a semiring-module
theory is again a semiring-module theory
This yields a complete axiomatization of Pω b Tstk :
uioi “ 1 uioj “ 0 uie “ 0 o1u1 ` . . .` onun ` e “ 1 eoi “ 0 ee “ e pi ‰ jq
where
epxq “ popp∅, . . . , ∅, xq oi pxq “ popp∅, . . . , x , . . . , ∅q pi pxq “ pushpxq
1Freyd 1966, Algebra valued functors in general and tensor products in particular
9/15
(Turing) Tape Monad
The tape monad Ttp is analogously defined as a submonad of
p--ˆZˆ ΓZqZˆΓZ(Z “ integer numbers)
Signature: rd (n-ary), wri (unary, 1 ď i ď n), mv1,mv-1 (unary)
Equations:
rdpwr1pxq, . . . ,wrnpxqq “ x
wri prdpx1, . . . , xnqq “ wri pxi q
mv-1pmv1pxqq “ x
mv1pmv-1pxqq “ x
wri pwrjpxqq “ wrjpxq
wri pmvkpwrjpmv-kpxqqqq “ mvkpwrjpmv-kpwri pxqqqq pk ‰ 0q
Proposition: Tape theory is not finitely axiomatizable
10/15
Combining Effects with
Reactivity
Reactive Expressions: Syntax
Reactive expressions EΣ,B0 are closed δ-expressions generated by the
grammar for a given signature Σ and generators B0 of a Σ-algebra B:
δ ::“ x | γ | f pδ, . . . , δq px P X , f P Σq
γ ::“ µx . pa1.δ& ¨ ¨ ¨&ak .δ&βq px P X , ai P Aq
β ::“ b | f pβ, . . . , βq pb P B0, f P Σq
Example (Nondeterministic Automata) (using 1` 1 “ 1 from B):
δ ::“ x | γ | ∅ | δ ` δ γ ::“ µx . pa1.δ& ¨ ¨ ¨&ak .δ&1q
Example (Deterministic PDA) (using pushi p1q “ 0 from B):
δ ::“ x | γ | pushi pδq | poppδ, . . . , δq
γ ::“ µx . pa1.δ& ¨ ¨ ¨&ak .δ&βq
β ::“ 0 | 1 | poppβ, . . . , βq
11/15
Reactive Expressions: Semantics
For e P EΣ,B0 , we define
• Brzozowski derivatives Bapeq P EΣ,B0 by induction, most notably
Bai
`
µx . pa1.t1& ¨ ¨ ¨&ak .tk&rq˘
“ ti re{xs
and by further induction, Bεpeq “ e, Baw peq “ BaBw peq
• final outputs opeq P B, again by induction, specifically
o`
µx . pa1.t1& ¨ ¨ ¨&ak .tk&rq˘
“ r
This induces the semantics JeK : A‹ Ñ B, by putting JeKpwq “ opBw peqq
Kleene Theorem: for any e P EΣ,B0 there is a T-automaton m over X
and a state x P X such that JeK “ JxKm and vice versa
12/15
Reactive Expressions: Semantics
For e P EΣ,B0 , we define
• Brzozowski derivatives Bapeq P EΣ,B0 by induction, most notably
Bai
`
µx . pa1.t1& ¨ ¨ ¨&ak .tk&rq˘
“ ti re{xs
and by further induction, Bεpeq “ e, Baw peq “ BaBw peq
• final outputs opeq P B, again by induction, specifically
o`
µx . pa1.t1& ¨ ¨ ¨&ak .tk&rq˘
“ r
This induces the semantics JeK : A‹ Ñ B, by putting JeKpwq “ opBw peqq
Kleene Theorem: for any e P EΣ,B0 there is a T-automaton m over X
and a state x P X such that JeK “ JxKm and vice versa
12/15
Reactive Expressions: Semantics
For e P EΣ,B0 , we define
• Brzozowski derivatives Bapeq P EΣ,B0 by induction, most notably
Bai
`
µx . pa1.t1& ¨ ¨ ¨&ak .tk&rq˘
“ ti re{xs
and by further induction, Bεpeq “ e, Baw peq “ BaBw peq
• final outputs opeq P B, again by induction, specifically
o`
µx . pa1.t1& ¨ ¨ ¨&ak .tk&rq˘
“ r
This induces the semantics JeK : A‹ Ñ B, by putting JeKpwq “ opBw peqq
Kleene Theorem: for any e P EΣ,B0 there is a T-automaton m over X
and a state x P X such that JeK “ JxKm and vice versa
12/15
Classes of Instances
• If B “ 2, e.g. for non-deterministic or alternating automata,
JeK P 2A‹ – PA‹ is the formal language of e
• For weighted automata, B is a semiring, T is the corresponding
B-module monad and we obtain standard semantics
• For PDA-like machines, JeK is not the ultimate semantics:
Theorem: if B is finite then JeK is regular regardless of the monad
We take B to be the quotient of Tstk2 by
pushi p0q “ pushi p1q “ 0,
equivalently, B consists of predicates Γ‹ Ñ 2 depending on a
bounded portion of stack, hence, JeK : Γ‹ Ñ PpA‹q• For valence automata, T “ Pω b p--ˆMq “ Pωp--ˆMq, B “ PωpMq
where M is a (polycyclic) monoid emulating storage mechanism
13/15
Expressivity of Stack Machines
Theorem: With m ranging over Tstk -automata, x0 over the respective
state spaces and γ0 over Γ, the sets!
w P A‹ | Jx0Km pwqpγ0q “ 1)
exhaustively cover all real-time deterministic context-free languages
Theorem: the classes of languages recognized by Pω b Tstk b . . .b Tstk ,
where m is the number of copies of Tstk , are as follows:
1. All context-free languages if m “ 1
2. All nondeterministic linear time languages if m ě 3
3. The case m “ 2 correspond to a language class properly between the
above two
Proof: Reduction to results on real-time machines from 60-s
14/15
Further Work
• Reactive expressions come with a canonical equational calculus,
which cannot always be complete (otherwise equivalence of PDA
would be decidable). Can one derive completeness from constraints
on the effect theory2?
• ε-elimination: for T-automata over a given B, can we obtain a useful
description of T1-automata over B 1 such that every original
non-real-time automaton is equivalent to a real-time automaton
w.r.t. T1 and B 1? In particular, when T “ T1 and B “ B 1? This is
very hard already for valence automata3
• Identifying further language and complexity classes. How to describe
the relation between the theory of effects and the complexity of
languages recognized?
2Bonsangue, Milius, and Silva 2013, Sound and Complete Axiomatizations of
Coalgebraic Language Equivalence3Zetzsche 2013, Silent Transitions in Automata with Storage
15/15
Thank You for Your Attention!
References I
References
Marcello M. Bonsangue, Stefan Milius, and Alexandra Silva. Sound and
complete axiomatizations of coalgebraic language equivalence. ACM
Trans. Comput. Log., 14(1):7:1–7:52, 2013.
Peter Freyd. Algebra valued functors in general and tensor products in
particular. Colloq. Math., 14:89–106, 1966.
Sergey Goncharov, Stefan Milius, and Alexandra Silva. Towards a
coalgebraic chomsky hierarchy. In TCS’14, volume 8705, pages
265–280. Springer, 2014.
Sergey Goncharov, Stefan Milius, and Alexandra Silva. Towards a
uniform theory of effectful state machines. CoRR, abs/1401.5277,
2018. URL http://arxiv.org/abs/1401.5277.
13/15
References II
Robert Myers. Rational Coalgebraic Machines in Varieties: Languages,
Completeness and Automatic Proofs. PhD thesis, Imperial College
London, 2013.
Jan J. M. M. Rutten. Automata and coinduction (an exercise in
coalgebra). In Davide Sangiorgi and Robert de Simone, editors,
CONCUR, volume 1466 of Lecture Notes in Computer Science, pages
194–218. Springer, 1998.
Alexandra Silva. Kleene coalgebra. PhD thesis, Radboud Univ. Nijmegen,
2010.
Alexandra Silva, Filippo Bonchi, Marcello M. Bonsangue, and Jan J.
M. M. Rutten. Generalizing the powerset construction, coalgebraically.
In FSTTCS, volume 8 of LIPIcs, pages 272–283, 2010.
14/15
References III
Joost Winter, Marcello M. Bonsangue, and Jan J. M. M. Rutten.
Coalgebraic characterizations of context-free languages. LMCS, 9(3),
2013.
Georg Zetzsche. Silent transitions in automata with storage. In Fedor V.
Fomin, Rusins Freivalds, Marta Z. Kwiatkowska, and David Peleg,
editors, Automata, Languages, and Programming - 40th International
Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings,
Part II, volume 7966 of Lecture Notes in Computer Science, pages
434–445. Springer, 2013.
15/15