CHR Operational Semantics

in Fluent Calculus (using Ramifications)

November, 2007

SimpleFluent Calculus (SFC)

Introduction

• A many-sorted first-order language with equality• Includes:

– Sorts: FLUENT < STATE, ACTION, SIT

– Functions:

– Predicate

Abbreviations

Foundational Axioms (Fstate)

SFC Domain Axiomatization

• State Constraints

• Unique simple Action Precondition Axiom for each function symbol with range ACTION

• A set of State Update Axioms

• Foundational Axioms (Fstate)

• Possibly further domain-specific axioms

Action Precondition Axiom

• Ex:

State Update Axiom

• Ex:

Ramifications in Fluent Calculus

Modeling Ramifications

Fluent Calculus with Ramifications

• Sorted second-order logic language• Reserved Predicates:

– Causes : STATE x STATE x STATE x STATE x STATE x STATE

• Causes(z1, e1+, e1-, z2, e2+, e2-)– If z1 is the result of positive effects e1+ and negative effects e1-,

then an additional effect is caused which leads to z2 (now the result of positive and negative effects e2+ and e2-, resp.)

– Ramify : STATE x STATE x STATE x STATE• Ramify(z, e+, e-, z’)

– z’ can be reached by iterated application of the underlying casual relation, starting in state z with momentum e+ and e-

Abbreviations

Foundational Axioms

(Reflexive and Transitive Closure of Causes)

State Update Axiomwith Ramifications

Causal Relations Axiomatization

• Relies on the assumption that the underlying Causes relation is completely specified

Fluent Calculus Domain Axiomatization

with Ramifications• State constraints• Causal Relations axiomatization• Unique action precondition axiom for each

function symbol with range ACTION• Set of state update axioms (possibly with

ramifications)• Foundational Axioms: Fstate and Framify• Domain Specific Axioms

CHR Operational Semantics in Fluent Calculus

Domain Sorts

• CONSTRAINT < FLUENT

• UDC < CONSTRAINT

• BIC < CONSTRAINT

• EQUATION < BIC

Domain Predicates

• entails : STATE x Set(EQUATION) x Set(BIC)– entails(s, h, g)– CT |= s \exists x(h ^ g)

Domain Actions

• AddConstraint : CONSTRAINT ACTION

Example

leq(X,X) <=> true. leq(X,Y), leq(Y,X) <=> X = Y. leq(X,Y), leq(Y,Z) ==> leq(X,Z).

Example

leq(X,X) <=> true.

leq(X,Y), leq(Y,Z) ==> leq(X,Z).

Example

leq(X,Y), leq(Y,Z) ==> leq(X,Z).

Example(Constraint Awakening)