This package draws on the SMT solver Z3 to provide a programmatic methodology for developing semantic theories and exploring their logics. Rather than computing whether a given sentence is a logical consequence of some set of sentences by hand, these resources allow users to find countermodels or establish logical consequence up to a finite level complexity specified by the user. Whereas logic has traditionally focused on small language fragments, this project develops a unified semantics for a Language of Thought (LoT) for AI agents to plan and reason with in order to promote rational transparency.

In addition to the unified semantics for the LoT, this package provides support for users to develop their own programmatic semantic theories. The TheoryLib includes the semantic theories that are available for users to import or use as a template to develop novel theories to be contributed to by pull request.

You can find more information about the background semantic theory provided for the LoT here.

The Language of Thought

Intensional action is predicated on forethought and planning, where this applies to AI agents as much as it does to human agents. Since strategic planning requires agents to contemplate nearby counterfactual possibilities, temporal eventualities, and causal and constitutive explanatory relationships, it is important to equip AI with the conceptual resources needed to think in these ways. I am working to extend this list to include indicative conditionals, epistemic modals, belief and revision operators, and probability operators for reasoning under uncertainty. Following these additions I aim to include deontic modal and normative explanatory operators for cooperating with other agents in optimizing preferences and values.

This package currently includes hyperintensional semantic clauses for the following operators:

• neg for negation
• wedge for conjunction
• vee for disjunction
• rightarrow for the material conditional
• leftrightarrow for the material biconditional
• boxright for the must counterfactual conditional
• diamondright for the might counterfactual conditional
• Box for necessity
• Diamond for possibility
• Future read 'it will always be the case that'
• future read 'it will be the case that'
• Past read 'it always has been the case that'
• past read 'it has been the case that'
• leq for ground read 'sufficient for'
• sqsubseteq for essence read 'necessary for'
• equiv for propositional identity read 'just is for'
• preceq for relevance

More specific details about the implementation of these semantic clauses can be found here.

TheoryLib

The model-checker includes a library of semantic theories, each of which:

• Introduces the semantic primitives that make up a frame
• Defines the propositions over a frame needed to interpret the language
• Defines the models of the language by assigning sentence letters to propositions
• Provides semantic clauses for the primitive operators included in the language
• Draws on the primitive operators to define a number of additional operators
• Includes a range of examples of logical consequences and countermodels

Once the extension of a semantic theory has been explored with adequate range of results, that theory can be included in the TheoryLib with a pull request. See the theory_lib/README.md for further details on the existing theories as well as contributing new theories.

Licensing

ModelChecker is licensed under the GNU General Public License v3.0 (GPL-3.0). This license ensures that the software remains free and open-source. Key aspects of the licensing include:

• Core Package: The main ModelChecker framework is licensed under GPL-3.0
• Theory Library: All theories in the theory_lib directory are also individually licensed under GPL-3.0
• Derivative Works: According to GPL-3.0 terms, any derivative works, including new theories based on existing ones, must maintain the same license
• Academic Attribution: Each theory includes a CITATION.md file with proper academic attribution information

Contributing New Theories

When contributing a new theory to the ModelChecker framework:

1. Your contribution must be compatible with the GPL-3.0 license
2. Your theory will be automatically licensed under GPL-3.0 when added to the theory library
3. You retain copyright for your specific implementation but grant license under GPL-3.0
4. Academic attribution for your theory will be maintained in CITATION.md

The licensing structure is designed to ensure that the ModelChecker ecosystem remains open and accessible while providing proper attribution to theory authors.

Installation

Basic Installation

For core functionality (command line interface and model checking):

pip install model-checker

This installs the essential z3-solver dependency needed for constraint solving.

Optional Features

For Jupyter notebook integration with interactive visualizations:

pip install model-checker[jupyter]

For development tools (running tests):

pip install model-checker[dev]

For a complete installation with all features:

pip install model-checker[all]

More installation information can be found in the accessible instructions.

Updating

Once installed, you can check the current version of the model-checker with:

model-checker -v

To update to the latest version, run:

model-checker -u

Usage

Run model-checker in the terminal without arguments to create a new project with the following modules:

• semantic.py specifies the Z3 primitives, frame constraints, models, theory of logical consequence, defined semantic terms, theory of propositions, and print instructions for displaying countermodels for the default semantics.
• operators.py specifies the semantic clauses for the primitive operators included in the default language along with a number of defined operators.
• examples/ includes modules with collections of examples and settings.
• notebooks/ includes Jupyter Notebooks for exploring theories and discussing findings.
• tests/ includes unit tests which aid in rapidly prototyping theories.

Alternatively, run model-checker -l THEORY_NAME to create a copy of the semantic theory with the name 'THEORY_NAME'. The library of available semantic theories can be found here. Additional theories can be added by submitting a pull request.

After changing to the project directory that you created, run model-checker project_examples.py to find a countermodel if there is any. The example settings specify the following inputs where the defaults are indicated below:

• The number of atomic states to include in each model: N = 3.
• An option to require all sentence letters to be contingent: contingent = False.
• An option to require all sentence letters to have at least one verifier and at least one falsifier: non_empty = False.
• An option to prevent sentence letters from having the null state as a verifier or a falsifier: non_null = False.
• An option to prevent sentence letters from having overlapping verifiers or falsifiers: disjoint = False.
• The maximum time in seconds to spend looking for a model: max_time = 1.

A number of general settings may also be specified with the following:

• An option to print impossible states: print_impssible = False.
• An option to print all Z3 constraints or unsatisfiable core constraints: print_constraints = False.
• An option to print the Z3 model if there is any: print_z3 = False.
• An option to prompt the user to append the output to the current file or to create a new file: save_output = False.

Examples are specified by defining a list as follows:

# CF_CM_1: COUNTERFACTUAL ANTECEDENT STRENGTHENING

CF_CM_1_premises = ['(A \\boxright C)']
CF_CM_1_conclusions = ['((A \\wedge B) \\boxright C)']
CF_CM_1_settings = {
    'N' : 3,
    'contingent' : True,
    'non_null' : True,
    'non_empty' : True,
    'disjoint' : False,
    'max_time' : 1,
}

CF_CM_1_example = [
    CF_CM_1_premises,
    CF_CM_1_conclusions,
    CF_CM_1_settings,
]

The example CF_CM_1_example includes:

• A list of zero or more premises that are treated conjunctively: premises = [].
• A list of zero or more conclusions that are treated disjunctively: conclusions = [].
• A dictionary of settings where the defaults are indicated above.

Alternatively, users can define a general stock of example_settings, reusing these for an number of examples. Users can override these settings from the command line by including the following flags:

• Include -c to set contingent = True.
• Include -d to set disjoint = True.
• Include -e to set non_empty = True.
• Include -i to set print_impossibe = True.
• Include -n to set non_null = True.
• Include -p to set print_constraints = True.
• Include -s to set save_bool = True.
• Include -z to set print_z3 = True.

Additional flags have been included in order to manage the package version:

• Include -h to print help information about the package and its usage.
• Include -v to print the installed version number.
• Include -u to upgrade to the latest version.

Jupyter Notebook Integration

ModelChecker can be used interactively in Jupyter notebooks, allowing for dynamic exploration of logical models with an interactive interface.

Installation

To use ModelChecker in Jupyter notebooks, install with optional dependencies:

pip install ipywidgets matplotlib networkx

Basic Usage

For quick formula checking, both LaTeX and some unicode characters are accepted:

from model_checker.notebook import check_formula

# Check a counterfactual formula
check_formula("(A \\vee B \\boxright C) \\rightarrow (A \\boxright C)")

# Check a modal formula
check_formula("□(p → q) → (□p → □q)")

# Check with premises
check_formula("q", premises=["p", "p \\diamondright q"])

Interactive Explorer

For interactive exploration with a UI:

from model_checker.notebook import InteractiveModelExplorer

# Create and display the explorer
explorer = InteractiveModelExplorer()
explorer.display()

The interactive explorer provides:

• Formula input with syntax highlighting
• Theory selection and settings customization
• Multiple visualization options
• Ability to find alternative models

For a demonstration, see the examples/jupyter_demo.ipynb notebook and the examples/README_jupyter.md documentation.

Hyperintensional Semantics

This section sketches the underlying semantics. More information can be found in the GitHub repository.

The semantics is hyperintensional insofar as sentences are evaluated at states which may be partial rather than total as in intensional semantic theories. States are modeled by bitvectors of a specified length (e.g., #b00101 has length 5), where state fusion is modeled by the bitwise OR operator |. For instance, #b00101 | #b11001 = #b11101. The atomic states have exactly one occurrence of 1 and the null state has no occurrences of 1. The space of states is finite and closed under fusion.

States are named by lowercase letters in order to print readable countermodels. Fusions are printed using . where a.b is the fusion of the states a and b. A state a is part of a state b just in case a.b = b. States may be either possible or impossible where the null state is required to be possible and every part of a possible state is possible. The states a and b are compatible just in case a.b is possible. A world state is any state that is both possible and includes every compatible state as a part.

Sentences are assigned verifier states and falsifier states where both the verifiers and falsifiers are required to be closed under fusion. A sentence is true at a world state w just in case w includes a verifier for that sentence as a part and false at w just in case w includes a falsifier for that sentence as a part. In order to ensure that sentence letters have at most one truth-value at each world state, a fusion a.b is required to be impossible whenever a is verifier for a sentence letter A and b is a falsifier for A. Additionally, sentence letters are guaranteed to have at least one truth-value at each world state by requiring every possible state to be compatible with either a verifier or falsifier for any sentence letter.

A negated sentence is verified by the falsifiers for the sentence negated and falsified by the verifiers for the sentence negated. A conjunctive sentence is verified by the pairwise fusions of verifiers for the conjuncts and falsified by falsifiers for either of the conjuncts or fusions thereof. A disjunctive sentence is verified by the verifiers for either disjunct or fusions thereof and falsified by pairwise fusions of falsifiers for the disjuncts. Conjunction and disjunction are dual operators obeying the standard idempotence and De Morgan laws. The absorption laws do not hold, nor does conjunction distribute over disjunction, nor vice versa. For a defense of the background theory of hyperintensional propositions, see this paper.

A necessity sentence Box A is true at a world just in case every world state includes a part that verifies A and a possibility sentence Diamond A is true at a world just in case some world state includes a part that verifies A. Given a world state w and state s, an s-alternative to w is any world state to include as parts both s and a maximal part of w that is compatible with s. A must counterfactual conditional sentences A boxright B is true at a world state w just in case its consequent is true at any s-alternative to w for any verifier s for the antecedent of the counterfactual. A might counterfactual conditional sentences A boxright B is true at a world state w just in case its consequent is true at some s-alternative to w for some verifier s for the antecedent of the counterfactual. The semantic theory for counterfactual conditionals is motivated and further elaborated in this accompanying paper. This account builds on Fine 2012 and Fine2012a.

A grounding sentence A leq B may be read 'A is sufficient for B' and an essence sentence A sqsubseteq B may be read 'A is necessary for B'. A propositional identity sentence A equiv B may be read 'A just is for B'. A relevance sentence A preceq B may be read 'A is wholly relevant to B'. The semantics for ground requires every verifier for the antecedent to be a verifier for the consequent, any fusion of a falsifier for the antecedent and consequent to be a falsifier for the consequent, and any falsifier for the consequent to have a part that falsifies the antecedent. The semantics for essence requires every fusion of a verifier for the antecedent and consequent to be a verifier for the consequent, any verifier for the consequent must have a part that verifies the antecedent, and every falsifier for the antecedent to be a falsifier for the consequent. The semantics for propositional identity requires the two arguments to have the same verifiers and falsifiers. The semantics for relevance requires any fusion of verifiers for the antecedent and consequent to be a verifier for the consequent and, similarly, any fusion of falsifiers for the antecedent and consequent to be a falsifier for the consequent. Whereas the first three constitutive operators are interdefinable, relevance is definable in terms of the other constitutive operators but not vice versa:

• A leq B := neg A sqsubseteq neg B := (A vee B) equiv B.
• A sqsubseteq B := neg A leq neg B := (A wedge B) equiv B.
• A equiv B := (A leq B) wedge (B leq A) := (A sqsubseteq B) wedge (B sqsubseteq A).
• A preceq B := (A wedge B) leq B := (A vee B) sqsubseteq B.

Instead of a Boolean lattice as in extensional and intensional semantics theories, the space of hyperintensional propositions forms a non-interlaced bilattice as described in this paper, building on Fine 2017.

More information can be found in the GitHub repository as well as in this recent manuscript.