-- needs ghc -fglasgow-exts

{- HetCATS/Logic.hs

$Id$

Till Mossakowski

Provides data structures for logics (with symbols)

and logic representations. Logics are

a type class with an "identitiy" type (usually interpreted

by a singleton set) which serves to treat logics as

data. All the functions in the type class take the

identity as first argument in order to determine the logic.

Logic representations are just collections of

functions between (some of) the types of logics.

References:

J. A. Goguen and R. M. Burstall

Institutions: Abstract Model Theory for Specification and

Programming

JACM 39, p. 95--146, 1992

(general notion of logic - model theory only)

J. Meseguer

General Logics

Logic Colloquium 87, p. 275--329, North Holland, 1989

(general notion of logic - also proof theory;

notion of logic representation, called map there)

T. Mossakowski:

Specification in an arbitrary institution with symbols

14th WADT 1999, LNCS 1827, p. 252--270

(treatment of symbols and raw symbols, see also CASL semantics)

T. Mossakowski, B. Klin:

Institution Independent Static Analysis for CASL

15h WADT 2001, LNCS

(what is needed for static anaylsis)

S. Autexier and T. Mossakowski

Integrating HOLCASL into the Development Graph Manager MAYA

FroCoS 2002, to appear

(interface to provers)

Todo:

ATerm, XML

Weak amalgamability, also for reprs

Metavars

repr maps

symbol maps for reprs

reprs out of sublogic relaionships

-}

module Logic where

import Id

import Anno

import Error

import Dynamic

-- maps and sets, just a quick thing

type Set a = [a]

type Map a = [(a,a)]

-- diagrams, nodes are just integers

type Node = Int

type Edge morphism = (Node,morphism,Node)

data Diagram object morphism = Diagram [(Node,object)] [Edge morphism]

empty_diagram :: Diagram o m

empty_diagram = Diagram [] []

add_node :: Diagram o m -> o -> (Node,Diagram o m)

add_node (Diagram nodes edges) obj =

(node,Diagram ((node,obj):nodes) edges) where

node = maximum (map fst nodes)

add_edge :: Diagram o m -> Edge m -> Diagram o m

add_edge (Diagram nodes edges) edge =

Diagram nodes (edge:edges)

object_at_node :: Node -> Diagram o m -> Maybe (Node,o)

object_at_node node (Diagram nodes edges) =

case lookup node nodes of

Just obj -> Just (node,obj)

Nothing -> Nothing

diagram_nodes :: Diagram o m -> [(Node,o)]

diagram_nodes (Diagram nodes edges) = nodes

diagram_edges :: Diagram o m -> [Edge m]

diagram_edges (Diagram nodes edges) = edges

-- Categories are given by a quotient,

-- i.e. we need equality

-- Should we allow arbitrary composition graphs and build paths?

class (Eq object, Eq morphism) =>

Category id object morphism | id ->, id -> morphism where

ide :: id -> object -> morphism

o :: id -> morphism -> morphism -> Maybe morphism

dom, cod :: id -> morphism -> object

legal_obj :: id -> object -> Bool

legal_mor :: id -> morphism -> Bool

-- abstract syntax, parsing and printing

class (Show basic_spec, Show sentence, Show symb_items,

Show symb_map_items) =>

Syntax basic_spec sentence symb_items symb_map_items

where

-- parsing

parse_basic_spec :: id -> String -> Result basic_spec

parse_sentence :: id -> String -> Result sentence

parse_symb_items :: id -> String -> Result [symb_items]

parse_symb_map_items :: id -> String -> Result [symb_map_items]

-- lattices (for sublogics)

class Ord l => LatticeWithTop l where

meet, join :: l -> l -> l

top :: l

-- theories and theory morphisms

data Theory sign sen =

Theory {sign_of :: sign,

ax_of :: [(String,sen)]

}

data TheoryMorphism sign sen mor =

TheoryMorphism {t_source, t_target :: Theory sign sen,

t_morphism :: mor

}

-- proofs and provers

type Rule = String

data Proof_tree sen = Axiom sen

| Branch (sen,Rule,[Proof_tree sen]) -- add substitutions here?

type Tactic_script = String -- the file name

data Proof_status sen = Open sen

| Disproved sen

| Proved(sen,[sen],Proof_tree sen,Tactic_script)

data Prover sen symbol =

Prover { prover_name :: String,

prover_sublogic :: String,

add_sym :: symbol -> IO(Bool), -- returns True if succeeded

remove_sym :: symbol -> IO(Bool), -- returns True if succeeded

prove :: sen -> IO([Proof_status sen]) -- proof status for goal and lemmas

}

data Cons_checker morphism =

Cons_checker {cons_checker_name :: String,

cons_checker_sublogic :: String,

cons_check :: morphism -> IO(Bool, Tactic_script)

}

-- logics

class (Syntax basic_spec sentence symb_items symb_map_items,

Show sign, Show morphism, Show symbol, Show raw_symbol,

Ord symbol, -- needed for efficient symbol tables

Category id sign morphism,

LatticeWithTop sublogics,

-- needed for heterogeneous coercions:

Typeable id, Typeable sublogics, Typeable sign, Typeable morphism, Typeable symbol, Typeable raw_symbol,

Typeable basic_spec, Typeable sentence, Typeable symb_items, Typeable symb_map_items) =>

Logic id sublogics

basic_spec sentence symb_items symb_map_items

local_env sign morphism symbol raw_symbol

| id -> sublogics, id -> basic_spec, id -> sentence, id -> symb_items,

id -> symb_map_items, id -> local_env,

id -> sign, id -> morphism, id ->symbol, id -> raw_symbol

where

logic_name :: id -> String

has_parser :: id -> Bool

has_printer :: id -> Bool

has_analysis :: id -> Bool

-- sentence translation

map_sen :: id -> morphism -> sentence -> Result sentence

-- static analysis of basic specifications and symbol maps

basic_analysis :: id ->

(basic_spec, -- abstract syntax tree

local_env, -- efficient table for env signature

[Anno]) -> -- global annotations

Result (sign,[(String,sentence)])

-- these include any new annotations

stat_symb_map_items :: id -> [symb_map_items] -> Result (Map raw_symbol)

stat_symb_items :: id -> [symb_items] -> Result [raw_symbol]

-- architectural sharing analysis for one morphism

ensures_amalgamability :: id ->

(Diagram sign morphism, Node, sign, Edge morphism, morphism) ->

Result (Diagram sign morphism)

-- do we need it also for sinks consisting of two morphisms?

-- symbols and symbol maps

symbol_to_raw :: id -> symbol -> raw_symbol

id_to_raw :: id -> Id -> raw_symbol

sym_of :: id -> sign -> Set symbol

symmap_of :: id -> morphism -> Map symbol

matches :: id -> symbol -> raw_symbol -> Bool

sym_name :: id -> symbol -> Id

-- operations on local envs, signatures and morphisms

empty_local_env :: local_env

add_sign :: sign -> local_env -> local_env

empty_signature :: id -> sign

signature_union :: id -> sign -> sign -> Result sign

final_union :: id -> sign -> sign -> Result sign

is_subsig :: id -> sign -> sign -> Bool

generated_sign, cogenerated_sign :: id -> [raw_symbol] -> sign -> Result morphism

induced_from_morphism :: id -> Map raw_symbol -> sign -> Result morphism

induced_from_to_morphism :: id -> Map raw_symbol -> sign -> sign -> Result morphism

extend_morphism :: Id -> sign -> morphism -> sign -> sign -> Result morphism

-- sublogics

sublogic_names :: id -> sublogics -> [String]

all_sublogics :: id -> [sublogics]

is_in_basic_spec :: id -> sublogics -> basic_spec -> Bool

is_in_sentence :: id -> sublogics -> sentence -> Bool

is_in_symb_items :: id -> sublogics -> symb_items -> Bool

is_in_symb_map_items :: id -> sublogics -> symb_map_items -> Bool

is_in_sign :: id -> sublogics -> sign -> Bool

is_in_morphism :: id -> sublogics -> morphism -> Bool

is_in_symbol :: id -> sublogics -> symbol -> Bool

min_sublogic_basic_spec :: id -> basic_spec -> sublogics

min_sublogic_sentence :: id -> sentence -> sublogics

min_sublogic_symb_items :: id -> symb_items -> sublogics

min_sublogic_symb_map_items :: id -> symb_map_items -> sublogics

min_sublogic_sign :: id -> sign -> sublogics

min_sublogic_morphism :: id -> morphism -> sublogics

min_sublogic_symbol :: id -> symbol -> sublogics

proj_sublogic_basic_spec :: id -> sublogics -> basic_spec -> basic_spec

proj_sublogic_symb_items :: id -> sublogics -> symb_items -> Maybe symb_items

proj_sublogic_symb_map_items :: id -> sublogics -> symb_map_items -> Maybe symb_map_items

proj_sublogic_sign :: id -> sublogics -> sign -> sign

proj_sublogic_morphism :: id -> sublogics -> morphism -> morphism

proj_sublogic_epsilon :: id -> sublogics -> sign -> morphism

proj_sublogic_symbol :: id -> sublogics -> symbol -> Maybe symbol

-- provers

provers :: [Prover sentence symbol]

cons_checkers :: [Cons_checker (TheoryMorphism sign sentence morphism)]

-- derived operations, need not to be given

-- printing, accessible via logic identity

show_basic_spec :: id -> basic_spec -> String

show_sentence :: id -> sentence -> String

show_symb_items :: id -> symb_items -> String

show_symb_map_items :: id -> symb_map_items -> String

show_anno :: id -> Anno -> String

show_sign :: id -> sign -> String

show_morphism :: id -> morphism -> String

show_symbol :: id -> symbol -> String

show_raw_symbol :: id -> raw_symbol -> String

show_basic_spec _ = show

show_sentence _ = show

show_symb_items _ = show

show_symb_map_items _ = show

show_sign _ = show

show_morphism _ = show

show_symbol _ = show

show_raw_symbol _ = show

-- Simple logic representations (possibly also morphisms via adjointness)

data (Logic id1 sublogics1

basic_spec1 sentence1 symb_items1 symb_map_items1

local_env1 sign1 morphism1 symbol1 raw_symbol1,

Logic id2 sublogics2

basic_spec2 sentence2 symb_items2 symb_map_items2

local_env2 sign2 morphism2 symbol2 raw_symbol2) =>

LogicRepr id1 sublogics1 basic_spec1 sentence1 symb_items1 symb_map_items1

local_env1 sign1 morphism1 symbol1 raw_symbol1

id2 sublogics2 basic_spec2 sentence2 symb_items2 symb_map_items2

local_env2 sign2 morphism2 symbol2 raw_symbol2

=

LogicRepr {repr_name :: String,

source :: id1, source_sublogic :: sublogics1,

target :: id2, target_sublogic :: sublogics2,

map_basic_spec :: basic_spec1 -> Maybe basic_spec2,

-- partial because the target may be a sublanguage

map_sentence :: sign1 -> sentence1 -> Maybe sentence2,

-- also cover semi-representations

map_sign :: sign1 -> Maybe sign2,

projection:: Maybe (sign2 -> sign1, morphism2 -> morphism1,

sign2 -> morphism2,

-- right adjoint and counit

basic_spec2 -> basic_spec1,

symbol2 -> Maybe symbol1),

map_morphism :: morphism1 -> Maybe morphism2,

map_symbol :: symbol1 -> Maybe symbol2

}